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Extending Classical Multirate Signal Processing
Theory to Graphs – Part II: M-Channel Filter Banks
Oguzhan Teke, Student Member, IEEE, P. P. Vaidyanathan, Fellow, IEEE
Abstract—This paper builds upon the basic theory of multirate
systems for graph signals developed in the companion paper
(Part I) and studies M -channel polynomial filter banks on graphs.
The behavior of such graph filter banks differs from that of
classical filter banks in many ways, the precise details depending
on the eigenstructure of the adjacency matrix A. It is shown that
graph filter banks represent (linear and) periodically shift-variant
systems only when A satisfies the noble identity conditions
developed in Part I. It is then shown that perfect reconstruction
graph filter banks can always be developed when A satisfies
the eigenvector structure satisfied by M -block cyclic graphs and
has distinct eigenvalues (further restrictions on eigenvalues being
unnecessary for this). If A is actually M -block cyclic then these
PR filter banks indeed become practical, i.e., arbitrary filter polynomial orders are possible, and there are robustness advantages.
In this case the PR condition is identical to PR in classical filter
banks – any classical PR example can be converted to a graph
PR filter bank on an M -block cyclic graph. It is shown that for
M -block cyclic graphs with all eigenvalues on the unit circle, the
frequency responses of filters have meaningful correspondence
with classical filter banks. Polyphase representations are then
developed for graph filter banks and utilized to develop alternate
conditions for alias cancellation and perfect reconstruction, again
for graphs with specific eigenstructures. It is then shown that
the eigenvector condition on the graph can be relaxed by using
similarity transforms.
Index Terms—Multirate processing, graph signals, filter banks
on graphs, aliasing on graphs, block-cyclic graphs.
I. I NTRODUCTION
In this paper we build upon the basic theory of multirate
graph systems developed in the companion paper [1] and study
M -channel filter banks on graphs. The M -channel maximally
decimated graph filter bank has the form shown in Fig. 1 where
Hk pAq and Fk pAq are polynomials in A, and D represents
the canonical decimator defined in [1]. As in [1]–[3] the graph
adjacency matrix A is assumed to be possibly complex and
non symmetric in general. We will see that the behavior of
such graph filter banks differs from that of classical filter
banks [4], [5] in many ways, the precise details depending
on the eigenstructure of the adjacency matrix. An outline of
the main results is given below. Needless to say, the problems
we address in this paper are inspired by the recent advances
reported in [6]–[11]. More can be found in [12]–[15].
Copyright (c) 2015 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to [email protected].
The authors are with the Department of Electrical Engineering, California Institute of Technology, Pasadena, CA, 91125, USA. Email:
[email protected], [email protected].
This work was supported in parts by the ONR grants N00014-11-1-0676
and N00014-15-1-2118, and the California Institute of Technology.
A. Scope and outline
In Sec. I-B we review some key equations from the companion paper [1] (renumbered here as (1)–(12)). With the
graph adjacency matrix regarded as a shift operator [9], [10],
it will be shown in Sec. II-A that the graph filter bank is a
shift-variant system, although it is in general not periodically
shift-variant as in classical time domain filter banks. We then
establish the conditions on the adjacency matrix A for the
periodically shift-varying property and show that it is exactly
identical to the conditions for the existence of graph noble
identities (Theorem 2).
Then in Sec. III we consider graphs that satisfy the eigenvector structure of (10) (Ω-graphs). These graphs are more
general than M -block cyclic graphs, which satisfy both the
eigenvalue and eigenvector conditions of (9) and (10). For such
graphs we define band-limited graph signals and polynomial
perfect interpolation filters for decimated versions of such
signals. This allows us to develop a class of perfect reconstruction filter banks for Ω-graphs (Theorem 4), similar to
ideal brickwall filter banks of classical sub-band coding theory.
Such graph filter bank designs are usually not practical because
the polynomial filters have order N -1 (where the graph has
N vertices and N can be very large). Furthermore these
specific filters for perfect reconstruction are very sensitive to
our knowledge of the graph eigenvalues.
In Sec. IV we develop graph filter banks on M -block cyclic
graphs (defined and studied in Sec. V of [1]). For such graphs
the eigenvalues and eigenvectors are both constrained as in (9),
(10). We show that for such graphs the condition for perfect
reconstruction is very similar to the PR condition in classical
filter banks (Theorem 6). For such graphs, it is therefore
possible to design PR filter banks by starting from any classical
PR system. In particular it is possible to obtain PR systems
with arbitrarily small orders (independent of the size of the
graph) for the polynomial filters. Furthermore the PR solutions
tHk pAq, Fk pAqu are not sensitive to graph eigenvalues.
In Sec. V we consider polyphase representations for graph
filter banks. This is useful to obtain alternative theoretical
representations, as well as in implementation. Unlike classical
filter banks, these polyphase representations are not always
valid. They are valid only for those graphs that satisfy the
noble identity requirements (Theorem 3 of [1]). For such
graphs, the PR condition and the alias cancellation condition
can further be expressed in terms of the analysis and synthesis
polyphase matrices if the graphs also satisfy the M -block
cyclic property (Theorems 10 and 11). Interestingly these alias
cancellation conditions are somewhat similar to the pseudo-
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circulant property developed for classical filter banks in [4].
circulant
property
developed
for classical
filter of
banks
in filter
[?].
In Sec. VI
we consider
frequency
responses
graph
In
Sec.
VI
we
consider
frequency
responses
of
graph
filter
banks inspired by similar ideas in [8], [10]. We will see that
banks
inspired
by meaningfully
similar ideas developed
in [?], [?].for
Wefilter
willbanks
see that
this
concept
can be
on
this
concept
can
be
meaningfully
developed
for
filter
banks
on
M -cyclic graphs with all eigenvalues on the unit circle, but
M
-cyclic
graphs
with
all
eigenvalues
on
the
unit
circle,
but
not for arbitrary graphs.
not
for arbitrary
Finally
in Sec. graphs.
VII we show that the eigenvector structure
Finally
in Sec. VIIcan
we be
show
that the
eigenvector
structure
in (10) (Ω-structure)
relaxed
simply
by considering
in
(10)
(Ω-structure)
can
be
relaxed
simply
by
considering
a transformed graph based on similarity transformations. This
a transformedtherefore
graph based
on similarity
transformations.
generalization
extends
many of the
results in this This
and
generalization
therefore
extends
many
of
the
results
this and
the companion paper [1] to more general graphs.
Ininshort,
all
the companion
paper [?] for
to more
general
graphs.
In short,
all
results
that we developed
Ω-graphs
(e.g.,
Theorems
3 and
results
that
we
developed
for
Ω-graphs
(e.g.,
Theorems
3
and
4) generalize to arbitrary graphs. Similarly all results which we
4) generalize
graphs.
Similarly
results which
we
developed
for to
Marbitrary
-block cyclic
graphs
(e.g.,allTheorems
5 and
developed
for
M
-block
cyclic
graphs
(e.g.,
Theorems
5
and
6) generalize to graphs that are subject only to the eigenvalue
6) generalize
to graphs
are subjectconstraint
only to the
eigenvalue
constraint
(9) and
not thethat
eigenvector
(10).
constraint
(9)
and
not
the
eigenvector
constraint
(10).
Section IX concludes the paper. This paper follows the
Section
IX concludes
the paper. paper
This [1].
paper follows the
notation
described
in the companion
notation described in the companion paper [?].
we showed that eigenvalues and eigenvectors of M -block
we showed
andrelation
eigenvectors of M -block
cyclic
graphs that
haveeigenvalues
the following
cyclic graphs have the following relation
λi,j+k wk λi,j ,
(9)
λi,j+k wkk λi,j ,
(9)
v i,j+k
Ωk vi,j ,
(10)
v i,j+k Ω v i,j ,
(10)
where
j2π {M
where
w e
,
(11)
j2π {M
w
e
, (11)
(12)
Ω diag 1 w-1
w-2 w-pM -1q b I N {M .
Ω diag 1 w-1 w-2 w-pM -1q
b I N {M . (12)
In the rest of the paper the eigenvector structure in (10) will
In the rest of the paper the eigenvector structure in (10) will
be referred to as the Ω-structure. A graph with eigenvectors
be referred to as the Ω-structure. A graph with eigenvectors
satisfying the condition in (10) will be referred to as an Ωsatisfying the condition in (10) will be referred to as an Ωgraph. Notice that an Ω-graph can have arbitrary eigenvalues.
graph. Notice that an Ω-graph can have arbitrary eigenvalues.
An M -block cyclic graph is also an Ω-graph since its eigenAn M -block cyclic graph is also an Ω-graph since its eigenvectors have the Ω-structure. Furthermore, any circulant graph
vectors have the Ω-structure. Furthermore, any circulant graph
is an Ω-graph since columns of the properly permuted DFT
is an Ω-graph since columns of the properly permuted DFT
matrix have the structure in (10).
matrix have the structure in (10).
B. Review from [1]
B. Review from [?]
We defined M -fold graph decimation operator by the matrix
We defined M -fold graph decimation operator by the matrix
N
D I N {M 0N {M 0N {M P C ppNN{{MMq
qN.. (1)
D I N {M 0N {M 0N {M P C
(1)
We then showed that the following noble identities
We then showed that the following noble identities
s
D H pAMMq H pA
(2)
DMH pAT q H
pAsqqD,
D,
(2)
T
s q,
H pA Mq D T D T H pA
(3)
s
H pA q D D H pAq,
(3)
are simultaneously satisfied for all polynomial filters H pq if
are simultaneously satisfied for all polynomial filters H pq if
and only if the following two equations are satisfied: A being
and only if the following two equations
are satisfied: A being
the adjacency matrix of the graph, AMM has the form
the adjacency matrix of the graph, A has the form
pAMMq1,1
0
M
p
A
q
0
A M
(4)
1,1
M
A (4)
00
ppAAMqq2,22,2 ,,
and
and
ss D AMMD TT P MNN{{MM,
A
(5)
A DA D P M
,
(5)
II.
F ILTER BANKS
II. G
GRAPH
RAPH F ILTER BANKS
In
found that
that multirate
multirate building
building
In the
the companion
companion paper
paper [1],
[?], we
we found
blocks
defined
on
graphs
satisfy
identities
similar
to
classical
blocks defined on graphs satisfy identities similar to classical
multirate
conditions on
on the
the adjaadjamultirate identities
identities only
only under
under certain
certain conditions
cency
matrix
A.
In
the
case
of
filter
banks,
even
the
simplest
cency matrix A. In the case of filter banks, even the simplest
maximally
filter bank
bank (the
(thelazy
lazyfilter
filterbank
bankofofFig.
Fig.??3
maximally decimated
decimated filter
of
[1])
may
or
may
not
satisfy
perfect
reconstruction
(unlike
of [?]) may or may not satisfy perfect reconstruction (unlike
in
These were
were elaborated
elaborated ininSec.
Sec.??
II of
of [?].
[1].
in the
the classical
classical case).
case). These
In
this
section
we
consider
filter
banks
in
greater
detail.
In this section we consider filter banks in greater detail.
Fig.
graph filter
filter bank
bank where
where
Fig. 11 shows
shows aa maximally
maximally decimated
decimated graph
the
analysis
filters
H
p
A
q
and
the
synthesis
filters
F
p
A
are
k
k
the analysis filters Hk A and the synthesis filters Fk A q are
polynomials
in
A,
and
D
is
as
in
(1).
In
the
classical
case,
polynomials in A, and D is as in (1). In the classical case, aa
maximally
is known
known to
to be
be aa periodically
periodically
maximally decimated
decimated filter
filter bank
bank is
time-varying
system
unless
aliasing
is
completely
canceled,
in
time-varying system unless aliasing is completely canceled, in
which
case
it
becomes
a
time-invariant
system.
It
is
possible
to
which case it becomes a time-invariant system. It is possible to
get
for filter
filter banks
banks on
on graphs,
graphs,
get aa somewhat
somewhat analogous
analogous property
property for
but
first develop
develop
but only
only under
under some conditions. In this section we first
these
banks that
that are
are
these results.
results. We then study a class of filter banks
analogous
filter banks
banks
analogous to
to ideal
ideal brickwall filters (bandlimited filter
in
reconstruction can
can
in the
the classical
classical case) and show that perfect reconstruction
be
eigen-structure of
of
be achieved
achieved under
under some constraints on the eigen-structure
the
the graph
graph A.
A.
M
N {M
where
wherepA
pAMq1,1
q1,1PPMMN {M. .We
Wenoted
notedthat
thatthe
the condition
condition in
in (4)
(4)
isisnot
trivial
in
the
sense
that
even
very
simple
graphs
not trivial in the sense that even very simple graphs can
can
fail
failtotosatisfy
satisfyit.it.We
Wealso
alsoshowed
showedthat
thatM
M-block
-block cyclic
cyclic graphs
graphs
satisfy
the
noble
identity
condition
(4).
We
called
satisfy the noble identity condition (4). We called aa graph
graph
(balanced)
(balanced)MM-block
-blockcyclic
cyclicwhen
whenitithad
had the
the form
form
00 00 00
A1
A1 00 00
0 A2
0 A2 00
.
AA 00 00 AA33 . . .
. .
.. .
.. . . .
.. .
..
. ..
. ..
.
00
00
00
00
00
00
.. .
. ..
A
AMM
00
00
.. .
. ..
00
A
AMM-1-1
00
N
N, (6)
M
M , (6)
00 PP
N {M
where
whereeach
eachAAj jPPM
MN {M. .For
For the
the eigenvalue
eigenvalue decomposition
decomposition
-1
ofofthe
adjacency
matrix
A
V
ΛV
the adjacency matrix A V ΛV -1, , when
when we
we have
have the
the
following
double
indexing
scheme
following double indexing scheme
ΛΛdiag
diag rλrλ1,1
1,M
NN{M,1
1,1 λλ
1,M λλ
{M,1 λλNN{{M,M
M,Mss ,,
VV vv1,1
1,M
1,1 vv
1,M vvNN{M,1
{M,1 vvNN{{M,M
M,M ,,
(7)
(7)
(8)
(8)
x
H0 pAq
D
..
.
..
.
DT
..
.
D
DT
HM -1 pAq
F0 pAq
y
..
.
FM -1 pAq
Fig.
filter bank
bank on
on aa graph
graph with
with
Fig. 1.
1. An
An M
M-channel
-channel maximally
maximally decimated
decimated filter
adjacency
Fk A
A are
are polynomials
polynomials in
in A
A (so
(so
adjacency matrix
matrix A.
A. Here
Here H
Hkk A
A and
and F
k
they
The decimation
decimation matrix
matrix D
D isis as
as in
in
they are
are linear
linear shift-invariant
shift-invariant systems
systems [1]).
[?]). The
(1)
of the
the filter
filter bank
bank is
is denoted
denoted
(1) with
with decimation
decimation ratio
ratio M
M .. Overall
Overall response
response of
as
as TT A
A ,, that
that is,
is, y
y T
T A
A x.
x.
pp qq
pp qq
pp qq
pp qq
A.
Shift-Varying Systems
Systems
A. Graph
Graph Filter
Filter Banks
Banks as
as Periodically
Periodically Shift-Varying
In
polynomials
In classical
classical filter banks where the filters are polynomials
in
decimated analanalin the
the shift
shift operator
operator z -1 , the maximally decimated
ysis/synthesis
shift-variant
ysis/synthesis system is a linear periodically shift-variant
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(LPSV(M )) system (i.e., the system from x to y in Fig. 1
is LPSV). This is always true regardless of what the filter
coefficients are. In particular, if the filter coefficients are such
that this LPSV(M ) system reduces to an LTI system, then
the system can be shown to be alias-free (and vice versa) [4].
Furthermore if this LTI system is a pure delay c z -n0 then the
system has the perfect reconstruction (PR) property. Since our
main goal is to get insights into a parallel theory for graph
filter banks, we now discuss the role of the periodically shiftvarying property for graph filter banks with polynomial filters.
For the filter bank in Fig. 1, let Tk pAq denote the response
of the k th channel. Therefore we have,
Tk pAq Fk pAq D T D Hk pAq,
(13)
Proof: The LPSV(M ) property, by Definition 1, is equivalent to
M
¸-1
T pAq k 0
Tk pAq M
¸-1
Fk pAq D T D Hk pAq.
M
¸-1
Tk pAq,
(16)
k 0
Since Tk pAq is as in (13), this is true for all polynomial filters
Hk pAq and Fk pAq if and only if
D T D AM
AM DT D.
(17)
Partition AM as in Eq. (18) of [1], and substitute into (17).
Then the result is pAM q1,2 0 and pAM q2,1 0, which is
the same as the condition (4). Conversely, if (4) holds it is
obvious that (17) holds (because D T D is as in Eq. (13) of
[1]), so the LPSV(M ) property (16) follows.
III. G RAPH F ILTER BANKS ON Ω-G RAPHS
(14)
k 0
Notice that Tk pAq is a linear operator. However, the DU
operator, D T D, does not commute with A in general, hence
response of the k th channel is shift-varying. That is to say,
Tk pAq A A Tk pAq,
AM
k 0
and the overall response from x to y is
M
¸-1
Tk pAq AM
(15)
for arbitrary analysis and synthesis filters. Therefore we have
proved:
Theorem 1 (Graph filter banks are shift-varying). An arbitrary
maximally decimated M -channel filter bank on an arbitrary
graph is in general a linear but shift-varying system (i.e., the
mapping from x to y in Fig. 1 is linear and shift-varying). ♦
This is similar to classical multirate theory where an arbitrary filter bank is a time-varying system [4]. In fact, filter
banks in the classical theory are periodically time-varying
systems with period M . This leads to the question: is the filter
bank on a graph periodically shift-varying?
In classical theory, when a system relates the input xpn°q to the output y pnq in the following way
y pnq k an pk q xpn-k q, a linear and periodically timevarying system with period M is defined by the defining
equation an+M pk q an pk q in [4]. It can be shown that a linear
system satisfies this relation if and only if the following is
true: if xpnq produces output y pnq, then xpn+M q produces
output y pn+M q. Motivated by this, we present the following
definition.
Definition 1 (Periodically shift-varying system). A linear
system H on a graph A is said to be periodically shift-varying
with period M , LPSV(M ), if AM H HAM . This reduces
to shift-invariance when M 1.
♦
Using this definition, we state the following result for the
periodically shift-varying response of a maximally decimated
M -channel filter bank on graphs.
Theorem 2 (Periodically shift-varying filter banks). The graph
FB in Fig. 1 is LPSV(M ) for all choices of the polynomial
filters tHk pAq, Fk pAqu, if and only if the adjacency matrix
of the graph satisfies the noble identity condition in (4). ♦
In this section we will construct maximally decimated M channel filter banks with perfect reconstruction (PR) property
on Ω-graphs. These filter banks are ideal in the sense that
each channel has a particular sub-band and there is no overlap
between different channels. The key point here is that this
construction uses analysis and synthesis filters that are both
alias-free (diagonal matrices in the frequency domain according to Def. 7 of [1]). If the filters do not have to satisfy
this restriction, then the problem of constructing PR filter
banks becomes rather trivial. (See Sec. III of [1].) In fact,
one can find low order polynomial filters by first designing
unconstrained filters (or, polynomial filters with high degree
as in [6], [12]) with PR property, then computing their low
order polynomial approximations. This approach results in a
filter bank with approximate PR property, with a trade-off
between the approximation error and the degree of the filters.
However, our approach here is to directly design alias-free
filters. Later in Sec. IV we will study how we can design low
order polynomial filters directly.
Remember that Ω-graphs do not have any constraints on
eigenvalues, however, the eigenvectors of Ω-graphs satisfy the
Ω-structure given in (10). In Sec. VII, we will show how this
constraint on eigenvectors can be removed by generalizing the
definition of the decimator D.
Remember from Eq. (68) of [1] that DU operation, D T D,
results in aliasing for an arbitrary graph signal. Nonetheless,
we can still recover the input signal from the output of D T D
if the input signal has zeros in its graph Fourier transform. To
discuss this further, we define band-limited signals on graphs
as follows:
Definition 2 (Band-limited graph signals on Ω-graphs). Let
A be the adjacency matrix of an Ω-graph with the following
eigenvalue decomposition A V ΛV -1 . A signal x on this
Ω-graph said to be k th -band-limited when its graph Fourier
p V -1 x, has zeros in the following way:
transform, x
x
pi,j
0,
1¤j
¤ M,
j
k,
1 ¤ i ¤ N {M, (18)
where we used the double indexing scheme similar to (7) and
pi,j .
(8) to denote the graph Fourier coefficients x
♦
In the literature, there are different notions and definitions
for band-limited graph signals [16]–[20]. Under an appropriate
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Transactions on Signal Processing
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re-indexing of the eigenvalues (and the eigenvectors), our
notion of k th -band-limited signal is similar to the one in [16]
with bandwidth N {M . This is consistent with our purpose of
constructing M -channel graph filter banks.
When a graph signal is k th -band-limited according to Definition 2, due to Eq. (68) of [1], the output spectrum of DU
operation becomes as follows.
1
x
pi,k .
ypi,1 ypi,2 ypi,M (19)
M
In this case we can recover the input signal from the output
of the DU operator since only one of the aliasing frequency
components is non-zero. For this purpose, let F be a linear
filter with frequency response V -1 F V I N {M b M ek eTk ,
where ek is the k th element of the standard basis for C M .
Then, consider the following system,
z
FD
T
(20)
D x.
Due to Eq. (66) of [1] and the construction of F above, in the
graph Fourier domain (20) translates to the following,
1
p p
z
I N {M b M ek eTk I N {M b
11T x
M
I N {M b ek 1T xp .
(21)
p is assumed to be k th -band-limited according to
Since x
px
p . That is, we can reconstruct the
Definition 2, we get z
original signal from its decimated version using the linear
reconstruction filter F .
Notice that the frequency response of F is a diagonal matrix
by its definition. Therefore F is an alias-free filter due to
Definition 7 of [1]. Furthermore, when the graph is assumed
to have distinct eigenvalues, F can be realized as a polynomial
filter due to Theorem 11 of [1]. We have therefore proved the
following theorem.
Theorem 3 (Polynomial interpolation filters for Ω-graphs).
Let A be the adjacency matrix of an Ω-graph. Let x be a
k th -band-limited signal on the graph. Then, there exists an
interpolation filter F that recovers x from Dx. When the
eigenvalues are distinct, F can be a polynomial in A, that is,
F F pAq.
♦
The recovery of missing samples in graph signals is also
discussed in various settings [16], [21], [22].
In the following, we will discuss how we can write an
arbitrary full-band signal as a sum of k th -band-limited signals.
For this purpose, consider the following identity,
IN
M̧
I N {M
b ek eTk .
(22)
k 1
p
Then we can write x
pk
x
°M
xp k , where
k 1
I N {M
b ek eTk
p.
x
(23)
p k is a k th -band-limited signal.
With the construction in (23), x
Notice that (23) is a frequency domain relation. In the graph
signal domain, consider a linear filter H k-1 with frequency
response
x k-1
H
V -1 H k-1 V I N {M b ek eTk .
(24)
°M
Then, we have x k1 xk , where xk H k-1 x. Since xk
is a k th -band-limited graph signal, we can reconstruct it from
its decimated version using the interpolation filter discussed
in Theorem 3. For this purpose, let F k-1 be a linear filter with
frequency response V -1 F k-1 V I N {M b M ek eTk . We will
then have:
xk
F k-1 DT D xk F k-1 DT D H k-1 x.
(25)
So we have proved the following theorem.
Theorem 4 (PR filter banks on Ω-graphs). Let A be the
adjacency matrix of an Ω-graph with the following eigenvalue
decomposition A V ΛV -1 . Now consider the maximally
decimated filter bank of Fig. 1 with analysis and synthesis
filters as follows:
H k-1 V
I N {M
b ek eTk
F k-1 M H k-1 (26)
V -1 ,
for 1 ¤ k ¤ M . This is a perfect reconstruction system, that
is, T pAq x x for all graph signals x. When the eigenvalues
are distinct, H k ’s can be designed to be polynomials in A,
that is, H k-1 Hk-1 pAq.
♦
Notice that the frequency response of each filter, (26), has
N {M nonzero values. These are similar to “ideal” bandlimited filters (with bandwidth 2π {M ) in classical theory. In
classical filter bank theory it is well known that an M -channel
maximally decimated filter bank has perfect reconstruction if
the filters are ideal “brickwall” filters chosen as
Hk pe
jω
q
#
1, 2πk {M ¤ ω
0, otherwise,
¤ 2πpk+1q{M,
(27)
and Fk pejω q M Hk pejω q. The result of Theorem 4 for graph
filter banks is analogous to that classical result. Notice that
in classical theory, ideal filters have infinite duration impulse
responses, whereas the graph filters are polynomials in A with
at most N taps.
Figure 2(a) shows the details of one channel of the brickwall
analysis bank, and Fig. 2(b) shows the corresponding channel
of the synthesis bank. The analysis filters have the form
H k V S k V -1 where V -1 is the graph Fourier transform
matrix and S k I N {M b ek eTk is a diagonal matrix (band
selector) which retains N {M outputs of V -1 and sets the rest
to zero. For example if N 6 and M 3 the three selector
matrices are
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
, S20
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
, S30
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
,
0 0
0 0
0 1
(28)
Once the appropriate outputs of the k th band have been
selected, the matrix V in H k is used to convert the subband
signal back to the “graph vertex domain.” (This is similar to
implementing the filtering operation in the frequency domain
and taking inverse Fourier transform to come back to time
domain.) This graph domain subband signal is then decimated
by D. For reconstruction, the synthesis filter F k is similarly
1
0
0
S1
0
0
0
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Transactions on Signal Processing
IEEE TRANSACTIONS ON SIGNAL PROCESSING
x
N
V
-1
(a)!
5
N/M
band selector k
V
D
x
k
H
k
x
k
N/M
D
T
V
-1
band selector k
N
V
some integer n as we shall see.
following result:
We begin by proving the
Theorem 5 (PR filter banks on M -block cyclic graphs).
Consider the graph filter bank of Fig. 1 and assume that
the adjacency matrix of the graph is diagonalizable M -block
cyclic. With no further restrictions on the graph, the system
has perfect reconstruction if and only if
y
k
(b)!
1
M
¸-1
Fk pλq Hk pwl λq M λn δ plq,
(29)
k 0
F
k
Fig. 2. (a) The kth channel of a graph filter bank, and (b) details of this
channel when the filters are brickwall filters as defined in Theorem 4.
used. Thus, the implementation of the brick wall filter bank
is not merely a matter of using the graph Fourier operator
V -1 , it also involves band selection, inverse transformation,
and decimation. Since V -1 is common to all analysis filters, it
contributes to a complexity of N 2 multiplications. The band
selector S k , and the matrices V and D can be combined
and implemented with pN {M q2 multiplications for each k,
so there is a total of N 2 {M multiplications for this part.
So the decimated analysis bank has complexity of about
N 2 N 2 {M which is OpN 2 q. Once again, by approximating
these brickwall filters with polynomial filters with order L we
can reduce the complexity to N L. (See Fig. 7 of [1].)
The perfect reconstruction result in Theorem 4 is restricted
to Ω-graphs. However this restriction can be removed by
applying a similarity transformation to the graph as described
later in Sec. VII.
As a final remark, filters in (26) are not unique in the sense
that we can design different filters and still have T pAq I in
the filter bank.
IV. G RAPH F ILTER BANKS ON M -B LOCK C YCLIC G RAPHS
In Theorem 4 aliasing was totally suppressed in each
channel by the use of ideal filters (26), which explains why
the filter polynomials had order N -1. But if we resort to
cancellation of aliasing among different channels, it is less
restrictive on the filters. While this idea is not easy to develop
for arbitrary graphs, the theory can be developed under some
further assumptions on the graph, namely that A be M -block
cyclic as we shall see. We will see that such filter banks
have many advantages compared to those given by Theorem 4,
which does not use the M -block cyclic assumption and applies
to more general class of Ω-graphs.
As shown in Theorem 4 of [1], M -block cyclic graphs satisfy the noble identity condition in (4). Therefore, Theorem 2
shows that a maximally decimated M -channel filter bank on
an M -block cyclic graph is a periodically shift-varying system
with period M . This can be interpreted as aliasing when
the graph A is diagonalizable (Theorem 10 and Fig. 10 of
[1]). As in the classical case, it is possible to cancel out
aliasing components arising from different channels, and in
fact, achieve perfect reconstruction, that is, T pAq An for
for some n, for all λ P C, and for all l in 0 ¤ l ¤ M -1, where
δ pq is the discrete Dirac function.
♦
Proof: The overall response of the system in Fig. 1 can
be written as
T pAq
M
¸-1
Fk pAq D T D Hk pAq,
k 0
M -1 M -1
1 ¸ ¸
Fk pAq Ωl Hk pAq,
M l0 k0
(30)
where we have used Eq. (58) of [1]. For simplicity, define:
Sl pAq M
¸-1
Fk pAq Ωl Hk pAq.
(31)
k 0
Therefore, the response of the system is:
T pAq 1
M
p q
p q p q
Somo
0 A
lo
on
S1 A
SM -1 A
loooooooooooooomoooooooooooooon
Polynomial
Alias components
. (32)
Notice that S0 pAq is the sum of products of polynomials in
A, therefore it is also a polynomial in A, hence it is alias-free
since A is assumed to be diagonalizable in [1]. However, for
l ¥ 1, there exists Ωl term in each Sl pAq in (31), which does
not commute with A in general. As a result Sl pAq is not shiftinvariant and results in aliasing (Theorem 10 of [1]). Perfect
reconstruction T pAq An can be achieved by imposing:
S0 pAq M An ,
M
¸-1
Sl pAq 0.
(33)
l 1
The second equation above is the alias cancellation condition.
Using the eigenvalue decomposition of the adjacency matrix,
A V ΛV -1 , the first condition in (33) reduces to:
V
M
¸-1
Fk pΛq Hk pΛq V -1
M V Λn V -1 .
(34)
k 0
Since Λ is a diagonal matrix consisting of eigenvalues, this
can be further simplified to
M
¸-1
Fk pλi,j q Hk pλi,j q M λni,j ,
(35)
k 0
for all eigenvalues, λi,j , of the adjacency matrix A.
pq
1 In analogy with classical filter banks where T z
z -n signifies the
PR property, we take T A
An to be the PR property. But this makes
practical sense only in situations where A is invertible so that the distortion
An can be canceled.
p q
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Transactions on Signal Processing
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6
Now consider the second condition in (33). With the eigenvalue decomposition, it reduces to
M
¸-1 M
¸-1
Fk pΛq V -1 Ωl V Hk pΛq 0.
(36)
l 1k 0
Since A is M -block cyclic, it satisfies the eigenvector structure
in (10). So we can use Eq. (63) of [1] and re-write (36) as:
M
¸-1 M
¸-1
Fk pΛq Πl Hk pΛq 0.
(37)
l 1k 0
Notice that the permutation matrix Πl is defined as the lth
power of the cyclic matrix of size M (Eq. (64) of [1]). Since
1
2
the supports of C lM
and C lM
have no common index for
different l1 and l2 , supports of Πl1 and Πl2 will also have
no common index. Furthermore, due to Λ being a diagonal
matrix, the support of the term Fk pΛq Πl Hk pΛq will be the
same as the support of Πl . Combining both arguments, we
can say that (37) holds if and only if the inner sum is zero for
each l, that is,
M
¸-1
Fk pΛq Πl Hk pΛq 0,
@ l P t1, , M -1u.
(38)
k 0
Since Λ is a diagonal matrix with the ordering scheme in
(7), the permutation Πl circularly shifts each eigenvalue of an
eigenfamily. Hence, (38) is equivalent to:
M
¸-1
Fk pλi,j+l q Hk pλi,j q 0,
(39)
k 0
for all 1 ¤ l ¤ M -1. Since A is M -block cyclic, eigenvalues
of A satisfy (9). Then (39) can be written as:
M
¸-1
Fk pλi,j+l q Hk pwM -l λi,j+l q 0.
(40)
k 0
By changing the index variables, (40) can be simplified as
follows:
M
¸-1
Fk pλi,j q Hk pwl λi,j q 0,
(41)
k 0
for all 1 ¤ l ¤ M -1 and for all eigenvalues, λi,j , of the
adjacency matrix A. Combining (35) and (41), if the filter
bank in Fig. 1 provides PR, (29) should be satisfied for all
eigenvalues of A. Since we want PR independent of the graph,
(29) should be satisfied for all λ P C. Hence, it is a necessary
condition.
Conversely, assume that the filters satisfy (29), hence (35)
and (41) are satisfied. Since the graph is assumed to be
diagonalizable M -block cyclic, (35) and (41) are equivalent to
(33). Therefore, the overall response of the filter bank is An ,
where n satisfies (35). So the filter bank has the PR property.
It is quite interesting to observe that the condition (29) on
the graph filters is the same as the PR condition in classical
multirate theory. We state this equivalence in the following
theorem.
Theorem 6 (PR condition equivalence). A set of polynomials,
tHk pλq, Fk pλqu, provides PR in maximally decimated M channel FB on all M -block cyclic graphs with diagonalizable
adjacency matrix if and only if tHk pz q, Fk pz qu provides PR
in the classical maximally decimated M -channel FB.
♦
Proof: The PR condition (29) is the same as the PR
condition for the classical maximally decimated M -channel
filter bank [4].
At the moment of this writing, we do not have examples of
graphs other than M -block cyclic for which such results can
be developed. At the expense of restraining the eigenvalues of
the adjacency matrix to have the structure in (9), Theorem 5
offers three significant benefits compared to construction of
PR filter banks on Ω-graphs discussed in Sec. III.
First of all, (29) puts a condition on the filter coefficients
independent of the graph as long as the graph is M -block
cyclic with diagonalizable adjacency matrix. A change in the
adjacency matrix does not affect the PR property. As a result,
the response of the overall graph filter bank is robust to
ambiguities in the adjacency matrix.
Secondly, eigenvalues of the adjacency matrix A do not
need to be distinct since the condition solely depends on the
filter coefficients.
Lastly, filter banks on M -block cyclic graphs are legitimate
generalization of the classical multirate theory to graph signals
due to Theorem 6. In order to design PR filter banks on an M block cyclic graph, we can use any algorithm developed in the
classical multirate theory [4], [23]. As an example, consider
the following set of polynomials,
H0 pλq 5+2λ+λ3 +2λ4 +λ5 ,
F0 pλq 3λ3 -2λ4 +λ5 , (42)
H2 pλq λ3 +2λ4 +λ5 ,
F2 pλq 1+13λ3 -8λ4 +3λ5 .
H1 pλq 2+λ+2λ3 +4λ4 +2λ5 , F1 pλq -8λ3 + 5λ4 -2λ5 ,
In classical theory, this is a 3-channel PR filter bank
[4]. Notice that these polynomials satisfy (29) with n 5.
Therefore, overall response of the filter bank constructed with
the polynomials in (42) on any 3-block cyclic graph will be
T pAq A5 .
For a randomly generated adjacency matrix of a 3-block
cyclic graph, channel responses of the filter bank utilizing the
filters in (42) are shown in Fig. 3. Response of each channel
has non-zero blocks with large values in it (relative to the
overall response of the FB), which results in large alias terms.
Notice that some blocks of the channel responses cancel out
each other exactly so that overall response, T pAq, is equal to
A5 (up to numerical precision) as seen from Fig. 3(f).
V. P OLYPHASE R EPRESENTATIONS
In [1], for any given polynomial filter H pAq on any graph
A, we wrote its Type-1 polyphase representation as
H pAq M
¸-1
Al El pAM q,
(43)
l 0
and its Type-2 polyphase decomposition as
H pAq M
¸-1
AM -1-l Rl pAM q,
(44)
l 0
1053-587X (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2016.2620111, IEEE
Transactions on Signal Processing
IEEE
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Fig.Fig.
3. 3.(a)(a)
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graph
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Fig.
maximally decimated
decimated M
M-channel
-channel
values
of
matrices.
values
of
thethe
matrices.
values of the matrices.
Fig.
4.bank
(a) in
Polyphase
of adjacency
the
maximally
decimated
M -channel
filter
bank
in Fig.
Fig. 11 on
onrepresentation
matrix
A
the
filter
aa graph with the
adjacency
matrix
A that
thatsatisfies
satisfies
the
filter
bank
in Fig.
1 on a graph
the adjacency
matrix A
satisfies
the
noble
identity
condition
representation
of
the
noble
identity
condition
(4). (b)with
Combined
representation
of that
the polyphase
polyphase
matrices.
The condition
decimation(4).
matrix
D is as in representation
(1)
transfer
noble
identity
(b) Combined
of the polyphase
matrices.
The
decimation
(1) and
and the
the polyphase
polyphase
transfer
whereEkEk ’s ’sand
andRR
’s
are
polynomials.
where
’s
are
polynomials.
k
k
s matrix
matrix isisThe
givendecimation
in (47).
(47). A
A
is as inD
(5).is as in (1) and the polyphase transfer
matrices.
matrix
given
in
whereFor
E
’spolyphase
and
Rk implementation
’simplementation
are polynomials.
kthe
For
the
polyphase
the filter
filter bank
bank inin matrix is given in (47). A
ofof the
s is as in (5).
p pq pq q
p p qp q q
p p pq qq
p pq pq q
pq
pqpq
pq
pqpq
For
the
polyphase
implementation
of filters
the filter
bank
in
Fig.
wedecompose
decompose
the analysis
analysis
filters
using
Type-1
Fig.
1, 1,we
the
using
Type-1
Fig.polyphase
1,
we
decompose
the
analysis
filters
using
Type-1
polyphasestructure
structureinin(43)
(43)and
andsynthesis
synthesisfilters
filterswith
with Type-2
Type-2 in
in Fig.
Fig. 11 has
has the
the polyphase
polyphase implementation
implementation given
given in
in Fig.
Fig.4(a)4(a)polyphase
structure
in(44).
(43)
andwe
synthesis
ssq,, is
decomposition
Then
we
have, filters with Type-2 in
Fig.
1 has
the polyphase
given
in(47),
Fig. and
4(a)decomposition
inin(44).
Then
have,
(b)
where
polyphase
matrix,
(b)
where
polyphase
transferimplementation
matrix, P
P pA
A
is as
as in
in
(47),
and
decompositionMin
s where
s(5).
sA
is
the
adjusted
shift
operator
given
in
♦
-1 (44). Then we have,
M
-1
(b)
polyphase
transfer
matrix,
P
p
A
q
,
is
as
in
(47),
and
A
is
the
adjusted
shift
operator
given
in
(5).
♦
M
-1
M
-1
¸
¸
¸
¸
M
-1-l
M
l l Ek,l pA
MM
-1-l
M
s is the adjusted shift operator given in (5).
M
-1
M
-1 A
q
p
A
q
A
R
p
A
q
.
A
♦
k
l,k
HkHpkApA
q
AA
Ek,l pAM q,q, FF
p
A
q
R
p
A
q
.
¸
¸
k
l,k
Since M
M-block
-block cyclic
cyclic matrices
matrices satisfy
l
M
M -1-l
Since
satisfy the
the noble
noble identity
identity
l00Ek,l pA q,
0
Hk pAq lA
Fk pAq ll0A
Rl,k pAM q.
condition
(4),
the polyphase
polyphase
representation
44 isisidentity
valid
(45) condition
Since M(4),
-block
cyclic matrices
satisfy of
the
noble
the
representation
of Fig.
Fig.
valid
(45)
l0
l0
for
graph
filter
banks
on
M
-block
cyclic
graphs.
However,
M
Using
the
polyphase
implementation
of
decimation
and
intercondition
(4),
the
polyphase
representation
of
Fig.
4
is
valid
(45)
for
graph
filter
banks
on
M
-block
cyclic
graphs.
However,
M-Using the polyphase implementation of decimation and interblock
cyclic
property
is
not
necessary
for
this.
The
condition
polation
filters
given
in
Eq.
(??)
and
Eq.
(??)
of
[1]
(they
for
graph
filter
banks
on
M
-block
cyclic
graphs.
However,
Mblock
cyclic
property
is
not
necessary
for
this.
The
condition
Using
the
polyphase
implementation
of
decimation
and
interpolation filters given in Eq. (??) and Eq. (??) of [1] (they
(4) isiscyclic
enough.
are
described
schematically
in
Fig.
??
and
Fig.
??
of
[1],
(4)
enough.
block
property
is
not
necessary
for
this.
The
condition
polation
filters
given
in
Eq.
(34)
and
Eq.
(35)
of
[1]
(they
are described schematically in Fig. ?? and Fig. ?? of [1],
In
Sec. III,
III, we
we showed
showed that
that it
respectively),
candefine
definethe
polyphase
component
matrices
it is
is possible
possible to
to construct
construct PR
PR
is Sec.
enough.
arerespectively),
described
schematically
inthepolyphase
Fig.
4 and
Fig. 5 matrices
of
[1], (4) In
wewecan
component
filter
banks
on
Ω-graphs.
In
order
to
talk
about
polyphase
as
follows:
filter
banksIII,onweΩ-graphs.
In order
talk about
polyphase
In Sec.
showed that
it is to
possible
to construct
PR
as follows: we can define the polyphase component matrices
respectively),
implementation of
of such
such filter
filter banks,
we
need
to
characterize
implementation
banks,
we
need
to
characterize
s
s
s
s
filter
banks
on
Ω-graphs.
In
order
to
talk
about
polyphase
as follows:
q i,j Ri-1,j-1 psAq, (46) the set of graph matrices A that satisfy both the noble identity
i-1,j-1
EE
pAspA
q qi,ji,jEEi-1,j-1
pAspAq,q, RRpAspA
the set of graph matrices
A thatbanks,
satisfywe
both
the to
noble
identity
q i,j Ri-1,j-1 pAq, (46) implementation
of such filter
need
characterize
s
s
s
s q, matrix
condition in (4) and have the Ω-structure in (10). From
N
s
Efor
p
A1q ¤i,ji,j E
p
Aq, EspA
RqpPAM
q
p
A
(46) condition
i-1,j-1
i-1,j-1
¤
M where
is R
the
polyphase
in
(4)
and
have
the
Ω-structure
in
(10).
From
Ni,j the
set
of
graph
matrices
A
that
satisfy
both
the
noble
identity
for 1 ¤ i, j ¤ M where E pAsq P M N is the polyphase matrix Theorem ?? and Theorem ?? of [1], we know that M -block
for the analysis filters, R
p
Aq P M
the polyphase matrix condition
Theorem ??
and
Theorem
??the
of [1],
we know in
that(10).
M -block
s
NN isisthe
in
(4)
and
have
Ω-structure
From
s
for
the
analysis
filters,
R
p
A
q
P
M
polyphase
matrix
for 1for
¤ i,thej ¤synthesis
M where
E pA
q PsAsMis the
is the
polyphase
cyclic graphs with diagonalizable adjacency matrices belong
filters,
and
adjusted
shift matrix
operator cyclic
graphs
with
diagonalizable
adjacency
matrices
belong
Nthe adjusted shift operator Theorem
4
and
Theorem
5
of
[1],
we
know
that
M
-block
s
for
the
synthesis
filters,
and
A
is
for the
analysis
pAeach
qPM
to that set. It is interesting to observe that any matrix that
given
in (5). filters,
Notice R
that
blockispEthe
psAsqqpolyphase
qqi,ji,jand
and pRmatrix
pAsAsqqqqi,ji,j cyclic
to thatgraphs
set. It is interesting
to observe
that any
matrixbelong
that
adjacency
matrices
s block
in (5). Notice
thats
each
p
E
p
A
p
R
p
belongs to thiswith
set isdiagonalizable
similar to an M
-block cyclic
graph. We
forgiven
the
synthesis
filters,
and
A
is
the
adjusted
shift
operator
is a polynomial insA, whereas overall polyphase matrices belongs
to
this
set
is
similar
to
an
M
-block
cyclic
graph.
We
thatthis
set.fact
It in
is the
interesting
observewhose
that any
that
is ainpolynomial
A, whereas
s qqi,j tostate
followingtotheorem
proofmatrix
is given
given
Notice
that
block overall
pE pAs inqqpolyphase
A
s(5).
sin
s and pRpmatrices
i,j
EspA
q
and
RspA
q
areeach
not polynomials
A.
state
this
fact
in
the
following
theorem
whose
proof
is
given
s
E
p
A
q
and
R
p
A
q
are
not
polynomials
in
A.
belongs
to
this
set
is
similar
to
an
M
-block
cyclic
graph.
We
s
in
Sec.
??
of
the
supplementary
document
[24].
is a polynomial
A, whereas
overall
polyphase
Using the in
polyphase
matrices
defined
in (46),matrices
we can in Sec. ?? of the supplementary document [24].
Using
the
polyphase
matrices
defined
in
(46),
we
can
state
this
fact
in
the
following
theorem
whose
proof
is
given
s
s
s
E pAimplement
q and RpAtheq are
notbank
polynomials
filter
in Fig. 1 in
as A.
in Fig. 4(a). This can Theorem 8 (The noble identity condition and the eigenvector
implement
bank
Fig. defined
1Pas
This
can
Sec. I of the
supplementary
document
[24]. eigenvector
sin
s q Ewe
Theorem
(The
identityeigenvalues
condition and
Using
the the
polyphase
matrices
inR(46),
be redrawn
asfilter
in Fig.
4(b)inwhere
p
A
q
Fig.
ps4(a).
A
pAsAsqq. .can
Due in
structure).8Let
A noble
have distinct
with the
the eigenvalue
s q RpA
be
redrawn
as
in
Fig.
4(b)
where
P
p
A
q
E
p
Due
-1
structure).
Let
A
have
distinct
eigenvalues
with
the
eigenvalue
to partitioning
the polyphase
in can
(46), Theorem
implement
the filterofbank
in Fig. 1 component
as in Fig. matrices
4(a). This
decomposition
A
V
ΛV
.
If
A
satisfies
the
noble
identity
8 (The noble identity
condition and the
eigenvector
-1
to
partitioning
of
the
polyphase
component
matrices
in
(46),
s
s
s
s
decomposition
A
V
ΛV
.
If
A
satisfies
the
noble
identity
we haveasthe
:
be redrawn
in following
Fig. 4(b) result
wherefor
P pthe
Aqpartitions
RpAqof
E pPApsA
q
. qDue
condition
in
(4)
and
has
the
Ω-structure
in
(10),
then
A is
structure). Let A have distinct eigenvalues with the eigenvalue
we have the following result for the partitions of P pAq:
condition
in
(4)
and
has
the
Ω-structure
in
(10),
then
A is
-1
to partitioning of the polyphase
component
matrices
in
(46),
similar
to
an
M
-block
cyclic
matrix.
More
precisely,
there
M
-1
decomposition A V ΛV . If A satisfies the noble identity
¸
there
-1
similar
to
an
M
-block
cyclic
matrix.
More
precisely,
M
-1
s
s
s
s
we have the following
pP pAqqresult
¸forRthe
pAq Ek,j-1ofpsAPq.pAq: (47) condition
exists a permutation
such that VinΠ(10),
Π -1Ais is
Λ Vthen
i-1,kpartitions
in (4) and matrix
has theΠΩ-structure
pP pAs qqi,ji,jM -1kR0 i-1,k
pAs q Ek,j-1
pAq.
(47) exists
a permutation
matrix Π such that V Π Λ V Π ♦is
M
-block
cyclic.
similar to an M -block cyclic matrix. More precisely,
¸
there
k 0
M -block cyclic.
♦
spolyphase
s q Eprovides
s the polyphase
Notice pthat
transfer
P pA
qq
Ri-1,kmatrix
p
A
(47) exists a permutation matrix Π such that V Π Λ V Π -1 is
i,j k,j-1 pAq.
Notice
that
polyphase
transfer
matrix
provides
the
polyphase
implementation of thekfilter
bank in Fig. 1, only if the graph MA.-block
0
Alias-free
and PR Property for M -Block Cyclic Graphs ♦
cyclic.
implementation
of the
filter condition
bank in Fig.
1, only
if the graph
satisfies the noble
identity
in (4).
Summarizing,
we A. Alias-free and PR Property for M -Block Cyclic Graphs
Notice
thatthe
polyphase
transfer
matrix provides
the polyphase
Returning to Fig. 4 for the polyphase representation of a
satisfies
noble identity
condition
in (4). Summarizing,
we
have proved:
Returning
to
for the
polyphase
representation
ofona
graph
filter bank,
we4Property
now
provide
a -Block
sufficient
condition
implementation
of
the
filter
bank
in
Fig.
1,
only
if
the
graph
A.
Alias-free
andFig.
PR
for M
Cyclic
Graphs
have proved:
s
graph
filter
bank,
we
now
provide
a
sufficient
condition
Theorem
7 (Polyphase
implementation
a filter bank). On
is
satisfies
the noble
identity condition
in (4).ofSummarizing,
we a P pAq for the PR property on M -block cyclic graphs. This on
4 for
theMpolyphase
representation
Theorem
7
(Polyphase
implementation
of
a
filter
bank).
On
a PanReturning
pAsextension
q for thetoofPRaFig.
property
on
-block
graphs.
Thisofis a
graph
with
the
adjacency
matrix
satisfying
the
noble
identity
similar
condition
(P pz qcyclic
I) that
guarantees
have proved:
bank,
webanks
now
provide
graph
with the
matrix
satisfying
nobleFB
identity
an
of afilter
similar
condition
(Papzsufficient
q I) thatcondition
guaranteeson
condition
(4), adjacency
the maximally
decimated
M the
-channel
given graph
PRextension
infilter
classical
[4].
s q for the PR property on M -block cyclic graphs. This is
Theorem
7
(Polyphase
implementation
of
a
filter
bank).
On
a
P
p
A
condition (4), the maximally decimated M -channel FB given PR in classical filter banks [4].
graph with the adjacency matrix satisfying the noble identity an extension of a similar condition (P pz q I) that guarantees
condition (4), the maximally decimated M -channel FB given PR in classical filter banks [4].
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Transactions on Signal Processing
IEEE TRANSACTIONS ON SIGNAL PROCESSING
8
Theorem 9 (Sufficiency condition for PR in polyphase filter
banks on M -block cyclic graphs). Consider the polyphase
implementation of a maximally decimated M -channel filter
bank in Fig. 4(b), and assume that the adjacency matrix of
the graph, A, is M -block cyclic. The system has PR property
if the polyphase transfer matrix has the following form:
s q IM
P pA
b As m ,
♦
s q has the form in (48). Then,
Proof: Assume that P pA
overall response of the filter bank in Fig. 4(b) is written as:
M
¸-1
k 0
M
¸-1
s m D Ak ,
AM -1-k D T A
A
(49)
AM -1-k D T D Ak AM m ,
k 0
M -1+M m
(50)
,
where we use the first noble identity (2) in (49) and the lazy
FB PR property Eq. (31) of [1] in (50), since M -block cyclic
matrices satisfy both of these properties.
Notice that when we let m 0 in Theorem 9, we get
s q I N which corresponds to the lazy filter bank strucP pA
ture given in Fig. 3(b) of [1]. This observation agrees with the
result given by Theorem 4 of [1] for M -block cyclic graphs.
For the most complete characterization of the PR property
on M -block cyclic graphs, we provide the following result,
whose proof is provided in Sec. II of the supplementary
document [24].
Theorem 10 (PR polyphase filter banks on M -block cyclic
graphs). Consider the polyphase implementation of a maximally decimated M -channel filter bank in Fig. 4(b), and
assume that the adjacency matrix of the graph, A, is an
invertible M -block cyclic matrix. The system has PR property
if and only if the polyphase transfer matrix has the following
form:
p q
s
P A
IM
b
sm
A
I M -n b I N {M
s
In b A
0
0
sq
P0 pA
s
sq
A PM -1 pA
sq P pA
..
.
sq
P1 pA
sq
P0 pA
..
.
p q
..
.
p q s P1 A
s
A
s P2 A
s
A
sq
PM -1 pA
s q
PM -2 pA
. (52)
..
.
sq
P0 pA
♦
Proof: Assume that the overall response of the filter bank
is alias-free, and the graph has distinct eigenvalues. Then,
due to Theorem 11 of [1], the response of the filter bank is
a polynomial filter. By re-indexing the coefficients, we can
decompose the polynomial response as follows:
T pAq AM -1-k D T D AM m Ak ,
k 0
M
¸-1
(48)
s is as in (5).
for some non-negative integer m, and A
T pAq
the following pseudo-block-circulant form:
, (51)
s is as in (5). ♦
for some integers m, n with 0 ¤ n ¤ M -1. A
When we waive the perfect reconstruction property and ask
for an alias-free response, we get a more relaxed condition on
the polyphase transfer matrix. The following theorem states
the necessary condition for this case. This is a generalization
of the classical pseudocirculant condition for alias cancellation
[4] to graph filter banks.
Theorem 11 (Alias-free polyphase filter banks on M -block
cyclic graphs). Consider the polyphase implementation of a
maximally decimated M -channel filter bank in Fig. 4(b), and
assume that the adjacency matrix of the graph, A, is M block cyclic with distinct eigenvalues. The system has aliasfree property if and only if the polyphase transfer matrix has
{
N¸
M -1 M
¸-1
m 0
αm,k AM -1+M m+k .
(53)
k 0
Since T pAq is a linear combination of PR systems, using
Theorem 10 and linearity, we can write the polyphase transfer
matrix that corresponds to (53) in the following way:
{
N¸
M -1 M
¸-1
p q
s
P A
m 0
αm,k
k 0
0
s m+1
Ik b A
s
I M -k b A
0
m
. (54)
looooooooooooooooooomooooooooooooooooooon
pseudo-block-circulant
By inspection, we can see that the block-matrix in (54) is
a pseudo-block-circulant matrix. Since a linear combination
of pseudo-block-circulant matrices is also a pseudo-blockcirculant, the polyphase transfer matrix has the form in (52)
for some polynomials.
s q has the form in (52),
Conversely, if the matrix P pA
then we can decompose it as in (54) for some set of αm,k .
Therefore, due to Theorem 10, the overall response of the filter
bank is a linear combination of perfect reconstruction systems.
It is therefore alias-free.
B. Importance of Polyphase Representations for Graph Filter
Banks
We know that the polyphase implementation in Fig. 4
is valid as long as the conditions (4) and (5) for noble identities are satisfied. Here each polyphase component
s q P MN {M is a polynomial in the matrix Ā:
Ek,l pA
Ek,l pĀq ek,l p0q I
s
ek,l p1q A
s , (55)
ek,l pK q A
K
s DAM D T as in (5), and K L{M . (L is the
where A
s is sparse and
order of the filters in Fig 1.) Assuming that A
can be implemented with negligible overhead, the complexity
s q (implemented similar to Fig. 7 of [1]) is about
for Ek,l pA
pL{M qpN {M q so that the entire polyphase matrix with M 2
such submatrices has complexity LN , identical to the non
polyphase implementation of polynomial filter banks (with
each filter implemented as in Fig. 7 of [1]). In fact even if A
is sparse, the matrix Ā DAM D T is in general not. So the
polyphase implementation may even have higher complexity
than direct implementation of polynomial filters as in Fig. 7
of [1]. Thus, while in classical filter banks the polyphase
implementation reduces the number of multiplications per unit
time, there is no such advantage in graph filter banks.
1053-587X (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2016.2620111, IEEE
Transactions on Signal Processing
IEEE TRANSACTIONS ON SIGNAL PROCESSING
9
Then the question is, what is the advantage of the polyphase
representation of Fig. 4 for graph filter banks? The answer lies
in the design phase rather than the complexity of implementation. To explain, suppose we want to optimize the polynomial
analysis filters to achieve certain properties of the decimated
subband signals tx0 , x1 , . . .u (e.g., sparsity), by using some
apriori information on the statistics of the graph signal x. If
we do this directly by optimizing the multipliers hk pnq in
the structure of Fig. 7 of [1] then it is not easy to constraint
these coefficients (during optimization) such that there will
exist a perfect reconstruction polynomial synthesis filter bank
tFk pAqu. But if we optimize the coefficients ek,l pnq of the
polynomials (55), then it can be shown that as long as the
equivalent classical polynomial matrix
E pz q E0,0 pz q
E1,0 pz q
..
.
E0,1 pz q
E1,1 pz q
..
.
...
...
..
.
E0,M -1 pz q
E1,M -1 pz q
..
.
(56)
EM -1,0 pz q EM -1,1 pz q . . . EM -1,M -1 pz q
has a polynomial inverse Rpz q (i.e., FIR inverse), there will
exist a polynomial graph synthesis bank tFk pAqu with perfect
reconstruction property. Now, the construction of classical
polynomial matrices E pz q with polynomial inverses Rpz q is a
well studied problem in filter bank theory and includes special
families such as FIR paraunitary matrices, FIR unimodular
matrices and so forth [4]. So, we can take advantage of
the results from classical literature to design optimal graph
filter banks that suit specific signal statistics, if we use the
polyphase representation of Fig. 4 in the design process. The
implementation of each filter in the bank could later be done
directly using Fig. 7 of [1], which is more economical.
VI. F REQUENCY D OMAIN A NALYSIS
In order to quantitatively interpret the amount of fluctuation
of a signal, the total variation of the signal x on a graph with
the adjacency matrix A is defined as [10]:
Spxq }x Ax}1 ,
(57)
where it is assumed that the adjacency matrix is normalized
such that the maximum eigenvalue has a unit magnitude [10].
With the definition in (57), the frequency of an eigenvector v
with the corresponding eigenvalue λ becomes
Spv q |1 λ|,
(58)
where all the eigenvectors are assumed to be scaled such that
they have unit `1 norm.
When the eigenvalues are real, from (58) it is clear that
two eigenvectors with distinct eigenvalues cannot have the
same amount of total variation. Hence, distinct eigenvalues
imply distinct total variations, and vice versa. However, for
complex eigenvalues this implication does not hold. As a
simple ?example
? consider two distinct complex eigenvalues
λ1 p 2-1q{ 2 and λ2 1{2+j {2. Clearly λ1 λ2 , yet
we have Spv 1 q Spv 2 q.
The problem with complex eigenvalues arises when we
want to describe the frequency domain behavior of a polynomial filter, H pλq. How the filter suppresses or amplifies
eigenvectors according to their total variation determines the
characteristics. When there are complex eigenvalues, a filter
may respond differently to eigenvectors with the same total
variation. More precisely, we may have |H pλ1 q| |H pλ2 q|
even when Spv 1 q Spv 2 q. As a result, we may not even be
able to define the behavior of the filter (low-pass, high-pass
etc.). Therefore, filters cannot be designed independent of the
graph spectrum when the spectrum has complex values.
The question we ask here is as follows: under what conditions can we design filters with meaningful behavior for all
graphs that satisfy such conditions? One obvious answer is the
case of real eigenvalues since Spv 1 q Spv 2 q implies λ1 λ2 ,
hence |H pλ1 q| |H pλ2 q|. As a result, a filter behaves in the
same way for all graphs with real eigenvalues.
For the complex case, we cannot answer the question in
the general sense for the time being. Nonetheless, when we
constrain the eigenvalues to be on the unit circle, we have the
following result.
Theorem 12 (Magnitude response of a polynomial filter,
and unit-modulus eigenvalues). Let H pλq be a polynomial
filter with real coefficients. Then H pλq has a well-defined
magnitude response w.r.t. the total variation for all graphs
with diagonalizable adjacency matrices with unit modulus
eigenvalues.
♦
Proof: Let eigenvalues of the adjacency matrix be on the
unit circle. Then we have λ ej θ for some 0 ¤ θ 2π.
Then, the total variation in (58) reduces to
Spv q |1 ej θ | 2 | sinpθ{2q|.
(59)
Due to symmetry of (59), eigenvectors will have the same
total variation if and only if the corresponding eigenvalues are
conjugate pairs. Assume that H pλq is a polynomial filter with
real coefficients. Then, its magnitude response will have the
conjugate symmetry, which is to say |H pλq| |H pλ q|, for all
|λ| 1. As a result, even though the same total variation may
correspond to a conjugate eigenvalue pair, those eigenvalues
will be mapped to the same magnitude. Hence, magnitude
response of the filter with respect to the total variation will
be well-defined and remain the same for all graphs with unit
modulus eigenvalues.
A drawback of (59) is the non-linearity of it in terms of
the phase angle. We will demonstrate this in the following
example. Consider the filter bank coefficients provided in
Table 6.5.1 on page 318 of [4]. These analysis filters are
optimized for perfect reconstruction in 3-channel filter banks
in the classical multirate theory with synthesis filters having
the following coefficients fk,l hk,14-l [4]. Furthermore, they
are designed such that each filter allows only one uniform
sub-band to pass. That is, Hk pz q passes frequencies in the
range |ω | P rπk {3 π pk+1q{3s. Their magnitude responses are
shown in Fig. 5(a). When all the eigenvalues of A are on the
unit circle, we can quantify the magnitude response of each
analysis filter w.r.t. the total variation due to Theorem 12.
However, the pass bands of H?
pAq and H2 pAq in
0 pAq, H1?
the total variation are r0, 1s, r1, 3s and r 3, 2s, respectively.
They are visualized in Fig. 5(b).
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Transactions on Signal Processing
0
0
−10
−10
Response (dB)
Response (dB)
IEEE TRANSACTIONS ON SIGNAL PROCESSING
−20
−30
−40
|H (ejω)|
−50
|H1(ejω)|
−60
0
0
0.2
0.3
0.4
Normalized Frequency (ω/2π)
0.5
onalizable M -block cyclic with unit magnitude eigenvalues.
If the analysis and the synthesis filters satisfy (29) with real
polynomial coefficients, then, the filter bank provides perfect
reconstruction, and each channel has a well-defined magnitude
response w.r.t. total variation spectra.
♦
−20
−30
|H0(λ)|
−40
|H1(λ)|
−50
|H2(ejω)|
0.1
10
−60
0
|H2(λ)|
0.5
1
1.5
2
Total Variation (|1−λ|)
(a)
(b)
Fig. 5. Magnitude response of the filters given in Table 6.5.1 of [4]. (a) is
the magnitude response in the classical theory. (b) is the magnitude response
w.r.t. the total variation in (58).
It is important to notice that magnitude responses of the
graph filters shown in Fig. 5(b) do not explicitly depend on
the eigenvalues of the graph as long as they are on the unit
circle. Even repeated eigenvalues are tolerated. Furthermore,
magnitude response and the length of the filters are unrelated
with the size of the graph. This makes the system robust to
ambiguities in the graph. Remember that for a given specific
graph with distinct eigenvalues, we can always construct
polynomial filters with the desired magnitude response using
Eq. (71) of [1]. However, those filters are unique to that
specific graph and quite sensitive to imperfections in the
eigenvalues.
Notice that for a maximally decimated filter bank as in
Fig. 1, having the PR property and having a well-defined
magnitude response are two different concepts, and they do
not imply each other. To see this, consider the polynomials in
(42). When the adjacency matrix is diagonalizable M -block
cyclic, those polynomials provide PR on the filter bank. Yet,
they may have a different frequency behavior w.r.t. the total
variation for different M -block cyclic graphs. Conversely, we
may select the filters with real coefficients. Then, for graphs
with unit modulus eigenvalues, we have a graph independent
characteristics due to Theorem 12. Yet, we cannot expect them
to provide the PR property. Therefore, we reach the following
conclusion: even if we assume unit-magnitude eigenvalues in
the adjacency matrix, filters given in Table 6.5.1 of [4] do
not provide perfect reconstruction on the filter bank. When
we further assume that the graph is 3-block cyclic, only
then 3-channel filter bank on the graph provides perfect
reconstruction with each channel allowing only a sub-band
of the total variation spectrum. Moreover, the characteristics
of the graph filter bank, PR and magnitude response, will be
independent of the actual values of the eigenvalues and the
size of the graph, provided that they satisfy the conditions.
The above results show that we can have robust filter
banks with practical use for signals on M -block cyclic graphs
with eigenvalues having unit magnitude. More importantly,
design of such multirate graph systems is not an issue. Due
to Theorem 6 and Theorem 12, any algorithm developed
for classical signal processing will serve the purpose. We
summarize this observation in the following theorem.
Theorem 13 (PR graph filter banks with well-defined magnitude response). Consider the graph filter bank of Fig. 1,
and assume that the adjacency matrix of the graph is diag-
As an application of Theorem 13, consider the cyclic graph
C N . Due to Fact 4 of [1], C N is an M -block cyclic graph.
Furthermore, its eigenvalues are in the form of λk ej2πk{N
for 0 ¤ k ¤ N -1, hence |λk | 1. As a result, Theorem 13
applies to C N . Remember that, when the adjacency matrix is
C N , graph signal processing reduces to the classical theory
[9]. This observation shows that Theorem 13 agrees with the
classical multirate theory and generalizes it to the graph case
with some restrictions on the graph.
VII. R EMOVING THE N EED FOR Ω-S TRUCTURE
In [1], we started with the canonical definition of the
decimator as in (1) and extended the classical multirate signal
processing theory to graphs. Even though we were able to
generalize a number of concepts from classical theory to
graph signals, many of the results require A to be an Ωgraph. In this section we will show that this requirement can
be relaxed completely under the mild assumption that A be
diagonalizable. The basic idea is to work with a similarityr QAQ-1 where Q is chosen
transformed graph matrix A
r
such that A has the Ω-structure as in (10). We will see that this
is always possible. However such a transformation changes the
underlying Fourier basis. Therefore, the matrix Q should be
selected carefully, as we will do throughout this section. Given
the original graph signal x we will then define a modified
signal
r Qx
x
(60)
r q, Fk pA
r qu with canonand use the modified filter bank tHk pA
ical decimator (1) to process it (see Fig. 6(a)). Then the filter
r is transformed back to y Q-1 y
r.
bank output y
r q, Fk pA
r qu sandwiched between Q
The filter bank tHk pA
r which satisfies the
and Q-1 operates on the modified graph A,
eigenvector structure (10). Also, it uses the standard canonical
decimator. So, many of the results developed in earlier sections
are applicable. We will show that the complete system from x
to y in Fig. 6(a) is equivalent to the graph filter bank system
shown in Fig. 6(b), which operates on the original graph.
Notice carefully that the canonical decimator and expander
have been replaced in this system by a new decimator and
expander (to be defined below in (63)).
The most important point is that the new filter bank
tHk pAr q, Fk pAp qu that transforms xr to yr and the original
filter bank that transforms x to y have the following close
relationship, as we shall show:
r q, Fk pA
r qu are polynomials in A
r if and
1) The filters tHk pA
only if the original filters tHk pAq, Fk pAqu are polynomials
in A.
r q, Fk pA
r qu is a perfect reconstruction
2) The filter bank tHk pA
r if and only if the original filter bank
system on the graph A
tHk pAq, Fk pAqu is a perfect reconstruction system on the
graph A.
1053-587X (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2016.2620111, IEEE
Transactions on Signal Processing
IEEE
TRANSACTIONSON
ONON
SIGNAL
PROCESSING
IEEE
PROCESSING
IEEE
TRANSACTIONS
PROCESSING
IEEETRANSACTIONS
TRANSACTIONSSIGNAL
ONSIGNAL
SIGNAL
PROCESSING
IEEE TRANSACTIONS ON SIGNAL PROCESSING
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that
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that
relations
in1,
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and
3follow
follow
from
the
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that
relations
from
the
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Notice
that
relations
in1,
33follow
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Notice
that
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inin1,
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and
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from
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which
holds
true
any
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and
for
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pp qq
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for
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invertible
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for
any
identity,
which
holds
true
for
any
invertible
Q
and
for
any
polynomial
H
pq
(Theorem
6.2.9
of
[25]):
polynomial
pq
(Theorem
6.2.9
[25]):
polynomial
HHpq
(Theorem
6.2.9
[25]):
polynomial
H pq
(Theorem
6.2.9
ofofof
[25]):
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r
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relation
follows
from
the
following
theorem.
(c)
(d)
TheThe
in in
4 follows
from
the
following
theorem.
r rsandFig.
6.
(a)
A
FB
on
the
similarity-transformed
adjacency
matrix
A
r sandFig.
6.
(a)
A
FB
on
the
similarity-transformed
adjacency
matrix
A
The relation in 4 follows from the following theorem.
Fig.6. 6. (a)(a)AAFB
FBononthe
thesimilarity-transformed
similarity-transformed
adjacency
matrix
A sandsandFig.
adjacency
matrix
A
-1 -1 .-1(b) The FB with generalized
wiched
between
Q
and
Q
decimator
Theorem
14
(Generalized
filter
banks).
Let
A
be
a
diagorr and
wiched
Q and
Q similarity-transformed
.Q
(b). The
FB with
generalized
decimator
and
Theorem
14
(Generalized
filter
banks).
Let
AAbebea a
diago6. between
(a)between
A FB
on
the
adjacency
matrix
A
sandwiched
Qand
and
(b)The
The
FB with
with
generalized
decimator
and
Theorem
14
(Generalized
filter
banks).Let
Let
diago- Fig.
-1
wiched
between
Q
Q
.
(b)
FB
generalized
decimator
and
Theorem
14
(Generalized
filter
banks).
A
be
a
diago-1
expander
on
the
original
adjacency
matrix
A.
(c)
and
(d)
are
input-output
expander
on the
original
A. with
(c)
(d)
between
Q original
andadjacency
Q adjacency
. (b) matrix
Thematrix
FB
generalized
decimator
and
nalizable
matrix
with
the
eigenvalue
decomposition
expander
onthe
the
A.and
(c)
andare
(d)input-output
are
input-output
Theorem
14 adjacency
(Generalized
filter
banks).
Let A decomposition
be
a diago- wiched
nalizable
adjacency
with
the
eigenvalue
nalizable
adjacency
matrix
with
the
eigenvalue
decomposition
r
expander
on
original
adjacency
matrix
A.
(c)
and
(d)
are
input-output
r
equivalent
of
the
systems
in
terms
of
A
and
A,
respectively.
Q
is
the
-1
nalizable
adjacency
matrix
with invertible
the eigenvalue
decomposition
equivalent
of the
systems
in
terms
ofmatrix
Aof and
is Q
theis the
randrespectively.
-1-1
expander
on
original
adjacency
A.A,
(c)respectively.
(d) areQ input-output
equivalent
the
systems
in terms
A
and
A
Vadjacency
EE bewith
matrix
with
the
ΩrA,
A
ΛV
VVΛV
ΛV
ananthe
invertible
matrix
with
nalizable
matrix
eigenvalue
decomposition
€
r rand
equivalent
ofofthe
systems
in
terms
of
A
respectively.
Q is
the
€D
rA
-1 .. . Let
A
ΛV
Let
be
invertible
matrix
withthe
theΩΩ- equivalent
similarity
transform.
D
is
as
in
(1).
and
U
are
as
in in
(63).
four
systems
rA,
similarity
transform.
D
isD
asis
inin
(1).
D
and
U
are
asare
in
(63).
All All
four
systems
€
oftransform.
the systems
terms
of
and
A,
respectively.
Q
is the
A structure
V
.
Let
E
be
an
invertible
matrix
with
the
Ωsimilarity
as
in
(1).
D
and
U
as
(63).
All
four
systems
-1
€
r
(10)
on
its
columns.
Set
the
similarity
transform
similarity
transform.
D
is
as
in
(1).
D
and
U
are
as
in
(63).
All
four
systems
(10)
Set
the
similarity
transform
shown
above
are
equivalent
to
each
other
in
terms
of
input-output
relations.
shown
above
are
equivalent
to
each
other
in
terms
of
input-output
relations.
A structure
V
ΛV
.
Let
E
be
an
invertible
matrix
with
the
Ω€
r
structure
(10)
on its columns.
thesimilarity
similaritytransform
transform similarity
shown transform.
above are equivalent
each
in are
terms
input-output
D is as into(1).
D other
and U
as of
in (63).
All fourrelations.
systems
-1
-1
-1-1 its columns.
-1
structure
on
SetSet
the
shown above are equivalent to each other in terms of input-output relations.
rr QAQ
as
Q
(10)
EV
.. .Hence
A
QAQ
Define
the
generalized
as
Q
(10)
EV
.. Define
the
generalized
shown above are equivalent to each other in terms of input-output relations.
structure
columns.
Set
the-1
similarity
transform
as
Q
EV
Hence
A
QAQ
.
Define
the
generalized
-1on its
-1
r
as decimator
Q EV-1D
Hence
QAQ-1expander
. DefineU
generalized
rrr. and
rrthe
generalized
expander
asgeneralized
decimator
and
generalized
rA
ras
as Q
EV r D
.D
Hence
A
QAQ
. expander
Define rU
the
decimator
andthe
the
generalized
U
as
-1 -1
decimatorrD
and the generalized
expander
U
as
flanked
6(a))
that
satisfies
flanked
bybyby
QQand
QQ
in-1in
Fig.
6(a))
that
satisfies
thethe
eigenvector
r as -1-1 TT
-1
-1
-1 TT U
flanked
Qand
and
Q-1
inFig.
Fig.
6(a))
that
satisfies
theeigenvector
eigenvector
rr rrexpander
decimator
DDQ
andthe
generalized
r
-1
-1
T
-1
T
D
DQ
,
U
Q
D
V
E
D
.
(63)
D
DEV
Q
D
V
E
D
.
(63)
flanked
by
Q
and
Q
in Fig. 6(a))
that
satisfies
thepossibly
eigenvector
-1Ω-structure
condition
(10).
This
Ω-structure
is
sufficient
(though
D
DQ
DEV
,
U
Q
D
V
E
D
.
(63)
condition
(10).
This
is
sufficient
(though
possibly
-1
-1
T
-1
T
r DQ DEV , U
r Q D V E D . (63) flanked
by
Q
and
Q
in
Fig.
6(a))
that
satisfies
the
eigenvector
condition
(10).
This
Ω-structure
is
sufficient
(though
possibly
D
-1 rT
-1 T
condition
(10).toto
This
Ω-structure
is sufficient
(though
possibly
rThen,
rpA
not
necessary)
be
able
to
use
some
of
the
filter
banks
wewe
r
not
necessary)
be
able
to
use
some
of
the
filter
banks
r-1rq,, FkU
r
D
DQ
Q
D
V
E
D
.
(63)
Then,
a FB
FBDEV
HkkpA
qu
on
A
that
uses
the
canonia
t
H
on
A
that
uses
the
canoni(10). This
Ω-structure
sufficient
(though
possibly
not necessary)
to be
able to useis some
of the
filter banks
we
r qu on A
r that uses the canoni- condition
Then, a FB tHrk pAq, Frk pA
not
necessary)
tothebe
able tofilter
use
some
ofof
the
filter4,banks
r6(a),
developed,
e.g.,
the
brickwall
bank
of
Theorem
andandwe
developed,
e.g.,
brickwall
filter
bank
Theorem
4,
Then,
adecimator
FB tHkand
p
A
q
,
F
p
A
qu
on
A
that
uses
the
canonical decimator
Fig.
is
equivalent
to
the
cal
expander,
Fig.
6(a),
is
equivalent
to
the
k
developed,
e.g.,
the
brickwall
filter
bank
of
Theorem
4,
and
not
necessary)
to
be
able
to
use
some
of
the
filter
banks
we
r
r
r
cal
decimator
and
expander,
Fig.
6(a),
is
equivalent
to
the
Then,
a FB tHFB
, Fk pA
onqu A
thatthat
uses
the
canonik pAqexpander,
alias
free filter
banks
II. Thus
the
similarity
transform
developed,
e.g.,
theofbrickwall
filter
bank
of Theorem
4, and
alias
ofSec.
Thus
the
transform
generalized
, FqukkpFig.
A
on
uses
generalized
calgeneralized
decimator
and
6(a),
equivalent
to the developed,
FB
H
ononA
AAis
that
uses
aliasfree
freefilter
filterbanks
banks
ofSec.
Sec.II.II.
Thus
thesimilarity
similarity
transform
e.g.,
the
brickwall
filter
bank
of
Theorem
4, and
generalizedand
FBtexpander,
t
HkkkpA
p
Aq q, F
that
usesgeneralized
generalized
k pAqu
cal
decimator
Fig.
6(a),
is
equivalent
to
the
Q
merely
sets
the
stage
for
that.
As
long
as
the
similarity
alias
free
filter
banks
of
Sec.
II.
Thus
the
similarity
transform
Q
merely
sets
the
stage
for
that.
As
long
as
the
similarity
decimator
and
expander,
Fig.
6(b).
Here
the
term
“equivalent”
generalized
FB
Hexpander,
FkFig.
pAqu6(b).
qu6(b).
onHere
A that
uses
generalized
decimator
andtexpander,
the
term
“equivalent”
Q free
merely
sets the of
stage
As
long
as the transform
similarity
filter
Sec.forII.that.
Thus
thebe
similarity
decimator
Here
the
termgeneralized
“equivalent” alias
generalized
FBand
texpander,
Hinput-output
pkApAq,qF,Fig.
onHere
A
that
uses
transform
isisbanks
selected
properly,
AAcan
treated
if itif it
k
k pA6(b).
Q
merelyQQ
sets
the
stage
for
that.
As
long
as
theassimilarity
transform
selected
properly,
can
be
treated
as
means that
that
the
input-output
behaviors
are
related
as
inin(61).
means
the
behaviors
are
related
as
(61).
decimator
and
the
term
“equivalent”
transform
Q
is
selected
properly,
A
can
be
treated
as if it
merely
sets even
the stage
for
that.
As
long as
the similarity
meansand
thatexpander,
the input-output
behaviors
are
related
as in (61). Q
istransform
an
Ω-graph
if ifit itisproperly,
not.
ForFor
example,
consider
the
decimator
Fig.
6(b).
Here
the
term
“equivalent”
is
an
Ω-graph
even
is
not.
example,
consider
♦
Q
is
selected
A
can
be
treated
as the
if it
♦♦ that the input-output behaviors are related as in (61). transform
means
is
an
Ω-graph
even
if
it
is
not.
For
example,
consider
Q
is selected
properly,
A can
be
treated
as
if When
it is the
an
spectrum
folding
phenomena
described
in in
Sec.
????
of of
[1].
When
means
that
the
input-output
behaviors
are
related
as
in
(61).
spectrum
folding
phenomena
described
Sec.
[1].
is
an
Ω-graph
even
if
it
is
not.
For
example,
consider
the
♦
spectrum
folding
phenomena
described
inasSec.
??the
ofwe
[1].
When
Proof: Let
Let tH
Hkk pA
Aq,, F
p
A
qu
be
the
generalized
FB
on
AA Ω-graph
k
even
if
it
is
not.
For
example,
consider
spectrum
Proof:
F
A
be
the
generalized
FB
on
the
decimator
and
the
expander
are
selected
in
(63),
can
k
♦
the
and
the
expander
are selected
asasin??
wewe
can
folding
phenomena
described
in Sec.
of
[1].
When
Proof:
tHk pAq,decimator
Fk pAqu be
theexpander.
generalized
FB on A spectrum
thedecimator
decimator
and
the
expander
selected
in(63),
(63),
can
that
uses
the Let
generalized
and
Then
phenomena
described
inare
Sec.
VII
of
[1].
When
the
that
uses
the
generalized
and
expander.
Then
remove
the
condition
onon
thetheeigenvectors
of of
thethe
adjacency
Proof:
Let
tgeneralized
Hk pAq, Fkdecimator
pdecimator
Aqu be the
generalized
FB
on A folding
remove
the
condition
eigenvectors
adjacency
that
uses
the
and
expander.
Then
the
decimator
and
the
expander
are
selected
as
in
(63),
we
can
remove
the
condition
on
the
eigenvectors
of
the
adjacency
Proof:
Let
t
H
p
A
q
,
F
p
A
qu
be
the
generalized
FB
on
A
M
-1
k
k
matrix.
We
state
this
result
as
follows.
decimator
and
thethis
expander
are
selected as in (63), we can
¸
that uses the generalized
decimator
and
expander.
Then
M
-1
matrix.
We
state
result
as
follows.
remove
the
condition
on theas eigenvectors
of the adjacency
M
matrix.the
Wecondition
state this on
result
follows.
¸-1F pA
that uses
decimator
pAqqgeneralized
-1¸
q Ur Dr H pand
Aq, expander. Then (64)
the and
eigenvectors
ofdecimation).
the adjacency
TTGGthe
(64) remove
15
(Spectrum
folding
generalized
matrix.
We
state
this
result
as
follows.
TGpA
pAMqM¸
-1kk00 FFkkkpApAqrqUrUrrDrDrHHkk pkApAq,q,
(64) Theorem
Theorem
15
(Spectrum
folding
and generalized decimation).
matrix.
We
state
this
result
as
follows.
Theorem
15
(Spectrum
folding
and
generalized
decimation).
¸
adjacency
of of
a graph
with
thethe
following
TG pAq M
F0k pAq U D Hk pAq,
(64) Let
k
-1
LetAAbebethe
the
adjacencymatrix
matrix
a -1graph
with
following
¸
r r Hk pA
M
-1 pA-1q U
Theorem
15
(Spectrum
folding
and
generalized
decimation).
TG pAq (64)
-1 q,T
-1
Let
A
be
the
adjacency
matrix
of
a
graph
with
the
following
k-1Q QFD
eigenvalue
decomposition
A
V
ΛV
.
Let
E
be
an
invertM
-1
k¸
0F
q
Q -1D TDQHk pAqQ -1Q, (65) Theorem
¸Q-1 QFk ppA
15
(Spectrum
folding
and
generalized
decimation).
eigenvalue
decomposition
A
V
ΛV
.
E
be
an
invert-1Letwith
A
q
Q
D
DQH
p
A
q
Q
Q,
(65)
-1
-1
T
-1
Let
A
be
the
adjacency
matrix
of
a
graph
the
following
k
k
k
0-1
decomposition
A
V ΛV
. columns.
Let E beDefine
an
invertM
ibleeigenvalue
matrix
with
the
Ω-structure
(10)
on
its
¸
k0 Q QFk pAqQ D DQHk pAqQ Q, (65) Let
A
be
the
adjacency
matrix
of
a
graph
with
the
following
-1
ible
matrix
with
the
Ω-structure
(10)
on
its
columns.
Define
-1 T
-1
eigenvalue
A expander
V ΛV
. inLet
E
be
an
invertible
matrixdecomposition
with
the Ω-structure
(10)-1
on
its
columns.
Define
-1 k pAqQ D DQHk pAqQ Q, (65) the
M¸-1kkQ-10-10-1MM¸
QF
generalized
decimator
and
as
(63).
Let
x
be
¸
-1 T T
-1
eigenvalue
decomposition
A
V (10)
ΛV .asLet
be Let
an invertthe generalized
decimator
andexpander
in E
(63).
x
be
r
r qk Q,
kQQ0Q-1 QF
q
Q
DQH
p
AqQ-1 Q, (65)
with
the
Ω-structure
its
columns.
Define
M
thematrix
generalized
decimator
as
in
(63).
x be
FpkA
p
A
q
DD
(66) aible
signal
on the
graph
A
and yand
be expander
the DUonversion
of x, Let
that
T D Hk pA
¸-1kF
r
r
p
A
q
D
D
H
p
A
q
Q,
(66)
a
signal
on
the
graph
A
and
y
be
the
DU
version
of
x,
that
ible
matrix
with
the
Ω-structure
(10)
on
its
columns.
Define
-1
T
k
k
r q D D Hk pA
r q Q,
ron the
k
0 Q
M
-1
the
generalized
decimator
and yexpander
asofversion
inx (63).
Let
be
k0 Fk pA
(66) is,
ay signal
graph
AFourier
and
be the DU
x,xthat
UrrDx.
graph
transform
and yof
are
¸
r Then,
0
is, generalized
yU
Then,
graphand
Fourier
transform
x and
arebe
-1M-1
the
decimator
expander
as version
inof(63).
Letyx,yx
-1k
r q D T D Hk pA
r q Q,
rDx.
r
r0kqpQ,
kp
Q-1Q
F
A
(66)
a
signal
on
the
graph
A
and
y
be
the
DU
of
that
is,
y
U
Dx.
Then,
graph
Fourier
transform
of
x
and
are
related
as
Q¸
T
A
(67)
rpqA
related as
(67) ais,
on
graph
A andFourier
y be the
version of x,ythat
1 graph
Q Q-1k-1T0FTpA
qQ,DT D Hk pAr q Q,
(66)
ras
rthe Then,
rrqQ,
k
T DU
related
yU
Dx.
transform
p
A
(67) signal
p
p, of x and
y
I
b
1
1
(68) are
1
M
N
{
M
M
r
r
Tx
where we use
(62)
in
(66).
Notice
that
the
generalized
is,
y
U
Dx.
Then,
graph
Fourier
transform
y are
-1k0
1
M
p
p
y
I
b
1
1
x
, of x and (68)
r q Q,
M MT
N {M
related
as
Q
T(62)
p
A
(67)
where
we
use
in
(66).
Notice
that
the
generalized
p M I N {M b 1M 1M x
p,
y
(68)
r which
-1use
FB
implicitly
operates
on
A,
is
an
Ω-graph
since
where
we
(62)
in
(66).
Notice
that
the
generalized
r
related
as
1
M
Q-1 Toperates
pEΛE
Aq Q,-1 on A,
r which is an Ω-graph (67)
T
but
the spectrum
folding
phenomena.
♦ (68)
FB
implicitly
since which is nothing y
I N {M b
r
p
p
1
1
x
,
r
A
QAQ
.
Hence,
the
eigenvector
condition
M
1
implicitly
on A,
whichthat
is an
since
where
we
use
in -1(66).
Notice
the Ω-graph
generalized
is nothing
but M
the Ispectrum
folding
♦
TMphenomena.
-1 (62)
rFB
p
p,
y
x
(68)
A
we
QAQ
operates
EΛE
. Hence,
eigenvector
condition which
which
is The
nothing
the N
spectrum
-1
M 1M phenomena.
{M b 1folding
rimplicitly
(10)
on
A is-1
implicitly
satisfied
onthe
the
generalized
FB.
rHence,
where
use
(62)
in (66).
Notice
that
theΩ-graph
generalized
Proof:
graphbut
Fourier
transforms
of x and y are given ♦
A
on
QAQ
EΛE
. A,
the
eigenvector
condition
M
FB(10)
operates
on
which
is
an
since
A-1 is implicitly
satisfied
on the
generalized
FB.
-1 spectrum
Fourier
transforms
of x
and y are
given♦
is -1nothing
folding
phenomena.
r which
-1
Theonidentity
in (62)
basically
says
that
instead
ofcondition
working
r p Proof:
pbut
x
V
xThe
and graph
y
the
Vthe
y,
respectively.
Therefore
FB
implicitly
on
A,
iseigenvector
an
Ω-graph
since aswhich
(10)
A operates
is EΛE
implicitly
satisfied
on the
generalized
FB.
Proof:
graph
Fourier
transforms
of
x andwe
y have
are given
A
QAQ
.basically
Hence,
the
-1 Thebut
-1
The
identity
in
(62)
says
that
instead
of
working
which
is
nothing
spectrum
folding
phenomena.
-1
-1
p
p
as
x
V
x
and
y
V
y,
respectively.
Therefore
we
have♦
the identity
given
adjacency
matrix,on
we
can
the condition
similarityr(10)
The
in (62)
basically
says
thatuse
instead
of working as x
A
ononQAQ
EΛE
. satisfied
Hence,
the
pProof:
p
V -1The
x and
y
V -1ry,rtransforms
respectively.
wegiven
have
on
isimplicitly
theeigenvector
generalized
FB.
graph
Fourier
of Therefore
x and y are
theAgiven
adjacency
matrix,
we
can
use
the
similarityp
p
V
y
U
D
V
x
,
(69)
transformed
adjacency
matrix
as
long
as
the
input
graph
signal
-1The graph Fourier
-1 rtransforms
theisgiven
matrix,
wethat
caninstead
use the
similarity- as x
(10)The
on A
implicitly
onsays
the
generalized
FB.
Proof:
ofTherefore
x and y are
given
r Vx
identity
in adjacency
(62) satisfied
basically
of
working
p
p
V
x
and
y
V
y,
respectively.
we
have
p
p
V
y
U
D
,
(69)
transformed
adjacency
matrix
as
long
as
the
input
graph
signal
rD
r Vx
istransformed
also
transformed
accordingly.
As
an as
example,
consider
the
p , Therefore we have
U
(69)
adjacency
matrixsays
aswe
long
the input
graph
signal
The
identity
in
(62) accordingly.
basically
instead
working
pis,
as
V -1 x and yp VV-1ypy,respectively.
on
the
given
adjacency
matrix,
canexample,
use
theof
similaritythatx
isgeneralized
also
transformed
Asthat
an
consider
the
FB.
The
matrix
Q
transforms
the
graph
signal
x
r
r
is also
transformed
accordingly.
As
an
example,
consider
the
ontransformed
the
given
adjacency
matrix,
we
can
use
the
similaritythat
is,
p
p
V
y
U
D
V
x
,
(69)
as
as the
input
graph
signal
generalized
FB. The
matrix
Q long
transforms
the of
graph
r as adjacency
into
x
shown
in matrix
Fig.
6(a).
Special
cases
this signal
can
bexx that is,
generalized
FB. The
matrix
QAs
transforms
the graph
graph
signal
p
pV, -1 V x
Vy
-1DUrTDrD
VEx
(69)
p V -1 V
p,
y
E -1
transformed
as long
as
the
input
signal
is also
transformed
accordingly.
an
example,
the
r in
into
x
asadjacency
shown
inmatrix
Fig.
6(a).
Special
cases
ofconsider
this
can
be
-1
T
-1
found
[9]
where
a
permutation
matrix
is
used
and
in
[7]
p
p,
y
V
V
E
D
D
E
V
V
x
-1
T
r asFB.
into
x
shown
in
Fig.
6(a).
Special
cases
of
this
can
be
that
is,
-1
-1
T
-1
isgeneralized
also
transformed
accordingly.
As
an
example,
consider
the
p
E
D
D
E
x
Thematrix
matrix
Q transforms
the
graph
signal
x
p V
p,
y
V E D D EV V x
found
[9] where
a permutation
matrix
is itused
in [7]
where in
a diagonal
is used.
Here
we use
for aand
different
that is,
found
[9]The
where
a permutation
matrix
isofused
and
pT1 1T x
1{EME-1-1-1DIDNT {TD
E
x
generalized
FB.
matrix
Qused.
the
r asa in
into
x
shown
inmatrix
Fig.
6(a).
Special
cases
this
caninx
be[7]
-1
-1,
where
diagonal
is
Here we
usegraph
it for
asignal
different
p
b
(70)
purpose,
namely
to
create
a transforms
hypothetical
system
(the
system
M
M
p
M
D
E
x
p V V E D D E TV V x
p,
y
where
a diagonal
is used.
Here
weisuse
for
different
r
into
x
as
in toFig.
6(a).
Special
cases
of itthis
found
in shown
[9]
where
amatrix
permutation
matrix
used
andacan
in be
[7]
-1
-1
T 1M 1MT-1 x
p
1
{
M
I
b
,
(70)
purpose,
namely
create
a hypothetical
system
(the
system
N
{
M
-1
T
p p
y
D1EMV1M Vx
px
,,
(70)
purpose,
namelya topermutation
a hypothetical
systemand
pb
{E
Mx
VE1-1{MVDTEIDND
found
ina [9]
where
matrix
is used
insystem
[7]
where
diagonal
matrixcreate
is used.
Here
we use
it for a(the
different
T
p
E
D
D
E
x
where
a diagonal
is used.
Here we use
it for (the
a different
1{M I N {M b 1M 1M xp ,
(70)
purpose,
namelymatrix
to create
a hypothetical
system
system
1{M I N {M b 1M 1TM xp ,
(70)
purpose, namely to create a hypothetical system (the system
1053-587X (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2016.2620111, IEEE
Transactions on Signal Processing
IEEE TRANSACTIONS ON SIGNAL PROCESSING
12
where the last equation follows from Eq. (60)-(66) of [1] since
E has the Ω-structure.
Even though results of Theorem 8 of [1] and Theorem 15
appear to be similar, the difference is that Theorem 8 of
[1] applies to diagonalizable Ω-graphs, whereas Theorem 15
applies to any graph with diagonalizable adjacency matrix.
Notice that Theorem 15 includes Theorem 8 of [1]: when the
eigenvectors of A have the Ω-structure, simply select E V ,
the decimator and the expander then reduce to canonical form
r D and U
r DT .
D
The application of a similarity transform QAQ-1 is therefore useful whenever the Ω-structure is necessary. For arbitrary
eigenvectors, we need to use a similarity matrix Q to adjust
them to the form (10), which, in turn, affects the choice of
decimation matrix: namely the non-canonical form (63) that
depends on Q has to be used. On the other hand, this technique
does not alter the eigenvalues of A. Remember from Eq. (71)
of [1] that coefficients of a polynomial filter and its frequency
response are related through the eigenvalues of the graph.
Therefore, whatever Q we choose, a polynomial filter on A,
r H pA
r q, have the
H pAq, and the same polynomial filter on A,
same frequency response in their corresponding graph Fourier
domains. In summary, for the desired multirate graph signal
processing, the coefficients of polynomial filters should be
designed according to the graph spectrum (eigenvalues of A);
the decimator and the expander should be designed according
to eigenvectors of the graph.
The generalized decimator and expander in (63) have two
important properties. Firstly, given a graph, they are not
unique: we can select E arbitrarily as long as its columns
have the Ω-property, and it is invertible. In fact, E can be
selected as a properly permuted version of the inverse DFT
matrix of size N . This follows from the fact that C N is an
M -block cyclic matrix for any M that divides N . Secondly,
the generalized decimator does depend on the eigenspaces of
the adjacency matrix due to presence of V in (63). Therefore,
we cannot talk about a “universal” decimator, and expander,
which can remove the eigenvector constraint in (10) for all
graphs.
It might appear that the use of Q and Q-1 involves additional complexity of the order of N 2 in Fig. 6(a) (where N is
the size of the graph). This can be avoided if we implement
Fig. 6(b) which is equivalent. In this implementation the
r Now, D
r is an
simple decimator D is replaced with D.
pN {M qN matrix with possibly non-zero entries everywhere,
and there are M such matrices in the figure. This gives the
impression that there is additional computational overhead of
r is not unique; it is defined
N 2 multiplications. But note that D
-1
r
as D DEV where E is an arbitrary matrix with the Ωstructure. The degrees of freedom in E can be exploited to
r relatively sparse to reduce the complexity. In fact D
r
make D
r rI N {M X s as shown next.
can have the form D
Assume that E has the Ω-structure on its columns. Then it
can be written as
E 1 pI N {M
..
E
.
E M pI N {M
b f H1 q
b
fH
M
q
,
(71)
for some E k where E k P MN {M and f k is the k th column
of the DFT matrix of size M . Let R V -1 and write it as
R rR1
RM s,
(72)
where Rk P C N N {M . Then we have the following for the
generalized decimator
r
D
D E R E 1 pI N {M b f H1 q R,
(73)
H
H
rE 1 pI N {M b f 1 qR1 E 1 pI N {M b f 1 qRM s.
Then, select E 1 as follows:
E1
I N {M
b f H1
R1
-1
.
(74)
As a result we get the decimator in the form of
r rI N {M X s where X depends on Rk ’s. This proves the
D
claim. In this construction the decimator and expander have
the form
r
D
r
U
V
b f H1 V -1 .
I N {M b f 1 E -1
1 .
E 1 I N {M
(75)
(76)
At the time of this writing, we do not know how to reduce
r and U
r simultaneously.
the complexities of both D
The similarity transform in (62) changes the eigenvectors,
and hence the graph Fourier basis that supports the filter
bank. So the transform matrix Q should be selected carefully,
otherwise the new Fourier basis may be of no use. Notice that
Fig. 6(a) and 6(b) are equivalent, hence the similarity transform can be integrated into the decimator and the expander.
At this point, notice the proposed decimator in (75) and the
expander in (76). They explicitly depend on the Fourier basis
of the original graph. As a result, the idea of a similarity
transform reduces to a carefully designed decimator, where
the decimator in (75) takes the Fourier basis of the original
graph into account and decimates the signal accordingly. The
same interpretation is valid for the expander in (76) as well.
Moreover, the filters in Fig. 6(b) are still polynomials in the
original graph, hence the original graph Fourier basis can still
be used to diagonalize the filter responses.
VIII. E XAMPLES
In this section, we will implement a 3-channel brickwall
filter bank for the famous Minnesota road graph [6], [7], [12].
With due thanks to the authors of [6] and [12], we use the data
publicly available in [26], [27]. This graph has 2642 nodes in
total where 2 nodes are disconnected to the rest of the graph.
Since a road graph is expected to be connected, we disregard
those two nodes. See Fig. 7 for the visual representation of
the graph. The adjacency matrix is extremely sparse with
only 0.1% non-zero entries. Since the edges are represented
with 1’s, the unit shift, Ax, merely requires addition and no
multiplication at all.
It is clear from Fig. 7 that the graph is not 3-block cyclic.
Therefore, we cannot apply Theorem 4 directly. However, we
can use the generalized decimator and expander as explained
in Sec. VII. Furthermore, we select the graph Laplacian to
be the unit shift element and find the graph Fourier basis,
V , as the eigenvectors of the graph Laplacian. This selection
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The reconstructed outputs from the channels, y 0 , y 1 , and y 2 ,
are given in Fig. 9(b), 9(c), 9(d), respectively. Similar to the
previous example, the low-pass channel captures most of the
energy. Even though the remaining two channels identify the
nodes where change in the signal is located, outputs are not
as strong as the previous example. This is due to smoother
character of the signal.
Geographic Coordinate (North)
49
48
47
46
45
44
-97
-96
-95
-94
-93
-92
-91
-90
Geographic Coordinate (West)
Fig. 7. Minnesota traffic graph which has N
2640 nodes, and 3302
undirected unweighted edges (courtesy of [6], [12]).
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
(a)
(the Laplacian but not the adjacency matrix) is consistent with
the development, since we do not put any specific meaning
to the unit shift operator. We precisely use (75) and (76) to
construct the generalized decimator and expander, respectively.
We construct the analysis and synthesis filters as in (26).
We denote the reconstructed output of the k th channel as
rD
r H k x for 0 ¤ k ¤ 2. (These are indicated
yk F k U
in Fig. 2(b)).
1
(b)
0.04
0.06
0.03
0.04
0.02
0.01
0.02
0
0
-0.01
-0.02
-0.02
-0.04
-0.03
-0.06
-0.04
(c)
(d)
Fig. 9. (a) Signal with exponential decay due to geographic distance, output
of (b) channel-0, (c) channel-1, (d) channel-2.
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
(a)
(b)
1
1
0.5
0.5
0
0
The Laplacian of the Minnesota traffic graph has repeated
eigenvalues. Furthermore, the distinct eigenvalues are closely
spaced. As a result, we cannot implement tH k u’s as polynomials, as low order polynomial approximations result in
poor performance. Hence, each filtering operation requires
N 2 7 107 multiplications. In the following, in order to
obtain filters with low complexity, we will replace the brickwall filters, tH k u, with their hard-thresholded counter-parts,
tH 1k u, that are constructed as follows:
pH 1 qi,j k
-0.5
-1
#
pH k qi,j , |pH k qi,j | ¥ γ
0,
otherwise
(77)
-0.5
-1
(c)
(d)
Fig. 8. (a) Signal consisting of only 1’s and -1’s as given in [7], output of
(b) channel-0, (c) channel-1, (d) channel-2.
In the first example, we consider the signal used in [6], [7]
as the filter bank input. A visual representation of this signal is
given in Fig. 8(a). The reconstructed outputs from subbands,
y 0 , y 1 , y 2 , are given in Fig. 8(b), 8(c), 8(d), respectively. We
observe that y 0 is a smooth approximation of the original
signal since it is the output of the low-pass channel. The
remaining two channels give information about nodes where
the discontinuity (high-frequency content) in the signal occurs.
In the second example, we select a smoother signal defined
as xi expp-0.5 d2i q where di denotes the geographic distance
between the ith node and the point r-93.5 45s. This signal,
which is used as the filter bank input, is visualized in Fig. 9(a).
for some threshold γ. It is clear that this non-linear operation
on the filters compromises the PR property of the filter bank.
However, it is interesting to investigate the trade-off between
the reconstruction error and the efficiency of the thresholded
filters. For this purpose, we define the°average
fraction
of non
M -1 zero elements of the filters as M 1N 2 k0 H 1k 0 where } }0
counts the number of non-zeros and define the reconstruction
error as }x x1 }22 {}x}22 where x1 is the output of the filter
bank with filters in (77). By sweeping over different thresholds, γ, we obtain the result in Fig. 10. Here xa denotes the
signal in Fig. 8(a), xb denotes the signal in Fig. 9(a), and xc
is a signal with i.i.d. Gaussian entries. These are the inputs
to the filter bank in the three cases. It is very interesting to
observe that when we tolerate 1% error in the reconstruction,
we can sparsify the filters significantly: we only need 7.6%,
12.8% and 3.5% of the non-zero values of the actual filters,
respectively. This is a huge saving, due to N 2 being very large.
We do not expect hard-thresholding to be optimal, and we are
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not suggesting this as a filter design approach. This example
merely shows that by replacing the PR property with near-PR
property, very efficient filters can be designed for M -channels.
Reconstruction error
10 -1
10 -2
10 -3
10 -4
xa
xb
10
-5
0.003
xc
0.01
0.03
0.08 0.13
0.5
Average fraction of non-zero elements
Fig. 10. Trade-off between the efficiency of the filters and the reconstruction
error. Trade-off depends on the input signal under consideration.
IX. C ONCLUDING R EMARKS
In this paper we extended the theory of M -channel maximally decimated filter banks, from classical time domain to the
domain of graphs. There are several important aspects in which
these filter banks differ from classical filter banks. We found
that perfect reconstruction (PR) can be achieved with polynomial filters for graphs which satisfy certain eigenstructure
conditions. Furthermore for M -block cyclic graphs, PR filter
banks can be built by starting from any classical filter bank.
We also developed polyphase representations for such filter
banks. Even though many of the results were developed for
graphs with a certain eigenstructure, it was shown in Sec. VII
that the eigenvector structure (10) can be relaxed, and only
the eigenvalue structure (9) remains to be satisfied.
This also brings up some practical questions which we have
not been able to address here: what are practical examples
of graphs which satisfy the eigenvalue constraints? How do
these filter banks perform for practical graph signals, and how
do they compare with alternative ways of processing these
graph signals? For example, imagine we build a compression
system (akin to a subband coder) based on the PR graph
filter bank with a certain order for the polynomial filters
tHk pAq, Fk pAqu. How does this compare with a brute-force
compression system that performs a large DFT or a graph
Fourier transform on the entire graph signal x, performs optimal bit allocation, and reconstructs with the inverse transform?
Many such practical questions remain to be addressed. In this
theoretically intense paper it has not been possible to address
these important practical issues, but we plan to explore these
aspects in future work.
X. ACKNOWLEDGMENTS
We wish to thank the anonymous reviewers for the encouragement, criticisms, thoughtful questions, and very useful suggestions. We are also grateful to Dr. Narang and Prof. Ortega
[6], [7] for the most informative website on the Minnesota
road graph [26] and for making important data available.
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Transactions on Signal Processing
IEEE TRANSACTIONS ON SIGNAL PROCESSING
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15
P. P. Vaidyanathan (S’80–M’83–SM’88–F’91) was
born in Calcutta, India on Oct. 16, 1954. He received
the B.Sc. (Hons.) degree in physics and the B.Tech.
and M.Tech. degrees in radiophysics and electronics,
all from the University of Calcutta, India, in 1974,
1977 and 1979, respectively, and the Ph.D degree
in electrical and computer engineering from the
University of California at Santa Barbara in 1982.
He was a post doctoral fellow at the University of
California, Santa Barbara from Sept. 1982 to March
1983. In March 1983 he joined the electrical engineering department of the Calfornia Institute of Technology as an Assistant
Professor, and since 1993 has been Professor of electrical engineering there.
His main research interests are in digital signal processing, multirate systems,
wavelet transforms, signal processing for digital communications, genomic
signal processing, radar signal processing, and sparse array signal processing.
Dr. Vaidyanathan served as Vice-Chairman of the Technical Program committee for the 1983 IEEE International symposium on Circuits and Systems,
and as the Technical Program Chairman for the 1992 IEEE International
symposium on Circuits and Systems. He was an Associate editor for the
IEEE Transactions on Circuits and Systems for the period 1985-1987, and is
currently an associate editor for the journal IEEE Signal Processing letters,
and a consulting editor for the journal Applied and computational harmonic
analysis. He has been a guest editor in 1998 for special issues of the IEEE
Trans. on Signal Processing and the IEEE Trans. on Circuits and Systems II,
on the topics of filter banks, wavelets and subband coders.
Dr. Vaidyanathan has authored nearly 500 papers in journals and conferences, and is the author/coauthor of the four books Multirate systems
and filter banks, Prentice Hall, 1993, Linear Prediction Theory, Morgan
and Claypool, 2008, and (with Phoong and Lin) Signal Processing and
Optimization for Transceiver Systems, Cambridge University Press, 2010, and
Filter Bank Transceivers for OFDM and DMT Systems, Cambridge University
Press, 2010. He has written several chapters for various signal processing
handbooks. He was a recipient of the Award for excellence in teaching at
the California Institute of Technology for the years 1983-1984, 1992-93 and
1993-94. He also received the NSF’s Presidential Young Investigator award
in 1986. In 1989 he received the IEEE ASSP Senior Award for his paper
on multirate perfect-reconstruction filter banks. In 1990 he was recepient
of the S. K. Mitra Memorial Award from the Institute of Electronics and
Telecommuncations Engineers, India, for his joint paper in the IETE journal.
In 2009 he was chosen to receive the IETE students’ journal award for his
tutorial paper in the IETE Journal of Education. He was also the coauthor
of a paper on linear-phase perfect reconstruction filter banks in the IEEE SP
Transactions, for which the first author (Truong Nguyen) received the Young
outstanding author award in 1993. Dr. Vaidyanathan was elected Fellow of
the IEEE in 1991. He received the 1995 F. E. Terman Award of the American
Society for Engineering Education, sponsored by Hewlett Packard Co., for
his contributions to engineering education. He has given several plenary
talks including at the IEEE ISCAS-04, Sampta-01, Eusipco-98, SPCOM-95,
and Asilomar-88 conferences on signal processing. He has been chosen a
distinguished lecturer for the IEEE Signal Processing Society for the year
1996-97. In 1999 he was chosen to receive the IEEE CAS Society’s Golden
Jubilee Medal. He is a recepient of the IEEE Signal Processing Society’s
Technical Achievement Award for the year 2002, and the IEEE Signal
Processing Society’s Education Award for the year 2012. He is a recipient
of the IEEE Gustav Kirchhoff Award (an IEEE Technical Field Award) in
2016, for “Fundamental contributions to digital signal processing.” In 2016
he also received the Northrup Grumman Prize for excellence in Teaching at
Caltech.
Oguzhan Teke (S’12) received the B.S. and the M.S.
degree in electrical and electronics engineering both
from Bilkent University, Turkey in 2012, and 2014,
respectively. He is currently pursuing the Ph.D.
degree in electrical engineering at the California
Institute of Technology, USA.
His research interests include signal processing,
graph signal processing, and convex optimization.
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