2011 IEEE International Symposium on Information Theory Proceedings
Computing Reliability Distributions of Windowed
Max-log-map (MLM) Detectors : ISI Channels
Fabian Lim1 and Aleksandar Kavcic2
1
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139
2
EE Department, University of Hawaii at Manoa, Honolulu, HI 96822
Abstract—In this paper, we consider sliding-window max-logmap (MLM) receivers, where for any integer m, the MLM detector is truncated to a length-m signaling neighborhood. For any
number n of chosen time instants, we provide exact expressions
for both i) the joint distribution of the MLM symbol reliabilities,
and ii) the joint probability of the erroneous MLM symbol
detections. The obtained expressions can be efficiently evaluated
using Monte-Carlo techniques. Comparisons performed with empirical distributions reveal good match. Dynamic programming
techniques are applied to simplify the procedures.
Index Terms—detection, intersymbol inteference, max-logmap, probability distribution, reliability
I. I NTRODUCTION
The max-log-map (MLM) detector has well-known applications to the intersymbol interefence (ISI) channel [1]. It is
the optimal sequence detector; it produces the same symbol
estimates as the Viterbi detector [2]. On the other hand, it
also additionally computes symbol reliabilities (also known as
soft-outputs, log-likelihood ratios, etc) to be used for coding
techniques, e.g. see [1]. The MLM is a well-accepted approximation of the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm [3].
In this paper, we consider a sliding-window implementation
of an MLM receiver. We consider an m-truncated MLM
receiver, i.e. only a signaling window of length m is considered around the time instant of interest. These truncations
are observed to well-approximate the actual receiver. The
analysis of truncated MLM receivers is tractable, and for
any number n of chosen time instants, we present exact,
closed-form expressions for both i) the joint distribution of
the symbol reliabilities, and ii) the joint probability that the
detected symbols are in error. While past work considered
only marginal distributions (for convolutional codes, see [4],
[5], and for approximations see [6], [7]), we provide analytic
expressions for joint MLM receiver statistics.
Notation: Deterministic vectors and matrices are denoted
using bold fonts (e.g., a and A, respectively). Deterministic
scalars are denoted using italic fonts (e.g., a). Random scalars
are denoted using upper-case italics (e.g., A) and random
vectors are denoted using upper-case bold italics (e.g., A ).
We do not reserve specific notation for random matrices. We
write max a to denote the maximum (vector) component in a.
A random sequence of symbols drawn from the set
{−1, 1}, denoted as · · · , A−2 , A−1 , A0 , A1 , A2 , · · · , is transmitted across the ISI channel. Let the following random
This work was performed when F. Lim was at the University of Hawaii.
This work was supported by the NSF under grant number CCF-1018984.
978-1-4577-0594-6/11/$26.00 ©2011 IEEE
Fig. 1. The m-truncated Max-Log-Map (MLM) detector, illustrated for the
case ℓ = 2. All 2ℓ = 4 possible states are shown. Channel states colored
black and white, correspond respectively to the symbol neighborhood A t , and
a candidate sequence a in the set M (see Definition 1). As shown, A t and a
may not have the same starting and/or end states.
sequence denoted as · · · , Z−2 , Z−1 , Z0 , Z1 , Z2 , · · · be the ISI
channel output sequence. Let h0 , h1 , · · · , hℓ denote the ISI
channel coefficients; the constant ℓ is the ISI channel memory
length. The input-output relationship of the ISI channel is
ℓ
X
hi At−i + Wt ,
(1)
Zt =
i=0
where the noise samples Wt are zero-mean jointly Gaussian.
Figure 1 depicts the time evolution of the ISI channel states.
The ISI channel state at time t equals the (length-ℓ) vector of
input symbols [At−ℓ+1 , At−ℓ+2 , · · · , At ]T .
A. The m-truncated max-log-map (MLM) detector
We proceed to describe the sliding-window MLM receiver.
At time instant t, the m-truncated MLM detector considers
△
the neighborhood of 2m + ℓ + 1 channel outputs Z t =
T
[Zt−m , Zt−m+1 , · · · , Zt+m+ℓ ] . Define the symbol neighborhood A t containing the following 2(m + ℓ) + 1 input symbols
△
A t = [At−m−ℓ , At−m−ℓ+1 , · · · , At+m+ℓ ]T . Both A t and Z t
are depicted in Figure 1. Let hi denote the following length(2m + ℓ + 1) vector
m+i
hi
△
=
m−i
z }| {
z }| {
[0, 0, · · · , 0, h0 , h1 , · · · , hℓ , 0, 0, · · · , 0]T ,
(2)
where i can take values |i| ≤ m. Let 0 denote an all-zeros
△
vector 0 = [0, 0, · · · , 0]T . Let both H and T denote the size
2772
2m + ℓ + 1 by 2(m + ℓ) + 1 matrices given as
ℓ
2m+1
by considering those candidates a ∈ M that compete, i.e.
satisfy a0 6= Bt , with the symbol decision Bt . Denote the
difference in the squared Euclidean distances
ℓ
z
}|
{ z
}|
{ z
}|
{
H = [ 0, 0, · · · , 0 h−m , h−m+1 , · · · , hm 0, 0, · · · , 0],
∆(a, ā) = ∆(a, ā; Z t )
h ,h ,· · · h

ℓ ℓ−1
1
△
Z t − T1 − Ha|2 − |Z
Z t − T1 − Hā|2 , (6)
= |Z
..


hℓ
.


h0 ,
△ 
 where a and ā are sequences in {−1, 1}2(m+ℓ)+1.
. . .. 0
0
···
0
T=
.
.. . .
.
.


.
.
h1

hℓ−2 · · · h0  Definition 3. Define the m-truncated MLM reliability
hℓ−1 · · · h1 h0
1
△
Rt = min
∆(a, B [t] ),
(7)
a∈M 2σ 2
(3)
△
a0 6=Bt
△
Using (3), rewrite Z t = [Zt−m , Zt−m+1 , · · · , Zt+m+ℓ ]T
using (1) into the following form
Zt
= (H + T) A t + W t ,
(4)
where here W t denotes the neighborhood of (zero-mean) noise
△
samples W t = [Wt−m , Wt−m+1 , · · · , Wt+m+ℓ ]T . We assume
stationary noise variance E{Wt2 } = σ 2 for all times t; this is
not a strict requirement, see [8].
Definition 1. Denote the set M that contains the m-truncated
MLM candidate sequences
n
o
△
M =
a ∈ {−1, 1}2(m+ℓ)+1 : ai = 1 for all |i| > m .
Each candidate a ∈ M has boundary1 symbols equal to
1; see Figure 1. Let the following sequence · · · , B−2 , B−1 ,
B0 , B1 , B2 , · · · denote symbol decisions on the channel inputs
· · · , A−2 , A−1 , A0 , A1 , A2 , · · · . Let 1 denote the all-ones vec△
tor 1 = [1, 1, · · · , 1]T . Let |a| denote the Euclidean norm of
the vector a.
Definition 2. The symbol decision Bt on channel input At , is
obtained by i) computing the sequence B [t] that achieves the
following minimum
B [t]
△
=
Z t − (H + T)a|2 ,
arg min |Z
=
Z t − T1 − Ha|2 ,
arg min |Z
B [t]
where ∆(a, B ) ≥ 0, see (6), is the difference in the obtained
squared Euclidean distances corresponding to candidates
a, B [t] ∈ M, and σ 2 is the noise variance.
Note that ∆(a, B [t] ) ≥ 0 for all a ∈ M, simply because B [t] achieves the minimum squared Euclidean distance
amongst all candidates in M, see (5).
II. D ISTRIBUTIONS & E RROR P ROBABILITIES
In this section, we give closed-form expressions for the joint
i) reliability distribution FRt1 ,Rt2 ,··· ,Rtn (r1 , r2 , · · · , rn ), and
ii) symbol error probability Pr {∩ni=1 {Bti 6= Ati }}. The result
holds for any number n of arbitrarily chosen time instants
t 1 , t2 , · · · , tn .
For all times t, define the following two random variables
Xt and Yt as
1
1
△
△
A t , a), and Yt = max ∆(A
A t , a) ≥ 0, (8)
Xt = max ∆(A
a∈M 4
a∈M 4
a0 =At
a0 6=At
At , a) is the difference in obtained squared Euwhere ∆(A
clidean distances, corresponding to the transmitted sequence
A t and a candidate a ∈ M, see (6). Note that Yt ≥ 0,
because there must exist a candidate a ∈ M that satisfies
A t , a) = 0, see (6); this particular candidate a ∈ M
∆(A
satisfies ai = At+i for all values of i satisfying |i| ≤ m.
a∈M
a∈M
(5)
and ii) setting the symbol decision Bt to the 0-th component
△
[t]
of B [t] in (5), i.e. set Bt = B0 where the sequence B [t] =
[t]
[t]
[t]
[t]
[t]
[t]
[1, B−m , B−m+1 , · · · , B−1 , B0 , B1 , · · · , Bm , 1]T .
The sequence B [t] in (5), and therefore the symbol decision
Bt , is obtained by considering the candidate sequences in the
set M, recall Definition 1 and refer to Figure 1. To obtain B [t] ,
we compare the squared Euclidean distances of each candidate
Ha from the received neighborhood Z t − T1.
In addition to computing hard, i.e., {−1, 1}, symbol decisions · · · , B−2 , B−1 , B0 , B1 , B2 , · · · , the m-truncated MLM
also computes the symbol reliabilities, denoted as · · · , R−2 ,
R−1 , R0 , R1 , R2 , · · · . The reliability Rt is computed by considering competing candidates in the set M; we compute Rt
Lemma 1. The m-truncated MLM reliability Rt in (7) satisfies
2
Rt =
|Xt − Yt |,
(9)
σ2
where both random variables Xt and Yt are given in (8).
The proof of Lemma 1 will appear in the journal version [8].
Lemma 1 is important as it relates Rt to random variables Xt
and Yt in (8); equation (9) allows us to obtain both reliability
distribution and error probability, from the distribution of Xt −
Yt (see [8] and upcoming Corollaries 1 and 2).
A. Closed-form expressions and evaluation procedure
For any n number of arbitrarily chosen time instants
t1 , t2 , · · · , tn , we wish to obtain the distribution of the vector
△
of reliabilities R tn1 = [Rt1 , Rt2 , · · · , Rtn ]T .
Definition 4. Define the binary vector ei of size 2(m + ℓ) + 1
1 Alternatively,
the boundary symbols can be specified to be any sequence
of choice in the set {−1, 1}ℓ ; here we choose the boundary sequence
[1, 1, · · · , 1] = 1 simply for clearer exposition.
2773
m+ℓ+i
ei
△
=
m+ℓ−i
z }| { z }| {
[0, 0, · · · , 0, 1, 0, 0, · · · , 0]T ,
(10)
where i can take values |i| ≤ m + ℓ. Further define the matrix
E of size 2(m + ℓ) + 1 by 2m as
△
E = [e−m , e−m+1 , · · · , e−1 , e1 , e2 , · · · , em ].
(11)
Definition 5. Define the matrix S of size 2m by 22m as
△
S = [s0 , s1 , · · · , s22m −1 ],
(12)
where the columns si make up all 22m possible, length-(2m)
binary vectors, i.e., {s0 , s1 , · · · , s22m −1 } = {0, 1}2m.
A t ) denote the diagonal matrix, whose diagonal
Let diag(A
equals A t . Recall the size 2m + ℓ + 1 by 2(m + ℓ) + 1 channel
At ) of size 2m +
matrix H given in (3). Define the matrix G(A
ℓ + 1 by 22m as
△
A t ) = H diag(A
A t )E.
G(A
(13)
of Q. Let diag(At1 , At2 , · · · , Atn ) denote a diagonal matrix
At ). Define the size n by 2mn matrix
similarly as diag(A
△
Atn1 ) = diag(At1 , At2 , · · · , Atn ) ⊗ hT0 · KW
F(A



SST Q1
At1 )
G(A
A t2 )
G(A

  SST Q2  †

Λ ,
·
..
.



..
.
T
Atn )
G(A
SS Qn
(19)
where h0 is given in (2), and Λ † is formed by reciprocating
only the non-zero diagonal elements of Λ . Define the following
A t ) and ν (A
At ) as
length-22m vectors µ (A
At ) =[µ1 , µ2 , · · · , µ22m −1 ]T
µ (A
△
At )S]T · T(1 − A t )
=[G(A
T
At )s0 |2 , |G(A
A t )s1 |2 , · · · , |G(A
At )s22m −1 |2 ,
− |G(A
(20)
Let W tn1 denote the concatenation
△
W Tt1 , W Tt2 , · · · , W Ttn ]T ,
W tn1 = [W
(14)
At ) =[ν1 , ν2 , · · · , ν22m −1 ]T
ν (A
where W t appears in (4). Note that (14) is a length n · (2m +
ℓ + 1) vector. Define the noise covariance matrix
KW
△
=
W tn1 W Ttn1 }.
E{W
(15)
Note, KW is generally not Toeplitz even if Wt is stationary.
Let A tn1 denote the concatenation of A t1 , A t2 , · · · , A tn , i.e.
similar to (14), we have A ti in place of W ti . Let I denote
the identity matrix; in particular I2m has size 2m by 2m. The
matrix SST can be verified to equal
SST
=
22m
−1
X
sk sTk = 22(m−1) · [I2m + 11T ], (16)
k=0
△
µ(A
At ) − 2At · hT0 G(A
A t )S,
=µ
At ) and νk = νk (A
At ) denote the k-th
where µk = µk (A
A t ) and ν k (A
At ) respectively, and T is
components of µ k (A
given in (3). Let ΦK (r) denote the distribution function of a
zero-mean Gaussian random vector with covariance matrix K.
△
Finally define the following length-n random vectors X tn1 =
△
[Xt1 , Xt2 , · · · , Xtn ]T and Y tn1 = [Yt1 , Yt2 , · · · , Ytn ]T , see
(8). Let R denote the set of real numbers.
Theorem 1. The distribution of X tn1 − Y tn1 equals
n
o
U , A tn1 ) − η (U
U , A tn1 )
FX tn −YY tn (r) = E ΦKV (A
A tn ) r + δ (U
1
△
where S is given in Definition 5 and 1 = [1, 1, · · · , 1] .
We denote the matrix Kronecker product using the opAt1 ), G(A
A t2 ), · · · , G(A
A tn )) denote
eration ⊗. Let diag (G(A
a block diagonal matrix, whose block-diagonal entries are
A t1 ), G(A
A t2 ), · · · , G(A
A tn ), see (13).
G(A
T
A tn1 ) of size 2mn
Definition 6. Let the square matrix Q = Q(A
by 2mn satisfy the following two conditions:
i) the matrix Q decomposes the following size 2mn matrix
1
1
(22)
for all r ∈ Rn , where the following random vectors and
matrices appear in (22)
• U is a standard zero-mean identity-covariance Gaussian
random vector of length-(2mn).
U , A tn1 ) = [δ1 , δ2 , · · · , δn ]T is a length-n vector in Rn ,
• δ (U
where
Λ2 QT = diag (G(A
At1 ), G(A
A t2 ), · · · , G(A
A tn ))T KW
QΛ
A t1 ), G(A
At2 ), · · · , G(A
Atn )) , (17)
· diag (G(A
△
U , Atn1 ) = max(ST Qi ΛU + µ(A
A ti ))
δi = δi (U
A ti )).(23)
− max(ST QiΛU + ν (A
•
A tn1 ) on the l.h.s. of (17) is a diagonal
where Λ = Λ (A
matrix. The number of non-zero diagonal elements in Λ ,
equals the rank of the matrix on the r.h.s. of (17).
ii) the matrix Q diagonalizes the matrix In ⊗ SST , i.e.,
QT (In ⊗ SST )Q = I.
(21)
(18)
See Appendix A on how to compute both matrices
Atn1 ) and Λ = Λ (A
A tn1 ) in (17). Denote matrices
Q = Q(A
Q1 , Q2 , · · · , Qn that vertically concatenate (similar to (14))
to form Q; each Qi is an equal-sized (2m by 2mn) partition
2774
U , A tn1 ) = [η1 , η2 , · · · , ηn ]T is a length-n vector in
η (U
n
R , where
U , A tn1 )
η (U
△
= diag(At1 , At2 , · · · , Atn ) · T
T
At1 , A t2 , · · · , A tn ] h0
· 1 · 1T − [A
A tn1 )U
U.
−|h0 |2 · 1 + F(A
(24)
•
A tn1 ) is the n by n matrix
KV (A
A tn1 )
KV (A
△
=
diag(At1 , At2 , · · · , Atn ) ⊗ hT0 · KW
· (diag(At1 , At2 , · · · , Atn ) ⊗ h0 )
Atn1 )F(A
Atn1 )T .
−F(A
(25)
Procedure 1: Evaluate Joint Distribution FX tn −YY tn (r)
1
1
2
3
4
k∈{0,1,··· ,2
−1}
max2m
k∈{0,1,··· ,2
5
6
1
Initialize: Set FX tn −YY tn (r) := 0 for all r ∈ Rn ;
1
1
while FX tn −YY tn (r) not converged do
1
1
Sample A tn1 = an1 using Pr A tn1 = an1 ; Sample the
length-n, standard zero-mean identity-covariance
Gaussian vector U = u;
Using the sampled realization A tn1 = an1 , obtain the
matrices Q = Q(an1 ) and Λ = Λ (an1 ) satisfying
Definition 6, see Appendix A;
Compute δi = δi (u, an1 ) for all i ∈ {1, 2, · · · , n}. For
δi compute
max2m
sTk QiΛ u + µk (a),
−1}
sTk QiΛ u + νk (a),
see (23). Here a is the sampled realization A ti = a,
and both µk (a) and νk (a) are the k-th components
of µ (a) and ν (a), see (20) and (21);
A tn1 ) in (19); Also compute η (u, an1 ) in
Compute F(A
(24) and KV (an1 ) in (25);
For all r ∈ Rn , update the result
FX tn −YY tn (r)
Fig. 2. Comparing the distributions FXt −Yt (σ2 /2 · r) across different
SNRs, for the PR1 channel and various truncation lengths m. Also shown are
comparisons with empirical MLM distributions.
where the probability


FX j −YY j (0) = Pr
τ
τ

1
1
\
{Xτ − Yτ ≤ 0}
τ ∈{τ1 ,τ2 ,··· ,τj }
has the similar closed form as in Theorem 1.



Denote the realizations of A tn1 , A t and U , as A tn1 = an1 ,
:= FX tn −YY tn (r) + ΦKV (an1 ) (r + δ (u, an1 ) − η (u, an1 )) and A t = a, and U = u. The Monte-Carlo Procedure 1
7
1
1
evaluates the closed-form of FX tn −YY tn (r) in Theorem 1.
1
1
8 end
Direct evaluations of the maximizations in Line 4 may be
The proof of Theorem 1 is lengthly and appears in [8]. difficult for large truncation length m; see Appendix B for an
△
Both i) the joint distribution of the reliabilities R tn1 = efficient dynamic programming technique for this task.
T
[Rt1T
, Rt1 , · · · , Rtn ] , and ii) the joint error probability
III. N UMERICAL C OMPUTATIONS
n
Pr { i=1 {Bti 6= Ati }}, follow as corollaries from Theorem
We now present numerical computations performed for
1; also see [8] for proofs. In the following we denote an index
various ISI channels. Due to space constraints, the results presubset {τ1 , τ2 , · · · , τj } ⊆ {t1 , t2 , · · · , tn } of size j, written
sented here are limited, see [8] for a more general discussion.
compactly in vector form as τ j1 = [τ1 , τ2 , · · · , τj ]T .
We consider only i.i.d. noise Wt and uniform input symbol
△
A t = a}, i.e. Pr {A
At = a} = 2−2(m+ℓ)−1 . The
X tn1 − Y tn1 |, distribution Pr {A
Corollary 1. The distribution of R tn1 = 2/σ 2 · |X
P
signal-to-noise (SNR) ratio is 10 log10 ( ℓi=0 h2i /σ 2 ).
see Proposition 1, is given as
2
Figure 2 shows the marginal distribution FXt −Yt (σ 2 /2 · r)
FR tn (r) = F|X
X tn −Y
Y tn | (σ /2 · r)
1
1
1
2
computed for the PR1 channel with memory ℓ = 1, i.e.
n
X
X
σ
j
j
h0 = h1 = 1. We fix a truncation length m = 4. As
=
(−1) · FX tn −YY tn
· α (ττ 1 , r)
1
1
2
SNR increases, the distributions FXt −Yt (σ 2 /2 · r) appear to
j=0 {τ1 ,τ2 ,··· ,τj }⊆
concentrate more probability mass over negative values of
{t1 ,t2 ,··· ,tn }
Xt −Yt . This is expected, because the symbol error probability
where the length-n vector α (ττ j1 , r) = [α1 , α2 , · · · , αn ]T Pr {B 6= A } = 1 − F
t
t
Xt −Yt (0) decreases as SNR increases.
satisfies
From
Figure
2,
the
(error)
probabilities Pr {Xt ≥ Yt } are
−ri if ti ∈ {τ1 , τ2 , · · · , τj },
αi = αi (ττ j1 , ri ) =
found
to
be
approximately
1.2
× 10−1 , 8 × 10−2 , 3 × 10−2 ,
ri otherwise ,
−2
and 1 × 10 , respectively for SNRs 3 to 10 dB. Finally,
and FX tn −YY tn (r) has closed form as in Theorem 1.
comparisons with empirical distributions, obtained from a non1
1
Tn
Corollary 2. The probability Pr { i=1 {Bti 6= Ati }} that all truncated MLM (see [8]) reveal a good match.
Figure 3 shows joint distributions FX t2 −YY t2 (σ 2 /2 · r)
symbol decisions Bt1 , Bt2 , · · · , Btn are in error, equals
1
1
( n
)
computed for two time lags |t1 − t2 | = 1 (i.e. neighboring
\
symbols) and |t1 − t2 | = 7. The SNR is moderate at 5 dB, and
Pr
{Bti 6= Ati } = Pr X tn1 ≥ Y tn1
truncation length m = 2. The difference between both cases
i=1
n
is
subtle (but nevertheless inherent); observe the differently
X
X
=1+
(−1)j · FX j −YY j (0),
labeled
points in Figure 3. The joint symbol error probability
τ
τ
1
1
j=1 {τ1 ,τ2 ,··· ,τj }⊆
Pr {Bt1 6= At1 , Bt2 6= At2 } is ≈ 6 × 10−2 and 2 × 10−2 for
{t1 ,t2 ,··· ,tn }
|t1 − t2 | = 1 and 7, respectively. Note that when |t1 − t2 | = 7,
1
1
2775
Procedure 2: Solve
Fig. 3. Joint reliability distribution FX
t2
1
Y 2 (σ
−Y
t
2 /2 · r)
computed for the
1
PR1 channel at SNR 5 dB, for a fixed truncation length m = 2.
1
2
both reliabilities Rt1 and Rt2 are independent; this is because
then |t1 − t2 | = 7 > 2(m + ℓ) = 6, refer to Figure 1.
IV. C ONCLUSION
In this paper, we presented closed-form expressions for both
i) the reliability distributions FX tn −YY tn (σ 2 /2 · r), and ii) the
1
Tn1
symbol error probabilities Pr { i=1 {Bti 6= Ati }}, for the mtruncated MLM detector. Our results hold jointly for any number n of arbitrarily chosen time instants t1 , t2 , · · · , tn . Efficient
Monte-Carlo procedures have been given; these procedures can
be used to numerically evaluate the closed-form expressions.
A PPENDIX
Lemma 2. Let S be given as in Definition 5. Let the size 2mn
by 2mn square matrix α diagonalize
(26)
Let β be the size 2mn by 2mn eigenvector matrix β in the
following decomposition
A t1 ), G(A
At2 ), · · · , G(A
Atn ))T
α T (In ⊗ SST ) · diag (G(A
A t1 ), G(A
A t2 ), · · · , G(A
Atn )) · (In ⊗ SST )α
α
· KW · diag (G(A
= β Λ 2β T ,
4
5
6
sT C − |G(a)s|2 by DP
Convention: Set C0 := −∞, and Cj := 0 for all |j| > m;
△
: Binary vector s̄ = [s̄ℓ−1 , s̄ℓ−2 , · · · , s̄0 ]T ;
Input: Matrix G(a); Vector of constants
C = [C−m , C−m+1 , · · · , C−1 , C1 , C2 , · · · , Cm ]T ;
Output: Value stored in βm+ℓ (s̄) = βm+ℓ (0);
Initialize: For all s̄ ∈ {0, 1}ℓ, set the values
n
0
if s̄ = 0,
f β−m−1 (s̄) := −∞ otherwise .
forall the τ ∈ {−m, −m + 1, · · · , m + ℓ } do
forall the s̄ ∈ {0, 1}ℓ do
Pℓ−1
Set the value α = α(s̄) := j=0 hj aτ −j s̄j . Set
the states s̄0 and s̄1 as
s̄0 := [0, s̄ℓ−1 , · · · , s̄2 , s̄1 ]T ,
;
s̄1 := [1, s̄ℓ−1 , · · · , s̄2 , s̄1 ]T .
2
Compute βτ (s̄) := max{−α + βτ −1 (s̄0 ), Cτ −ℓ −
[hℓ aτ −ℓ + α]2 + βτ −1 (s̄1 )};
end
end
where i can take values 1 ≤ i < 2m, and the last eigenvector
is simply 1/|1| = 1/(2m).
B. On efficient execution of Line 4 of Procedure 1
A. Computing the matrix Q = Q(Atn1 ) in Definition 6
In this appendix, we show that the size 2mn square matrix
Q with both properties i) and ii) as stated in Definition 6,
can be easily found. We begin by noting from (16) that
rank(SST ) = 2m, therefore the matrix In ⊗ SST has rank
2mn and is positive definite.
α = I.
α T (In ⊗ SST )α
3
max
s∈{0,1}2m
(27)
Λ2
see (17), and Λ is the eigenvalue matrix of (27), therefore Λ 2
in (27) is diagonal of size 2mn. Then Q = αβ satisfies both
properties i) and ii) stated in Definition 6.
See [8] for the proof. To summarize Lemma 2, the matrix
Atn1 ) in Definition 6, is obtained by first computing
Q = Q(A
two size 2mn matrices α and β respectively satisfying (26)
and (27), and then setting Q = αβ . The matrix β is obtained
from an eigenvalue decomposition of the size-2mn matrix
(27), and clearly β depends on the symbols A tn1 .
Remark 1. The matrix α in (26) is obtained from the
eigenvectors of the matrix SST in (16). The first 2m − 1
eigenvectors of SST are
2m−(i+1)
i
z }| {
z }| {
2 − 21
(i + i ) · [1, 1, · · · , 1, −i, 0, 0, · · · , 0]T
△
Define the length-(2m) vector C = [C−m , C−m+1 , · · · ,
C−1 , C1 , C2 , · · · , Cm ]T . Set C0 := −∞ and Cτ := 0 for all
|τ | > m. It can be verified from (20) and (21), that by setting
C
C
:=
:=
QiΛ u + [G(a)]T · T(1 − a) and
QiΛ u + [G(a)]T · [T(1 − a) − 2a0 · h0 ],
respectively, we can solve both maximizations in Line 4,
Procedure 1 as
maxs∈{0,1}2m sT C − |G(a)s|2 .
(28)
Matrix G(a) is (ℓ + 1)-banded, see [8], and therefore (28)
is solved using the (dynamic programming) Procedure 2.
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