PROCEEDINGS
653
LETTERS
pulse, i t is possible to produce a pseudorandom signal coinciding
with the clocking rate of Mi.'This fact makes i t possible to achieve
output ratesof the order of 1CP numbers/s.
ACKNOWLEDGMENT
The author wishes to thank Prof. J. Kawata and M. Ohyama for
their useful suggestions during thiswork.
-
-1
Isare]
I
1
1
7
5
3 -1
1
-3
3
1
-1-3
-3
-1
1
3
-9
-17
17
9
1
-7
1
0
0
0
0 -1
0-1
1
7
1
REFERENCES
111 S. Ayanaga and M. Sugiura. Algebra. Japan: 1-mi.
I21 C. M. Rader el
"A fast method ofgenerating
ol..
1
- 1 - 1
0
1
1
1
1
1
-5
-7
I
X
O X K
1971.
digital m d o m numbers.'
K
b
Ins4901
Bell Surf. Tech. J . , vol. 49, pp. 2303-2310. 1970.
(31 B. R.Gaines. "Stochasticcomputing,"in Proc. Sprirg J o i d Comgvkr C a f , .
1967. pp. 149-156.
(0)
r
Slant Haar Transform
1
1
1
1
7
5
3 -1
AND
1
1
1
-3
-5
1
-7 x
3
1
-1 -3 -3
-1
1
3x
7
-1
-9 17-17
9
1
-7 x
-1
-1
1
1
1
-1
I s w e r e l -1
-1 -1/T 1
1
1
1 -1 -11
BERNARD J. FINO
1
-1
V. RALPH ALGAZI
Abstracf-The slant Haar transform (SHT) is defined and related to .the slant Walsh-Hadamardtransform (SWHT). A fast
algorithm for the SHT is presented and its computational complexity
computed. In most applications, the SHT is faster and performs as
well as the SWHT.
(bl
Fig. 1.
Ordered slant transforms of order 8. (a) SHT. (b) SWHT.
INTRODUCTION
The
slant
\Valsh-Hadamard transform (SWHT) (originally
called slant transform) has been proposed by Enomoto and Shibata
[ l ] for the order 8 and used in TV image encoding. Pratt et al. (21
and Chen [3] have generalized this transform to any order 2" and
compared its performance with other transforms. In [4], we have
given a simpler definition of the SLVHT as a particular case of a unified treatment of fast unitary transforms and computed the number
of elementary operations required by its fast algorithm.' The interesting feature of the S\VHT is the presence of a slant vector with
linearly decreasing components in its basis. On the other hand, n e
have found that locally dependent basis vectors, such as in the Haar
transform (HT),.are of interest [SI. In this letter, we define a composite fast unitary transform: the slant Haar transform (SHT). We
show that its relations to the SWHT parallel the relations between
theHTandthe
LValsh-Hadamard transform (WHT) 161. This
over
previous work leads u s to expect that the SHT has an advantage
the SWHTbecause of its speed and comparable performance.
DEFINITION
The generalized Kronecker product of the set {a>of n matrices
[.4j](j=0, . . . , n - 1 ) of order m and the set {a}of m matrices
[ B k ](k = 0 , . . . , m - 1) of order n is the matrix [C] of order mn such
=A..,mBm~u'
when u , u ' < n and w , w ' < m . [C] can
that Cum+r.u,m+r,
be factorized and has a fast algorithm [4]. Thegeneralized Kronecker
product provides a simple way to recursively define fastunitary
transforms. Consider the matrix of order 2 :
Then the matrix of order 2", denoted ( T p ] , is obtained from the
{ T p - I ] , [Tp-I])
where @ denotes a
matrix of order 2"-' by {a).@
generalized Kronecker product. R l t h this recursive notation, the
H T is obtained for {a].= [ K ( r / 4 ) ] , [Iz],. * . , [I,]and the LVHT
Manuscript received December 6 , 1973. This work was supported in part by the
Sational Aeronautics and Space Administration under Grant NASA-NGR-05403404 and in part hy the Xational Science Foundation under Grant NSF-GK37282.
The authors are with the Electronics Research Lahoratory. CoUege of Engineering. University of California. Berkeley, Calif. 94720.
S o t e that the number of operations given in (31 seems in error and that fewer
operations may in fact he needed in a different organization 141.
with 0. < r / 2 given by
COS
en =
~~
2"- I
--
+/T$
.
This rotationintroduces the ''slant vector,P thecomponents of which
are linearly decreasing (see [ 7 ] for a complete description of slant
vectors).
ORDERING
OF THE BASISVECTORS
In signal processing practice, the vectors of the HT are used i n
rank order [4] and those of the WHT are used in sequency (number
of sign changes) order. There is an ordering of the slant transforms
consistent with these orderings;
the basis vectors obtained by the
previously given recursive definitions are ordered by: a) decreasing
number of nonzero elements (from globally to locally dependent
basis vectors); b) increasing number of zero crossings; and c) from
left to right. The ordered matrices [SHTR] and [S\VHTs] are shown
i n Fig. 1.
RELATIONS
BETWEEN SHT AND SLVHT
The SHT and SLVHT have relations similar to the relations between the HT and the
LVHT [6]. Partition theordered SHT (SWHT)
into 2n rectangular submatrices, denoted [MSH,,k-'] ([MSWH,,L-']),
h = O , . . . , n - 1 and i = O , 1. Fork, i = O , 1, [MSH,,o*o],["SH,,o~l],
["SH..L.a], and [MSH,.1.1] are the four first rows of the ordered
matrix- [SHT,.]. For k, i > 1, [MSHtmk.']is formed from the rows of
ranks 2 k - 1 + i 2 k - 2 ~ r < 3 X 2 C * + i 2 C I . The matrices [MSWH,,L.']
are similarly defined. The submatrices for the order 8 are shown i n
Fig. 1. It can be shown, following the proof given in [6], that
[?V~SU'H~,'~.']
= [S?k-?Ii[WHT,c-?][?VfSH,n"i]
where
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654
PROCEEDINGS OF THE
stonds for ka + l b
IEEE,MAY 1974
By another algorithm, the SHT of order 4, which is also the
SWHT of the same order, can be performed with 8 additions and 2
multiplications [3], compared with 10 additions and 2 shifts as
given by the above formulas. This order-4 algorithm can be intrcduced i n the recursive definition to trade 2*-1 additions and 2”-1 shifts
for Zn-’ multiplications in the previously given results.
The SHThas the same number of multiplications and shifts than
the S\t’HT, but has (n-3)2R+4 fewer additions and 1 more normalization; therefore the SHTis faster than the SWHT.
TRANSFORMS
OTHERSLANT
The two slant transforms considered in this letter are only two
members of a large family of slant transforms; this family has been
studied in some detail in [7].
CONCLUSIONS
We have defined the SHT and presented some properties of this
transform, mainly as compared to the SWHT, which has been successfully considered for image encoding. The SHT is faster and preserves some local properties of the signal and we believe i t should be
preferred over the SWHT.
REFERENCES
-I
-I
-I
Fig. 2. Fast algorithm for the SHT of order 8 .
(a) Unordered rows. (b) Ordered rows.
[I] H. Enomoto and K. Shihata. ‘Orthogonal transform coding system for television
signals,” in Prw. 1971 Symp. Appliutionr o f f k Wdsh Furclions. A B 7 2 7 OOO,
pp. 11-17.
[2j W. K. Pratt. L. R Welch. and W. H. Chen. “Slant transform for image coding,’
in Prw. 1972 Sump. Applicalions o f f h e W d r h Func&ms. AD-744 650. pp. 229234.
131 W. H. Chen. “Slant transform image coding,’Electron. Science Lab..Univ.
Southern California. Tech. Rep. 441, May 1973.
[4] B. J. Fino and V. R. Alga& ‘A unified treatment for fast unitary transforms.”
to be published.
[SI B. J. Fino, ‘Experimental study of image coding by Haar and complex Hadamard transforms” (inFrench).Arx. Tckcummur., pp. 185-208, May-June 1972.
(61
*Relations betweenHaar and Walsh/Hadamard transforms,. Proc. IEEE
(Lett.). voL 6 0 , pp. 647-648. May 1972.
171 -,
“Recursive definition and computation of fastunitary
transforms.’
Ph.D. dissertation. Univ. California, Berkeley, Nov. 1973.
-.
Phaseplane Analysis of Phase-Locked Loops
with Rapidly Varying Phase Error
and [WHT,,] is the ordered WH matrix of order 2*.
C. H. CHEN
As for the H T and WHT, these relations imply ‘zonal” relations
between the components of the representation of a vectorby the SHT
Abstract-Nonlinear behavior of phase-locked loops with rapidly
and SWHT and,
in particular, an identical energy partition according varying phase error is examined by using computer phase-plane
to these zones.
analysis. The phase variation is modeled by a sinusoidal function.
Threshold loopparameters are presented for both sinusoidal and
FASTALGORITHM
sawtooth phase comparators.
The framework developed in [4] gives the fast algorithm preI. INTRODUCTION
normalsented inFig. 2(a) for the order 8. Note that the stepby-step
izations inthe rotations by the
matrix [ R ( 8 , ) ] can be delayed so that
The performance of command and telemetry systems, useful in
the
the rows 1 and 2*-1 are in fact rotated by matrix
deepspace communications, is frequently affected by the R F phase
error which is introduced a t the point of reception by means of the
carrier tracking loop. In low data-rate communications, the phase
error may vary rapidly over the duration of the signaling interval.
Causes of this type of behavior in planetary entry are turbulence,
L
dispersion, attenuation, and residual Doppler. The phase variations
which requires only 2 shifts, 2 additions, and 1 multiplication. This cannot be tracked by a phase-locked loop of lower bandwidth, while
algorithm can be reorganized as shown in Fig. 2(b) to give the or- the signal-to-noise ratio (SNR) in this minimum loop bandwidth is
dered transform.
too low.
The relations (1) also yield a decomposition of the SWHT algcWhen the ratio of the system data rate to carrier tracking loop
rithm into an SHT and WHT
of lower orders.
bandwidth is less than one, the problem of power allocation between
the carrier and the data hasbeen considered by Hayes and Lindsey
COMPUTATIONAL COMPLEXITY
[l 1. In this letter thephase variation is characterizedby a sinusoidal
The required number of each elementary operation can be com- input phase, k sin ( o $ + r / 6 ) , which models a typical phase variation
puted with general formulas [4] and the previous definition of the in communication over turbulent media such as the Venus atmoSHT. We find that the computation of the SHT of a vector of order sphere. Conditions for synchronization stability and the acquisition
2“ requires 2”+’-6 additions, 2*-f-l multiplications, 2”-2 shifts, behaviors of the loops in the absence of noise are examined from an
and 3(2n-*) normalizationsf at the last stage
of computation.
* W e assume that the transform coeflicients are scaled to the most commonly
encountered normalization factor;thusthe
transform isunitarywithinascale
factor.
Manuscript received January 3, 1974. This work was supported by NASA under
Grant NGR 22-031402.
The author is with Southeastern Massachusetts University. North Dartmouth.
M w . 02747.
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