I ORIGINA TING
ACTIVITY (Corporalte author)
of report and incsive
REFS
functions, close
briefly, and
the com-
applications are then discussed.
It is concluded that
pattern
recognition,
B LINK C
Haar functions
Walsh functions
Orthogonal functions
Series convergence
and
Coefficients .................................
4
3.2 Complete
Haar Transform ......................
4.5 Information
suggests
that
Haar
be
Haar f£wctions are
Haar
cluded that whereas
useful
multiplexing.
PROBLEM
STATUS
AUTHORIZATION
Project XF 53-241-003
suggest-
ing
One such
The purpose of this report
is to
seem
their potential
stracted
mathematical details may be found. Several as-
pects of the Haa, transform,
including computational
cussed in
Section 2. Applications of Haar functions are discussed in Section 4.
2. RELEVANT
posed of func ,ions labeled by two indices:
S=
4
 PW
(cont'4)
1
<x < 1
approached
in
Fig.
1.
The Haar functionis are a complete orthonormal basis of L
2
tions f(x) that are defined
over [0, 1] with f
2
(x)
In
this
report,
infinite series in terms of Haar
functions:
0
2n-I
?~,no
f(x) =
discussed
N
2
n-1
= /1 hr ,o1,2'.,2.Tiscnegn
for continuous
functions, dthesequecethe
k = 0, 1, 2, . ...
are at binary-rational points. •
associated
an interval can usually
at binary-rational points.
longer
irrational
discontinuities. This means that given an approximation accuracy e that must be
satisfied at a particular point x
1
all N >M we have
ISN(xl) - (X
guarantee
tained simultaneously at all points in [0,1].
2.3
Several
aspects of the potential utility of Haar functions derive from an important
property of the partial
Nth Haar partial sum SN(x) is
a step
function with
on
in the interval
2
N steps that is the best approximation to &(x) in
the mean-square-error sense. This mean-value property of SN
is also true for the Walsh
series
approximations to
the function
the
exparided as
Now
the coefficients
in the Haar Series also have a simple relationship with the mean
value of f(x)
010-
Ov 01 02 Q3 04 05 06 Q7 08 09 1t0 0
01 02 03
05- 05-
05
09 to
(6a)
SL.2m-2)12
2
I2m-1),
= -,+1/21m
(6b
in the interval (a, b). Thus cq is proportional to
the difference in ih,. mean
value of f(x)
differently,
to the difference of two adjacent steps of S2(x), namely
the
x
functions with a bounded first derivative that exists everywhere, there
is
any Haar partial
[0,1],
the
If x is restricted
SN,
then
restricted to
N,
the
3.1
TI.
We
2-4
equal
1
approximation of
f(t) in
T/2N.
are chosen
so that the integral after a period T/2N is the mean value of f(t) during that
period. The
and held for a period of T/2N. The
output of
Haar-series partial
and hold may
pass sequency
applications of
Walsh functions
of a low-pass
tionship
between Walsh functions and Haar functions. (Walsh functions may be written
as simple linear
values of
f(t) in
Eqs. (6a)
is easily obtained
from them. We
if it did not involve multiplications by
the variable
can use
by a constant.
fied
Haar
the leading factor in
responds to replacing
2
23 8
points. The
+ x8
k
1
=x
1
+ x
2
+ x
3
+ x
4
-x
5
-x
6
-x
7
-x8
transform.
as additions.
Walsh
proper
case of the modified Haar transform, sums and differences are
calculated at each
is 2(n -
arithmetic element that produces both sum and difference
reduces this to n
transform
tion. No information is lost by modifying the leading factors
in
Eqs.
(6a)
involve
in many cases we should be able to
analyze the problem in terms of the unnormalized set of functions
fo
=ý0
tM
re-
on
use of
the modified
transform can
energy may be used to
transmit different
obtained
by
that if u is odd,
then p = (2N - u + 1)/2 is an integer. The multiplication by 1/2p can I
therefore be accomplished
representation. J
this case, multiplication
shift
following
0
is neces-
sary only
We can take
advantage of the
are by a constant factor, namely
V1 We note that
which is within about
2% of V"i. Thus, kVT'can be approximated by k + (k/4) + (k/8) •
+ (k/16), which can be obtained with
three additions
accomplished
with
2(n - 1) adds and 2(n - 1) shifts. The (n - 2)/3 multiplications required
for
requirements for the Walsh and
Haar transforms
Walsh and Haar Transforms
- 1) 3(n - 1)
- 4 2n - 3
It is important to
note that in the fast Haar transforras, the average number of opera-
tions per point is independent of the transform size. For example,
only
two
the
fast
increases as
the
arithmetic
the
fast
on transform
is such that the
the
transform,
then
these n locations are sufficient to complete the transform. If the samples
are accepted one
transformed coefficients can be put out
im-
data are present. For example, the order of calculation in
the 23 = 8-point example
at one
2
, a
2
log
2
4. IMPLICATIONS FOR
the ease by which they
can be
operations that involve
them. Mathematically, this comes from the fact that Walsh functions have a constant value
of plus or minus one on each of
2
N equal subintervals and that the sequence of values may
be d2erived froai
the character group of the dyadic grollp. Haar functions are
also constant
constants,
on each
interval they may have one of three values, plus one, minus one, or zero. Thus
binary
operations involving Haar functions are not
likely to
This that Haar functions not have as potential
for practical applica-
likely to be convenient in applica-
tions requiring manipulation
Other
possibilities
are
us to
Haar partial sums and coefficients. This
brings to
is to encode the coefficients of an expansion in terms
)f some set of basis functions. If
convergence is
rapid, many coefficients are small. and it may be possible to reduce
the
transmission bandwidth from that required to send the time -domain
signal
on all
points, as in the trigonometric Fourier
series, then a certain immunity to channel errors results. This was pointed out by Pratt,
et al. (11).
given Haar-series coefficient cm contains information from the interval ((2m - 2)/2n,
2m/2n)
the full set of
to co and c
2
depends on a smaller region of f(x).
To
reduction can result, consider the example shown in Fig. 3.
The
function
in
The nth partial Haar sum is shown in Fig. 3b.
Of the 2n coefficients in Sn,, only
two, co
transform, if a function is
constant
where 2
reduction is summarized
in a general
(6b).
over adjacent
may be particularly
areas of constant or slowly changing tone.
Another pos-
data
for
is an application in which
the disadvantages
the study of
relation-
systems.
b
o o~
Fig. 3a-A function f(x) that is constant Fig. 3b-A Haar-function approximation
everywhere except in one subinterval to f(x)
 
two
between
the counter is fully decoded into
2
k
lines,
each
wave of
a logical AND
gate, so that
the combination of
that
0 and
V. When the input from the decoder is a logical zero,
the
levels from
(0, V)
to (-V',
V'). Desired
normclization is
possible
CG
circuit
Haar fune- Fig. 5--A possible conversion-gate circuit. Here
tion. Bockaked"D"hale
he c ockis adjusted to provide the
desired gai.n. The use of
rate. Blocks
decoder lines be
pulse. Blocks marked DECODER decode inverted.
the k outputs of a modulus-2k counter into
2houtput lines.
technique
2
filter
Section 3.1. The output
function
with
output from the are added to form the multiplexed siglial.
An alternative to
using the low-pass
sequency filters on each channel is to sample the input waveform di-
rectly. However, the uutput of each sequency filter is the
best step-function approximation
to the input waveform in the mean-square-error sense, whereas the step functions produced
by direct sampling is not. Since
the approximation
will be
corrupted by
outputs.
a signal plus zero mean noise, then the low-pass sequency
filters will integrate the
Fig. 6-A
a low-pass sequency filter of the type described in
Section
3.
multi-
plied
normal and have period T . The outputs of the
multipliers are added to form the multiplexed signal.
Demultiplexing is performed by
chronous
The multiplexing functions fi are
periodic in T' are are orthonormal
in a single frame
it to be demultiplexed. Normalization
results in
segment of time during which the filter
outputs remain constant, the
The coefficients ci
and sequency-division
multiplexing (SDM) are the results of specific choices of the multiplexing functions fi. In
fact, we can choose functions that result in a combination
of
The block
functions in Fig. 8
functions in Fig. 9 will
result in something
be
This is an
example of the lesson, first learned in connection with pulse-compression radar, that the
coding
of
situation.
4
4
o
4.
83~
o1
4-
184
o
1
Fig.
7-Four
block
 
Their
advantages of
pends on our
Aew of nature.
In any case,
rms-voltage ratio
Beginning
is given by
ii
to
of width T'/2N, the energy transmitted in
one frame is
(112))T'/2N. Thus
2N
i=I
The power averaged over many frames depends on the statistics of the input
channels ci.
the same
the
Vrms=
I•-
Z
F1c/j
(18)
To compare
the peak voltage and power, we must determine for each set of functions
fi
The TDM orthonormal
block functions satisfy
the highest signal level of all the channels;
the peak power is proportional to the square. If we assume that the input
channels have
V(TDM) =
2
2
n=
2
each
are at
so
that
V(SDM)
2
N
(28)
ax
and
p(SDM) =
2
2N.
(29)
in
n=1
m=1
In any frame the peak is reached in the first slot where
all functions that
contribute to the
sum
have a positive sign. The peak voltage in any frame is therefore
N
SDM
average-power rati3; TDM has the
lowest,
those for Haar multiplexing are in between. For large N
n
.5=
V((TDM)
Hmax
-. max
(35)
the
2
result, in SDM the average
value
is
further
Haar-function multiplexer
channel
results is that for a given signal-to-noise ratio, mul-
tiplexing with Haar functions requires less dynamic range than with Walsh functions.
In
As a
final point, we note that demultiplexing is equivalent to recovering the coeffi-
cients
6, this can be ac-
complished
the multiplexed signal in terms of the func-
tions
fi. This is particularly easy with Haar functions, as discussed in
Section
3.
The
result
digital transform if a computer is
already
communication systems this is often the
case.
be
As a
simple example,
Fig. 10,
the point x
coefficients cm is nonzero.
locates
1/2n-1. Taking
the sign of the coefficient into account improves the resolution to
1/2n.
=
n > N. For
coefficient
theory is sug-
sampling
2
that a func-
tion with a
1 12
transform
related to changes in the amplitude
of the
Eq.
36
bf
whereb-a c-b=L=1/2N anda, b, and c are functions of m.
This
intervals of width
can be
well-suited
 
for helpful discussions relating to Section
4.3. The circuit in Fig. 5 is due to W. Smith, and the expansion in Eq. (9) was suggested
by
grateful to J. S. Lee
for
(1923).
FunctionsSymposium
and
Workshop,
March
Laboratory.
I'D-727 000,
Washington, D.C.
Annalen
Pratt, "Digital Image
Laboratory, Washington, D.C., 1970, p. 183.
7. H.C. Andrews, "Walsh
Workshop,
8. J.E. Shore
NRL
Report
7470,
(Available as
1163.)
in Proceedings
and Workshop,
Naval Research
11.
(1969).