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Discrete Walsh- Hadamard Transform

Contents

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Introduction

Walsh (or Hadamard) Functions

Discrete Walsh- Hadamard Transform

Walsh- Transform Applications

References

Introduction

The Walsh- Hadamard transform (WHT) is a suboptimal, non- sinusoidal,

orthogonal transformation that decomposes a signal into a set of

orthogonal, rectangular waveforms called Walsh functions. The

transformation has no multipliers and is real because the amplitude of

Walsh (or Hadamard) functions has only two values, +1 or -1.

WHTs are used in many different applications, such as power spectrum

analysis,

filtering,

processing

speech

and

medical

signals,

multiplexing

and

coding

in

communications,

characterizing

non-linear

signals,

solving

non-linear differential equations, and logical design and analysis.

This demo provides an overview of the Walsh-Hadamard transform and some

of its properties by showcasing two applications, communications using

spread spectrum and processing of ECG signals.

Walsh (or Hadamard) Functions

Walsh

functions

are

rectangular

or

square

waveforms

with

values

of

-1

or

+1. An important characteristic of Walsh functions is sequency which is

determined

from

the

number

of

zero-crossings

per

unit

time

interval.

Every

Walsh function has a unique sequency value.

Walsh functions can be generated in many ways (see [1]). Here we use the

hadamard function

in MATLAB? to generate Walsh functions. Length eight

Walsh functions are generated as follows.

N = 8; % Length of Walsh (Hadamard) functions

hadamardMatrix = hadamard(N)

hadamardMatrix =

1 1 1 1 1 1 1 1

1 -1 1 -1 1 -1 1 -1

1 1 -1 -1 1 1 -1 -1

1 -1 -1 1 1 -1 -1 1

1 1 1 1 -1 -1 -1 -1

1 -1 1 -1 -1 1 -1 1

1 1 -1 -1 -1 -1 1 1

1 -1 -1 1 -1 1 1 -1

The rows (or columns) of the symmetric hadamardMatrix contain the Walsh

functions. The Walsh functions in the matrix are not arranged in

increasing order of their sequencies or number of zero-crossings (i.e.

'sequency

order')

but

are

arranged

in

'Hadamard

order'.

The

Walsh

matrix,

which contains the Walsh functions along the rows or columns in the

increasing order of their sequencies is obtained by changing the index

of the hadamardMatrix as follows.

HadIdx = 0:N-1; % Hadamard index

M = log2(N)+1; % Number of bits to represent

the index

Each column of the sequency index (in binary format) is given by the

modulo-2

addition

of

columns

of

the

bit-reversed

Hadamard

index

(in

binary

format).

binHadIdx

=

fliplr(dec2bin(HadIdx,M))-'0';

%

Bit

reversing

of

the

binary

index

binSeqIdx = zeros(N,M-1); % Pre-allocate memory

for k = M:-1:2

% Binary sequency index

binSeqIdx(:,k) = xor(binHadIdx(:,k),binHadIdx(:,k-1));

end

SeqIdx = binSeqIdx*pow2((M-1:-1:0)'); % Binary to integer sequency

index

walshMatrix = hadamardMatrix(SeqIdx+1,:) % 1-based indexing

walshMatrix =

1 1 1 1 1 1 1 1

1 1 1 1 -1 -1 -1 -1

1 1 -1 -1 -1 -1 1 1

1 1 -1 -1 1 1 -1 -1

1 -1 -1 1 1 -1 -1 1

1 -1 -1 1 -1 1 1 -1

1 -1 1 -1 -1 1 -1 1

1 -1 1 -1 1 -1 1 -1

Discrete Walsh-Hadamard Transform

The forward and inverse

Walsh transform pair

for a signal

x(t) of length

N are

Fast algorithms, similar to the Cooley-Tukey algorithm, have been

developed to implement the Walsh-Hadamard transform with complexity

O(NlogN)

(see

[1]

and

[2]).

Since

the

Walsh

matrix

is

symmetric,

both

the

forward and inverse transformations are identical operations except for

the scaling factor of 1/N. The functions fwht and ifwht implement the

forward and the inverse WHT respectively.

Example 1

Perform WHT on the Walsh matrix. The expected result is an

identity matrix because the rows (or columns) of the symmetric Walsh

matrix contain the Walsh functions.

y1 = fwht(walshMatrix) % Fast Walsh-Hadamard transform

y1 =

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

Example

2

Construct

a

discontinuous

signal

by

scaling

and

adding

arbitrary

columns

of

the

Hadamard

matrix.

This

signal

is

formed

using

weighted

Walsh

functions,

so

the

WHT

should

return

non-zero

values

equal

to

the

weights

at

the

respective

sequency

indices.

While

evaluating

the

WHT,

the

ordering

is specified as 'hadamard', because a Hadamard matrix (instead of the

Walsh matrix) is used to obtain the Walsh functions.

N = 8;

H = hadamard(N); % Hadamard matrix

% Construct a signal by adding a few weighted Walsh functions

x = 8.*H(1,:) + 12.*H(3,:) + 18.*H(5,:) + 10.*H(8,:);

y = fwht(x,N,'hadamard')

y =

8 0 12 0 18 0 0 10

WHT is a reversible transform and the original signal can be recovered

perfectly using the inverse transform. The norm between the original

signal and the signal obtained from inverse transformation equals zero,

indicating perfect reconstruction.

xHat = ifwht(y,N,'hadamard');

norm(x-xHat)

ans =

0

The Walsh-Hadamard transform involves expansion using a set of

rectangular waveforms, so it is useful in applications involving

discontinuous signals that can be readily expressed in terms of Walsh

functions. Below are two applications of Walsh-Hadamard transforms.

Walsh-Transform Applications

ECG

signal

processing

Often,

it

is

necessary

to

record

electro-cardiogram

(ECG)

signals

of

patients

at

different

instants

of

time.

This

results

in

a

large

amount

of

data,

which

needs

to

be

stored

for

analysis,

comparison,

etc.

at

a

later

time.

Walsh-Hadamard

transform

is

suitable

for

compression

of ECG signals because it offers advantages such as fast computation of

Walsh-Hadamard coefficients, less required storage space since it

suffices

to

store

only

those

sequency

coefficients

with

large

magnitudes,

and fast signal reconstruction.

An

ECG

signal

and

its

corresponding

Walsh-Hadamard

transform

is

evaluated

and shown below.

x1 = ecg(512); % Single ecg wave

x = repmat(x1,1,8);

x = x + 0.1.*randn(1,length(x)); % Noisy ecg signal

y = fwht(x); % Fast Walsh-Hadamard transform

figure('Color','white');

subplot(2,1,1);

plot(x);

xlabel('Sample index');

ylabel('Amplitude');

title('ECG Signal');

subplot(2,1,2);

plot(abs(y))

xlabel('Sequency index');

ylabel('Magnitude');

title('WHT Coefficients');

As

can

be

seen

in

the

above

plot,

most

of

the

signal

energy

is

concentrated

at

lower

sequency

values.

For

investigation

purposes,

only

the

first

1024

coefficients are stored and used to reconstruct the original signal.