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Open walshhadamarddemo.m in the EditorRun in the Command WindowDiscrete Walsh-Hadamard Transform Contents•Introduction•Walsh (or Hadamard) Functions•Discrete Walsh-Hadamard Transform•Walsh-Transform Applications•ReferencesIntroductionThe 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) FunctionsWalsh 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) functionshadamardMatrix = hadamard(N)hadamardMatrix =1 1 1 1 1 1 1 11 -1 1 -1 1 -1 1 -11 1 -1 -1 1 1 -1 -11 -1 -1 1 1 -1 -1 11 1 1 1 -1 -1 -1 -11 -1 1 -1 -1 1 -1 11 1 -1 -1 -1 -1 1 11 -1 -1 1 -1 1 1 -1The 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 indexM = log2(N)+1; % Number of bits to represent the indexEach 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 indexbinSeqIdx = zeros(N,M-1); % Pre-allocate memoryfor k = M:-1:2% Binary sequency indexbinSeqIdx(:,k) = xor(binHadIdx(:,k),binHadIdx(:,k-1));endSeqIdx = binSeqIdx*pow2((M-1:-1:0)'); % Binary to integer sequency indexwalshMatrix = hadamardMatrix(SeqIdx+1,:) % 1-based indexing walshMatrix =1 1 1 1 1 1 1 11 1 1 1 -1 -1 -1 -11 1 -1 -1 -1 -1 1 11 1 -1 -1 1 1 -1 -11 -1 -1 1 1 -1 -1 11 -1 -1 1 -1 1 1 -11 -1 1 -1 -1 1 -1 11 -1 1 -1 1 -1 1 -1Discrete Walsh-Hadamard TransformThe forward and inverse Walsh transform pair for a signal x(t) of length N areFast algorithms, similar to the Cooley-Tukey algorithm, have been developed to implement the Walsh-Hadamard transform with complexityO(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 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1Example 2Construct 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 functionsx = 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 10WHT 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 =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 ApplicationsECG 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 wavex = repmat(x1,1,8);x = x + 0.1.*randn(1,length(x)); % Noisy ecg signaly = 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.Truncating the higher sequency coefficients also helps with noise suppression. The original and the reproduced signals are shown below.y(1025:length(x)) = 0; % Zeroing out the higher coefficients xHat = ifwht(y); % Signal reconstruction using inverse WHTfigure('Color','white');plot(x);hold onplot(xHat,'r');xlabel('Sample index');ylabel('ECG signal amplitude');legend('Original Signal','Reconstructed Signal');x=ecg(512);x1=repmat(x,1,8);x1=x1+0.1*randn(1,length(x1));figure('Color','white');y1=fft(x1);subplot(2,1,1)plot(abs(y1))y1(201:length(x1))=0;xhat=ifft(y1);plot(abs(xhat))x2 = ecg(512); % Single ecg wavex = repmat(x2,1,8);x = x + 0.1.*randn(1,length(x));plot(x)y1(1025:length(x1))=0;xhat=ifft(y1);plot(abs(xhat))The reproduced signal is very close to the original signal.To reconstruct the original signal, we stored only the first 1024 coefficients and the ECG signal length. This represents a compression ratio of approximately 4:1.req = [length(x) y(1:1024)];whos x reqName Size Bytes Class Attributesreq 1x1025 8200 doublex 1x4096 32768 doubleCommunication using Spread Spectrum Spread spectrum-based communication technologies, like CDMA, use Walsh codes (derived from Walsh functions) to spread message signals and WHT transforms to despread them. Since Walsh codes are orthogonal, any Walsh-encoded signal appears as random noise to a terminal unless that terminal uses the same code for encoding. Below we demonstrate the process of spreading, determining Walsh codes used for spreading, and despreading to recover the message signal.Two CDMA terminals spread their respective message signals using two different Walsh codes (also known as Hadamard codes) of length 64. The spread message signals are corrupted by a additive white Gaussian noise of variance 0.1.At the receiver (base station), signal processing is non-coherent and the received sequence of length N needs to be correlated with 2^N Walsh codewords to extract the Walsh codes used by the respective transmitters. This can be effectively done by transforming the received signals to sequency domain using the fast Walsh-Hadamard transform. Using the sequency location at a which a peak occurs, the correspondingWalsh-Hadamard code (or the Walsh function) used can be determined. The plot below shows that Walsh-Hadamard codes with sequency (with ordering = 'hadamard') 60 and 10 were used in the first and the second transmitter, respectively.load mess_rcvd_signals.matN = length(rcvdSig1);y1 = fwht(rcvdSig1,N,'hadamard');y2 = fwht(rcvdSig2,N,'hadamard');figure('Color','white');plot(0:63,y1,0:63,y2,'r');xlabel('Sequency index');ylabel('WHT of the Received Signals');title('Walsh-Hadamard Code Extraction');legend('WHT of Tx - 1 signal','WHT of Tx - 2 signal');Despreading (or decoding) to extract the message signal can be carried out in a straightforward manner by multiplying the received signals by the respective Walsh-hadamard codes generated using the hadamard function. (Note that the indexing in MATLAB® starts from 1, hence Walsh-Hadamard codes with sequency 60 and 10 are obtained from by selecting the columns (or rows) 61 and 11 in the Hadamard matrix.)N = 64;hadamardMatrix = hadamard(N);codeTx1 = hadamardMatrix(:,61); % Code used by transmitter 1 codeTx2 = hadamardMatrix(:,11); % Code used by transmitter 2The decoding operation to recover the original message signal isxHat1 = codeTx1 .* rcvdSig1; % Decoded signal at receiver 1 xHat2 = codeTx2 .* rcvdSig2; % Decoded signal at receiver 2The recovered message signals at the receiver side are shown below and superimposed with the original signals for comparison.subplot(2,1,1);plot(x1);hold onplot(xHat1,'r');legend('Original Message','Reconstructed Message','Location','Best'); xlabel('Sample index');ylabel('Message signal amplitude');subplot(2,1,2);plot(x2);hold onplot(xHat2,'r');legend('Original Message','Reconstructed Message','Location','Best'); xlabel('Sample index');ylabel('Message signal amplitude');References1.K.G. Beauchamp, Applications of Walsh and Related Functions - Withan Introduction to Sequency Theory, Academic Press, 19842.T. Beer, Walsh Transforms, American Journal of Physics, Vol. 49,Issue 5, May 1981Copyright 2008-2010 The MathWorks, Inc.Published with MATLAB® 7.13MATLAB and Simulink are registered trademarks of The MathWorks, Inc. Please see /trademarks for a list of other trademarks owned by The MathWorks, Inc. Other product or brand names are trademarks or registered trademarks of their respective owners.。