2 LEVEL WAVELET TRANSFORM USING MATLAB

To conduct the Wavelet transform on a signal with different harmonic content 

Software used: MATLAB R2018a

To know how to download MATLAB R 2018a Click Here to see

Theory:

It consists of a set of basic functions that can be used to analyze signals in both time and frequency domains simultaneously. This analysis is accomplished by the use of a scalable window to cover the time and frequency plane, providing a convenient means for analyzing the non-stationary signal. Wavelet has infinite functions that can be used for many applications. The main characteristic of these functions is that they have a limited time duration. This characteristic allows Wavelet to have good time localization and compact support. It consists of 2 decompositions and reconstruction of the wavelet transform that is shown by the below figures.

CODE FOR WAVELET TRANSFORM:

% WAVELET EXTENTION 

clc

clear all

t=0:0.0001:0.1;

N=6; % No Of Sample

f=50;

w=2*pi*f;

A=sin(w*t)+(1/3)*sin(3*w*t)+(1/5)*sin(5*w*t)+(1/7)*sin(7*w*t); %DISTORTED INPUT SIGNAL

figure(1)

plot(A)

grid on;

[a3 b3]=freqz(A); %FFT ANALYSIS OD INPUT SIGNAL

figure(2),plot(b3, abs(a3));

grid on;

 

n=0:1:N-1;

ho=[1 -1 -8 -8 -1 1]; %LOW PASS FILTER

h1=((-1).^n).* ho; %HIGH PASS FILTER

go=((-1).^n).* h1; % SECOND BLOCK UPPER LINE

g1=((-1).^n).* ho; % SECOND BLOCK LOWER LINE

A1=conv(A,ho);%CONVOLUTION WITH LPF

figure(3)

plot(A1)

grid on;

 

[a3 b3]=freqz(A1); %FFT ANALYSIS OF LOW PASS FILTER

figure(4),plot(b3, abs(a3));

grid on;

 

 

B1=downsample(A1,2);%DOWN SAMPLE BY 2 OF A1

[a3 b3]=freqz(B1); %FFT ANALYSIS

figure(5),plot(b3, abs(a3));

grid on;

 

% UPPER AREM UPPER PATH

C1=conv(B1,ho);%CONVOLUTION of B1 WITH LOW PASS FILTER

figure(6)

plot(C1)

grid on;

 

[a3 b3]=freqz(C1); %FFT ANALYSIS

D1=downsample(C1,2);%DOWN SAMPLE BY 2 OF C1

[a3 b3]=freqz(D1); %FFT ANALYSIS

figure(7),plot(b3, abs(a3));

grid on;

 

E1=upsample(D1,2); %UP SAMPLE BY 2 OF D1

[a3 b3]=freqz(E1); %FFT ANALYSIS

figure(8),plot(b3, abs(a3));

grid on;

 

F1=conv(E1,go);

[a3 b3]=freqz(F1); %FFT ANALYSIS

figure(9),plot(b3, abs(a3));

grid on;

 

% UPPER ARM LOWER PATH

C2=conv(B1,h1);%CONVOLUTION of B1 WITH LOW PASS FILTER

figure(10)

plot(C2)

grid on;

 

[a3 b3]=freqz(C2); %FFT ANALYSIS

 

D2=downsample(C2,2);%DOWN SAMPLE BY 2 OF C1

[a3 b3]=freqz(D2); %FFT ANALYSIS

figure(11),plot(b3, abs(a3));

grid on;

 

E2=upsample(D2,2); %UP SAMPLE BY 2 OF D1

[a3 b3]=freqz(E2); %FFT ANALYSIS

figure(12),plot(b3, abs(a3));

grid on;

 

F2=conv(E2,g1);

[a3 b3]=freqz(F2); %FFT ANALYSIS

figure(13),plot(b3, abs(a3));

grid on;

 

G1=(F1+F2)/(2) %OUTPUT SINAL AT G1

figure(14)

plot(G1)

grid on;

 

H1=upsample(G1,2);

figure(15)

plot(H1)

grid on;

 

I1=conv(H1,go);%CONVOLUTION of B1 WITH LOW PASS FILTER

figure(16)

plot(I1)

grid on;

[a3 b3]=freqz(I1); %FFT ANALYSIS

% LOWER ARM

 

A2=conv(A,h1);%CONVOLUTION WITH LOW PASS FILTER

figure(17)

plot(A2)

grid on;

 [a3 b3]=freqz(A2); %FFT ANALYSIS

figure(18),plot(b3, abs(a3));

grid on;

B2=downsample(A1,2);%DOWN SAMPLE BY 2 OF yA

[a3 b3]=freqz(B2); %FFT ANALYSIS

figure(19),plot(b3, abs(a3));

grid on;

 

%LOWER ARM UPPER PATH

C3=conv(B2,ho);%CONVOLUTION of B2 WITH LOW PASS FILTER

figure(20)

plot(C3)

grid on;

 [a3 b3]=freqz(C3); %FFT ANALYSIS

D3=downsample(C3,2);%DOWN SAMPLE BY 2 OF C1

[a3 b3]=freqz(D3); %FFT ANALYSIS

figure(21),plot(b3, abs(a3));

grid on;

E3=upsample(D3,2); %UP SAMPLE BY 2 OF D1

[a3 b3]=freqz(E3); %FFT ANALYSIS

figure(22),plot(b3, abs(a3));

grid on;

F3=conv(E3,go);

[a3 b3]=freqz(F3); %FFT ANALYSIS

figure(23),plot(b3, abs(a3));

grid on;

C4=conv(B2,h1);%CONVOLUTION of B1 WITH LOW PASS FILTER

figure(24)

grid on;

plot(C4)

[a3 b3]=freqz(C2); %FFT ANALYSIS

D4=downsample(C4,2);%DOWN SAMPLE BY 2 OF C1

[a3 b3]=freqz(D4); %FFT ANALYSIS

figure(25),plot(b3, abs(a3));

grid on;

E4=upsample(D4,2); %UP SAMPLE BY 2 OF D1

[a3 b3]=freqz(E4); %FFT ANALYSIS

figure(26),plot(b3, abs(a3));

grid on;

 

F4=conv(E4,g1);

[a3 b3]=freqz(F1); %FFT ANALYSIS

figure(27),plot(b3, abs(a3));

G2=(F3+F4)/(2^7) %OUTPUT SINAL AT G2

figure(28)

plot(G2)

grid on;

 

H2=upsample(G2,2);

figure(29)

plot(H2)

grid on;

 

I2=conv(H2,g1);%CONVOLUTION of B1 WITH LOW PASS FILTER

figure(30)

plot(I2)

grid on;

 

[a3 b3]=freqz(I2); %FFT ANALYSIS

J=((0.9/60)*(I1+I2))/(2^7)

figure(31)

 

plot(J)

[a3 b3]=freqz(J); %FFT ANALYSIS OF THE OUTPUT SIGNAL

figure(32),plot(b3, abs(a3));

WAVEFORMS:

1. Distorted Input Signal

2. FFT Analysis of Input Signal

3.  Convolution with LPF

4. FFT Analysis of Convolution in LPF

5. FFT Analysis of Down sampler by 2

6. Convolution with LPF

7. FFT Analysis of Down sampler by 2

8. FFT Analysis of upsampler by 2

9. FFT Analysis of Convolution of upsampler and LPF 

10. FFT Analysis of HPF with Down sampled signal

11: FFT Analysis of the upsampler by 2

12. FFT Analysis of the downsampler by 2

13. FFT Analysis of Convolution of upsampler and HPF inverse

14. Addition of the 2 output Signals

15. Up sampled by 2 Signal

16. Convolution of upsampler and LPF

 

17. FFT Analysis of HPF 

18. Convolution of LPF

19. FFT Analysis of downsampler by 2

20. FFT Analysis of the downsampler

21. FFT Analysis of downsampler with LPF

22. FFT Analysis of the upsampler

23: FFT Analysis of down sampler by 2

24. Convolution with Up Sampler with inverse LPF

25. Output Signal 2

26. The output of Up Sampler

27. Output of Inverse High Pass Filter

28. Final Output Signal

29. FFT Analysis of the Output Signal

Conclusion:

Wavelet Transform is studied and different waveforms are plotted by decomposition and then reconstruction of the signal. And hence at the end, the same signal with harmonic content is retrieved