I was trying to decompose an EMG signal, which was sampled at 200 Hz, into 7 levels of decomposition using DWT. And the result is shown in the figure( only upto d3 is shown here) However I would like to know , how to see in MATLAB that with each level , the frequency is split into high and low?. How to find the power spectrum of the same(if its possible) ?
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https://dsp.stackexchange.com/questions/88698/how-can-we-prove-using-matlab-that-after-each-level-of-decomposition-of-a-signal
Each level of decomposition produces two sets of coefficients: the approximation coefficients (also called the low-pass coefficients) and the detail coefficients (also called the high-pass coefficients). The approximation coefficients capture the coarse-scale information in the signal, while the detail coefficients capture the fine-scale information in the signal.
To demonstrate that the frequency content of a signal is split into high and low after each level of decomposition using the discrete wavelet transform (DWT), you can plot the power spectral density (PSD) of the signal and its wavelet coefficients at each level of decomposition. The PSD represents the distribution of power over frequency in the signal, while the wavelet coefficients represent the frequency content of the signal at different scales.
Here's an example code snippet that demonstrates the frequency splitting property of the DWT:
```matlab
% Load a sample signal
load mtlb
% Set the wavelet and decomposition level
wname = 'db4';
level = 4;
% Perform the DWT
[c, l] = wavedec(mtlb, level, wname);
% Compute the PSD of the signal
Fs = 200;
N = length(mtlb);
f = Fs*(0:(N/2))/N;
psd = abs(fft(mtlb)./N).^2;
psd = psd(1:N/2+1);
% Compute the PSD of the wavelet coefficients at each level
for i = 1:level
% Extract the coefficients at the current level
a_i = appcoef(c, l, wname, i);
d_i = detcoef(c, l, i);
% Compute the PSD of the approximation coefficients
a_i_psd = abs(fft(a_i)./N).^2;
a_i_psd = a_i_psd(1:N/2+1);
% Compute the PSD of the detail coefficients
d_i_psd = abs(fft(d_i)./N).^2;
d_i_psd = d_i_psd(1:N/2+1);
% Plot the PSDs
subplot(level+1, 2, 2*i-1);
plot(f, a_i_psd);
title(sprintf('Approximation coefficients (level %d)', i));
xlabel('Frequency (Hz)');
ylabel('Power spectral density');
subplot(level+1, 2, 2*i);
plot(f, d_i_psd);
title(sprintf('Detail coefficients (level %d)', i));
xlabel('Frequency (Hz)');
ylabel('Power spectral density');
end
% Plot the PSD of the original signal
subplot(level+1, 2, 2*level+1);
plot(f, psd);
title('Original signal');
xlabel('Frequency (Hz)');
ylabel('Power spectral density');
```
Load a sample signal (`mtlb`) and set the wavelet and decomposition level (`db4` and 4, respectively). We then perform the DWT using the `wavedec` function, which returns the wavelet coefficients `c` and the lengths of the coefficient vectors `l`. We compute the PSD of the original signal using the `fft` function, and plot it alongside the PSDs of the approximation and detail coefficients at each level of decomposition. We use the `appcoef` and `detcoef` functions to extract the approximation and detail coefficients at each level, respectively.
If the DWT is splitting the frequency content of the signal into high and low at each level of decomposition, we should observe that the PSD of the approximation coefficients becomes smoother and shifts towards lower frequencies as the decomposition level increases, while the PSD of the detail coefficients becomes spikier and shifts towards higher frequencies. This is because the approximation coefficients capture the coarse-scale information in the signal, while the detail coefficients capture the fine-scale information, which has higher frequencies. The plot should demonstrate the frequency splitting property of the DWT, which is a fundamental property of the wavelet transform.
Hope it helps!!
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