The frequency response between the components obtained from wavelet decomposition and the original signal

Why does the fft of the component corresponding to the last approximation coefficient obtained after wavelet decomposition not match the fft of the original signal in the low-frequency region, resulting in an unsmooth frequency response between the two and varying with the original signal. Is using FFT to solve frequency response unreliable?

clearvars;close all;clc;

fs=10;

dt=1/fs;

t=dt:dt:200;

N=length(t);

signal=(0.2)*randn(1,N);

Max_level=wmaxlev(length(signal),'db10');

[C,L]=wavedec(signal,Max_level,'fk8');

level=6;

xL_DWT = wrcoef('a',C,L,'fk8',level);

xH_DWT=signal-xL_DWT;

%fft

Nfft=length(t);

f_DWT = (1:Nfft/2)*fs/(Nfft);

xL_DWT_fft=fftshift(fft(xL_DWT,Nfft));

xH_DWT_fft=fftshift(fft(xH_DWT,Nfft));

signal_fft=fftshift(fft(signal,Nfft));

wn_low_DWT=xL_DWT_fft./signal_fft;

wn_high_DWT=1-wn_low_DWT;

figure

plot(f_DWT,wn_low_DWT(Nfft/2+1:end));

hold on

plot(f_DWT,wn_high_DWT(Nfft/2+1:end));

ylim([-0.5 2]);

yticks([-0.5:0.5:2])

xlim([0 fs/2^(level-1)]);

xticks([0 fs/2^(level+1) fs/2^(level) fs/2^(level-1)])

xticklabels({'0','fs/2^{i+1}','fs/2^{i}','fs/2^{i-1}'})

Answer 1

[Umar]

Hi 志鹏 汪,

The discrepancy in the FFT of the component corresponding to the last approximation coefficient obtained after wavelet decomposition, compared to the FFT of the original signal in the low-frequency region, leading to an unsmooth frequency response between them is primarily due to the nature of wavelet decomposition and its effects on signal processing.

Wavelet decomposition is a technique used to analyze signals at different resolutions, decomposing the original signal into approximation and detail coefficients across multiple levels. In this process, high-frequency components are captured in the detail coefficients, while low-frequency components are retained in the approximation coefficients.

As a result, when you perform the inverse transform to reconstruct the signal using only the last approximation coefficient (xL_DWT), you may lose some high-frequency information present in the original signal. This loss of high-frequency information can lead to discrepancies in the frequency domain when comparing the FFT of xL_DWT with that of the original signal. The unsmooth frequency response between them can be attributed to this missing high-frequency content, causing variations in the FFT results.

Using FFT to solve frequency response is not inherently unreliable; however, it is essential to consider the limitations and assumptions of both wavelet decomposition and FFT when analyzing signals. While FFT provides a powerful tool for frequency domain analysis, wavelet decomposition introduces complexities related to multi-resolution analysis that can affect the accuracy of frequency responses, especially when dealing with signals containing significant high-frequency components.

To address these discrepancies and improve the consistency of frequency responses between wavelet decomposed components and the original signal, it may be beneficial to explore alternative methods or combine different signal processing techniques to enhance signal reconstruction fidelity and preserve critical frequency information across all components. While using FFT for frequency response analysis is valuable, understanding how wavelet decomposition impacts signal processing and its implications on frequency domain representation is crucial for achieving accurate and reliable results in complex signal analysis scenarios.

It is recommended to carefully evaluate the trade-offs between resolution levels and frequency content preservation to optimize signal reconstruction and maintain fidelity in frequency responses.