【Practical Exercise】Time-Frequency Analysis of Signals Using MATLAB

# 2.1 Principles and Methods of Time-Frequency Analysis

Time-frequency analysis is a technique that represents signals simultaneously in the time and frequency domains. It reveals the frequency components of a signal that change with time, providing a more comprehensive perspective for signal analysis and processing.

### 2.1.1 Short-Time Fourier Transform (STFT)

STFT is a classical method for time-frequency analysis. It divides the signal into a series of overlapping time windows and then performs a Fourier transform on each window. By connecting the spectra of each time window, the time-frequency distribution of the signal is obtained.

% Signal
x = sin(2*pi*100*t) + sin(2*pi*200*t);

% STFT
[S, F, T] = spectrogram(x, 256, 128, 512, 1000);

% Plotting time-frequency distribution
surf(T, F, abs(S), 'EdgeColor', 'none');
xlabel('Time (s)');
ylabel('Frequency (Hz)');
zlabel('Magnitude');

# 2. MATLAB Time-Frequency Analysis Toolbox

### 2.1 Principles and Methods of Time-Frequency Analysis

Time-frequency analysis is a powerful technique in signal processing used to analyze the time-frequency characteristics of signals. It decomposes signals into a joint representation of time and frequency, thereby revealing hidden patterns and trends in the signal.

#### 2.1.1 Short-Time Fourier Transform (STFT)

STFT is one of the most commonly used methods in time-frequency analysis. It divides the signal into a series of overlapping windows and then applies a Fourier transform to each window. This results in a time-frequency diagram, where the time axis represents the window position, and the frequency axis represents the frequency components of the Fourier transform.

% Import signal
x = load('signal.mat');

% Set STFT parameters
windowSize = 256;
overlap = 0.5;

% Calculate STFT
[S, F, T] = spectrogram(x, windowSize, overlap);

% Plot time-frequency diagram
imagesc(T, F, abs(S));
colorbar;
title('STFT Time-Frequency Diagram');
xlabel('Time');
ylabel('Frequency');

#### 2.1.2 Wavelet Transform

The wavelet transform is a multi-scale analysis technique that decomposes signals using a set of basis functions called wavelets. Wavelets have localization properties, which enable them to capture transient and non-stationary features of a signal.

% Import signal
x = load('signal.mat');

% Set wavelet parameters
waveletName = 'db4';
scales = 1:10;

% Calculate wavelet transform
[C, L] = wavedec(x, scales, waveletName);

% Plot wavelet coefficient graph
figure;
for i = 1:length(scales)
     subplot(length(scales), 1, i);
     plot(C{i});
     title(['Wavelet Coefficient Graph: Scale', num2str(scales(i))]);
end

#### 2.1.3 Hilbert-Huang Transform (HHT)

HHT is a nonlinear time-frequency analysis method that decomposes signals into a series of components called intrinsic mode functions (IMFs). IMFs are localized and represent different frequency components in the signal.

% Import signal
x = load('signal.mat');

% Calculate HHT
imfs = emd(x);

% Plot HHT time-frequency diagram
figure;
for i = 1:length(imfs)
     subplot(length(imfs), 1, i);
     plot(x, imfs{i});
     title(['Intrinsic Mode Function: ', num2str(i)]);
end

# 3. MATLAB Time-Frequency Analysis Practice

### 3.1 Time-Frequency Feature Extraction of Signals

#### 3.1.1 Power Spectral Density (PSD)

The power spectral density (PSD) is a function that describes how signal power is distributed across frequencies. It can reveal the spectral characteristics of a signal and is used to identify periodic components and noise in a signal.

**Calculation Method:**

% Signal x
x = randn(1000, 1);

% Calculate PSD
psd = pwelch(x, [], [], [], 1024);

% Plot PSD
figure;
plot(psd);
xlabel('Frequency (Hz)');
ylabel('Power Spectral Density');
title('Power Spectral Density of Signal');

**Logical Analysis:**

* The `pwelch` function calculates the PSD of a signal, where:
* `x`: Input signal
* `[]`: Specifies the use of the default window size
* `[]`: Specifies the use of the default overlap rate
* `[]`: Specifies the use of the default sampling rate
* `1024`: Specifies the frequency resolution of the PSD

#### 3.1.2 Time-Frequency Distribution (TFD)

The time-frequency distribution (TFD) is a function that describes the energy distribution of a signal on the time-frequency plane. It can reveal the time-varying characteristics of a signal and is used for analyzing transient and non-stationary signals.

**Calculation Method:**

% Signal x
x = chirp(0:0.001:10, 0, 1000, 2000);

% Calculate time-frequency distribution (using STFT)
tfd = spectrogram(x, 256, 128, 512, 1000);

% Plot time-frequency distribution
figure;
imagesc(tfd);
xlabel('Time (s)');
ylabel('Frequency (Hz)');
title('Time-Frequency Distribution of Signal');

**Logical Analysis:**

* The `spectrogram` function calculates the time-frequency distribution of a signal, where:
* `x`: Input signal
* `256`: Specifies the window size
* `128`: Specifies the overlap rate
* `512`: Specifies the frequency resolution
* `1000`: Specifies the sampling rate

#### 3.1.3 Coherence

Coherence is a function that describes the correlation between two signals. It can reveal the similarity and time-frequency relationship between signals.

**Calculation Method:**

% Signals x and y
x = randn(1000, 1);
y = randn(1000, 1);

% Calculate coherence
coh = mscohere(x, y, [], [], [], 1000);

% Plot coherence
figure;
plot(coh);
xlabel('Frequency (Hz)');
ylabel('Coherence');
title('Coherence between Signals');

**Logical Analysis:**

* The `mscohere` function calculates the coherence of signals, where:
* `x`: Input signal 1
* `y`: Input signal 2
* `[]`: Specifies the use of the default window size
* `[]`: Specifies the use of the default overlap rate
* `[]`: Specifies the use of the default sampling rate
* `1000