MATLAB Code Implementation for White Noise Testing

White noise testing is a fundamental and crucial aspect of signal processing, primarily used to verify whether a signal's randomness and frequency-domain characteristics align with theoretical expectations. MATLAB, as a powerful numerical computing tool, offers various built-in functions and methods for generating and analyzing white noise signals. ### White Noise Generation White noise is a random signal with uniformly distributed power spectral density. In MATLAB, Gaussian white noise sequences can be generated using built-in random number functions like `randn`. For uniformly distributed white noise, the `rand` function can be utilized. These functions generate signals with specific statistical properties, facilitating subsequent testing procedures. Implementation details: Use `noise_signal = randn(1,N)` to generate Gaussian white noise of length N, or `noise_signal = rand(1,N)-0.5` for zero-mean uniform white noise. ### Time-Domain Testing of White Noise In the time domain, white noise typically exhibits zero mean and an autocorrelation function approximating a unit impulse function. Calculating the signal's mean, variance, and autocorrelation function provides preliminary validation of its white noise characteristics. MATLAB functions such as `mean`, `var`, and `xcorr` can perform these computations. Implementation approach: Compute autocorrelation using `[acf,lags] = xcorr(signal,'unbiased')` and verify if it approximates a delta function at zero lag. ### Frequency Domain Analysis The frequency-domain characteristics are central to white noise identification, where the power spectral density should ideally be flat across all frequencies. MATLAB provides several methods for spectral analysis: Periodogram method: The `periodogram` function estimates power spectral density directly from the signal. Welch's method: Implemented via `pwelch`, this approach reduces variance through segment averaging, improving spectral estimate smoothness. Fast Fourier Transform (FFT): Direct FFT transformation followed by magnitude squared calculation provides spectral approximation. Code implementation: Use `[pxx,f] = pwelch(signal,[],[],[],fs)` to obtain power spectral density with specified sampling frequency fs. ### Statistical Testing Beyond time and frequency domain analysis, statistical methods can further validate white noise properties. For instance, the Ljung-Box test examines whether signal autocorrelations are statistically significant, while the Kolmogorov-Smirnov test verifies distribution conformity to Gaussian characteristics. MATLAB implementation: Conduct Ljung-Box test using `lbqtest` function and distribution tests with `kstest`. Through these comprehensive analysis steps, signals can be thoroughly evaluated for white noise characteristics. MATLAB's extensive toolbox provides efficient implementation pathways suitable for various engineering and research applications in signal processing.