Comparing FFT and STFT Algorithm Differences Using MATLAB Tools

In this document, we will utilize MATLAB tools to compare the differences between FFT and STFT algorithms. First, let's examine the FFT algorithm. The Fast Fourier Transform (FFT) algorithm is a computational method that converts time-domain signals into frequency-domain representations. Using MATLAB's fft() function, this algorithm efficiently decomposes signals into different frequency components by implementing the Cooley-Tukey algorithm with O(n log n) complexity, allowing analysis of spectral characteristics across the entire signal duration. In contrast, the Short-Time Fourier Transform (STFT) algorithm, implemented through MATLAB's spectrogram() function, decomposes signals into frequency components within sliding time windows using windowing functions like Hamming or Hanning windows. This approach enables better understanding of time-varying signal characteristics by maintaining both time and frequency information simultaneously.

By comparing these two algorithms through MATLAB implementations, we can observe that FFT algorithm is suitable for spectral analysis of entire signals using commands like Y = fft(x) followed by P2 = abs(Y/L) for power spectrum calculation, while STFT algorithm is more appropriate for analyzing time-varying characteristics through [s,f,t] = spectrogram(x,window,noverlap,nfft,fs) which generates time-frequency representations. This distinction helps in selecting the appropriate algorithm for specific applications. When comparing using MATLAB tools, we can visually observe these differences by plotting spectrograms using imagesc(t,f,abs(s)) for STFT and power spectral density plots using plot(f,P1) for FFT, clearly demonstrating how STFT preserves temporal information that FFT sacrifices.

In summary, by employing MATLAB tools for comparison with proper parameter selection (window sizes, overlap percentages, and FFT lengths), we can better understand the fundamental differences between FFT and STFT algorithms and select the most suitable approach for specific signal processing applications, whether for stationary signal analysis or non-stationary, time-varying signal examination.