The DFT and FFT are powerful tools for analyzing signals in the frequency domain. They transform time-domain data into a spectrum, revealing hidden patterns and frequencies. This process is crucial for various applications, from audio processing to image analysis.

Spectral analysis using DFT/FFT allows us to interpret frequency components, understand signal characteristics, and perform tasks like filtering and denoising. It's a fundamental technique that bridges the gap between time and frequency representations of signals.

Spectral Analysis with DFT and FFT

Discrete Fourier Transform (DFT)

• Mathematical tool converts finite sequence of equally-spaced samples of a function into same-length sequence of equally-spaced samples of discrete-time Fourier transform (DTFT), a complex-valued function of frequency
• Defined by formula $X[k] = \sum_{n=0}^{N-1} x[n] * e^{-j*2*\pi*kn/N}$, where:
  • $x[n]$ input signal
  • $X[k]$ output spectrum
  • $N$ number of samples
  • $k$ frequency index

Fast Fourier Transform (FFT)

• Efficient algorithm computes DFT, reducing computational complexity from $O(N^2)$ to $O(N \log N)$
• Based on divide-and-conquer approach
  • Recursively breaks down DFT into smaller DFTs
  • Exploits symmetry and periodicity properties of complex exponential term
• Requires number of samples $N$ to be power of 2 for optimal performance
  • Zero-padding can extend signal length to nearest power of 2 ($2^n$, where $n$ is an integer)

Frequency Spectrum Interpretation

Frequency Spectrum Components

• DFT/FFT output $X[k]$ represents frequency spectrum of input signal $x[n]$
  • Each value corresponds to specific frequency component
• Magnitude spectrum $|X[k]|$ provides information about amplitude or strength of each frequency component in signal
• Phase spectrum $\angle X[k]$ represents phase shift of each frequency component relative to origin

Frequency Resolution and Symmetry

• Frequency resolution of spectrum given by $f_s/N$, where:
  • $f_s$ sampling frequency
  • $N$ number of samples
• Determines spacing between frequency bins in spectrum
• Spectrum symmetric around Nyquist frequency ($f_s/2$) for real-valued input signals
  • Second half is complex conjugate of first half

Frequency Resolution and Leakage

Frequency Resolution

• Ability to distinguish between closely spaced frequency components in spectrum
• Determined by number of samples $N$ and sampling frequency $f_s$
• Increasing number of samples $N$ improves frequency resolution
  • Allows for better separation of nearby frequency components

Spectral Leakage

• Occurs when input signal is not periodic within DFT/FFT window
  • Causes energy from one frequency component to leak into adjacent frequency bins
• Caused by implicit rectangular windowing of input signal by DFT/FFT
  • Introduces discontinuities at boundaries
• Windowing techniques (Hann, Hamming, Blackman) can reduce spectral leakage
  • Applied to input signal before DFT/FFT
  • Smooths signal at boundaries

DFT/FFT Applications

Filtering

• Achieved by modifying DFT/FFT coefficients and performing inverse DFT/FFT to obtain filtered signal
• Low-pass filtering removes high-frequency components by setting corresponding DFT/FFT coefficients to zero
• High-pass filtering removes low-frequency components by setting corresponding DFT/FFT coefficients to zero
• Band-pass and band-stop filtering achieved by selectively retaining or removing specific frequency ranges

Denoising

• Performed by identifying and suppressing noise components in frequency domain
• Thresholding techniques (hard or soft thresholding) applied to DFT/FFT coefficients to remove noise while preserving signal components
• Choice of threshold depends on noise characteristics and desired trade-off between noise reduction and signal preservation

Feature Extraction

• Involves identifying and extracting relevant frequency-domain features from DFT/FFT spectrum for further analysis or classification
• Spectral features computed from DFT/FFT coefficients
  • Peak frequencies
  • Spectral centroid
  • Spectral bandwidth
  • Spectral entropy
• Features used for tasks such as:
  • Audio classification
  • Speech recognition
  • Signal characterization