〰️Signal Processing Unit 14 – Fourier Analysis in Signal Processing

Foundations of Signals and Systems

• Signals represent physical quantities that vary with one or more independent variables (time, space, frequency)
• Systems process input signals to produce output signals based on specific rules or operations
• Continuous-time signals have values defined at every point in time (analog signals)
  • Examples include audio signals, voltage levels, and temperature readings
• Discrete-time signals have values defined at discrete points in time (digital signals)
  • Consist of a sequence of numbers obtained by sampling continuous-time signals at regular intervals
  • Examples include digital audio, images, and sensor data
• Linear systems satisfy the properties of superposition and homogeneity
  • Superposition: Output of a sum of inputs equals the sum of outputs for each input
  • Homogeneity: Scaling the input by a constant factor scales the output by the same factor
• Time-invariant systems produce the same output regardless of when the input is applied (shift-invariant)

Introduction to Fourier Analysis

• Fourier analysis decomposes signals into a sum of sinusoidal components with different frequencies, amplitudes, and phases
• Enables the study of signals in both time and frequency domains
• Fourier series represents periodic signals as a sum of harmonically related sinusoids
  • Expresses the signal as a sum of complex exponentials with discrete frequencies
• Fourier transform extends the concept to non-periodic signals
  • Represents the signal as a continuous spectrum of frequencies
• Fourier analysis has wide-ranging applications in signal processing, communications, and system analysis
  • Allows for efficient computation, filtering, and manipulation of signals in the frequency domain
• Inverse Fourier transform reconstructs the time-domain signal from its frequency-domain representation

Time-Domain vs. Frequency-Domain Representations

• Time-domain representation shows how a signal varies with time
  • Plots the signal amplitude or value against time
  • Provides information about the signal's temporal characteristics (duration, shape, peaks)
• Frequency-domain representation shows the signal's frequency content
  • Plots the signal's amplitude or power against frequency
  • Reveals the presence and strength of different frequency components within the signal
• Fourier transform converts a signal from the time domain to the frequency domain
  • Decomposes the signal into its constituent frequencies and their respective amplitudes
• Inverse Fourier transform converts a signal from the frequency domain back to the time domain
  • Reconstructs the time-domain signal from its frequency components
• Time and frequency domains provide complementary perspectives on the signal
  • Time domain focuses on the signal's behavior over time
  • Frequency domain emphasizes the signal's spectral content and composition

Fourier Series for Periodic Signals

• Fourier series represents a periodic signal as a sum of sinusoids with harmonically related frequencies
• Fundamental frequency $f_0$ is the reciprocal of the signal's period $T$: $f_0 = \frac{1}{T}$
• Harmonics are integer multiples of the fundamental frequency: $f_n = n \cdot f_0$, where $n = 1, 2, 3, ...$
• Fourier series expansion of a periodic signal $x(t)$ with period $T$:
  • $x(t) = \sum_{n=-\infty}^{\infty} c_n e^{j2\pi nf_0 t}$
  • $c_n$ are the complex Fourier series coefficients, representing the amplitude and phase of each harmonic
• Coefficients $c_n$ are calculated using the formula:
  • $c_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-j2\pi nf_0 t} dt$
• Fourier series allows for the analysis and synthesis of periodic signals
  • Analysis: Decomposing the signal into its harmonic components (Fourier series coefficients)
  • Synthesis: Reconstructing the signal from its Fourier series coefficients

Fourier Transform for Non-Periodic Signals

• Fourier transform extends the concept of Fourier series to non-periodic signals
• Represents a non-periodic signal as a continuous spectrum of frequencies
• Forward Fourier transform of a signal $x(t)$:
  • $X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt$
  • $X(f)$ is the Fourier transform of $x(t)$, representing the signal's frequency-domain representation
• Inverse Fourier transform reconstructs the time-domain signal from its Fourier transform:
  • $x(t) = \int_{-\infty}^{\infty} X(f) e^{j2\pi ft} df$
• Fourier transform exists for signals that satisfy certain conditions (absolute integrability)
  • Signals with finite energy (square-integrable) have a well-defined Fourier transform
• Fourier transform reveals the signal's frequency content and spectral characteristics
  • Amplitude spectrum: $|X(f)|$ represents the magnitude of each frequency component
  • Phase spectrum: $\angle X(f)$ represents the phase of each frequency component

Properties and Applications of Fourier Transform

• Linearity: Fourier transform of a linear combination of signals equals the linear combination of their Fourier transforms
• Time shifting: Shifting a signal in time introduces a linear phase shift in its Fourier transform
  • $x(t-t_0) \leftrightarrow e^{-j2\pi ft_0} X(f)$
• Frequency shifting: Multiplying a signal by a complex exponential shifts its Fourier transform in frequency
  • $x(t)e^{j2\pi f_0 t} \leftrightarrow X(f-f_0)$
• Scaling: Scaling a signal in time inversely scales its Fourier transform in frequency
  • $x(at) \leftrightarrow \frac{1}{|a|}X(\frac{f}{a})$
• Convolution: Convolution in the time domain corresponds to multiplication in the frequency domain
  • $x(t) * y(t) \leftrightarrow X(f)Y(f)$
• Parseval's theorem: Energy of a signal is preserved in both time and frequency domains
  • $\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df$
• Applications of Fourier transform include filtering, modulation, demodulation, and spectral analysis
  • Filtering: Modifying the signal's frequency content by multiplying its Fourier transform with a filter's frequency response
  • Modulation: Shifting the signal's frequency content for transmission or multiplexing

Discrete Fourier Transform (DFT) and FFT

• Discrete Fourier Transform (DFT) is the discrete-time equivalent of the continuous-time Fourier transform
• Operates on discrete-time signals with a finite number of samples
• DFT of a discrete-time signal $x[n]$ with $N$ samples:
  • $X[k] = \sum_{n=0}^{N-1} x[n] e^{-j2\pi kn/N}$, for $k = 0, 1, ..., N-1$
  • $X[k]$ represents the frequency-domain representation of $x[n]$
• Inverse DFT (IDFT) reconstructs the time-domain signal from its DFT:
  • $x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j2\pi kn/N}$, for $n = 0, 1, ..., N-1$
• Fast Fourier Transform (FFT) is an efficient algorithm for computing the DFT
  • Reduces the computational complexity from $O(N^2)$ to $O(N \log N)$
  • Exploits the symmetry and periodicity properties of the DFT
• FFT algorithms include Cooley-Tukey, Radix-2, and Split-Radix
  • Divide-and-conquer approach, recursively dividing the DFT into smaller sub-problems
• DFT and FFT are widely used in digital signal processing applications
  • Spectrum analysis, filtering, convolution, correlation, and data compression

Practical Applications in Signal Processing

• Audio and speech processing: Fourier analysis for frequency-domain filtering, equalization, and compression
  • Removing noise, enhancing specific frequency bands, and applying audio effects
• Image processing: Fourier transform for image filtering, enhancement, and compression
  • Low-pass filtering for smoothing, high-pass filtering for edge detection, and image compression (JPEG)
• Radar and sonar: Fourier analysis for target detection, ranging, and Doppler processing
  • Detecting moving targets, estimating target velocity, and suppressing clutter
• Communications: Fourier transform for modulation, demodulation, and channel equalization
  • Orthogonal Frequency Division Multiplexing (OFDM), channel estimation, and equalization
• Biomedical signal processing: Fourier analysis for EEG, ECG, and EMG signal analysis
  • Extracting frequency-domain features, removing artifacts, and identifying patterns
• Vibration analysis: Fourier transform for machinery condition monitoring and fault diagnosis
  • Detecting and diagnosing faults based on the frequency content of vibration signals
• Seismic data processing: Fourier analysis for seismic data filtering and interpretation
  • Removing noise, enhancing specific frequency bands, and extracting subsurface information