4.3 Signal processing and filtering

Signal processing and filtering are crucial tools in Fourier analysis. They allow us to manipulate and analyze signals in both time and frequency domains. By applying filters, we can remove unwanted noise, isolate specific frequencies, and extract valuable information from complex signals.

These techniques have wide-ranging applications, from audio processing to image enhancement. Understanding how to use Fourier transforms, convolution, and filtering methods empowers us to tackle real-world signal analysis problems effectively.

Fourier Analysis

Fourier Transform and Domains

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• Fourier transform converts signals between time and frequency domains
• Time domain represents a signal as a function of time, showing how the signal changes over time
• Frequency domain represents a signal as a function of frequency, revealing the frequency components that make up the signal
• Fourier transform decomposes a signal into its constituent frequencies, allowing for analysis and manipulation in the frequency domain

Fast Fourier Transform (FFT)

• Fast Fourier Transform (FFT) is an efficient algorithm for computing the Discrete Fourier Transform (DFT)
• FFT reduces the computational complexity of the DFT from $O(N^2)$ to $O(N \log N)$, making it practical for large datasets
• Commonly used in digital signal processing, image processing, and audio analysis
• Enables rapid computation of frequency spectra, convolution, and filtering operations

Signal Filtering

Filter Types

• Low-pass filter attenuates high-frequency components while allowing low frequencies to pass through
  • Removes noise and smooths signals (audio de-noising)
• High-pass filter attenuates low-frequency components while allowing high frequencies to pass through
  • Removes DC offset and baseline drift (EEG signal processing)
• Band-pass filter allows a specific range of frequencies to pass through while attenuating frequencies outside the range
  • Isolates specific frequency bands of interest (extracting voice frequencies from audio)

Convolution

• Convolution is a mathematical operation that combines two functions to produce a third function
• In signal processing, convolution is used to apply filters to signals
• The input signal is convolved with the filter's impulse response to obtain the filtered output
• Convolution in the time domain is equivalent to multiplication in the frequency domain, enabling efficient filtering using the FFT

Sampling Theory

Sampling Theorem and Nyquist Frequency

• Sampling theorem, also known as the Nyquist-Shannon sampling theorem, states that a signal can be perfectly reconstructed from its samples if the sampling frequency is at least twice the highest frequency component in the signal
• The minimum sampling frequency required to avoid aliasing is called the Nyquist frequency
• Sampling at a rate lower than the Nyquist frequency results in aliasing, where high-frequency components are misinterpreted as low-frequency components

Aliasing and Signal Reconstruction

• Aliasing occurs when a signal is sampled at a rate insufficient to capture its highest frequency components
• Aliased frequencies appear as lower-frequency components in the sampled signal, causing distortion and loss of information
• To avoid aliasing, the signal must be bandlimited to frequencies below half the sampling frequency (low-pass filtering before sampling)
• Signal reconstruction involves interpolating between the sampled values to recover the original continuous-time signal
• Ideal reconstruction requires a sinc interpolation kernel, which is practically approximated using various interpolation methods (linear, spline)