20.1 Sampling theorem and aliasing

Sampling theorem and aliasing are crucial concepts in signal processing. They explain how to accurately convert continuous signals to digital form and avoid distortion. Understanding these ideas is key to working with real-world data in engineering applications.

The Nyquist-Shannon sampling theorem sets the minimum sampling rate needed to capture a signal's information. Aliasing occurs when sampling is too slow, causing high frequencies to be misinterpreted as lower ones. These concepts are essential for proper signal analysis and reconstruction.

Sampling Theorem

Nyquist-Shannon Sampling Theorem and Bandlimited Signals

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• Nyquist-Shannon sampling theorem establishes the minimum sampling rate required to accurately represent a continuous-time signal in the discrete-time domain without losing information
• Theorem applies to bandlimited signals, which are signals whose frequency content is limited to a specific range
  • Bandlimited signals have a maximum frequency component called the bandwidth
  • Signals with frequency components above the bandwidth are considered non-bandlimited
• Examples of bandlimited signals include audio signals (20 Hz to 20 kHz) and video signals (4.2 MHz for standard definition)

Sampling Rate and Nyquist Frequency

• Sampling rate, denoted as $f_s$, is the number of samples taken per second when converting a continuous-time signal to a discrete-time signal
  • Measured in samples per second (Hz) or hertz (Hz)
  • Higher sampling rates capture more information about the original signal
• Nyquist frequency, denoted as $f_N$, is half the sampling rate and represents the maximum frequency that can be accurately represented in the sampled signal without aliasing
  • Mathematically, $f_N = \frac{f_s}{2}$
• To accurately represent a bandlimited signal, the sampling rate must be at least twice the maximum frequency component (bandwidth) of the signal
  • If $B$ is the bandwidth of the signal, then $f_s \geq 2B$
  • This minimum sampling rate is known as the Nyquist rate

Aliasing and Undersampling

Aliasing

• Aliasing occurs when a signal is sampled at a rate lower than the Nyquist rate, causing high-frequency components to be misinterpreted as lower-frequency components
• Aliased frequencies "fold back" into the frequency spectrum below the Nyquist frequency, creating distortion and loss of information
  • Aliased frequencies appear as mirror images of the original frequencies across the Nyquist frequency
• Examples of aliasing include wagon wheel effect in videos (spokes appearing to rotate backwards) and stroboscopic effect in audio (high-pitched sounds becoming lower-pitched)

Undersampling and Frequency Spectrum

• Undersampling is the process of sampling a signal at a rate lower than the Nyquist rate
  • Results in aliasing and loss of high-frequency information
• Frequency spectrum represents the distribution of frequency components in a signal
  • Undersampling causes the frequency spectrum to be distorted, with high-frequency components folding back and overlapping with low-frequency components
• To avoid aliasing, the sampling rate must be chosen such that the Nyquist frequency is higher than the maximum frequency component of the signal
  • Low-pass filtering can be applied before sampling to remove high-frequency components and prevent aliasing

Oversampling and Filtering

Oversampling

• Oversampling is the process of sampling a signal at a rate higher than the Nyquist rate
  • Provides a more accurate representation of the original signal
  • Reduces quantization noise and improves signal-to-noise ratio (SNR)
• Oversampling allows for the use of simpler, less steep anti-aliasing filters
  • Relaxes the requirements for the filter design
• Examples of oversampling include audio CD sampling at 44.1 kHz (higher than the 40 kHz Nyquist rate for 20 kHz audio) and sigma-delta analog-to-digital converters (ADCs)

Anti-Aliasing Filter

• Anti-aliasing filter is a low-pass filter applied to a signal before sampling to remove frequency components above the Nyquist frequency
  • Prevents aliasing by ensuring the signal is bandlimited
• Ideal anti-aliasing filter has a sharp cutoff at the Nyquist frequency, completely removing higher frequencies
  • Practical filters have a transition band and may allow some aliasing
• Oversampling relaxes the requirements for the anti-aliasing filter, allowing for a wider transition band and less steep rolloff
• Examples of anti-aliasing filters include analog RC filters and digital finite impulse response (FIR) filters