9.1 Sampling theorem and aliasing

The Nyquist-Shannon sampling theorem is a cornerstone of digital signal processing. It states that to accurately capture a signal, you need to sample it at least twice as fast as its highest frequency component. This rule helps prevent aliasing, a distortion that occurs when signals are undersampled.

Aliasing can lead to weird effects like pitch shifting in audio or moiré patterns in images. To avoid it, you need to choose the right sampling frequency. Oversampling can improve signal quality, but it comes with trade-offs like increased data storage and processing needs.

Sampling Theorem and Aliasing

Nyquist-Shannon sampling theorem significance

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• States a band-limited continuous-time signal can be perfectly reconstructed from its samples if the sampling frequency is at least twice the signal's maximum frequency (bandwidth)
  • Mathematically expressed as $f_s \geq 2f_{max}$, where $f_s$ is the sampling frequency and $f_{max}$ is the maximum frequency of the signal
• Provides a fundamental guideline for selecting an appropriate sampling frequency to accurately represent and reconstruct a continuous-time signal (audio, video)
• Ensures the original signal can be recovered from its discrete-time representation without loss of information
• Helps prevent aliasing, which can cause distortion and loss of information in the reconstructed signal (distorted audio, visual artifacts)

Aliasing and sampling frequency

• Occurs when a signal is sampled at a frequency lower than twice its maximum frequency (Nyquist rate)
  • Results in high-frequency components of the signal being misinterpreted as lower-frequency components in the sampled signal (audio pitch shifting, visual moiré patterns)
  • Causes distortion and loss of information in the reconstructed signal
• If the sampling frequency is less than twice the maximum frequency of the signal, aliasing will occur
• Increasing the sampling frequency above the Nyquist rate reduces the risk of aliasing (44.1 kHz for audio CDs, 48 kHz for professional audio)
• Sampling at exactly the Nyquist rate ($f_s = 2f_{max}$) theoretically prevents aliasing, but practical considerations often require a higher sampling frequency (anti-aliasing filters, noise)

Minimum sampling frequency calculation

• To avoid aliasing, the minimum sampling frequency ($f_s$) must be at least twice the signal's bandwidth ($B$)
  • Mathematically expressed as $f_s \geq 2B$
• Calculate the minimum sampling frequency:
  1. Identify the signal's bandwidth ($B$), which is the difference between the maximum and minimum frequencies present in the signal
  2. Multiply the bandwidth by 2 to obtain the minimum sampling frequency required to avoid aliasing
• If a signal has a bandwidth of 100 Hz, the minimum sampling frequency to avoid aliasing would be $f_s \geq 2 \times 100 = 200$ Hz

Effects of under and oversampling

• Undersampling (sampling below the Nyquist rate)
  • Results in aliasing, causing high-frequency components to be misinterpreted as lower-frequency components (audio pitch shifting, visual moiré patterns)
  • Leads to distortion and loss of information in the reconstructed signal
  • Cannot accurately reconstruct the original signal from the sampled data
• Oversampling (sampling above the Nyquist rate)
  • Helps reduce aliasing and improve signal reconstruction accuracy
  • Provides more samples than the minimum required, allowing for better representation of the original signal (higher resolution, smoother curves)
  • Enables the use of simpler anti-aliasing filters with lower cutoff frequencies
  • Increases the signal-to-noise ratio (SNR) of the sampled signal (reduced quantization noise)
  • Requires more storage space and processing power compared to sampling at the Nyquist rate