Perfect Reconstruction from Samples

July 8, 2019

During sampling, the original waveform can theoretically be perfectly recovered from the samples if the following conditions are met:

  1. The waveform is band-limited.
  2. The sample rate Fs is at least twice the highest non-zero magnitude frequency (fh).
  3. There is no quantization noise.

Nyquist Rate

The Nyquist rate is a sample rate of Fs = 2 x fh. The higher the sample rate, the easier a perfect reconstruction becomes. Sampling higher than the Nyquist rate is called over-sampling. Sampling lower than the Nyquist rate is called under-sampling, which can result in a disastrous effect called aliasing, where high-frequency content can masquerade as low-frequency content.

In general, the above three conditions are theoretic ideals and don’t exist in the real world. In practice, signals are, more or less, band-limited by the use of a low-pass anti-aliasing filter. Additionally, quantization noise is always present, as it is caused by forcing a continuous waveform onto a set of discrete values. In a real-world application, the signal must be sufficiently strong enough relative to the quantization noise.