Sampling Theorem

Fourier Analysis · Axiom Academy

LESSON The Nyquist-Shannon Sampling Theorem The fundamental theorem that bridges continuous and discrete signal processing 1. The Sampling Theorem Statement A continuous signal x(t) that is bandlimited to frequencies below f max can be perfectly reconstructed from its samples if the sampling frequency satisfies: The critical frequency f Nyquist = 2f max is called the Nyquist rate . This is the minimum sampling rate needed for perfect reconstruction. 2. Why 2f max ? Understanding the Frequency Domain The key insight comes from the frequency domain. A continuous signal x(t) has a frequency spectrum X(f). When we sample at rate f s , something remarkable happens: the spectrum replicates at every multiple of f s . The sampled signal's spectrum consists of infinite copies of the original spectrum, centered at 0, ±f s , ±2f s , etc. 3. The Nyquist Condition: No Spectral Overlap For perfect reconstruction, we need to recover the original spectrum X(f) from the sampled spectrum X s (f). This is only possible if the replicated spectra don't overlap . If f s > 2f max , the copies are separated and we can use an ideal lowpass filter to extract the original spectrum. If f s max , the copies overlap, causing aliasing - information is irreversibly lost. The theorem requires that x(t) is bandlimited , meaning its frequency content is restricted to |f| ≤ f max . In other words:

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