Why do multiple copies appear in the spectrum of a sampled signal?

Why do multiple copies appear in the spectrum of a sampled signal?

• A. The spectrum is convolved with a rectangular window in time, producing copies.
• B. The spectrum is unaffected by sampling.
• C. The spectrum is replicated at multiples of Fs due to sampling, with equal amplitude.
• D. The spectrum is inverted in frequency.

Answer: (C)

Sampling a signal in time multiply it by a train of impulses. In frequency, that multiplication becomes a convolution with a Dirac comb: an impulse train spaced by Fs. Convolving the original spectrum with this comb places shifted copies of the spectrum at every multiple of the sampling frequency, so you see replicas at 0, ±Fs, ±2Fs, and so on. In ideal impulse sampling, each copy has the same amplitude as the original spectrum because the Dirac comb just shifts and repeats the spectrum without changing its scale. If those copies overlap, they add together and cause aliasing; keeping Fs high enough or filtering beforehand prevents that. In real-world sampling, the replicas appear with the same basic spacing, but their shapes can be modulated by the pulse used for sampling, whereas the essential reason for multiple copies is the periodic replication in frequency due to the impulse train in time.