Spectrum of a discrete-time signal

Summary

A sampled (discrete-time) signal has a periodic spectrum covering the whole frequency axis.
Adding zero samples does not alter the spectrum, but allows independent manipulation of spectral replicas.
The continuous-time signal can be reconstructed using a lowpass filter.

Figure 1 shows the spectrum of a continuous-time signal.

Figure 1: Spectrum of a continuous-time signal

It is sampled at a given rate.
Figure 2 shows the spectrum of the sampled signal.

Figure 2: Spectrum of the sampled signal

The time-discrete signal is a series of dirac delta pulses, and the spectrum of a dirac pulse is infinitely wide.
Therefore, the resulting spectrum must cover the whole frequency axis.
It contains an infinite number of replicas of the original signal.

Any manipulation on the signal, such as filtering, will affect all replicas at the same time (Figure 3).

Figure 3: Spectrum after some filtering

A lowpass filter can reconstruct the original continous-wave signal by filtering the pulse stream (Figure 4)

Figure 4: Reconstruction using a lowpass filter on the pulse stream

Note: todo make this an own topic The dirac pulse is a mathematical construct.In an electrical circuit (D/A converter) it is not practical to reconstruct the signal using dirac pulses.Instead, one uses a sample-and-hold circuit.From a mathematical point-of-view, sample-and-hold is the same as convolving a rectangular pulse with the dirac delta pulse stream.As a result, the spectrum of the resulting signal is multiplied with the spectrum of the rectangular pulse.

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