Sampling, aliasing and reconstruction

Sample one or two tones and watch the spectrum copy itself to every multiple of fs. The alias frequency is worked out the GATE way, the low-pass output is listed line by line for any cut-off, and a band-limited message shows its copies overlapping below the Nyquist rate.

Time domainsignalsampleslow-pass output

Spectrum after samplingoriginal tone (±f)copies at k·fs ± fshaded: what the low-pass filter keeps

Orange dots are the components that survive the low-pass filter.

f_1 \text{ (tone, Hz)}

3 Hz add a second tone

f_s \text{ (sampling rate)}

8 Hz low-pass cut-off at

f_s/2

(ideal reconstruction)

Nyquist rate 2·f_max6 Hz

Tone 3 Hz aliases to3 Hz

Low-pass output (f_c = 4 Hz)3 Hz

Sampled above the Nyquist rate

What to notice

Sampling copies every spectral line to $k f_s \pm f$ . The alias is the copy that lands in $[0, f_s/2]$ : $f_a = |f - k f_s|$ with $k$ the nearest whole number to $f/f_s$ . Here $|3 - 0\times8| = 3$ Hz.
$f_s = 8 > 2f_{max} = 6$ Hz, so the baseband copy is the original and the low-pass output (orange) retraces the signal.
A tone at exactly $f_s/2$ is ambiguous: a cosine survives, a sine samples to zero. That is why the sampling theorem needs $f_s > 2f_{max}$ , strictly.

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