Sampling & Aliases

Along the way

1
Sampling replicates the spectrum. Impulse sampling copies \(X(f)\) to every \(k f_s\). If anything already lives above \(f_s/2\), a replica lands in baseband and masquerades as a lower tone.
2
Time-domain twin. The rose curve is the baseband impostor. Same blue dots, different continuous story. DSP after the ADC only ever sees the dots.
3
Shannon is the other side of the same coin. A signal bandlimited to \(B < f_s/2\) can be reconstructed. Aliasing is what happens when that bandlimit is violated — or the anti-alias filter fails (Part 10).
4
Undersampling is deliberate aliasing. IF sampling uses a higher Nyquist zone on purpose — but then the filter must select that zone, not “no filter.”
5
Noise folds too. Broadband noise from many zones piles into baseband (Part 5). Tones are just the easy picture.

Complex baseband, I/Q, aperture jitter, and nonuniform sampling. The twin-sine picture is the lunch-and-learn; periodic spectrum replication is the deeper course topic.

Next: Part 3 — discretize amplitude. The stairs, the LSB, and \(\sigma_e=\Delta/\sqrt{12}\).

Same samples, different truths

Left: time — gray true tone, rose alias twin, blue samples · Right: spectrum fold into \([0,f_s/2]\)

\(f_s\) (Hz) True \(f_{\mathrm{in}}\) (Hz)

True \(x_c(t)\) Alias twin Samples

True tone In-band alias Nyquist

Nyquist f_s/2 —

In-band f_alias —

Zone / fold —

Status —

\[ f_{\mathrm{Nyq}} = f_s/2 \]

= —

\[ f_{\mathrm{alias}} = \big|\,f_{\mathrm{in}} - m\,f_s\,\big|_{\text{folded to }[0,f_s/2]} \]

= —

For real sampling, spectrum replicas appear at \(\pm f_{\mathrm{in}} + k f_s\). The baseband observer only sees the replica that lands in \([0,f_s/2]\).

Try it: set \(f_{\mathrm{in}}=26\) Hz with \(f_s=20\) Hz. The blue dots lie on both the gray 26 Hz wave and a rose ~6 Hz wave. After sampling you cannot tell which was true — that is aliasing.

Analog Intuition

Same samples, different truths