Given. Signal frequency $f_0 = 100\text{ Hz}$; sampling frequency $f_s = 500\text{ Hz}$; peak-to-peak amplitude $10\text{ V}$.
Find. Two other sinusoid frequencies whose samples at $f_s$ are indistinguishable from the samples of the $100\text{ Hz}$ tone; (1) why this happens; (2) the name of the phenomenon; (3) how to guarantee unique sampled representations.
Approach. A sampled sinusoid is indistinguishable from any other sinusoid whose frequency differs by an integer multiple of the sampling frequency ($f = n f_s \pm f_0$); the two smallest such "images" of $f_0=100\text{ Hz}$ at $n=1$ are the ones asked for, and the Nyquist criterion is what turns this ambiguity into a guaranteed-unique reconstruction.
State the aliasing (image-frequency) relation. For a real sinusoid sampled at $f_s$, every frequency $f = n f_s \pm f_0$ ($n=1,2,3,\ldots$) reproduces the same sequence of sample MAGNITUDES as $f_0$, because $\sin\!\big(2\pi (n f_s \pm f_0) t\big)\big|_{t=k/f_s} = \sin\!\big(2\pi(\pm f_0)k/f_s + 2\pi nk\big)$, and the added $2\pi nk$ term (an integer multiple of $2\pi$) vanishes at every sample instant.
Evaluate the two smallest images ($n=1$).
$$f_{\text{alias},1} = f_s - f_0 = 500 - 100 = \boxed{400\text{ Hz}}$$
$$f_{\text{alias},2} = f_s + f_0 = 500 + 100 = \boxed{600\text{ Hz}}$$
The $600\text{ Hz}$ tone reproduces the $100\text{ Hz}$ samples with no phase adjustment; the $400\text{ Hz}$ tone reproduces them only after a $180^{\circ}$ phase flip (equivalently, opposite amplitude sign) — a $400\text{ Hz}$ sinusoid of the opposite sign lands on exactly the same sample points as the $100\text{ Hz}$ wave every $T_s = 1/500 = 2\text{ ms}$, as plotted below.
(1) Why this happens. This is aliasing (frequency folding): sampling a continuous signal is equivalent, in the frequency domain, to replicating its spectrum at every multiple of $f_s$. Because $f_s=500\text{ Hz}$ is being asked to represent tones spaced $500\text{ Hz}$ apart, the sampler cannot tell $100\text{ Hz}$, $400\text{ Hz}$ and $600\text{ Hz}$ apart — all three collapse onto the same discrete-time sequence of numbers.
(2) Name of the phenomenon. The phenomenon is called aliasing (also "frequency folding" or "spectral folding"), because the higher-frequency tone "impersonates" (takes on the alias of) a lower-frequency one once it is sampled.
(3) Guaranteeing a unique representation. Satisfy the Nyquist sampling criterion for every frequency component actually present in the signal ($f_s > 2f_{\max}$, i.e. sample at more than twice the highest frequency of interest), and, since real signals always carry some out-of-band noise or interference, place an anti-aliasing low-pass filter ahead of the sampler/ADC to attenuate any energy above $f_s/2$ before it can fold back into the band of interest. Only when both conditions hold is the mapping from continuous signal to sample sequence one-to-one, so the original waveform can be perfectly reconstructed (e.g. via ideal sinc interpolation).
Fig. Q1 — the 100 Hz signal (solid blue) and its two 500 Hz aliases (600 Hz same-phase, 400 Hz phase-flipped, both dashed) coincide at every sample instant (black dots), so the sampler cannot tell them apart.
Quantity
Value
Alias frequency 1
400 Hz
Alias frequency 2
600 Hz
Phenomenon
Aliasing (frequency/spectral folding)
Uniqueness guarantee
Nyquist criterion $f_s>2f_{\max}$ plus an anti-aliasing low-pass filter before the ADC
