THE ALIASING undersample and a frequency wears a mask — f and f+fs are twins
Aliasing is what Nyquist warns against, seen from the other side. Sample too slowly and every frequency has infinitely many aliases: f, f+fs, f+2fs, … all produce the identical sample sequence, because sampling folds the frequency axis modulo the sample rate. It is the wagon-wheel that spins backward in a film, the moiré on a screen, the false low tone in a badly-sampled recording. The phase advances by (f·n) mod fs, so any two frequencies congruent mod fs are indistinguishable — a pure fact of modular arithmetic.
THE TECHNIQUE sampled phase = (f·n) mod fs; f and f+fs are identical
The demo shows frequencies 3 and 11 (=3+8) sampled at fs=8 produce the identical phase sequence, while 3 and 2 differ: live demo
HISTORY & CREDIT aliasing · the wagon-wheel effect
“A high frequency just looks noisy when undersampled.” — it looks like a specific lower frequency, exactly — its alias. cited
the fold · sampling maps f → f mod fs — the frequency axis wraps. the twins · f and f + k·fs give identical samples for all integer k. in the world · the backward wagon wheel, screen moiré, the anti-alias filter that prevents it.
Frequencies folded onto each other by too-slow a clock — a high tone wearing a low one’s face. modular
RECOMMEND FOR I-13 the fold, on the compiler
On the canonical compiler, frequencies 3 and 11 sampled at fs=8 give the identical phase sequence; 3 and 2 differ:
$ i13 run cr_aliasing.i13 # phase = (f*n) mod 8
RUN OK · 384 step(s) · peak stack 5 · call depth 9
alias_3_11 = 1 -- freq 3 and freq 11 (=3+8): identical samples
distinct_3_2 = 0 -- freq 3 and freq 2: different
Recommend as a NULL — modular arithmetic, a theorem. Aliasing is the statement that sampled phase is (f·n) mod fs, so frequencies congruent mod fs are identical — a fact of modular arithmetic every sampler obeys (B39), the mirror of dart 448. i13 grounds it (3 aliases 11 at fs=8). NULL — the failure mode the Nyquist rate is set to avoid.