Let's look at some numerical examples, then generalize. Suppose the real waveform (observed by high frequency sampling) is known to be at 1000 Hz. At what frequency would it appear if sampled at 1 Hz? First of all, if there is jitter (variation in frequency over the course of an experiment), we may just see a blur. But if everything is stable, We will see what looks like a single value -- all zeros (if the sampling is synchronized to the zero crossing at each half-millisecond interval) or all some other value. 1000 Hz is the 1000 multiple or 1000th harmonic of 1 Hz. What if we sample at 1.1 Hz? If the first point is taken at a positive-going zero crossing of the 1000 Hz waveform, we will take 11 points during the ensuing 10 s before hitting the next mutual point of positive zero crossing. The apparent frequency of the sampled waveform will be 0.1 Hz. Sample at 202 Hz, and we see one sampled cycle in 0.1 s.

Is there a pattern emerging? An integer times the sampling rate differs from the actual signal frequency by the observed, aliased frequency. We'll put the integer in green to make it obvious. 1000 - 5*202 = 1000 - 1010 = -10 Hz (period = 0.1 s, as seen in the above figure). 0.9 Hz - 1*1 Hz = - 0.1 Hz (several previous examples). 1000 - 909*1.1 = 0.1 Hz.

The missing mathematical concept here is modulus. The modulus is the remainder from a division problem. 7 modulus 4 = 3. 19 modulus 6 = 2. So in determining the effect of aliasing, the apparent frequency is determined by finding which harmonic is closest to the actual frequency, then subtracting the harmonic number times the sampling frequency from the actual frequency. Thus, at the end of the previous paragraph, we subtracted the 909th harmonic of 1.1 Hz from 1000 Hz. What would have happened if we'd used the 908th or 910th harmonic? 1000 - 908*1.1 = 1000 - 998.8 = 1.2 Hz, and 1.2 is bigger (in absolute value) than the Nyquist frequency (1.1 Hz/2 = 0.55 Hz). 1000 - 910*1.1 = 1000 - 1001 = -1 Hz, which is also (in absolute value) bigger than the Nyquist freqency. So choose the harmonic such that, when the harmonic frequency is subtracted from the true frequency, the result lies between 1 Nyquist frequency (in absolute value) of 0.

Exercise: A 1.275 MHz signal is sampled at 50.000 KHz. At what frequency does the aliased signal appear?