Sampling a sinusoidal signal smaller than Nyquist rate

Question

If we have a sinusoidal signal at 50 kHz and we sample it by an ADC with a sampling rate of 7 kHz. What would be the output of the ADC? Since the sampling rate is way less than the Nyquist rate (100KHz), we definitely have aliasing. When I write down the frequency domain signals, I will end up with this sequence:

$$y(n) = \sum_{i} sin(2\pi f_s(i)n)$$

Where $$f_s(i) =[1,6,8,13,15,20,22,27,29,34,36,41,43,48,50,\dots] \times 50\text{KHz}$$ We can see that $$f_s(i)$$ consists of multiples of $7 \pm 1$. But when I do this on MATLAB(sampling at the rate of 7KHz on a 50KHz signal, I get this, which seems like a simple sinusoidal signal. What am I missing? Is the summation above results in a single-tone sinusoidal signal? I tried drawing it, but it doesn’t give me the expected signal.

Answer 1

As per the suggestion by @Dan Boschen, I am converting my comments into a short answer. For simplicity, ignore units, so the 50 kHz signal frequency becomes 50, the 7 kHz sampling frequency becomes 7 and so on. Then the Nyquist frequency is 3.5, i.e., half the sampling frequency.

Now consider a simple demo like this: take a piece of stiff cardboard 3.5 inches long and attach a thread 50 inches long to one end. Start spooling the thread around the card. The thread will make 7 complete loops, leaving 1 inch left. That is the alias. Note that 50 = 1 + (14 x 3.5) and 14 is even, so the modulus is the alias, i.e., 50 mod 3.5 = 1. In other words, the alias is what was left over after the integer number of complete spooling loops were done.

If the signal was at 54, for example, then 54 mod 3.5 = 1.5, so the alias would be at 2, i.e., 3.5 - 1.5. This is because 54 = 1.5 + (15 x 3.5) and 15 is odd. So there were 7 complete loops plus half a loop back to the Nyquist value and then the leftover part (the modulus) heading back toward the start, i.e., zero frequency.

The following steps illustrate how to find the alias frequency. For brevity, the word frequency is omitted throughout.

1. Calculate the modulus as signal mod Nyquist.
2. Calculate the integer N, where N = (signal - modulus)/Nyquist.
3. If N is even, then the alias equals the modulus.
4. If N is odd, then the alias equals (Nyquist - modulus)