After I posted the last two parts of this series (which I thought wrapped it up…) I received an email asking about whether there’s a similar thing happening if you remove the reconstruction (low-pass) filter in the digital-to-analogue part of the signal path.
The answer to this question turned out to be more interesting than I expected… So I wound up turning it into a “Part 3” in the series.
Let’s take a case where you have a 1 kHz signal in a 48 kHz system. The figure below shows three plots. The top plot shows the individual sample values as black circles on a red line, which is the analogue output of a DAC with a reconstruction filter.
The middle plot shows what the analogue output of the DAC would look like if we implemented a Sample-and-hold on the sample values, and we had an infinite analogue bandwidth (which means that the steps have instantaneous transitions and perfect right angles).
The bottom plot shows what the analogue output of the DAC would look like if we implemented the signal as a pulse wave instead, but if we still we had an infinite analogue bandwidth. (Well… sort of…. Those pulses aren’t infinitely short. But they’re short enough to continue with this story.)
If we calculate the spectra of these three signals , they’ll look like the responses shown in Figure 2.
Notice that all three have a spike at 1 kHz, as we would expect. The outputs of the stepped wave and the pulsed wave have much higher “noise” floors, as well as artefacts in the high frequencies. I’ve indicated the sampling rate at 48 kHz as a vertical black line to make things easy to see.
We’ll come back to those artefacts below.
Let’s do the same thing for a 5 kHz sine wave, still in a 48 kHz system, seen in Figures 3 and 4.
Compare the high-frequency artefacts in Figure 4 to those in Figure 2.
Now, we’ll do it again for a 15 kHz sine wave.
There are three things to notice, comparing Figures 2, 4, and 6.
The first thing is that artefacts for the stepped and pulsed waves have the same frequency components.
The second thing is that those artefacts are related to the signal frequency and the sampling rate. For example, the two spikes immediately adjacent to the sampling rate are Fs ± Fc where Fs is the sampling rate and Fc is the frequency of the sine wave. The higher-frequency artefacts are mirrors around multiples of the sampling rate. So, we can generalise to say that the artefacts will appear at
n * Fs ± Fc
where n is an integer value.
This is interesting because it’s aliasing, but it’s aliasing around the sampling rate instead of the Nyquist Frequency, which is what happens at the ADC and inside the digital domain before the DAC.
The third thing is a minor issue. This is the fact that the level of the fundamental frequency in the pulsed wave is lower than it is for the stepped wave. This should not be a surprise, since there’s inherently less energy in that wave (since, most of the time, it’s sitting at 0). However, the artefacts have roughly the same levels; the higher-frequency ones have even higher levels than in the case of the stepped wave. So, the “signal to THD+N” of the pulsed wave is lower than for the stepped wave.