What is meant by the signal energy to alias ratio and how is it quantified?

Question

Below solid black curve is the bandwidth of a signal(a 4kHz RC low pass filter). And when we sample the signal at 8kHz rate due to folding the blue dashed line shows the aliasing:

1-) It looks very symmetric but how can we calculate the overall shape of the original bandwidth plus the aliasing? Should we add or multiply these curves?

2-) And from here I found the above curve which the author is able to quantify 50:1 signal energy to alias ratio. What is meant by that and how can that be calculated?

Answer 1

You add the energies in the signal and aliased curves.

Now if you assume the aliasing is completely de-correlated with the signal, you can treat them as two independent sources (like a "random walk" with 2 steps in statistics) you can use an RMS summation of the amplitudes.

Since the crossover at 4 kHz has both amplitudes as 0.7, or -3dB, the result is simply a flat line at 1 independent of frequency.

But it's a dangerous assumption and probably wrong for certain signals, so it's worth considering a worst case scenario, where the aliasing is perfectly correlated and in phase with the signal : in this case you can add amplitudes, giving a peak of 1.4 (+3dB) at 4 kHz.

With regard to:

Can you fix this problem by increasing the filter order? Not really. Figure 7 on page 10 shows the result of using a 6th-order Butterworth low pass filter instead of a 1st-order filter. We’ve improved the situation

Well he's right : you can't fix the problem simply by increasing filter order. You can improve it (decreasing the "alias" energy below 4 kHz and narrowing the +3dB peak in the worst case, thus decreasing its area). Thus you have improved the situation somewhat, but not fixed it.

A fix : if you need to reduce the aliasing energy below some level : would involve reducing the filter cutoff frequency - as well as optionally increasing the order - such that the attenuation at Fs/2 (4 kHz here) met your target.

And regarding comments to the other answer : aliasing below 4 kHz is the system response (due to the sampling process) to energy after the filter above 4 kHz. (which btw sounds nasty).

Aliasing at 7 kHz is huge because the filter has no attenuation at 1 kHz. It would also sound nasty but will be removed by the DAC's reconstruction filter.

Answer 2

It looks very symmetric but how can we calculate the overall shape of the original bandwidth plus the aliasing? Should we add or multiply these curves?

Neither!

This is just an amplitude spectrum plot. There's nothing said about the phase of the signals at the individual frequencies.

For example, if the signal at originally 12 kHz was highly correlated, in that it was the same, but with a negative sign, as the signal at 4 kHz, the sum of nyquist zone + alias would cancel out. If it was in-phase, it would constructively add.

If we assume the signal to be uncorrelated, e.g. by being white noise, OK; the powers would add. Powers adding up means you get the square root of the resulting power as amplitude; so not even in that case could you just add the amplitude graphs.

(That's one of the practical reasons why we use power spectral densities and not amplitude spectral densities to describe stochastic signals, by the way.)