Test Your EQ (Engineering Quotient)

EQ #47

In Figure 1, why does this reconstruction of a 20-kHz sinewave sampled at 44.1 kHz show ripple in its amplitude?

Figure 1

The actual sampled data, represented by the square dots in the diagram, contains equal levels of Fsignal (the sine wave) and Fsample-Fsignal (one of the aliases of the sinewave). Any reconstruction filter is going to have difficulty passing the one and eliminating the other, so you inevitably get some of the alias signal, which, when added to the desired signal, produces the “modulation” you see.

In the case of a software display of a waveform on a computer screen (e.g., such as you might see in software used to edit audio recordings), they’re probably using an FIR low-pass filter (sin(x)/x coefficients) windowed to some finite length. A shorter window gives faster drawing times, so they’re making a tradeoff between visual fidelity and interactive performance. The windowing makes the filter somewhat less than brick-wall, so you get the leakage of the alias and the modulation.

In the case of a real audio D/A converter, even with oversampling you can’t get perfect stopband attenuation (and you must always do at least some of the filtering in the analog domain), so once again you see the leakage and modulation.

In this example, Fsignal = 0.9×Fnyquist, so Falias = 1.1×Fnyquist and Falias/Fsignal = 1.22. To eliminate the visible artifacts, the reconstruction filter would need to have a slope of about 60dB over this frequency span, or about 200 dB/octave.


Advertise Here

Don't miss out on upcoming issues of Circuit Cellar. Subscribe today!

Note: We’ve made the October 2017 issue of Circuit Cellar available as a free sample issue. In it, you’ll find a rich variety of the kinds of articles and information that exemplify a typical issue of the current magazine.

Would you like to write for Circuit Cellar? We are always accepting articles/posts from the technical community. Get in touch with us and let's discuss your ideas.