We look at what Pulse Amplitude Modulation added to our analog source audio.
What did PAM add?
Earlier, we noted that the PAM signal represents the the source signal plus some additional high frequency content that we need to remove with a lowpass filter before we listen back.
Again, PAM is amplitude modulation of the source signal with a pulse train. Mathematically, we know precisely what amplitude modulation produces—the sums and differences of every frequency component between the two input signals. That is, if you you multiply a 100 Hz sine wave by a 6 Hz sine wave, the result is the sum of 106 Hz and 94 Hz sine waves. For signals with more frequency components, there are more sums and differences in the result.
To answer our question, “What got added?”, we need to understand the frequency content of a pulse train. One way to know that would be to use an Fourier Transform on the pulse train. But I want to use intuitive reasoning to eliminate as much math as possible. Fortunately, I already know what the extra frequency content is—it’s the spectral images in sampled systems, as described in classic DSP textbooks. That coupled with knowledge of amplitude modulation tips me off that we’ll need a frequency component at 0 Hz (DC—we need that to keep our original source band), at the sample rate, and at every integer multiple of the sample rate. Through infinity.
OK, we’ll lighten up on the infinity requirement. We can’t produce a perfect impulse in the analog world anyway. And we don’t need to. However, once in the digital domain, samples represent perfect impulses. While their values may have deviated slightly from a perfect representation of the analog signal, due to sampling time jitter and quantization, any math we do to them is “perfect” (again, subject to quantization and any other approximations). In the digital realm, the images do go to infinity.
Indeed, as you add cosine waves of 0, 1, 2, 3, 4…times the sample rate, the result gets closer and closer to the shape of an impulse. (Cosine instead of sine so that the peaks of the different frequencies line up.)
And that means we’ll have a copy of the source signal mirrored around 0 Hz, around the sample rate, twice the sample rate, three times the sample rate…to infinity. (In both directions, but we can ignore negative frequencies—for real signals, the negative spectrum mirrors the positive.)
What we’ve learned
Revisiting my “secrets”, with added comments:
1. Individual digital samples are impulses. Not bandlimited impulses, ideal ones.
Bothered that ideal impulses are impossible? Only in the physical world. There, we accept limitations. For instance, gather together infinity of something. Anything—I’ll wait. Meanwhile, in the mathematical world, infinity fits easily on this page: ∞
2. We know what lies between samples—virtual zero samples.
Think there’s really a continuous wave, implied, between samples? If so, you probably think it’s because samples represent a bandlimited impulse. No—you’re getting confused with what will come out of the DAC’s lowpass filter later, when we play back audio.
3. Audio samples don’t represent the source audio. They represent a modulated version of the audio. We modulated the audio to ensure points #1 and #2.
This is a frequency-domain observation that follows from the first two points, which are time domain. If you understand this point, you’ll never be confused about sample rate conversion.