Chapter 5 Digitalization: 5.3 and Review on 5.1 Part I

5.1.2.2 Sampling and Aliasing

Mathematically, this part is very confusing to me, but with the help of the Flash Tutorial, I can visualize the phenomenon of aliasing better, but not sampling.

On page 4, under Figure 5.3, it stated “Figure 5.3 shows eight samples being taken for each cycle of the sound wave”, and then it talks about the frequency of the sound wave and sampling rate. And the next paragraph gave an example on aliasing. But I feel the process of sampling and phenomenon of aliasing could be explained better. I’d like to emphasize on the SAMPLING. If we want 8 samples in one cycle, we chose a sampling rate that is 8 times as high as the frequency of sound wave, thus given frequency of 440Hz, we need sampling rate of 8*440Hz, which is 3520Hz. We do NOT need to mention that the samples are stored as binary numbers right now since binary numbers will not be talked about until quantization, and it might cause confusion here. The reconstruction of sound wave is a process of the DAC fitting a sine wave on the sampled points (I’m not 100% sure on this part, but it must be some kind of fitting through some numerical methods, say, starting from fitting the sine wave of 20Hz, the lowest frequency human ears can hear, and increasing the frequency until all the samples can be fitted), which means, we need at least 2 samples per cycle to reconstruct the original sound wave; that is a sampling rate of at least twice as high as the frequency of the original sound wave; this is also the core meaning of the Nyquist Theorem. THEN we can say that IF the sampling rate is not high enough, aliasing will happen, followed by an example of aliasing.

In the figures, I hope the fact that the samples are measured at “equally-spaced moments in time“ will be emphasized. Figure 5.3 is definitely not only a graph of a 440Hz sound wave, it also shows the samples measured at every 1/(440*8) seconds.

At the end of 5.1.2.2 we can say refer to 5.3.1 for the algorithms for aliasing.

5.3.1 Mathematics and Algorithms for Aliasing

I have to admit that I see the algorithms, but the examples given are just NOT helpful since they only demonstrate how WE draw the graphs, but not how the graphs are actually generated in the process of sampling. The algorithm is NOT what causes the problem. It is the omission of necessary sampled points that lead to the error.

Figure 5.28 should be called 1760Hz wave aliased to a 240Hz wave due to being sampled at 1000Hz, because obviously there are two waves in it, and the 1760Hz one is not as important as the 240Hz one, plus there is a typo that it’s Hz not kHz.

5.1.2.3 Bit Depth and Quantization Error

At the beginning it says the bit depth determines the precision of sample, but HOW? It would be nice to mention that bit depth is actually the memory it takes to store this sample. It is like image resolution; the higher the bit depth is, the less quantization error is, the more accurate the sound is after ADC, meanwhile the more memory it takes to save this sample, giving the readers a general idea of what bit depth is. Also I believe most readers need to recognize the tradeoff between quality and audio file size, but not how bit depth actually works.

The example given on quantized wave explains itself pretty well if standing on itself, away from sampling; and the figures, as well as Max Demo: Bit Depth, also go along with the example (quantization by rounding down). The Flash Tutorial: Quantization, which is made from Audition, however, gives an example of rounding to the nearest integers (for a 4-bit one, it has 15 levels, 7 positive, 0, and 7 negative). I am not sure whether we should keep it consistent through all our examples, or at least mention it somewhere that the example given in the Flash Tutorial rounds to the nearest integers.

The description for Figure 5.7 is just WRONG! I do not see why we subtract the stair-step wave form the true sine wave, since we DO NOT hear the original sound wave, and the unwanted part should be the part which the stair-step wave has but the original wave does not; instead, we need to subtract the sine wave FROM the stair-step Wave, and we will get negative amplitude, which doesn’t matter as long as it is right. OR we can just say the green wave is the difference of those two waves, making it okay to be a positive amplitude.

(Part II coming up...)