Recording - February 2016 - 72

In fact, to sample (and digitize) a waveform at a given frequency, the
samples have to be taken at a minimum of twice that frequency, so that
each cycle of the signal (or any of its harmonic components), gets sampled
at least two times. If you look at Figure 2, the samples happen less often
than that; in fact, in that example, the sampling frequency is actually lower
than the signal frequency.
This principle-that a sampling-based system can only work properly for
signals less then half the sampling frequency-was shown mathematically
in 1928 by the engineer Harry Nyquist, and the principle is generally
known as the Nyquist Theorem (though, to be fair to the other engineers
and mathematicians who developed it independently, it could well be
called the Whittaker-Nyquist-Küpfmüller-Kotel'nikov-Whittaker-Raabe-GaborShannon Theorem.) And the frequency which is half of the "sampling frequency" is known as the Nyquist frequency.
Names aside, let's state the basic principle again, because it's really
important: you can never successfully sample signals of higher than half of
the "sampling frequency". Higher frequency signals are out of bounds, like
a wayward soccer ball. That's why audio engineers keep calling it the
"sampling frequency", rather than the more correct "sampling rate"; it
helps when thinking about the Nyquist frequency to know that it's always
half of the "sampling frequency". And that's why it makes a certain bassackwards sense to think of the sampling rate as a frequency; half of a frequency is another frequency.

4
There's a practical takeaway from all this: a digital recording system
needs to keep out-of-bounds signals from entering the analog-to-digital converter, which means the designers need to include a very steep lowpass filter just below the Nyquist frequency. So if the sampling rate is 44,100
samples per second, the Nyquist frequency will be 22,050 Hz; the filter
needs to have its cutoff frequency around 21,000 Hz, and it needs to be
so steep that the response will be attenuated by roughly 90 dB by the time
you get up to the Nyquist Frequency of 22,050 Hz. These are (reasonably)
known as anti-aliasing filters.
That's one attraction of higher sampling rates like 96 kHz and more: the
anti-aliasing filters can be less drastic. Extremely steep filters make a mess
of the phase and impulse response of signals, and a sample rate that lets
you use a less steep filter mangles the waveform less. (Note for the nitpickers: these days the filtering is implemented digitally, in a way that minimizes mangling, but the principle still applies.)
Besides, a higher sample rate lets you record higher frequency sounds.
Conventional wisdom states that humans, even young, healthy humans,
can't hear above 20 kHz, but there are some sounds up there (mostly harmonics of lower-frequency sounds), and reports suggest that some listeners,
though they don't actually hear those sounds, can tell when they're missing.
So higher sample rates (96,000 samples/sec. and up) remain popular.
Some engineers, if they know they'll be mixing on an analog console, will
track at 96k, then mix to a format with a lower sample rate for eventual
72

RECORDING February 2016

issue on CD or downloads. A 96,000 samples/
second rate will let you record to just below its
Nyquist frequency of 48 kHz. (As Flanders &
Swann once put it, "The ear can't hear as high as
that/Still it ought to please any passing bat.")
If not now, when?
While we're talking about when sampling takes
place, it's really important that the samples are
taken at the right time. Go back to Figure 1, where
the sampled waveform produces the numbers
269...1,209...5,414...1,332. Let's say the first
sample happens a little early; the waveform is
increasing at that point, so an early sample might
measure 268 instead of 269. Imagine that the last
sample is also a little early; because the waveform
is decreasing at that point, an early sample might
come out as 1,334 instead of 1,332. Put them all
together, and the sequence becomes
268...1,209...5,414...1,334. And that's distortion; the first and last numbers are wrong, because
the samples were done at the wrong times.
Variation in sampling time is a very real phenomenon in digital sampling systems, and it's
called jitter. What keeps jitter to a minimum?
First and foremost, an accurate clock; in fact,
one of the big differences between consumergrade and professional-grade converters is that
the pro units are specified as having lower jitter,
sometimes a lot lower. We'll talk about that a little next month.
And it comes out here
When a PCM-encoded signal comes out of a
digital to analog converter, it looks like Figure 4.
You can see the signal is coming out in discrete
units, quantized in time (the horizontal axis), corresponding to the individual samples, Its levels
have also been quantized (the vertical axis). For
obvious reasons, engineers call this a stair-step
waveform. The rapid transitions between levels
are ultrasonic signals (too high to hear), which
can cause distortion later in the audio chain.
Most D/A converters will filter the ultrasonic stuff
out using a lowpass filter with a cutoff frequency
just above the audio range, leaving a smooth
waveform. These are known as reconstruction filters. Again, if the sample rate is higher, the
reconstruction filter can be less drastic (usually a
less-steep slope, which will have better phase
and transient response... all to the good.)
So that's the signal chain for a minimal digital
audio system:
Input > Anti-Aliasing Filter > A/D converter >
Storage/transmission > D/A converter >
Reconstruction Filter > Output
Of course, there's a lot more going on; we'll
talk about that in the coming months. To begin
with, in the next installment we'll look at those
converters that live as hardware in the real
world. Stay tuned!
Paul J. Stamler (stamler@recordingmag.com) is
a recording engineer, musician, and educator,
living in St. Louis.



Table of Contents for the Digital Edition of Recording - February 2016

Contents
Recording - February 2016 - Cover1
Recording - February 2016 - Cover2
Recording - February 2016 - 1
Recording - February 2016 - 2
Recording - February 2016 - 3
Recording - February 2016 - 4
Recording - February 2016 - 5
Recording - February 2016 - 6
Recording - February 2016 - 7
Recording - February 2016 - 8
Recording - February 2016 - 9
Recording - February 2016 - Contents
Recording - February 2016 - 11
Recording - February 2016 - 12
Recording - February 2016 - 13
Recording - February 2016 - 14
Recording - February 2016 - 15
Recording - February 2016 - 16
Recording - February 2016 - 17
Recording - February 2016 - 18
Recording - February 2016 - 19
Recording - February 2016 - 20
Recording - February 2016 - 21
Recording - February 2016 - 22
Recording - February 2016 - 23
Recording - February 2016 - 24
Recording - February 2016 - 25
Recording - February 2016 - 26
Recording - February 2016 - 27
Recording - February 2016 - 28
Recording - February 2016 - 29
Recording - February 2016 - 30
Recording - February 2016 - 31
Recording - February 2016 - 32
Recording - February 2016 - 33
Recording - February 2016 - 34
Recording - February 2016 - 35
Recording - February 2016 - 36
Recording - February 2016 - 37
Recording - February 2016 - 38
Recording - February 2016 - 39
Recording - February 2016 - 40
Recording - February 2016 - 41
Recording - February 2016 - 42
Recording - February 2016 - 43
Recording - February 2016 - 44
Recording - February 2016 - 45
Recording - February 2016 - 46
Recording - February 2016 - 47
Recording - February 2016 - 48
Recording - February 2016 - 49
Recording - February 2016 - 50
Recording - February 2016 - 51
Recording - February 2016 - 52
Recording - February 2016 - 53
Recording - February 2016 - 54
Recording - February 2016 - 55
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Recording - February 2016 - 57
Recording - February 2016 - 58
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Recording - February 2016 - 60
Recording - February 2016 - 61
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Recording - February 2016 - 76
Recording - February 2016 - 77
Recording - February 2016 - 78
Recording - February 2016 - 79
Recording - February 2016 - 80
Recording - February 2016 - Cover3
Recording - February 2016 - Cover4
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