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SYED AZEEM AHMED
SYED AZEEM AHMED
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SIGNAL SYSTEM’S
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Quantization Error
The fundamental difference between digital audio and analog audio is one of resolution.
Analog representations of analog signals have a theoretically infinite resolution in both
level and time. Digital representations of an analog sound wave are discretized into
quantifiable levels in slices of time. We've already talked about discrete time and
sampling rates a little in the previous section and we'll elaborate more on it later, but for
now, let's concentrate on quantization of the signal level.
As we've already seen, a PCM-based digital audio system has a finite number of levels
that can be used to specifiy the signal level for a particular sample on a given channel.
For example, a compact disc uses a 16-bit binary word for each sample, therefore there
are a total of 65,536 (or
) quantization levels available. However, we have to always
keep in mind that we only use all of these levels if the signal has an amplitude equal to
the maximum possible level in the system. If we reduce the level by a factor of 2 (in other
words, a gain of -6.02 dB) we are using one fewer bits worth of quantization levels to
measure the signal. The lower the amplitude of the signal, the fewer quantization levels
that we can use until, if we keep attenuating the signal, we arrive at a situation where the
amplitude of the signal is the level of 1 Least Significant Bit (or LSB).
Let's look at an example. Figure 10.1 shows a single cycle of a sine wave plotted with a
pretty high degree of resolution (well... high enough for the purposes of this discussion).
Figure 8.13: A single cycle of a sine wave. We'll consider this to be the analog input
signal to our digital converter.
Let's say that this signal is converted into a PCM digital representation using a converter
that has 3 bits of resolution - therefore there are a total of 8 different levels that can be
used to describe the level of the signal. In a two's complement system, this means we
have the zero line with 3 levels above it and 4 below. If the signal in Figure 10.1 is
aligned in level so that its positive peak is the same as the maximum possible level in the
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PCM digital representation, then the resulting digital signal will look like the one shown
in Figure 8.14.
Figure 8.14: A single cycle of a sine wave after conversion to digital using 4-bit, PCM,
two's complement where the signal level is rounded to the nearest quantization level at
each sample. The blue plot is the original waveform, the red is the digital representation.
Not surprisingly, the digital representation isn't exactly the same as the original sine
wave. As we've already seen in the previous section, the cost of quantization is the
introduction of errors in the measurement. However, let's look at exactly how much error
is introduced and what it looks like.
This error is the difference between what we put into the system and what comes out of
it, so we can see this difference by subtracting the red waveform in Figure 8.14 from the
blue waveform.
Figure 8.15: A plot of the quantization error generated by the conversion shown in
Figure 8.14.
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There are a couple of characteristics of this error that we should discuss. Firstly, because
the sine wave repeats itself, the error signal will be periodic. Also, the period of this
complex waveform made up of the will be identical to the original sine wave - therefore it
will be comprised of harmonics of the original signal. Secondly, notice that the maximum
quantization error that we introduce is one half of 1 LSB. The significant thing to note
about this is its relationship to the signal amplitude. The quantization error will never be
greater than one half of an LSB, so the more quantization levels we have, the louder we
can make the signal we want to hear relative to the error that we don't want to hear. See
Figures 8.16 through 8.18 for a graphic illustration of this concept.
Figure 8.16: A combined plot of the original signal, the quantized signal and the
resulting quantization error in a 3-bit system.
Figure 8.17: A combined plot of the original signal, the quantized signal and the
resulting quantization error in a 5-bit system.
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Figure 8.18: A combined plot of the original signal, the quantized signal and the
resulting quantization error in a 9-bit system.
As is evident from Figures 8.16, 8.17 and 8.18, the greater the number of bits that we
have available to describe the instantaneous signal level, the lower the apparent level of
the quantization error. I use the word ``apparent'' here in a strange way - no matter how
many bits you have, the quantization error will be a signal that has a peak amplitude of
one half of an LSB in the worst case. So, if we're thinking in terms of LSB's - then the
amplitude of the quantization error is the same no matter what your resolution. However,
that's not the way we normally think - typically we think in terms of our signal level, so,
relative to that signal, the higher the number of available quantization levels, the lower
the amplitude of the quantization error.
Given that a CD has 65,536 quantization levels available to us, do we really care about
this error? The answer is ``yes'' - for two reasons:
1. We have to always remember that the only time all of the bits in a digital system
are being used is when the signal is at its maximum possible level. If you go
lower than this - and we usually do - then you're using a subset of the number of
quantization levels. Since the quantization error stays constant at +/- 0.5 LSB and
since the signal level is typically lower, then the relative level of the quantization
error to the signal is typically higher. The lower the signal, the more audible the
error. This is particularly true at the end of the decay of a note on an instrument or
the reverberation in a large room. As the sound decays from maximum to nothing,
it uses fewer and fewer quantization levels and the perceived quality drops
because the error becomes more and more evident because it is less and less
masked.
2. Since the quantization error is periodic, it is a distortion of the signal and is
therefore directly related to the signal itself. Our brains are quite good at ignoring
unimportant things. For example, you walk into someone's house and you smell a
new smell - the way that house smells. After 5 minutes you don't smell it
anymore. The smell hasn't gone away - your brain just ignores it when it realizes
that it's a constant. The same is true of analog tape noise. If you're like most
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people you pay attention to the music, you stop hearing the noise after a couple of
minutes. Your brain is able to do this all by itself because the noise is unrelated to
the signal. It's a constant, unrelated sound that never changes with the music and
is therefore unrelated - the brain decides that it doesn't change so it's not worth
tracking. Distortion is something different. Distortion, like noise, is typically
comprised entirely of unwanted material (I'm not talking about guitar distortion
effects or the distortion of a vintage microphone here...). Unlike noise, however,
distortion products modulate with the signal. Consequently the brain thinks that
this is important material because it's trackable, and therefore you're always
paying attention. This is why it's much more difficult to ignore distortion than
noise. Unfortunately, quantization error produces distortion - not noise.
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