Uniform Quantization

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Uniform Quantization
It was discussed in the previous lecture that the
disadvantage of using uniform quantization is
that low amplitude signals are drastically
effected. This fact can be observed by
considering the simulation results in the next
four slides. In both cases two signals with a
similar shape, but different amplitudes, are
applied to the same quantizer with a spacing of
0.0625 between two quantization levels. The
effects of quantization on the low amplitude
signal are obviously more significant than on the
high amplitude signal.
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Uniform Quantization
Input Signal 1.
3
Uniform Quantization
Quantized Signal 1. ?v0.0625
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Uniform Quantization
Input Signal 2.
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Uniform Quantization
Quantized Signal 2. ?v0.0625
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Uniform Quantization
Figure-1 Input output characteristic of a uniform
quantizer.
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Uniform Quantization
Recall that the Signal to Quantization Noise
Ratio of a uniform quantizer is given
by This equation verifies the discussion on
slide-1 that SNqR for a low amplitude signal is
quite low. Therefore, the effect of quantization
noise on such audio signals should be noticeable.
Lets consider the case of voice signals (see next
slide)
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Uniform Quantization
Click on the following links to listen to a
sample voice signal. First play voice file-1
then play voice file-1 Quantized. Do you notice
the degradation in voice quality? This
degradation can be attributed to uniformly spaced
quantization levels.
Voice file-1
Voice file-1. Quantized (uniform)
Note You may not notice the difference between
the two clips if you are using small laptop
speakers. You should use either headphones or
larger speakers.
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Uniform Quantization
More insight into signal degradation can be
gained by looking at the voice signals
Histogram. A histogram shows the distribution of
values of data. Figure-2 below shows the
histogram of the voice signal-1. Most of the
values have low amplitude and occur around zero.
Therefore, for voice signals uniform quantization
will result in signal degradation.
Figure-2 Histogram of voice signal-1
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Non-Uniform Quantization
The effect of quantization noise can be reduced
by increasing the number of quantization
intervals in the low amplitude regions. This
means that spacing between the quantization
levels should not be uniform. This type of
quantization is called Non-Uniform
Quantization. Input-Output Characteristics shown
below.
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Non-uniform Quantization
Non-uniform quantization is achieved by, first
passing the input signal through a compressor.
The output of the compressor is then passed
through a uniform quantizer. The combined effect
of the compressor and the uniform quantizer is
that of a non-uniform quantizer. (see figure
3.) At the receiver the voice signal is restored
to its original form by using an expander. This
complete process of Compressing and Expanding the
signal before and after uniform quantization is
called Companding.
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Non-uniform Quantization (Companding)
yg(x)
1
-1
xm(t)/mp
1
-1
Input output relationship of a compressor.
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Non-uniform Quantization (Companding)
A-Law (USA)
Where,
The value of µ used with 8-bit quantizers for
voice signals is 255
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Non-uniform Quantization (Companding)
The µ-law compressor characteristic curve for
different values of µ.
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Non-uniform Quantization (Companding)
Click on symbols to listen to voice signal at
each stage
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16
Non-uniform Quantization (Companding)
Click on symbols to listen to voice signal at
each stage
The 3 stages combine to give the characteristics
of a Non-uniform quantizer.
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17
Non-uniform Quantization (Companding)
Click on symbols to listen to voice signal at
each stage
A uniform quantizer with input and output voice
files is presented here for comparison with
non-uniform quantizer.
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Non-Uniform Quantization
Lets have a look at the histogram of the
compressed voice signal. In contrast to the
histogram of the uncompressed signal (figure-2)
you can see that the values are now more
distributed. Therefore, it can be said that the
compressor changes the histogram/ pdf of the
voice signal from gaussian (bell shape) to a
uniform distribution (shown below).
Figure-3 Histogram of compressed voice signal
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Non-Uniform Quantization
Where is the Compression..??? The compression
process in Non-uniform quantization demands some
elaboration for clarity of concepts. It should be
noted that the compression mentioned in previous
slides is not the time or frequency domain
compression which students are familiar with.
This can be verified by looking at the time
domain waveforms at the input and output of the
compressor. Note that both the signals last for
3.75 seconds. Therefore, there is no compression
in time or frequency.
Fig-4-a Signal at Compressor Input
Fig-4-b Signal at Compressor Output
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Non-Uniform Quantization
Where is the Compression..??? The compression
here occurs in the amplitude values. An intuitive
way of explaining this compression in amplitudes
is to say that the amplitudes of the compressed
signal are more closely spaced (compressed) in
comparison to the original signal. This can also
be observed by looking at the waveform of the
compressed signal (fig-4-b). The compressor
boosts the small amplitudes by a large amount.
However, the large amplitude values receive very
small gain and the maximum value remains the
same. Therefore, the small values are multiplied
by a large gain and are spaced relatively closer
to the large amplitude values. A parameter which
can be used to measure the degree of compression
here is the Dynamic range. The Dynamic Range is
the ratio of maximum and minimum value of a
variable quantity such as sound or light
. In the simulations the Dynamic Range (DR) of
the compressor input 41.45 dB Whereas Dynamic
Range (DR) of compressor output 13.95 dB
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Reference Text Books

• Lecture Notes Advanced Digital Communications
  by Dr. Norbert Goertz. MSc Signal Processing
  Communication January 2007, The University of
  Edinburgh.
• Modern Digital Analog Communications 3rd
  Edition by B. P. Lathi.
• Digital Analog Communication Systems 6th
  Edition by Leon W. Couch, II.
• Communication Systems 4th Edition by Simon
  Haykin.
• Analog Digital Communication Systems by
  Martin S. Roden.
• Sample voice file taken from CD of Digital Signal
  Processing a Computer Based Approach By S. K.
  Mitra.
• Note With the exception of figures on slides 06
  and 14 all figures have been sketched by Hassan
  Aqeel Khan. The voice files have been generated
  by using Matlab 7.

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