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Quantization Error Analysis
Author: Anil Pothireddy
12/10/2002
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Organization of the Presentation
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Introduction to Quantization.
Quantization Error Analysis.
Quantization Error Reduction Techniques.
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QUANTIZATION
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Definition : The transformation of a signal x[n]
into one of a set of prescribed values.
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Quantization converts a Discrete-Time Signal to
a Digital Signal.
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MATHEMATICAL REPRESENTATION:
xq[n] = Q( x[n] )
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EXAMPLE OF QUANTIZATION
(a) Unquantized samples of x[n] = 0.99cos(n/10). (b) with a 3-bit quantizer.
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QUANTIZER
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Quantizers can be defined with either uniformly
or non uniformly spaced quantization levels.
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Quantizers can also be customized to work on
either uni-polar or bipolar signals.
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TYPICAL QUANTIZER
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QUANTIZATION LEVELS
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In the previous figure, the 8-quantization levels,
can be labeled using a binary code of 3–bits.
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In general, to represent B-quantization levels we
need log2B(rounded to next highest integer) bits.
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The step size of the quantizer will be:
∆ = 2Xm / 2B
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ADVANTAGES OF QUANTIZATION
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The quantized signal, which is an approximation
of the original signal, can be more efficiently
separated from ADDITIVE NOISE. (by using
repeaters).
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Transmission bandwidth can be controlled by
using an appropriate number of quantization
levels (and hence the bits to represent them).
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QUANTIZATION ERROR

The quantized sample will generally differ from
the original signal. The difference between them
is called the quantization error.
e[n] = xq[n] - x[n]
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For a 3-bit Quantizer, if ∆/2 < x[n] =< 3 ∆/2, then
xq[n] = ∆, and it follows that:
-∆/2 < e[n] =< ∆/2
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QUANTIZER MODEL

In this model, the quantization error samples are
thought of as an ADDITIVE NOISE SIGNAL. (The
model is exactly equivalent to a Quantizer if e[n] is
exactly known).
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STATISTICAL REPRESENTATION OF
QUANTIZATION ERRORS
ASSUMPTIONS
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e[n] is a sample sequence of a stationary random
process.
e[n] is uncorrelated with the sequence x[n].
The random variables of the error process are
uncorrelated.
The probability distribution of the error process is
uniform over the range of quantization error.
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QUANTIZATION ERROR (3-BIT & 8-BIT)
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STATISTICAL REPRESENTATION OF
QUANTIZATION ERRORS (2)
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We know that : -∆/2 < e[n] =< ∆/2
For small ∆,it is reasonable to assume that e[n]
is a Random variable uniformly distributed from -∆/2 to ∆/2.
Thus e[n] is a uniformly distributed white-noise
sequence
The mean value of e[n] = 0.
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STATISTICAL REPRESENTATION OF
QUANTIZATION ERRORS (3)
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STATISTICAL REPRESENTATION OF
QUANTIZATION ERRORS (4)
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OBSERVATIONS
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We see that the signal-to-noise ratio increases
approximately 6dB for each bit added to the
word length of the Quantized samples.
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If σx = Xm / 4 then: SNR (approx) = 6B – 1.25dB.
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Obtaining a 90-96dB SNR for use in HighQuality audio requires a 16-bit Quantization.
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QUANTIZATION ERROR REDUCTION
TECHNIQUES.
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INCREASING THE SAMPLING RATE.
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DIFFERENTIAL QUANTIZATION.
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NON UNIFORM QUANTIZATION
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INCREASING THE SAMPLING RATE
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It has been proved that: for every doubling of the
oversampling ratio M, we need ½ bit less to
achieve a given Signal-to-Quantization-Noise
ratio.
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If we oversample by a factor M=4, we need one
less bit to achieve a desired accuracy in
representing a signal. (i.e. M = 4(no of bits reduced)).
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This technique is of little practical importance, as
it involves a rather high overhead
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DIFFERENTIAL QUANTIZATION
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In many practical situations, due to the statistical
nature of the message signal, the sequence x[n]
will consist of samples that are correlated with
each other.
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For a given number of levels per sample,
differential quantization schemes yield a lower
value of quantizing noise value than direct
quantizing schemes.
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DIFFERENTIAL QUANTIZING SCHEME
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The error reduction is possible as long as the sample
to sample correlation is non-zero.
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EXAMPLE PROBLEM
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SOLUTION
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NON UNIFORM QUANTIZATION
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The qunatization error (noise) depends on the
step size ∆. Hence if the steps are uniform in
size, small-amplitude signals will have a poor
Signal-to-Quantization-Noise ratio.
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To illustrate this effect, assume a full scale voltage of 10V
and that the actual resolution is +/- 4mV (i.e. ∆ = 8mV).
When the signal is close to 10V, the peak quantization
error is in the neighborhood of (4mV / 10V ) * 100% =
0.04%.
When the signal level hovers around 10mV , the error is in
the vicinity of (4mV / 10mV) * 100% = 40% !!!
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NON UNIFORM QUANTIZATION (2)
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The severity of this problem depends on the
dynamic range of the signal and the number of
bits used in encoding (quantizing).
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In theory, a sufficient number of bits could be
added to decrease the peak quantization error to
a more tolerable level, but this is an inefficient
and often impractical process.
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COMPANDING
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To correct this situation within the constraint of
fixed number of levels, it is advantageous to
taper the step size so that the steps are close
together at low signal amplitudes and further
apart at large amplitudes
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This leads to the SNR improvement for small
signals, but the strong signals will be impaired.
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However the Inst. speech signal amplitude < ¼
rms signal value, for more than 50% of the time.
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COMPANDER (2)
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While it is possible to build a quantizer with
tapered steps, it is more feasible/practical to
achieve an equivalent effect by distorting the
signal before quantizing.
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An inverse distortion is introduced at the
receiving end so that the overall transmission is
distortionless.
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COMPANDER (3)
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COMPANDER (4)
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At low amplitudes the slope is larger than at high
amplitudes.
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Consequently a given signal change at low
amplitude will carry the quantizer through more
steps than will be the case at large amplitudes.
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This network is called a COMPRESSOR. The
inverse operation is performed by a
EXPANDER. The combination is called a
COMPANDER.
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REFERENCES
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Discrete Time Signal Processing.
Oppenheim and Schaffer,Prentice Hall.
Digital and Analog Communication Systems.
K. Shanmugam, John Wiley.
Principles of Communication Systems.
Taub and Shilling, McGraw-Hill.
Web Sources:
 www.dspguru.com
 www.ece.utexas.edu
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