QUANTIZATION ERROR AND NOISE

QUANTIZATION ERROR AND NOISE

1. DIGITAL COMMUNICATION
SYSTEMS (ECE4001)
Dr. Arvind Kumar| School of Electronics Engineering, VIT Vellore |

2. PULSE CODE MODULATION
(PCM)
Dr. Arvind Kumar| School of Electronics Engineering, VIT Vellore |

3. Quantization Noise
Inevitably, the use of quantization introduces an error defined as the difference
between the continuous input sample mand the quantized output sample v. The
error is called quantization noise.
Let the quantizerinput mbe the sample value of a zero-mean random variable M.
Let the quantization error be denoted by the random variable Qof sample value
q.

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Illustration of the quantization process.(Ref. S. Haykin, Page no. 280)
midtreadtype

5. •Quantizerscan be of a uniform or nonuniformtype. In a uniform quantizer, the
representation levels are uniformly spaced; otherwise, the quantizeris nonuniform.
•The quantizercharacteristic can also be of midtreador midrisetype.

6. Two types of quantization:
Midtreadtype
which is so called because the origin lies in the middle of a tread of the staircaselike
graph.
Midrise type,
In which the origin lies in the middle of a rising part of the staircaselikegraph.

7. Cont’d
•With the input Mhaving zero mean and the quantizerassumed to be symmetric as
in, it follows that the quantizeroutput Vand, therefore, the quantization error Q
will also have zero mean.
•Thus, for a partial statistical characterization of the quantizerin terms of output
signal-to-(quantization) noise ratio, we need only find the mean-square value of
the quantization error Q.
•Consider, then, an input mof continuous amplitude, which, symmetrically,
occupies the range [–m
max, m
max].
•The step size of the quantizeris given by:
•whereL is the total number of representation levels.
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8. Cont’d
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Four-bit coding (16 discretelevels)

9. Cont’d
•Step Size
•For a uniform quantizer, the quantization error Qwill have its sample values
bounded by:
•If the step size is sufficiently small (i.e., the number of representation levels Lis
sufficiently large).
•It is reasonable to assume that the quantization error Qis a uniformly distributed
random variable and the interfering effect of the quantization error on the
quantizerinput is similar to that of thermal noise, hence the reference to
quantization error as quantization noise.
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10. Cont’d
•Probability density function of the quantization noise as:
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11. •For this to be true, however, we must ensure that the incoming continuous sample
does not overload the quantizer. Then, with the mean of the quantization noise
being zero, its variance is the same as the mean-square value; that is,
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12. •Let R denote the number of bits per sample used in the construction of the binary
code. We may then write
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13. Output SNR
•Let Pdenote the average power of the original message signal m(t).
•output signal-to-noise ratio(SNR) of a uniform quantizeras(Book S Haykinpage
no.281):
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Output SNR increases exponentially with increasing number of bits per sample R,

14. Sinusoidal Modulating Signal
•Sinusoidal modulating signal of amplitude A
m, which utilizes all the representation
levels provided. The average signal power is (assuming a load of 1 ohm)
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17. Signal-to-(quantization) noise ratio for varying number of
representation levels for sinusoidal modulation
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18. Question
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Hint

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A sinusoidal signal of amplitude A is quantized by a uniform quantizerAssume that the signal
utilizes all the representation levels of the quantizer. If the signal to quantization noise ratio is
31.8 dB the number of levels in the quantizeris _________-
a)32
b)65
c)80
d)33

20. Compute Signal to Noise Ratio
Dr. Arvind Kumar, SENSE, VIT VELLORE 20
•Signal energy = ; Noise energy = ;
•Noise energy =
•Signal to noise =
•Every bit increases SNR by ~ 6 decibels12
2
 N
A2
 2
2
3N
A
 2
3
log10
2
N 2
2
A

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What is the type of quantizer, if a Zero is assigned a quantization level?
a) Midrise type
b) Mid tread type
c) Mistreat type
d) None of the mentioned
.
3. What is the type of quantizer, if a Zero is assigned a decision level?
a) Midrise type
b) Mid tread type
c) Mistreat type
d) None of the mentioned

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What is the type of quantizer, if a Zero is assigned a quantization level?
a) Midrise type
b) Mid tread type
c) Mistreat type
d) None of the mentioned
View Answer
Answer: b
Explanation: If a zero is assigned a quantization level, the quantizeris of the mid treat type.
3. What is the type of quantizer, if a Zero is assigned a decision level?
a) Midrise type
b) Mid tread type
c) Mistreat type
d) None of the mentioned
View Answer
Answer: a
Explanation: If a zero is assigned a decision level, the quantizeris of the midrise type.

23. Compute Signal to Noise Ratio
Dr. Arvind Kumar, SENSE, VIT VELLORE 23
•Signal energy = ; Noise energy = ;
•Noise energy =
•Signal to noise =
•Every bit increases SNR by ~ 6 decibels12
2
 N
A2
 2
2
3N
A
 2
3
log10
2
N 2
2
A

24. Next PCM
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