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 m and the quantized output sample v.The
error is called quantization noise.
 Let the quantizer input m be the sample value of a zero-mean random variable M.
 Let the quantization error be denoted by the random variable Q of sample value
q.

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Illustration of the quantization process.(Ref. S. Haykin, Page no. 280)
midtread type

5. • Quantizers can be of a uniform or nonuniform type. In a uniform quantizer, the
representation levels are uniformly spaced; otherwise, the quantizer is nonuniform.
• The quantizer characteristic can also be of midtread or midrise type.

6. Two types of quantization:
Midtread type
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 staircaselike graph.

7. Cont’d
• With the input M having zero mean and the quantizer assumed to be symmetric as
in, it follows that the quantizer outputV and, therefore, the quantization error Q
will also have zero mean.
• Thus, for a partial statistical characterization of the quantizer in 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 m of continuous amplitude, which, symmetrically,
occupies the range [–mmax, mmax].
• The step size of the quantizer is given by:
• where L is the total number of representation levels.
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8. Cont’d
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Four-bit coding (16 discrete levels)

9. Cont’d
• Step Size
• For a uniform quantizer, the quantization error Q will have its sample values
bounded by:
• If the step size is sufficiently small (i.e., the number of representation levels L is
sufficiently large).
• It is reasonable to assume that the quantization error Q is a uniformly distributed
random variable and the interfering effect of the quantization error on the
quantizer input 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 P denote the average power of the original message signal m(t).
• output signal-to-noise ratio(SNR) of a uniform quantizer as(Book S Haykin page
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 Am, 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 quantizer Assume 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 quantizer is _________-
a) 32
b) 65
c) 80
d) 33

20. Compute Signal to Noise Ratio
Dr.Arvind Kumar, SENSE,VITVELLORE 20
• Signal energy = ; Noise energy = ;
• Noise energy =
• Signal to noise =
• Every bit increases SNR by ~ 6 decibels
12
2

N
A2

2
2
3 N
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 quantizer is 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 quantizer is of the midrise type.

23. Compute Signal to Noise Ratio
Dr.Arvind Kumar, SENSE,VITVELLORE 23
• Signal energy = ; Noise energy = ;
• Noise energy =
• Signal to noise =
• Every bit increases SNR by ~ 6 decibels
12
2

N
A2

2
2
3 N
A

2
3
log10
2
N
2
2
A

24. Next PCM
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