Agenda
Review of the last lecture
Quantization
PCM & Line Coding
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Quantization, PCM and Line Coding-Digital Communications-Lecture Slides, Slides of Digital Communication Systems
This lecture was delivered by Mr. Sujay Rangarajan at Birla Institute of Technology and Science. Its part of lecture series on Digital Communications course. It includes: Quantization, PCM, Line, Coding, Sampling, Flat-top, Gating, Impulse, Spectral, overlapping, Aliasing, Uniform, ADC, Mean-squared, Value
Typology: Slides
2011/2012
1 / 30
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Agenda^ ^ Review of the last lecture^ ^ Quantization^ ^ PCM & Line Coding
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Summary Of Sampling ^ Ideal Sampling(or Impulse Sampling) ^ Natural Sampling(or Gating) ^ Flat-Top Sampling ^ For all sampling techniques^
^ If^ fs > 2B then we can recover x(t) exactly^ ^ If^ fs < 2B
)^ spectral overlapping
known as
a liasing will
occur
( )^ ( )
( )^
( )^ (
) ( )^ (^
)
s^
s n^ s^
s n x^ t^ x t x
t^ x t
t^
nT x nT^
t^ nT δ
∞ δ =−∞∞ δ =−∞ =^
=^
− =^
−
2
( )^ ( )
( )^
( )^
j^ nf ts
s^
p^
n n
x^ t^
x t x^ t
x t^
∞^ π c e =−∞ =^
=^ ∑
( )^ '( ) *
( )^
( )^ (
s^
s n
x^ t^ x
t^ p t
x t^
t^ nT^
p t ∞^ δ =−∞ ⎡^
=^
=^
⎢^
⎣^
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^ The mean-squared value (noise variance) of the quantizationerror is given by:
/ 2^
/ 2^
/ 2
2
2
2
/ 2^
/ 2^
/ 2 1
1
( ) q^ 2
q^
q
q^
q^
q
e p e de
e de
e de q^
q
σ^
−^
−^
− ⎛^ ⎞ =^
=
∫^
∫^
∫ ⎜^ ⎟ ⎝^ ⎠
=
/ 2 3 / 2
2 1 3
12 q q
q e = q −
=
Signal to Quantization Noise Ratio
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^ The peak power of the analog signal (normalized to 1Ohms)can be expressed as: ^ Therefore the Signal to Quatization Noise Ratio is given by:
(^22)
2 2 2
V^ ppp V^
L q P^
⎛^ ⎞^
⎛^ ⎞
⎜^ ⎟^
⎜^ ⎟⎜^ ⎟
⎜^ ⎟^
⎝^ ⎠
⎝^ ⎠
=^ =^
= 2 2 / 4 2 / 1 2
(^23) L qS N R qq
L =
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^ The level of quantization noise is dependent on howclose any particular sample is to one of the
L^ levels in
Signal to Quantization Noise Ratio the converter^ ^ For a speech input, this quantization error resembles a noise-like disturbance at the output of a DAC converter
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Sources of Corruption in the sampled,
quantized signal
^ Sampling and Quantization Effects^ ^ Quantization (Granularity) Noise: Results whenquantization levels are not finely spaced apartenough to accurately approximate input signalresulting in truncation or rounding error.^ ^ Quantizer Saturation or Overload Noise: Resultswhen input signal is larger in magnitude thanhighest quantization level resulting in clipping ofthe signal.^ ^ Timing Jitter: Error caused by a shift in thesampler position. Can be isolated with stableclock reference.
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Center for Advanced Studies inEngineering
11
Nonuniform Quantization ^ Nonuniform quantizers
have unequally spaced levels
^ The spacing can be chosen to optimize the Signal-to-Noise Ratio for a particular type of signal It is characterized by: ^ Variable step size ^ Quantizer size depend on signal size
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^ Many signals such as speech have a nonuniform distribution^ ^ See Figure on next slide ^ Basic principle
is to use more levels at regions with large probability density function (pdf)^ ^ use fine quantization (small step size) for weak signals andcoarse quantization (large step size) for strong signals
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Center for Advanced Studies inEngineering
14
Companding
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^ A=87.6 is used as a standard value
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Center for Advanced Studies inEngineering
18
law vs A-Law characteristics μ
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Center for Advanced Studies inEngineering
20
Input/Output Relationship of a
compander
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Transmission BW and the output SNR^ ^ In PCM we assign distinct group of
“n”^ binary
digits to each of the
“L”^ quantization levels where, ^ Each quantized sample is thus encoded into
“n”^ bits
^ The signal
m(t)^ having Bandwidth
“B”^ Hz requires
minimum of
“2B”^ samples per second ^ Hence, we require a total of
2nB bps
^ If^ 1 Hz^
can transmit a max of
2 bits^ of information per
second, then we require a minimum channel of
BW=nB Hz
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