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Practical Sampling Rates Speech
- Telephone quality speech has a bandwidth of 4 kHz(actually 300 to 3300Hz)
- Most digital telephone systems are sampled at 8000samples/sec
Audio:
- The highest frequency the human ear can hear isapproximately 15kHz
- CD quality audio are sampled at rate of 44,000
samples/sec Video
- The human eye requires samples at a rate of at
least 20 frames/sec to achieve smooth motion
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2.6 Pulse Code Modulation (PCM)
Pulse Code Modulation refers to a digital baseband signal that is
generated directly from the quantizer output
Sometimes the term PCM is used interchangeably with quantization
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See Figure 2.16 (Page 80)
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Advantages of PCM:
Relatively inexpensive
Easily multiplexed: PCM waveforms from different
sources can be transmitted over a common digital
channel (TDM)
Easily regenerated: useful for long-distance
communication, e.g. telephone
Better noise performance than analog system
Signals may be stored and time-scaled efficiently (e.g.,
satellite communication)
Efficient codes are readily available
Disadvantage:
Requires wider bandwidth than analog signals
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2.5 Sources of Corruption in the sampled,quantized and transmitted pulses Sampling and Quantization Effects
Quantization (Granularity) Noise: Results when
quantization levels are not finely spaced apart enoughto accurately approximate input signal resulting intruncation or rounding error.
Quantizer Saturation or Overload Noise: Results when
input signal is larger in magnitude than highestquantization level resulting in clipping of the signal.
Timing Jitter: Error caused by a shift in the sampler
position. Can be isolated with stable clock reference. Channel Effects
Channel Noise
Intersymbol Interference (ISI)
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The level of quantization noise is dependent on how close any
particular sample is to one of the L levels in the converter
For a speech input, this quantization error resembles a noise-like disturbance at the output of a DAC converter
Signal to Quantization Noise Ratio
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2.7 Uniform and Nonuniform Quantization A quantizer with equal quantization level is a Uniform Quantizer Each sample is approximated within a quantile interval
Uniform quantizers are optimal when the input distribution isuniform
i.e. when all values within the range areequally likely
Most ADCs are implemented using uniform quantizers
Error of a uniform quantizer is bounded by
2 2
q qe <
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The mean-squared value (noise variance) of the quantization error is
given by:
/ 2 / 2 / 22 2 2
/ 2 / 2 / 2
1 1( )2
q q q
q q q
e p e de e de e deq q
= =
=
3 / 2
/ 2
21 3 12
q
qqeq = =
Signal to Quantization Noise Ratio
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The peak power of the analog signal (normalized to 1Ohms )can beexpressed as:
Therefore the Signal to Quatization Noise Ratio is given by:
22 2 2
2 41
pppVV L q
P
= = =
2 2
2
/ 4
/ 12
23q
L q
qS N R L==
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where L = 2n
is the number of quantization levels for the converter.(n is the number of bits).
Since L = 2n
, SNR = 22n
or in decibels
p pVq =
210log (2 ) 6
10
nS n dB
N dB
= =
Ifq is the step size, then the maximum quantization error that canoccur in the sampled output of an A/D converter is q
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Nonuniform Quantization Nonuniform quantizers have unequally spaced levels
The spacing can be chosen to optimize the Signal-to-Noise Ratiofor 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 page (Fig. 2.17)
Basic principle is to use more levels at regions with large probability
density function (pdf)
Concentrate quantization levels in areas of largest pdf
Or use fine quantization (small step size) for weak signals and
coarse quantization (large step size) for strong signals
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Statistics of speech Signal Amplitudes
Figure 2.17: Statistical distribution of single talker speech signal
magnitudes (Page 81)
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Nonuniform quantization using companding
Companding is a method of reducing the number of bits required inADC while achieving an equivalent dynamic range or SQNR
In order to improve the resolution of weak signals within a converter,
and hence enhance the SQNR, the weak signals need to be
enlarged, or the quantization step size decreased, but only for theweak signals
But strong signals can potentially be reducedwithout significantly
degrading the SQNR or alternatively increasing quantization step size
The compression process at the transmitter must be matched with anequivalent expansion process at the receiver
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The signal below shows the effect of compression, where theamplitude of one of the signals is compressed
After compression, input to the quantizer will have a more uniform
distribution after sampling
At the receiver, the signal isexpanded by an inverseoperation
The process of COMpressingand exPANDING the signal iscalled companding
Companding is a techniqueused to reduce the number of bitsrequired in ADC or DAC whileachieving comparable SQNR
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Basically, companding introduces a nonlinearity into the signal This maps a nonuniform distribution into something that more
closely resembles a uniform distribution
A standard ADC with uniform spacing between levels can be used
after the compandor (or compander)
The companding operation is inverted at the receiver
There are in fact two standard logarithm based compandingtechniques
US standard called-law companding
European standard calledA-law companding
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Input/Output Relationship of Compander
Logarithmic expression Y = log Xis the most commonly
used compander This reduces the dynamic range ofY
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Types of Companding -Law Companding Standard (North & SouthAmerica, and Japan)
where x and y represent the input and output voltages
is a constant number determined by experiment In the U.S., telephone lines uses companding with = 255
Samples 4 kHz speech waveform at 8,000 sample/sec
Encodes each sample with 8 bits, L = 256quantizer levels
Hence data rate R = 64 kbit/sec
= 0corresponds to uniform quantization
[ ]maxmax
log 1 (| | /
sgn( )log (1 )
e
e
x x
y y x
+
= +
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A-Law Companding Standard (Europe, China, Russia,
Asia, Africa)
where
x and y represent the input and output voltages A = 87.6
A is a constant number determined by experiment
maxmax
max
max
max
max
| |
| | 1sgn( ), 0
(1 )( )| |
1 log1 | |
sgn( ), 1(1 log )
e
e
xA
x xy x
x Ay xx
Ax x
y xA A x
<
+= +
< +
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2.8 Baseband Modulation Recall that analog signals can be represented by a sequence of discrete
samples (output of sampler) Samples are converted into bits. But bits are just abstract entities that
have no physical definition
We use pulses to convey a bit of information, e.g.,
In order to transmit the bits over a physical channel they must betransformed into a physical waveform
A line coder or baseband binary transmittertransforms a stream of bitsinto a physical waveform suitable for transmission over a channel
Line coders use the terminology markfor 1 and space to mean 0
In baseband systems, binary data can be transmitted using many kinds of
pulses
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There are many types of waveforms. Why? performance criteria! Each line code type have merits and demerits
The choice of waveform depends on operating characteristics of a
system such as:
Modulation-demodulation requirements
Bandwidth requirement
Synchronization requirement
Receiver complexity, etc.,
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Goals of Line Coding (qualities to look for)
A line code is designed to meet one or more of the following goals: Self-synchronization
The ability to recover timing from the signal itself
That is, self-clocking (self-synchronization) - ease of clock lock
or signal recovery for symbol synchronization
Long series of ones and zeros could cause a problem
Low probability of bit error
Receiver needs to be able to distinguish the waveform associatedwith a markfrom the waveform associated with a space
BER performance
relative immunity to noise
Error detection capability
enhances low probability of error
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Spectrum Suitable for the channel
Spectrum matching of the channel
e.g. presence or absence of DC level
In some cases DC components should be avoided
The transmission bandwidth should be minimized
Power Spectral Density
Particularly its value at zero
PSD of code should be negligible at the frequency near zero
Transmission Bandwidth Should be as small as possible
Transparency
The property that any arbitrary symbol or bit pattern can be
transmitted and received, i.e., all possible data sequence should
be faithfully reproducible
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Line Coder The input to the line encoder is
the output of the A/D converter
or a sequence of values an that
is a function of the data bit
The output of the line encoderis a waveform:
where f(t) is the pulse shape and Tb is the bit period (Tb=Ts/n fornbit quantizer)
This means that each line code is described by a symbol mappingfunction an and pulse shape f(t)
Details of this operation are set by the type of line code that isbeing used
( ) ( )n bn
s t a f t nT
=
=
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Commonly Used Line Codes
Polar line codes use the antipodal mapping
Polar NRZ uses NRZ pulse shape Polar RZ uses RZ pulse shape
, 1
, 0
n
n
n
A w h e n Xa
A w h e n X
+ ==
=
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Unipolar NRZ Line Code
Unipolar non-return-to-zero (NRZ) line code is defined byunipolar mapping
In addition, the pulse shape for unipolar NRZ is:
where Tb is the bit period
, 1
0, 0
n
n
n
A when Xa
when X
+ ==
=Where Xn is the n
th data bit
( ) , NRZ Pulse Shapeb
tf t
T
=
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Bipolar Line Codes
With bipolar line codes a space is mapped to zero anda mark is alternately mapped to -A and +A
It is also called pseudoternary signaling oralternate mark inversion
(AMI)
Either RZ or NRZ pulse shape can be used
, when 1 and last mark
, when 1 and last mark
0, when 0
n
n n
n
X A
a A X A
X
+ =
= = + =
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Manchester Line Codes
Manchester line codes use the antipodal mappingand the following split-phasepulse shape:
4 4( )
2 2
b b
b b
T Tt t
f t T T
+ =
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Summary of Line Codes
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Comparison of Line Codes Self-synchronization
Manchester codes have built in timing information because they
always have a zero crossing in the center of the pulse
Polar RZ codes tend to be good because the signal level alwaysgoes to zero for the second half of the pulse
NRZ signals are not good for self-synchronization
Error probability
Polar codes perform better (are more energy efficient) than
Unipolar or Bipolar codes
Channel characteristics
We need to find the power spectral density (PSD) of the linecodes to compare the line codes in terms of the channel
characteristics
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Comparisons of Line Codes Different pulse shapes are used
to control the spectrum of the transmitted signal (no DC value,
bandwidth, etc.)
guarantee transitions every symbol interval to assist in symbol timing
recovery1. Power Spectral Density of Line Codes (see Fig. 2.23, Page 90)
After line coding, the pulses may be filtered or shaped to further
improve there properties such as
Spectral efficiency
Immunity to Intersymbol Interference
Distinction between Line Coding and Pulse Shaping is not easy
2. DC Component and Bandwidth DC Components
Unipolar NRZ, polar NRZ, and unipolar RZ all have DC components
Bipolar RZ and Manchester NRZ do not have DC components
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First Null Bandwidth
Unipolar NRZ, polar NRZ, and bipolar all have 1st null bandwidths
ofRb = 1/Tb Unipolar RZ has 1st null BW of2Rb
Manchester NRZ also has 1st null BW of2Rb, although thespectrum becomes very low at 1.6Rb
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Generation of Line Codes
The FIR filter realizes the different pulse shapes
Baseband modulation with arbitrary pulse shapes can bedetected by
correlation detector
matched filter detector (this is the most common detector)
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Section 2.8.4: Bits per PCM Word and Bits per Symbol L=2l
Section 2.8.5: M-ary Pulse Modulation Waveforms
M = 2k
Problem 2.14: The information in an analog waveform, whosemaximum frequency fm=4000Hz, is to be transmitted using a 16-level
PAM system. The quantization must not exceed 1% of the peak-to-peak analog signal.
(a) What is the minimum number of bits per sample or bits per PCMword that should be used in this system?
(b) What is the minimum required sampling rate, and what is theresulting bit rate?
(c) What is the 16-ary PAM symbol Transmission rate?
Bits per PCM word and M-ary Modulation
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max
2 2
2
2
| | | 0.01 | | |2
122
1
log log (50) 62
8000 48000 16
4800012000 / sec
log ( ) 4
pp
pp lpp
qe pV p e
VV Lq q L
L p
l lp
fs Rs M
RR symbols
M
= =
= = =
=
= = =
= = =
Solution to Problem 2.14
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