pulse code modulation & source coding sampling theory 1
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PULSE CODE MODULATION & SOURCE CODING
Sampling Theory
1
Sampling Theory
Signal Reconstruction
Aliasing
LEARNING OBJECTS
2
Basic elements of a PCM systemBasic elements of a PCM system
3
Sampling Theory
In many applications, e.g. PCM, it is useful to represent a signal in terms of sample values taken at appropriately spaced intervals.
The signal can be reconstructed from the sampled waveform by passing it through an ideal low pass filter.
In order to ensure a faithful reconstruction, the original signal must be sampled at an appropriate rate as described in the Nyquist’s sampling theorem. ◦A real-valued band-limited signal having no spectral
components above a frequency of B Hz is determined uniquely by its values at uniform intervals spaced no greater than seconds apart.
4
Sampling Theory
)]([*)(2
1)( tFGG
sTs
5
Impulse Sampling
6
Impulse Sampling
7
Sampling Visualized in Frequency Domain
8
Interpolation
From the spectrum of the sampled signal, we can see that the original signal can be recovered by passing its samples through a LPF
9
Ideal Interpolation
10
Ideal Interpolation
11
Ideal Interpolation
12
Practical Considerations in Nyquist Sampling
13
Gradual Roll-Off Low Pass Filter
14
Gradual Roll-Off Low Pass Filter
15
Aliasing
Resultantly, they will be not band limited.
16
Aliasing
17
A Solution: The Antialiasing Filter
The anti-aliasing, being an ideal filter, is unrealizable. In practice we use a steep cutoff which leaves a sharply attenuated residual spectrum beyond the folding frequencies.
18
Practical Sampling
19
Practical Sampling
20
Some Applications of Sampling TheoremSampling theorem is very important in signal
analysis, processing and transmission because it allows to replace a continuous time signal by a discrete sequence of numbers. This leads into the area of digital filtering.
In communication, the transmission of continuous-time message reduces to the transmission of a sequence of numbers. This opens the doors to many new techniques of communicating continuous-time signals by pulse trains.
The continuous-time signal g(t) is sampled, and sampled values are used to modify certain parameters of a periodic pulse train.
21
The sampled value can be used to vary amplitude, width or position of the pulse in proportion to the sample values of the signal g(t). Accordingly we get
Samplingg(t) Pulse
Modulation
Value of the sample
Some Applications of Sampling Theorem
[22]
Pulse Modulated Signals
23
Some Applications of Sampling Theorem• Pulse modulation permits simultaneous transmission of several
signals on a time-sharing basis: Time Division Multiplexing. Because a pulse modulated signal occupies only a part of the channel time, therefore several pulse modulated signals can be transmitted on the same channel by interweaving.
• Similarly several baseband signals can be transmitted simultaneously by frequency division multiplexing where spectrum of each message is shifted to a specific band not occupied by any other signal.
24
Time Division Multiplexing
25
Pulse Code ModulationMost useful and widely used of all the pulse modulations.PCM is a method of converting an analog signal into a digital signal
(A/D conversion). An analog signal’s amplitude can take on any value over a continuous
range while digital signal amplitude can take on only a finite number of values.
An analog signal can be converted into a digital signal by means of three steps:◦ sampling ◦quantizing, that is, rounding off its value to one of the
closest permissible numbers (or quantized levels) ◦Binary coding, that is conversion of quantized samples to
0s and 1s.
26
Amplitude Quantization
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27
• (a) Mid-tread • (b) Mid-rise
Scalar Quantizer
28
Quantization
29
Quantization
30
Quantization Error
31
Quantization Error
32
Quantization Noise
33
Quantization Noise
34
Quantization Noise
12
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35
Quantization SNR
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)23
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23
1
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P
PSNR
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L
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R
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, 6dB per bit36
Non-uniform Quantization• Motivation
– Speech signals have the characteristic that small-amplitude samples occur more frequently than large-amplitude ones
– Human auditory system exhibits a logarithmic sensitivity
• More sensitive at small-amplitude range (e.g., 0 might sound different from 0.1)
• Less sensitive at large-amplitude range (e.g., 0.7 might not sound different much from 0.8)
histogram of typical speech signals
[37]
Non-uniform Quantization
38
Non-uniform QuantizationNon-uniform Quantization
39
Non-uniform Quantization
Non-uniform Quantization = Compression + Uniform quantization
40
Non-uniform Quantization
41
Law / A LawThe μ-law algorithm (μ-law) is a companding algorithm, primarily
used in the digital telecommunication systems of North America and Japan.
Its purpose is to reduce the dynamic range of an audio signal. In the analog domain, this can increase the SNR achieved during
transmission, and in the digital domain, it can reduce the quantization error (hence increasing signal to quantization noise ratio).
A-law algorithm used in the rest of worlds. A-law algorithm provides a slightly larger dynamic range than the
μ-law at the cost of worse proportional distortion for small signals. By convention, A-law is used for an international connection if at
least one country uses it.
42
Law Compression
43
A-Law Compression
44
Binary Coding
From practical viewpoint, a binary digital signal (a signal that can take on only two values) is very desirable because of its simplicity, economy, and ease of engineering. We can convert an L-ary signal into a binary signal by using pulse coding.
This code, formed by binary representation of the 16 decimal digits from 0 to 15, is known as the natural binary code (NBC).
Each of the 16 levels to be transmitted is assigned one binary code of four digits. The analog signal m(t) is now converted to a (binary) digital signal. A binary digit is called a bit for convenience.
45
Binary CodingNow each sample is encoded by four bits. To transmit this binary data, we need to
assign a distinct pulse shape to each of the two bits.
One possible way is to assign a negative pulse to a binary 0 and a positive pulse to a binary 1 so that each sample is now transmitted by a group of four binary pulses (pulse code). The resulting signal is a binary signal.
46
Sigma-Delta ADC
47
Pulse Code Modulation Examples
48
Transmission Bandwidth and SNRFor a binary PCM, we assign a distinct group of n binary digits (bits) to each of the L quantization levels. Because a sequence of n binary digits can be arranged in distinct 2n patterns,
L=2n or n=log2LEach quantized sample is, thus, encoded into n bits. Because a signal m (t) band-limited to B Hz requires a minimum of 2B samples per second, we require a total of 2nB bits per second (bps), that is, 2nB pieces of information per second. Because a unit bandwidth (1 Hz) can transmit a maximum of two pieces of information per second, we require a minimum channel of bandwidth Hz, given by
BT=nB HzThis is the theoretical minimum transmission bandwidth required to transmit the PCM signal.
49
Transmission Bandwidth and SNR• We know that L2 = 22n, and the output SNR can be expressed
as
where
Lathi book
50
Transmission Bandwidth and SNRWe observe that the SNR increases exponentially with the transmission bandwidth BT. This trade of SNR with bandwidth is attractive. A small increase in bandwidth yields a large benefit in terms of SNR. This relationship is clearly seen by rewriting using the decibel scale as
51
Transmission Bandwidth and SNRThis shows that increasing n by 1 (increasing one
bit in the code word) quadruples the output SNR (6-dB increase).
Thus, if we increase n from 8 to 9, the SNR quadruples, but the transmission bandwidth increases only from 32 to 36 kHz (an increase of only 12.5%).
This shows that in PCM, SNR can be controlled by transmission bandwidth.
Frequency and phase modulation also do this. But it requires a doubling of the bandwidth to quadruple the SNR. In this respect, PCM is strikingly superior to FM or PM.
52
Differential PCM
53
Differential Pulse Code Modulation (DPCM)
If [ ] is the th sample, instead of transmitting [ ],
difference [ ] [ ] [ 1] is transmitted.
At the receiver, knowing of the difference [ ] and the
previous sample value [ 1], we can construc
m k k m k
d k m k m k
d k
m k
t [ ].
Difference between successive samples is generaly much
smaller than the sample values.
m k
54
Differential Pulse Code Modulation (DPCM)
2
Therefore the peak amplitude of the transmitted
value reduces considerably. Hence quantization interval
for a given (or ) by .12
p
p
m
vmv L nL
For a given (transmission bandwidth), we can
increase the SNR, or for a given SNR, we can reduce
(transmission bandwidth).
n
n
55
DPCMThis scheme by estimating
(predicting) the value of th sample [ ] from the
knowledge of the previous sample val
can
u
further e
.
b
es
improved
k m k
At the receiver also we determine the estimate m[k],
from the previous sample values and generate [ ],
by adding the received [ ] to the estimate [ ].
m k
d k m k
][ˆ km][ˆ][][ kmkmkd
If the estimate is , then the difference is transmitted.
][ˆ km
56
][ˆ km
DPCM
Since difference between the predicted value and
the actual value will be even smaller than
the difference between the actual values, this scheme
is kn Differential Pulse Code Modulationown as (DPCM).
57
How Does the Predictor Works ?
2 3.. ...
Suppose for a signal, we can express the as
(
Taylor
) ( ) ( ) ( ) ( ) .............2! 3!
( ) ( ) for
Serie
s all
s
m
S SS s
s S
T Tm t T m t T m t m t m t
m t T m t T
.
If we know the ( ), we can predict the future signal
value from knowledge of signal and its derivative.
Let us denote the th sample of ( ) by [ ], that is
[ ] [ ], and ( ) [ 1] and so oS S S
m t
k m t m k
m kT m k m kT T m k
S
n
setting t=kT 58
The Predictor.
[ ] [ ( ) ( )]/ , then we obtain
[ 1] ( ) [ ( ) [ 1) / ]
2 [ ] [ 1]
S S S S S
S S
m kT m kT m kT T T
m k m k T m k m k T
m k m k
Crude prediction of [ 1] can be made by obtaining
two previous samples. This approximation can be
further imrpoved as we add more stages in the series.
m k
.
59
The Linear Predictor
1 2 3
1 2 3
In general, we can express the prediction formula
[ ] [ 1] [ 2] [ 3] ... [ ]
and the predicted value of [ ] is
[ ] [ 1] [ 2] [ 3] .... [ ]
Therefore larger N would resu
N
N
m k a m k a m k a m k a m k N
m k
m k a m k a m k a m k a m k N
lt in better prediction value.
][ˆ km
A tapped delay-line (transversal) filter used as a linear predictor
with tap gains equal to prediction coefficients 60
Linear Prediction Coding (LPC)Consider a finite-duration impulse response (FIR) discrete-time filter which consists of three blocks :
1. Set of p ( p: prediction order) unit-delay elements (z-1) 2. Set of multipliers with coefficients w1,w2,…wp
3. Set of adders ( )
61
DPCM
The DPCM transmitter
62
SNR Improvement
2
Peak value of ( ) be and ( ) be ( ),
For same value of , quantization step v in DPCM
is reduced by the factor .
Becuase the quantization noise power is ( ) /12,
the quatization noi
p p
p
p
m t m d t d difference
L
d
m
v
2
se in DPCM reduces by the factor
( ) , and the SNR increases by the same factor.p
p
m
d
63
By exploiting redundancies from the speech signal, prediction can be improvedPredictor coefficients are derived from the sampled signal and transmitted along with the signalPrediction can be so good that after some time only the predictor coefficients are sent.We get transmission at 8-16 kbps with the same quality of PCM
Coded Excited Linear Prediction (CELP)Coded Excited Linear Prediction (CELP)
64
Delta Modulation
65
Delta Modulation (DM)
Sample correlation used in DPCM is further
exploited in delta modulation (DM) by
over sampling (typically 4 times the Nyquist rate)
the baseband signal.
This increases the correlation between adjacent
samples, which results in a small prediction error
that can be encoded using only one bit (L = 2).
DM is basically a 1-bit DPCM, that is, a DPCM
that uses only two levels (L = 2) for quantization
of the [ ] [ ]. qm k m k
In DM, we use a first-order predictor which is just a delay.
66
Delta Modulation (DM)In comparison to PCM (and DPCM), it is a very
simple and inexpensive method of A/D conversion.
A 1-bit code word in DM makes word framing
unnecessary at the transmitter and the receiver.
This strategy allows us to use fewer bits per sample
for encoding a baseband signal.
67
Delta Modulation (DM)
68
DM System: Transmitter and Receiver
69
DM System: Transmitter and Receiver
k
mqq
qqq
qqq
mdkm
kdkmkm
kdkmkm
0
121
Hence
1
( Integrator)
( differentiator )
70
Delta Modulation (DM)
71
Slope Overload Distortionand Granular Noise
72
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size step when occurs noisegranular hand,other On the
)(
max
require we, distortion overload-slope avoid To
tm
dt
tdm
Ts
Slope Overload Distortion and Granular Noise
73
Adaptive Delta Modulation
Slope overload and granular noise reduce the dynamic range of DM
Adaptive DM adjusts the step size according to frequency
Output SNR is proportional to◦(For single integration case) (BT/B)^3◦(For double integration case) (BT/B)^5
Comparison with PCM: at low BT/B, DM is superior; at high BT/B, the advantage is reversed
74
Comparison with PCM
Single Integration
Double Integration
Performance Comparison: Performance Comparison: PCM Vs DPCM/DMPCM Vs DPCM/DM
75
Line Coding
76
Digital Data Transmission
Source
Input to a digital system is in the form of sequence of digits. It could be from a data set, computer, digitized voice signal (PCM or DM), digital camera, fax machine, television, telemetry equipment etc.
77
Line Coding and Decoding
78
Data Rate Vs. Signal Rate• Data rate: the number of data elements (bits) sent in 1sec (bps). It’s
also called the bit rate or transmission rate.• Signal rate: the number of signal elements sent in 1sec. It’s also
called the pulse rate, the modulation rate, symbol rate or the baud rate.
• Transmission bandwidth is related to baud rate.• We wish to:
– increase the data rate (increase the speed of transmission)
– decrease the signal rate (decrease the bandwidth requirement)
79
Line Codes
• Output of the transmitter is coded into electrical pulses or waveforms for the purposes of transmission over the channel or to modulate a carrier.
• This process is called line coding or transmission coding.
• There are many possible ways to assign a waveform (pulse) to a digital data based of various desirables.
80
Line coding schemes
81
On-Off Return to Zero (RZ)
1 1111 10 0 0
t
1 is encoded with p(t) and 0 is encoded with no pulse. Pulse returns to zero level after every 1.
82
Polar Return to Zero (RZ)1 is encoded with p(t) and 0 is encoded with –p(t). Pulses returns to zero level after every 1 and 0.
1 1111 1
0 0 0
t
83
Bipolar Return to Zero (RZ)1 is encoded with p(t) or –p(t) depending on whether previous 1 is encoded p(t) or –p(t)
while 0 is encoded with no pulse. Pulses returns to zero level after every 1 and 0.
Also known as Pseudoternary or Alternate Mark Inversion (AMI)
1 1
1
1
1 10 0 0
t
84
On-Off Non Return to Zero (NRZ)1 is encoded with p(t)
while 0 is encoded with no pulse. Pulses do not return to zero level after every 1 and 0.
1 1111 10 0 0
t
85
Polar Non Return to Zero (NRZ)1 is encoded with p(t)
while 0 is encoded with –p(t). Pulses do not returns to zero level after every 1 and 0.
1 1111 10 0 0
t
86
Desirable Properties of Line Codes• Transmission bandwidth
• Power efficiency
• Error detection and correction capability
• Favorable power spectral density
• Adequate timing content
• Transparency
87
Desirable Properties of Line Codes
Transmission bandwidthIt should be as small as possible.
Power efficiencyFor a given bandwidth and specified detection error probability, transmitted power should be as small as possible.
Error detection and correction capabilityIt should be possible to detect and if possible to correct detected errors.
88
Desirable Properties of Line Codes
Favorable power spectral densityIt is desirable to have zero PSD at =0 (dc) as ac coupling and transformers are used at the repeaters.
Adequate timing contentIt should be possible to extract timing or clock information from the signal.
TransparencyIt should be possible to transmit a digital signal correctly regardless of the pattern of 1’s and 0’s.
89
PSD of Various Line Codes:Assumptions
• Pulses are spaced Tb seconds apart. Consequently, the transmission rate is Rb=1/ Tb pulses per second.
• The basic pulse used is denoted by p(t) and its Fourier transform is P().
• The PSD of the line code depends upon that of the pulse shape p(t). We assume p(t) to be a rectangular pulse of width Tb/2 i.e.
90
PSD of Polar Signaling
91
Polar Signaling
• Essential bandwidth of the signal is 2Rb Hz. – This is four times the theoretical BW (Nyquist)
• Polar signaling has no error detection capability.
• It has non-zero PSD at =0.• Polar signaling is the most power-efficient
scheme.• Transparent
92
93
PSD On-Off Signaling
94
On-Off Signaling
• For a given transmitted power, it is less immune to noise interference than polar scheme.
• Made up of a polar signal plus periodic signal; hence, BW is similar to polar signaling (Fig 7.2. Page 296, Lathi).
• Contains a discrete component of clock frequency (Eq 7.19, Lathi).• PSD of On-Off signaling is ¼ of that of polar signaling (Eq 7.19, Lathi).• Non-transparent.• All the disadvantages of polar schemes such as:
– Excessive transmission bandwidth– Non-zero power spectrum at =0– No error detection capability.
95
PSD of Bipolar Signaling
96
Advantages of Bipolar (Pseudoternary or AMI) Signaling
• Spectrum has DC null.
• Bandwidth is not excessive
• Has single error detection capability (If error
then violation of AMI rule).
• If rectified, an off-on signal is formed that has
a discrete component at clock frequency.97
Disadvantages of Bipolar (Pseudoternary or AMI) Signaling
• Required twice (3db) as much power as polar signal.
• Not transparent (long strings of zeros problematic)– Various substitution scheme are used to prevent
long strings of zeros
98
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