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Chapter 10Applications in Communications
School of Information Science and Engineering, SDU.
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Introduction
Some methods for digitizing analog waveforms:
Pulse-code modulation (PCM)Differential PCM (DPCM) Adaptive differential PCM (ADPCM)Delta modulation (DM)Adaptive delta modulation (ADM)Linear predictive coding (LDC)
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Pulse-Code Modulation (PCM)
PCM is used for quantizing an analog signal to transmit storing the signal in digital formSpeech transmission Telemetry systems
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μ- law compressor
A logarithmic compressor employed in U.S. and Canadian telecommunications systems
s : the mormalized input;y : the normalized output;
sgn(·) : the sign funciton;μ : a parameter that is selected to give
the desired compression characteristic.
( )( ) ( )
ln 1 sy sgn s ; s 1, y 1
ln 1+µ
= ≤ ≤+µ
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A – law compressor
The logarithmic compressor standard used in European telecommunication systems:
where A is chosen as 87.56.
( ) ( )
( )
1 ln A s 1sgn s , s 11 ln A AyA s 1sgn s , 0 s
1 ln A A
⎧ +≤ ≤⎪⎪ += ⎨
⎪ ≤ ≤⎪⎩ +
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Figure 10.1 Comparison of μ-law and A-law nonlinearities
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Project 10.1: PCM
Purpose of this project:To gain an understanding of PCM comprssion (linear-to-logarithmic) PCM expansion (logaithmic-to-linear).
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Three Matlab functions are needed:
A μ-law compressor functionA quantizer function
A μ-law expander funciton
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Figure 10.2 PCM project
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The signal-to-quantization noise ratio (SQNR) in dB is:
( )
( ) ( )( )
N2
n 110 N 2
qn 1
s nSQNR 10log
s n s n
=
=
⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟−⎜ ⎟⎝ ⎠
∑
∑
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Differential PCM (DPCM)
s(n): the current sample of speech: the predicted value of s(n)
a(i): the predictor coefficients
( ) ( ) ( )p
i 1s n a i s n i
=
= −∑
( )s n
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The error function is the sum of squared errors, so we select the a(i) to minimize:
where rss(m) is the autocorrlation function of s(n)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
2pN N2
pn 1 n 1 n 1
p p p
ss ss ssn 1 i 1 j 1
e n s n a i s n i
r 0 2 a i r i a i a j r i j
= = =
= = =
⎡ ⎤ε = = − −⎢ ⎥
⎣ ⎦
= − + −
∑ ∑ ∑
∑ ∑∑
( ) ( ) ( )N
ssi 1
r m s i s i m=
= +∑
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Figure 10.3 Block diagram of a DPCM transcoder
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The output of the predictor is
The difference
is the input to the quantizer.
( ) ( )p
i 1s a i s n i
=
= −∑
( ) ( ) ( )e n s n s n= −
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The estimate value of s(n) is obtained by taking a linear combination of past values , k=1,2,…,p.The estimate of s(n) is
( )s n
( )s n k−
( ) ( ) ( ) ( ) ( )p p
i 1 i 1
s n a i s n i b i e n i= =
= − + −∑ ∑
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Project 10.2: DPCM
Generate correlated random sequences using a pole-zero signal model of the form:
where x(n) is a zero-mean unit variance Gaussian sequence.
( ) ( ) ( ) ( ) ( )0 1s n a 1 s n 1 b x n b x n 1= − + + −
filter function
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Some modules for this project:A model predictor functionA DPCM encoder function A DPCM decoder function
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Figure 10.4 DPCM modified by the linearly filtered error sequence
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Adaptive PCM (ADPCM) and DPCM
Adaptive quantizer:feedforward adaptive quantizer
Adjust its step size for each signal sample
feedback adaptive quantizerEmploy the output of the quantizer in the adjustment of the step size.
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Figure 10.5 Example of a quantizer with an adaptive step size
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Table 10.1 Multiplication factors for adaptive step size adjustment
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Figure 10.6 ADPCM block diagram (Encoder part)
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Figure 10.6 ADPCM block diagram (Decoder part)
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Project 10.3: ADPCM
Figure 10.7 ADPCM interface to PCM system
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Delta Modulation (DM)
DM may be viewed as asimplified form of DPCM in which a two-level (1-bit) quantizer is used in conjunction with a fixed first-order predictor.
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We note that
Since
It follows that
( ) ( ) ( ) ( )s n s n 1 s n 1 e n 1= − = − + −
( ) ( ) ( ) ( ) ( ) ( )q n e n e n e n s n s n⎡ ⎤= − = − −⎣ ⎦
( ) ( ) ( )s n s n 1 q n 1= − + −
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Figure 10.8 Block diagram of a delta modulation system
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Figure 10.9 An equivalent realization of a delta modulation system
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Adaptive Delta Modulation (ADM)
Figure 10.10 Two types of distortion in the DM encoder
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Figure 10.11 An example of a delta modulation system with adaptive step size
Encoder part
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Decoder part
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Project 10.4: DM and ADM
A Hanning filter that has the impulse response
may be used, where the length N may be selected in the range .
( ) 1 2 nh n 1 cos , 0 n N 12 N 1⎡ π ⎤⎛ ⎞= − ≤ ≤ −⎜ ⎟⎢ ⎥−⎝ ⎠⎣ ⎦
5 N 15≤ ≤
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Linear Predictive Coding (LPC) of Speech
The LPC method is based on modeling the vocal tract as a linear all-pole filter.The system function:
p : the number of poles;G : the filter gain;
{ap(k)}: parameters that determine the poles.
( )( )
pk
pk 1
GH z1 a k z−
=
=+∑
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Figure 10.12 Block diagram model for the generation of a speech signal
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Figure 10.13 Encoder and decoder for LPC
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Project 10.5: LPC
The encoder divides speech signals into short-time segments, and process each segment, separately.The decoder that performs the synthesis is an all-pole lattice filter.The output is a synthetic speech signal.
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Dual-Tone Multi-frequency (DTMF) Signals
DTMF is the generic name for push-button telephone signaling.DTMF also finds widespread use in electronic mail systems and telephone banking systems.A combination of a high-frequency tone and low-frequency tone represent a specific digit or the characters * and #.
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Figure 10.14 DTMF digits
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The Goertzel Algorithm
The Goertzel algorithm exploits the periodicity of the phase factors and allows us to express the computation of the DFT as a linear filtering operation.Since , we can multiply the DFT by this factor. Thus
kNW
kNNW 1− =
( ) ( ) ( ) ( )N 1
k N mkNN N
m 0X k W X k x m W
−− −−
=
= = ∑
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Figure 10.15 Realization of two-pole resonator for computing the DFT
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Project 10.6: DTMF Signaling
Design the following Matlab modules:A tone generation functionA dial-tone generator A decoding funciton
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Binary Digital Communications
A binary digital communications system employs two signal waveforms:
s1(t)=s(t)s2(t)=-s(t)
To measure the performance, we normally use the average probability of error, which is often called the bit error rate (BER).
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Project 10.7: Binary Data Communications System
Five Matlab functions are required:A binary data generator moduleA modulator module A noise generator A demodulator moduleA detector and error-counting module
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Figure 10.16 Model of binary data communications system
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Spread-Spectrum Communications
Figure 10.17 Basic spread spectrum digital communications system
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Project 10.8: Binary Spread-Spectrum Communications
Figure 10.18 Block diagram of binary PN spread-spectrum system for simulation experiment
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That’s all!