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3 Multiplexing of PCM Signals3 Multiplexing of PCM Signals

In TDM, a given time interval Ts is selected as a frame. - Each frame is subdivided into N subintervals of duration

Ts/N where N corresponds to the number of users that will use the common communication channel.

- Each user is assigned a subinterval within each frame.

The method for simultaneous transmission of several signalover a common communication channel is called multiplexing. If the signals from different users are multiplexed in time, then it is called time-division multiplexing (TDM).

::

voice 1LPF

voice 24

LPF

:Q(.)

sf

encoder

PCM

quantizing

256 levels (8 bits) 82=L

sfTkHzf sss μ 1258000/1/18 ===⇒=

Example. 24 speech channels, each LPF filtered with cutoff at 4 kHz and sampled at 8000 samples/sec.

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1)-(DS Mbps544.1)648.0/(1/1 === sTR bb μ

sTT sb μ648.0193/125193/ ===

1931)17(24 =++×

bT

sμ 125 frame dmultiplexe =

signalinginformationbits

marker pulsefor framing

The total number of pulses sent over Ts is

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User 1 User 2 User 3 User 24

1b 1b 1b1b7 bits 7 bits … 7 bits

mark pulse for frame

Informationbits

Signaling bit

64 kbpseach

…

12

24

1.544 MbpsMulti-plexer

1st levelmulti-plexerDS-1

64 kbpseach

12

24

1.544 Mbpseach

2nd levelmulti-plexerDS-2

3rd levelmulti-plexerDS-3

4th levelmulti-plexerDS-4

5th levelmulti-plexerDS-5

234

…

1

6.312 Mbpseach

1

2 2

27 6

1 1

… …

44.736 Mbpseach

674.176 Mbpseach

Output 750.16 Mbps

Signals from other DS-1

units

Signals from other DS-2

units

Signals from other DS-3

units

Signals from other DS-4

units

Figure 3.1. Digital TDM hierarchy for North American telephone communication system

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Objective: To use finite discrete amplitude levels to represent infinite analog amplitude levels and to minimize the distortion introduced.

4 Uniform Quantization and Signal-to-Noise Ratio4 Uniform Quantization and Signal-to-Noise Ratio

Figure 4.1. Quantization Mapping g(x)

Δ

signal amplituderange

representation levels

Decision thresholds

1y 2y 3y 4y 5y 6y 7y 8y

1x 2x 3x 4x 5x 6x 7x 8x0x

1ℜ 2ℜ 3ℜ 4ℜ 5ℜ 6ℜ 7ℜ 8ℜ

Δ

Step size

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which is the quantized version of x in . yk is then represented by a binary sequence and transmitted. The quantization function g(x) is defined as

{ }kkkkk xxxxxyxg ≤<=ℜℜ∈= −1|, allfor ,)(

Remark. Q(x) is non-invertible, thus the information lost in the process of quantization is unrecoverable. In the words, the price that we have paid for a decrease in rate from infinity to logN (the number of bits required to transmit each source output) is the introduction of distortion.

ℜThe set of real numbers is partitioned into N disjoint subsets denoted by

where xk is called a decision threshold and Δ, the step size. Corresponding to each subset , we choose a representation level yk (usually yk ∈) such that

kℜ

{ } Nkxxxxxx kkkkk ≤≤Δ=−≤<=ℜ −− 1for , ,| 11

kℜ

Nkyy kk ≤≤Δ=− − 1for , 1

kℜ

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Figure 4.2 Quantizer of midriser type

L,2 ,1 ,0 , ±±=Δ= kkxk

Δ−

=+

= −

212

21 kxxy kk

k

],(,)( 1 kkk xxxyxg −∈=

Peak-to-peakexcursion

Output

inputΔ4Δ3Δ2Δ

2/7Δ

2/5Δ

2/3Δ

2/Δ

2/7Δ−

2/5Δ−

2/3Δ−

2/Δ−

Δ− 3 Δ− 2 Δ−Δ− 4

Overload level = (peak-to-peak

range of sample)/2

Quantization error

input

2/Δ

2/Δ−,71for ≤≤ k

{ } ,| 11 xxx ≤<−∞=ℜ

Figure 4.2 gives an example of an 8-level quantization scheme. In this scheme, the eight regions are defined as

{ }kkk xxxx ≤<=ℜ −1|

),( 78 ∞=ℜ x

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Figure 4.3 Quantizer of midtread type

L,2 ,1 ,0 ,2

12±±=Δ

+= kkxk

Δ=+

= − kxxy kkk 2

1

],(,)( 1 kkk xxxyxg −∈=

2/7Δ2/5Δ2/3Δ2/Δ

2/7Δ− 2/5Δ− 2/3Δ− 2/Δ−

Δ−

Δ−2

Δ−3

Δ3

Δ2

Δ

Output

input

Peak-to-peakexcursion

Overload level

Figure 4.3 is an example of 7-level quantization scheme.

Quantization error

input

2/Δ

2/Δ−

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12/)var( ,0)][

22 Δ==

=⇒

QQE

Qσ

⎩⎨⎧ Δ+≤≤Δ−Δ

=otherwise ,0

2/2/ ,/1)(

qqfQ

If the step-size is sufficiently small (or N is large), it is reasonable to assume that Q is uniformly distributed over [−Δ/2 , Δ/2], then the pdf of Q is given by

Let xmax be the magnitude of the peak value of an input x. The step-size of the quantizer is given by

Nxmax2

=Δ if , kk xxyq ℜ∈−=

We consider the quantizer input as a random variable X with zero mean and variance . Then both the quantization error Q and the quantized output Y = g(X) are random variables. In other words, we have

2Xσ

Let q represent the quantization error of a data sample x,

Quantization error:

]2/,2/[ where ΔΔ−∈q

XXgQ −= )(

for ),( kk xxgy ℜ∈=

where

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]))([(][ 22 XXgEQE −=

The mean square of Q

is called mean squared distorsion, or quantization noise.

2

2

)(Q

XOSNR

σσ

=

Uniformquantizer

X

Figure 4.4

g(X) = X + Q

signal error

The quantizer’s performance can be described by the output signal-to-quantization noise ratio (SNR) defined as

][][)( 2

2

QEXESNR O = 2

2

Q

X

σσ

=

.12/

)( 2

2

Δ= X

OSNR σ

Since Q is uniformly distributed, from the above computation,

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1281

256)1(1 =

−−=Δ

Solution. The step size is the ratio of peak-to-peak value of the signal of the number of quantization on levels,

)(16.486553612/)128/1(

3/112/

)( 22

2

dBSNR Xo ≈==

Δ=⇒

σ

Since signal X is uniformly distributed over [−1, 1], its pdfis given by

⎩⎨⎧ ≤≤−

=otherwise ,0

11 if ,2/1)(

xxf X

31

12))1(1(

22 =

−−=⇒ Xσ

Example 4. 1 What is the resulting (SNR)o for a signal uniformly distributed on [−1, 1] when uniform quantizer with 256 levels is employed?

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Motivation: using the correlation between data samples to predict the future value of the signal or to consider how to remove correlation between samples.

5. Differential Pulse-Code Modulation (DPCM)5. Differential Pulse-Code Modulation (DPCM)

A. Structure of DPCM

Set: A baseband signal x(t) is sampled at the rate (> 2W).

, a sequence of correlated samples Ts seconds apart where n is an integer. For simplicity, we write

xn is a sample of the random sequence Xn with zero mean.

ss T

f 1=

)}({ snTx

,...2 ,1 ,0 ),( ±±== nnTxx sn

Purpose: reduce the transmission rate (or bit rate).

Fig. 5.2. Receiver:

++

reconstructionfilter

x*(t)

+PCM

decoder

predictionfilter

ny

nx̂

*nxDPCM

sequence

N-level

+

Fig. 5. 1 A general DPCM transmitter

sample

x(t) *nx+

+

+

ny

predictionfilter

−+

nx̂

nxne PCM

encoderquantizer

Q(x)+DPCMwave form

nnn xxe ˆ−= nnn qey +=

nnn xyx ˆ* +=

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PCM Decoder: to recover ynfrom the DPCM sequence.

Predictor: the same as that in the transmitter.

nnn qey +=

Quantizer input:

nnn xxe ˆ−=

en is called prediction error.Quantizer output:

qn, quantization error

Prediction filter is usually implemented by a linear predictor in which the last r samples are used to predict the values of the next sample.

Predictor input:

nx̂

nnn xyx ˆ* +=

Predictor output:

Legend in Transmitter:

Legend in Receiver:

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nnn qxx +=⇒ *

nnnn xqex ˆ * ++=⇒

nnnnn eyxxq −=−= ∗

Remark 1. Usually, the range of variation of en is much smaller compared to that of xn and, therefore, en can be quantized with fewer bits. Note that qnis independent of the predictor. Thus, DPCM can achieve performance levels compared to PCM at lower bit rates, which finds wide applications in speech and image compression.

Property. For the DPCM structure in Figures 5.1 and 5.2, the quantization error between xn and its reconstructed value is the same as the quantization error between the input and the output of the quantizer.

∗nx

nnnn xqxx ˆ)ˆ( ++−=

Reconstructed signal symbol:

Thus

nnn xyx ˆ* +=

predicted value of xnoutput of the quantizer

This result can be shown as follows.

Remark 2. The above result shows that in the absence of channel noise, the difference between x(t) and x*(t) is due to the quantization error qn.

Linear Predictor:

A linear predictor is a finite-duration impulse response (FIR) discrete-time filter: • it contains a set of r unit-delay elements, each of which is represented by D, • the output of the filter is the convolution sum:

rnrnnn xcxcxcx −−− +++= L2211ˆ

where r, the number of unit-delay elements, is called the predictor order, and ci are coefficients of the filter.

Figure 5.3. Linear Predictor

xn D

cr-1 cr

+ +

D D

c1 c2

+

xn-1 xn-2 xn-r+1

...

...

...

xn-r

…

nx̂

The design objective: to choose the filter coefficients c1, c2, …, cr so as to minimize the mean-square error

where

both En and Xn are the random sequences of zero mean, and en and xn are samples of En and Xn, , respectively. In this case, the filter is called the optimal predictor.

])ˆ[(][ 22nnn XXEEED −==

∑=

−−=−=r

iininnnn XcXXXE

1

ˆQuestion: How does one find the values of the filter coefficients such that the resulting filter is optimal ? Think about the predictor of order 1 (i.e., r = 1).

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How to obtain the coefficients ci’s such that D is minimized?

∑∑∑= ==

−+−=r

i

r

jXji

r

iXiX jiRcciRcRD

1 11)(2)(2)0(

where RX denotes the autocorrelation function of the process X = {Xn}. To minimize D, we differentiate with respect to the ci’s and find the roots. After differentiating, we have

rjjRjiRc X

r

iXi ≤≤=−∑

=

1 ),()(1

Expanding that, we obtain

Solving the above set of linear equations, one can find the optimal set of predictor coefficients.

Figure 5.5 A simple DPCM system

∑=

∗−=

r

iinin xcx

1

ˆ

…

Figure 5.4 A general DPCM system with a linear predictor

+ Q

+Predictor

nx ne ny

∗nx

+

Predictor

∑=

∗−=

r

iinin xcx

1

ˆ

∗nx

ny

-

+

+ Q

+D

nx ne nv

∗nx

+

D

∗nxnv

∗−1nx

∗−1nx

...-

+

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How to compute signal-to-quantization noise ratio of DPCM?

PPQ

E

E

X SNRG )(2

2

2

2

⋅=⋅=σσ

σσ

,2Qσ

,2Eσ

,)( oSNR

,σσSNR

Q

EP 2

2

)( =

, 2

2

E

XPG

σσ

=

2

2

)(Q

XoSNR

σσ

=We can write

where

the variance of the zero-mean prediction error en,

the variance of the zero-mean quantization error qn,

the ratio for the DPCM transmitter,

the ratio for the quantizer, called the prediction signal-to-quantization noise ratio

the prediction gain produced by DPCM.

Thus the signal-to-quantization noise ratio of DPCM is determined by the product of the signal-to-quantization noise ratio and the prediction gain.

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Remark. For a given baseband signal x(t), the variance is fixed, thus

2Xσ

minimized maximized 2 ↓↔↑ EPG σ

If the predictor is good, then . In other words, for a given number of quantization levels, the variance of the quantization error qnfor en is smaller than the variance of the quantization error qn for xn as in PCM.

22XE σσ <

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Example 5.1.Message sources are normally correlated. To conserve transmission bandwidth, it is desirable to remove the correlation prior to transmission over the communications channel. Differential pulse code modulation(DPCM) is a relatively simple technique for correlation removal.

DPCM involves the quantization and prediction encoding at the transmitter, and the use of feedback decoding at the receiver. How does DPCM remove correlation? Why does not feedback decoding at the receiver cause quantization error propagation?

Solution.

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nnn xxy ˆ* −=

In the DPCM system, the correlation of the input sampled data {xn} is removed by subtracting the estimate (based on previous samples) from the current input, and quantize the prediction error

instead of quantizing the input xn itself. In the case of an optimal predictor, the prediction error en is uncorrelated with previous quantized signal samples . At the receiver, the input signal is the quantized prediction error

. The output signal of the receiver is

*nx

nnn xxe ˆ−=

},,,{ *2

*1 L−− nn xx

nn

nnnnnnnnn

qxqexqexyxx

+=++=++=+=

)ˆ()(ˆˆ*

where qn is the quantization error of the prediction error en of current data sample. Therefore, the receiver outputs the quantized valueof the information sample xn with quantization error qn. Because the error qn depends only on the current prediction error en, there is no quantization error propagation in the DPCM system.

nx̂

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1ˆ −= nn cxx

2Eσ

Example 5.2. Predictor of Order One

D

c

nX

nX̂

D

c

nX

nX̂+ ne

Predictor of order one Predictor-error filter of order one

Consider the first-order predictor defined by

where xn is a sample of a stationary signal of zero mean and c is a constant.

(a) Determine the optimal value of c which minimizes and find the value of the minimized .

(b) Find the optimal prediction gain Gp.

2Eσ

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Solution:

1 ˆ −−=−= nnnnn cxxxxe

have wemean, zero a has Since ly.respective ˆ and ,,

sequences random ingcorrespond the of samples are ˆ and ,, where

nnnn

nnn

XXEX

xex

0][ ][ ][][

1

1

=−=−=

−

−

nn

nnn

XcEXEcXXEEE

⎥⎦

⎤⎢⎣

⎡−+= 2

22 )1(21X

XX

cRcσ

σ

Therefore, the variance of the predictor error is

])[( ][ 21

22−−== nnnE cXXEEEσ

]2[ 12

122

−− −+= nnnn XcXXcXE

)1(2)0()0( 2XXX cRRcR −+=

0=∂c

E2∂σ

For finding the optimal value for c, let

222 )1(0 )1(22

X

Xopt

X

XX

RcRcσσ

σ =⇒=⎥⎦

⎤⎢⎣

⎡−

i.e.,

Thus, the minimized mean square of the prediction error is

Note that

(b)

22

min,

2

, 1 optE

XoptP cG −==

σσ

[ ]222 21 optoptX cc −+=σ 2min,Eσ

)1( 22optX c−=σ

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6. Delta Modulation (DM)6. Delta Modulation (DM)

Delta modulation is a simplified version of the simple DPCM shown in Sec. 5. In DM, the quantizer is a 1-bit (two level) quantizer with magnitudes ± δ. In other words, in DM,

- Quantizer: 2 level, i.e., L = 2- Prediction filter: a pure delay device, i. e.,

a simple DPCM described in Sec. 5

*1ˆ −= nn xx

Justification: By over sampling the input message signal x(t) (i.e., at a rate much higher than the Nyquist rate), the correlation between adjacent samples is increased purposely. In this way, the prediction error, so that the quantization error, can be very small.

Remark. In DM, 1 bit/sample, the quantization noise ↑, unless the dynamic range of en ↓ ⇒ the correlation of ↑.

1 and −nn xx

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DM Encode DM Decode

*1

*1ˆ ˆ where

−

−

−=

=−=

nn

nnnnn

xx

xxxxe

+ 2-levelQ

+D: Ts

nx ne ny∗nx

+

D: Ts

∗nxny

∗−1nx

∗−= 1ˆ nn xx

...

Accumulator

−+

Accumulator

Fig. 6.1 DM System

A. Structures of DM Encode and Decode

Low-pass filter

Output

Special property of DM:

The 2-level quantizer:output

input

δ+

δ−

0

δ2 size-step =Δ

)sgn( nn ey δ=

)sgn(1∑=

=n

iieδ*

01

1*

2*

1* )( xyyyxyxx

n

iinnnnnn +==++=+= ∑

=−−− L

Accumulative property: if we choose the initial state , then we have 0 *

0 =x

∗nx

2-levelQ

This means that to obtain , one only to accumulate the values of . This simplifies the block diagram Fig. 6.1 to Figure 6.2.

ny

+nx ne δ±=ny ∗nx

ny...−+

Accumulator

Accumulator

Fig. 6.2. DM with integrators

Low-pass filter

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ˆ*1 ⇒=− nn xx

sTδ

Staircase approximation x*(t)

x(t)

t

B. Staircase Approximation

nx a staircase approximation to

Fig. 6.3. Small δ and slope-overload distortion

C. Quantization Noise

1. Slope-overload distortion

}{ *nx

The step size is too small for the staircase to follow a steep segment of x(t).

2δ=Δ

Fig. 6.4. Large δ and granular noise

Staircase approximation x*(t)

x(t)

2. Granular noiseThe step size Δ is too large relative to the local slope characteristics of the input waveform x(t), therefore causing the staircase approximation to hunt around a relative flat segment of the input waveform. This type of noise is analogus to quantization noise in PCM.

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nnn qxx +=*

*1−−= nnn xxe

11)( −− −−= nnn qxx

Notice that

Then

discrete approximation to the derivative of the message signal

scc TfAfAdt

tdx πδπ 2 2)(max ≥⇒=

Example 6.1. For sinusoidal signal , we have)2cos()( θπ += tfAtx c

dttdx

Ts

)(max≥δ

To avoid the slope-overload distortion, we require:

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(ADM) modulation delta adaptive : noiseon quantizati thereduce toTechniques

⇒

)( ofsegment flat a during decrease ii));( ofsegment steep a during increase i)

: Principle

txtx

δδ

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Example 6.2. For the DM depicted in Fig 6.1., find the maximum output signal-to-quantization noise ratio under the assumption of no slope-overload for sinusoidal signal )2cos()( θπ += tfAtx c

312)2(

12

2222 δδσ ==

Δ=Q

Solution. Assume that quantization error is uniformly distributed over [-δ, δ].

Then

1,1 ⎥⎦

⎤⎢⎣

⎡−

ss TTWe also assume that the psd of the error is flat in the interval: . The filter in the DM decoder is a low-pass filter whose bandwidth is equal to the message bandwidth W. Then the average noise power at the filter output is given by

sQQ T

Wout /2

2 22 σσ =3

22 δσ sQs WTWT ==

222

22

max 822

scsc Tf

APT

fAπ

δδπ ==⇒≤ 3222max

maxo, 83SNR

scQ TWfP

outπσ

==⇒

9dBby increased is SNR , doublingBy maxo,⇒sfRemark.

For a sinusoidal input, cfAdt

tdx π2)(Max = Since there is no slope-overload, we have

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Rx.at )( Tx toat )( from diagramblock :signals PCM ofon Transmissi .1

* txtx

Summary of Chapter 3 ( Chapter 5 in the text)

2. Binary signal detection in AWGN: block diagram of the Tx and Rx, functionalities in each block, decision variables, error probabilities, the match filter as an optimal detector, computation of average probability of bit error for the on-off signaling and antipodal signaling, SNRo and ratio E/N0.

3. Time division multiplexing (TDM) of PCM signals

2

222 and ,

12 :on quantizati Uniform.4

Q

XoQ SNR

σσσ =

Δ=

dttdx

Ts

)(max :overload slope avoid to(c)

ionapproximat staircase (b) structuresreceiver andtter transmi(a)

: DM .6

≥δ

5. DPCM: transmitter and receiver structures, linear predictor and optimal predictor in the sense of minimizing the mean square error.