I. Previously on IET

I. Previously on IET

Introduction to Digital Modulation: Pulse Code Modulation

© Tallal Elshabrawy 3

Digital Communication Systems

Source of Information

User of Information

Source Encoder

Channel Encoder

Modulator

Source Decoder

Channel Decoder

De-Modulator

Channel


Pulse Code Modulation An analog message signal is converted to discrete

form in both time and amplitude and then represented by a sequence of coded pulses


Source of continuous-time (i.e., analog) message signal

Low pass Filter

Sampler Quantizer Encoder

Pulse Code Modulation

PCM Signal

Low Pass Filter Confining the frequency content of the message signal

Sampling To ensure perfect reconstruction of message signal at the receiver, the

sampling rate must exceed twice the highest frequency component of the message signal (Sampling Theorem)

Quantization Converting of analog samples to a set of discrete amplitudes

Encoding Translating the discrete set of samples in a form suitable for digital

transmission

Analog-to-Digital Converter


Sampling Process: Introductory Note

Periodic signal in the time domain

Sampling of the signal spectrum in the frequency domain

By Duality

Sampling of the signal in the time domain

Making the spectrum of the signal periodic in the frequency domain


Sampling Process Basic operation for digital communications

Converts an analog signal into a corresponding sequence of samples (usually spaced uniformly in time)

Questions What should be the sampling rate? Can we reconstruct the original signal after the

sampling process?


Effect of Sampling on Frequency Content of Signals

m(t)

t (sec.)Representation of analog signal m(t) in time domain

f (Hz)

M(f)

W-W

Let assume that the frequency content of analog signal in the frequency domain is confined with W

Define TS as the sampling interval Define fS as the sampling frequency


fS>2W

m(t)

f (Hz)

M(f)

W-W

TS=1/fSt (sec)

0 fS-fS fS-W fS+W-fS+W-fS-W

By using a LPF with W<fcutoff<fS-W at the receiver, it is possible to reconstruct the original signal from received samples

LPF

fcutoff


fS=2W

m(t)

f (Hz)

M(f)

W-W

TS=1/fSt (sec)

0 fS=2W-fS=-2W 3W-3W

By using a LPF with fcutoff=W at the receiver, it is possible to reconstruct the original signal from received samples

LPF

fcutoff


fS<2W

m(t)

f (Hz)

M(f)

W-W

TS=1/fSt (sec)

0 fS-fS fS+W-fS-W

It is no longer possible to reconstruct the original signal from received samples

-fS+W fS-W


Sampling Theorem Sampling Theorem states that

A band-limited signal of finite energy which has no frequency components higher than W Hz is completely described by specifying the values of the signal at instants of time separated by 1/2W seconds

A band-limited signal of finite energy which has no frequency components higher than W Hz may be completely recovered from knowledge of its samples taken at the rate of 2W samples per second

fS=2W is called the Nyquist Rate

tS=1/2W is called the Nyquist interval


Pulse Code Modulation Revisited

Source of continuous-time (i.e., analog) message signal

Low pass Filter

Sampler Quantizer Encoder PCM Signal


Let TQ represent the time interval between two consecutive quantized representation levels

Let TS represent the time interval between two consecutive m-ary encoded symbols

Representation Levels (vj)j=1,2,…,L

m-ary SymbolEncoder

Transmitting Filter

PCM Signalsk(t)

(uk)k=1,2,…,logmL


M-ary Encoder Examples

Binary SymbolEncoder

Binary Code ukk=1,2Symbol Rate = 1/TS=6/TQ

64 Quantized representation levels vkk=1,2,…,64Sampling Rate = 1/TQ

4-ary SymbolEncoder

4-ary Code ukk=1,2,3,4Symbol Rate = 1/TS=3/TQ

64 Quantized representation levels vkk=1,2,…,64Sampling Rate = 1/TQ


Transmitting Filter The output from the m-ary encoder is still a logical variable rather than an actual signal The transmitting filter converts the output of the m-ary encoder to a pulse signal Example:

Square pulse transmitting filter

1

TS

ku 1, 1

Binary Code

0

PCM SignalTS TS TS

t=0t=TSt=2TSt=3TSt=0t=TSt=2TSt=3TSt=4TS

1

TS

ku 3, 1, 1, 3 4-ary Code

0

PCM Signal

TS TS TS

t=0t=TSt=2TSt=3TS

t=0t=TSt=2TSt=3TSt=4TS

+1 -1 +1 +1

+3 -3 +1 +3


Optimal Receiving Filter

k k k S

k k k S

k k k S

k k k k

x t =s t +w t ,0 t T

y t =s t *h t +w t *h t ,0 t T

y t =r t +n t ,0 t T

r t s t h t , n t w t h t

Transmitting Filter g(t)

sk(t)+

wk(t)

Receiving Filter h(t)

xk(t) yk(t) yk(TS)ku

Sample at t=TS

Optimality At sampling Instantt=TS

2

k

2k

r Tη=

E n t

is maximized


Matched Filter

Objective: Design the optimal receiving filter to minimize the

effects of AWGN

Matched Filter1. h(t)=g(TS-t), i.e., H(f)=G*(f )2. Sample the output of receiving filter every TS


sk(t)+

wk(t)

Receiving Filter h(t)

xk(t) yk(t) yk(TS)Sample at t=TS

PCM Signal


xk(t)


Assume AWGN Noise wk(t) is negligible, binary symbols +1,+1,-1,+11

TS0Transmitting Filter g(t)

sk(t)+wk(t)

xk(t)

1

TS0Matched Filter g(TS-t)

yk(t)

yk(TS)

Sample at t=TS

yk(t)


TSTS

-TS

TS

yk(iTS)


TSTS

-TS

TS

Matched Filter: Square Pulse Transmitting Filter


Basic Blocks of Digital Communications

Source of continuous-time

(i.e., analog) message signal

Low pass Filter

Sampler Quantizer Encoder Band Pass Modulated Signal


m-ary SymbolEncoder

Transmitting Filter Modulation


Time-Limited Signal = Frequency Unlimited Spectrum

TS0

Fourier Transfor

m

It is desirable for transmitted signals to

be band-limited (limited frequency

spectrum)

Guarantee completely

orthogonal channels for pass-band signals

WHY?

Square Pulse is a Time-Limited Signal

1/TS 2/TS 3/TS-1/TS-2/TS-3/TS

0


Inter-symbol Interference (ISI)

Sampling Instants

yk(t) yk(iTS)

Frequency Limited Spectrum=Time-Unlimited Signals

A time unlimited signal means inter-symbol interference (ISI)

Neighboring symbols affect the measured value and the corresponding decision at sampling instants


Nyquist Criterion for No ISI For a given symbol

transmitted at iTS

S

A k=1z kT

0 o/w


sk(t)+

wk(t)

Receiving Filter g (TS-

t)

xk(t)yk(t) yk(TS)

kuSample at t=TS

Assume AWGN Noise wk(t) is negligible


Receiving Filter g (TS-

t)

yk(t) yk(TS)Sample at t=TS

z(t)=g(t)* g(TS-t)

ku


z(t): Impulse

Response

Z(f): Spectrum(Transfer Function)

T: symbol interval

RS: symbol rater: roll-off factor

Z(f)

Raised Cosine Filter Bandwidth = RS(1+r)/2

Pulse-shaping with Raised-Cosine Filter


Examples An analog signal of bandwidth 100 KHz is sampled

according to the Nyquist sampling and then quantized and represented by 64 quantization levels. A 4-ary encoder is adopted and a Raised cosine filter is used with roll off factor of 0.5 for base band transmission. Calculate the minimum channel bandwidth to transfer the PCM wave

An analog signal of bandwidth 56 KHz is sampled, quantized and encoded using a quaternary PCM system with raised-cosine spectrum. The rolloff factor is 0.6. If the total available channel bandwidth is 2048 KHz and the channel can support up to 10 users, calculate the number of representation levels of the Quantizer.

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