I Previously on IET Introduction to Digital Modulation

I. Previously on IET

Introduction to Digital Modulation: Pulse Code Modulation

Digital Communication Systems Source of Information User of Information Source Encoder Source Decoder Channel Encoder Channel Decoder Modulator De-Modulator Channel © Tallal Elshabrawy 3

Pulse Code Modulation l An analog message signal is converted to discrete form in both time and amplitude and then represented by a sequence of coded pulses © Tallal Elshabrawy 4

Pulse Code Modulation Source of continuoustime (i. e. , analog) message signal Low pass Filter Sampler Quantizer Encoder PCM Signal Analog-to-Digital Converter l Low Pass Filter l l Sampling l l 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 l l Confining the frequency content of the message signal Converting of analog samples to a set of discrete amplitudes Encoding l Translating the discrete set of samples in a form suitable for digital transmission © Tallal Elshabrawy 5

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 © Tallal Elshabrawy Making the spectrum of the signal periodic in the frequency domain 6

Sampling Process l Basic operation for digital communications l Converts an analog signal into a corresponding sequence of samples (usually spaced uniformly in time) l Questions l What should be the sampling rate? l Can we reconstruct the original signal after the sampling process? © Tallal Elshabrawy 7

Effect of Sampling on Frequency Content of Signals m(t) M(f) t (sec. ) Representation of analog signal m(t) in time domain l l l -W W f (Hz) Let assume that the frequency content of analog signal in the frequency domain is confined with W Define TS as the sampling interval Define f. S as the sampling frequency © Tallal Elshabrawy 8

f. S>2 W m(t) t (sec) TS=1/f. S M(f) LPF -f. S-W -f. S+W -W 0 W f. S-W f. S+W f (Hz) fcutoff l By using a LPF with W<fcutoff<f. S-W at the receiver, it is possible to reconstruct the original signal from received samples © Tallal Elshabrawy 9

f. S=2 W m(t) t (sec) TS=1/f. S M(f) LPF -3 W -f. S=-2 W -W 0 W f. S=2 W 3 W f (Hz) fcutoff l By using a LPF with fcutoff=W at the receiver, it is possible to reconstruct the original signal from received samples © Tallal Elshabrawy 10

f. S<2 W m(t) TS=1/f. S t (sec) M(f) -f. S-W l -f. S -W -f. S+W 0 f. S-W W f. S+W f (Hz) It is no longer possible to reconstruct the original signal from received samples © Tallal Elshabrawy 11

Sampling Theorem l Sampling Theorem states that l 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 1/2 W seconds l 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 2 W samples per second l f. S=2 W is called the Nyquist Rate l t. S=1/2 W is called the Nyquist interval © Tallal Elshabrawy 12

Pulse Code Modulation Revisited Analog-to-Digital Converter Source of continuoustime (i. e. , analog) message signal Representation Levels (vj) j=1, 2, …, L Low pass Filter Quantizer Sampler m-ary Symbol Encoder Transmitting Filter (uk) k=1, 2, …, logm. L Encoder PCM Signal sk(t) l Let TQ represent the time interval between two consecutive quantized representation levels l Let TS represent the time interval between two consecutive m-ary encoded symbols © Tallal Elshabrawy 13

M-ary Encoder Examples 64 Quantized representation levels vk k=1, 2, …, 64 Sampling Rate = 1/TQ © Tallal Elshabrawy Binary Symbol Encoder Binary Code uk k=1, 2 Symbol Rate = 1/TS=6/TQ 4 -ary Symbol Encoder 4 -ary Code uk k=1, 2, 3, 4 Symbol Rate = 1/TS=3/TQ 14

Transmitting Filter l l l 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: l Square pulse transmitting filter Binary Code TS PCM Signal TS TS 1 t=3 TS t=2 TS t=0 +1 -1 +1 +1 0 TS t=4 TS 4 -ary Code TS t=2 TS t=0 t=TS t=0 PCM Signal 1 TS TS t=3 TS t=2 TS t=0 +3 -3 +1 +3 © Tallal Elshabrawy t=3 TS 0 TS t=4 TS t=3 TS t=2 TS 15

Optimal Receiving Filter Transmitting Filter g(t) sk(t) + xk(t) Receiving Filter h(t) yk(TS) Sample at t=TS wk(t) Optimality © Tallal Elshabrawy At sampling Instant t=TS is maximized 16

Matched Filter PCM Signal Transmitting Filter g(t) sk(t) + xk(t) Receiving Filter h(t) yk(TS) Sample at t=TS wk(t) Objective: l Design the optimal receiving filter to minimize the effects of AWGN l Matched Filter 1. h(t)=g(TS-t), i. e. , H(f)=G*(f ) 2. Sample the output of receiving filter every TS © Tallal Elshabrawy 17

Matched Filter: Square Pulse Transmitting Filter Assume AWGN Noise wk(t) is negligible, binary symbols +1, -1, +1 1 Transmitting Filter g(t) xk(t) TS 0 sk(t) wk(t) t=4 TS t=3 TS t=0 + xk(t) Matched Filter g(TS -t) t=2 TS yk(t) TS 1 TS 0 Sample at t=TS t=4 TS t=3 TS t=2 TS TS t=0 -TS yk(t) yk(TS) TS TS t=4 TS yk(i. TS) TS t=3 TS t=2 TS TS t=0 -TS © Tallal Elshabrawy 18

Basic Blocks of Digital Communications Analog-to-Digital Converter Source of continuoustime (i. e. , analog) message signal Low pass Filter Sampler m-ary Symbol Encoder © Tallal Elshabrawy Quantizer Transmitting Filter Encoder Band Pass Modulated Signal Modulation 19

Square Pulse is a Time-Limited Signal = Frequency Unlimited Spectrum Fourier Transform 0 TS -3/TS -2/TS -1/TS 0 1/TS 2/TS 3/TS It is desirable for WHY? Guarantee completely transmitted signals to be orthogonal channels for band-limited (limited pass-band signals frequency spectrum) © Tallal Elshabrawy 20

Inter-symbol Interference (ISI) Frequency Limited Spectrum=Time-Unlimited Signals l A time unlimited signal means inter-symbol interference (ISI) l Neighboring symbols affect the measured value and the corresponding decision at sampling instants Sampling Instants yk(t) © Tallal Elshabrawy yk(i. TS) 21

Nyquist Criterion for No ISI l For a given symbol transmitted at i. TS Transmitting Filter g(t) sk(t) + xk(t) yk(t) Receiving Filter g (TS-t) yk(TS) Sample at t=TS wk(t) Assume AWGN Noise wk(t) is negligible Transmitting Filter g(t) yk(t) Receiving Filter g (TS-t) z(t)=g(t)* g(TS-t) © Tallal Elshabrawy yk(TS) Sample at t=TS 22

Pulse-shaping with Raised-Cosine Filter z(t): Impulse Response Z(f): Spectrum (Transfer Function) Z(f) T: symbol interval RS: symbol rate r: roll-off factor Raised Cosine Filter Bandwidth = RS(1+r)/2 © Tallal Elshabrawy 23

Examples l 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 l 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. © Tallal Elshabrawy 24