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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

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

Pulse Code Modulation Low Pass Filter Sampling Quantization Encoding Source of continuous-time (i.e., analog) message signal Low pass Filter Sampler Quantizer Encoder PCM Signal Analog-to-Digital Converter 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

Sampling Process: Introductory Note Sampling of the signal spectrum in the frequency domain Periodic signal in the time 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) M(f) t (sec.) -W W f (Hz) Representation of analog signal m(t) in time domain 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) t (sec) TS=1/fS M(f) LPF -fS-W -fS -fS+W -W W fS-W fS fS+W f (Hz) fcutoff By using a LPF with W<fcutoff<fS-W at the receiver, it is possible to reconstruct the original signal from received samples

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

fS<2W m(t) t (sec) TS=1/fS M(f) -fS-W -fS -W -fS+W fS-W W fS fS+W f (Hz) It is no longer possible to reconstruct the original signal from received samples

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 Analog-to-Digital Converter Source of continuous-time (i.e., analog) message signal Low pass Filter Sampler Quantizer Encoder PCM Signal Representation Levels (vj) j=1,2,…,L m-ary Symbol Encoder Transmitting Filter PCM Signal sk(t) (uk) k=1,2,…,logmL 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

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

Transmitting Filter Binary Code PCM Signal 4-ary Code PCM Signal 1 TS 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 Binary Code PCM Signal TS TS TS 1 TS t=4TS t=3TS t=2TS t=TS t=0 t=3TS t=2TS t=TS t=0 +1 -1 +1 +1 4-ary Code PCM Signal 1 TS TS TS TS t=4TS t=3TS t=2TS t=TS t=0 t=3TS t=2TS t=TS t=0 +3 -3 +1 +3

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

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

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

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

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

Inter-symbol Interference (ISI) 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 Sampling Instants yk(t) yk(iTS)

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

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

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.