1. I. Previously on IET

2. Introduction to Digital Modulation: Pulse Code Modulation

3. Digital Communication Systems
Source of Information
User of Information
Source Encoder
Source Decoder
Channel Encoder
Channel Decoder
Modulator
De-Modulator
Channel

4. 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

5. 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

6. 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

7. 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?

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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)

22. 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)

23. 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

24. 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.