1. I. Previously on IET

2. 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

3. 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

4. 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)

5. 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)

6. 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

7. 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.

8. Phase Shift Keying (PSK) Modulation1 1 1 1
Base band Signal
X(t)
Band Pass Signal
Y(t)

9. PSK Demodulation x X(t)[2cos2(2πfct)] =X(t)[1+cos(4πfct)]
Low Pass Filter x
X(t)
2cos(2πfct)
X(t)[2cos2(2πfct)] =X(t)[1+cos(4πfct)]
X(t)[2cos2(2πfct)]=X(t) +X(t)cos(4πfct)]
Base band Signal
(i.e., low frequency content)
High frequency content

10. Orthogonality of sin and cos Functions
X(t)[2sin(2πfct)cos(2πfct)]
X(t)cos(2πfct) x
Low Pass Filter
2sin(2πfct)
X(t)[2sin(2πfct)cos(2πfct)]=X(t) sin(4πfct)]
High frequency content

11. Quadrature- PSK Modulation (QPSK)
XI(t)cos(2πfct)
XI(t) x
Y(t)
cos(2πfct) +
X(t)
Serial-to-Parallel
XQ(t)
XQ(t)sin(2πfct) x
sin(2πfct)

12. QPSK Demodulation Parallel-to-Serial X (t ) x Low Pass Filter X(t) Y(t
2cos(2πfct) x
Low Pass Filter X (t ) Q
2sin(2πfct)

13. Modulation in Time-Limited Communications
Binary
Encoder
Transmitting
Filter
Cosine
Modulation
Binary
Symbols
Rectangular Filter
In Phase Modulation 
Time Representation
ES=(1)2×1=1 1 TS 1
Frequency Representation TS f f
-fc fc
Time Representation
ES=(-1)2×1 TS -1
Frequency Representation
-fc fc f f
-TS

14. Modeling of In phase Modulation
Binary
Encoder
Transmitting
Filter
Cosine
Modulation
ES=A2 -A A

15. Modulation in Band-Limited Communications
Binary
Encoder
Transmitting
Filter
Cosine
Modulation
Binary
Symbols
Raised Cosine Filter
In Phase Modulation 
Time Representation
ES=(1)2×1=1 1 t 1 t
Frequency Representation
1/RS f f
-RS/2
RS/2
-fc- RS/2
-fc+ RS/2
fc- RS/2
fc+ RS/2
-fc fc
Time Representation
ES=(-1)2×1 t t
Bit Rate = RS
Bandwidth = RS
1 b/s/Hz -1
Frequency Representation
-RS/2
RS/2
-fc- RS/2
-fc
-fc+ RS/2
fc- RS/2 fc
fc+ RS/2 f
-1/RS 15

16. Modeling of In phase Modulation
Binary
Encoder
Transmitting
Filter
Cosine
Modulation
ES=A2 -A A

17. Modulation in Time-Limited Communications
Binary
Encoder
Transmitting
Filter
Sine
Modulation
Binary
Symbols
Rectangular Filter
In Quadrature Modulation 
Time Representation
ES=(1)2×1=1 1 TS 1
Frequency Representation TS fc f
-fc f
Time Representation
ES=(-1)2×1 TS -1
Frequency Representation
-fc f f fc
-TS 17

18. Modeling of In phase Modulation
Binary
Encoder
Transmitting
Filter
Sine
Modulation
ES=A2 jA
-jA

19. Modulation in Band-Limited Communications
Binary
Encoder
Transmitting
Filter
Sine
Modulation
Binary
Symbols
Raised Cosine Filter
In Quadrature Modulation 
Time Representation
ES=(1)2×1=1 1 t 1 t
Frequency Representation
1/RS fc
fc- RS/2
fc+ RS/2 f f
-RS/2
RS/2
-fc- RS/2
-fc+ RS/2
-fc
Time Representation
ES=(-1)2×1 t t
Bit Rate = RS
Bandwidth = RS
1 b/s/Hz -1
Frequency Representation
-RS/2
RS/2
-fc- RS/2
-fc
-fc+ RS/2 f
fc- RS/2 fc
fc+ RS/2
-1/RS 19

20. Modeling of In phase Modulation
Binary
Encoder
Transmitting
Filter
Sine
Modulation
ES=A2 jA
-jA

21. Modulation Constellations
BPSK
QPSK
1 b/s/Hz
2 b/s/Hz
8-QPSK
16 QAM
3 b/s/Hz
4 b/s/Hz

22. Basic Communication Model in AWGNR S* TX RX
Detection +
Channel Model
R=S+N
Detection Performance:
Correct Detection
S = S*
Erroneous Detection
S ≠ S*

23. BPSK Modulation over AWGN Channels
ES  Energy per Symbol

24. BPSK Modulation over AWGN Channels
Gaussian Noise

25. BPSK Modulation over AWGN Channels
Received signal distribution given transmitted

26. BPSK Modulation over AWGN Channels
Error Calculation given transmitted
Symmetry of Gaussian Distribution
Let

27. BPSK Modulation over AWGN Channels
Received signal distribution given transmitted

28. BPSK Modulation over AWGN Channels
Error Calculation given transmitted
Let

29. BPSK Modulation over AWGN Channels
Signal Power & Symbol Error Performance

30. BPSK Modulation over AWGN Channels
Signal Power & Symbol Error Performance

31. BER of PSK over AWGN Channels
Notes:
Define N0 Total Noise Power
N0/2  Noise Power over Cosine axis, i.e., σ2=N0/2
Each symbol corresponds to a single bit
Eb = ES
Pb = Pe