Outline

1
Outline

• Transmitters (Chapters 3 and 4, Source Coding and
  Modulation) (week 1 and 2)
• Receivers (Chapter 5) (week 3 and 4)
• Received Signal Synchronization (Chapter 6) (week
  5)
• Channel Capacity (Chapter 7) (week 6)
• Error Correction Codes (Chapter 8) (week 7 and 8)
• Equalization (Bandwidth Constrained Channels)
  (Chapter 10) (week 9)
• Adaptive Equalization (Chapter 11) (week 10 and
  11)
• Spread Spectrum (Chapter 13) (week 12)
• Fading and multi path (Chapter 14) (week 12)

2
Digital Communication System
Transmitter
Receiver
3
Receivers (Chapter 5) (week 3 and 4)

• Optimal Receivers
• Probability of Error

4
Optimal Receivers

• Demodulators
• Optimum Detection

5
Demodulators

• Correlation Demodulator
• Matched filter

6
Correlation Demodulator

• Decomposes the signal into orthonormal basis
  vector correlation terms
• These are strongly correlated to the signal
  vector coefficients sm

7
Correlation Demodulator

• Received Signal model
• Additive White Gaussian Noise (AWGN)
• Distortion
• Pattern dependant noise
• Attenuation
• Inter symbol Interference
• Crosstalk
• Feedback

8
Additive White Gaussian Noise (AWGN)
i.e., the noise is flat in Frequency domain
9
Correlation Demodulator

• Consider each demodulator output

10
Correlation Demodulator

• Noise components

nk are uncorrelated Gaussian random variables
11
Correlation Demodulator

• Correlator outputs

Have mean signal
For each of the M codes
Number of basis functions (2 for QAM)
12
Matched filter Demodulator

• Use filters whose impulse response is the
  orthonormal basis of signal
• Can show this is exactly equivalent to the
  correlation demodulator

13
Matched filter Demodulator

• We find that this Demodulator Maximizes the SNR
• Essentially show that any other function than
  f1() decreases SNR as is not as well correlated
  to components of r(t)

14
The optimal Detector

• Maximum Likelihood (ML)

15
The optimal Detector

• Maximum Likelihood (ML)

16
Optimal Detector

• Can show that

so
17
Optimal Detector

• Thus get new type of correlation demodulator
  using symbols not the basis functions

18
Alternate Optimal rectangular QAM Detector

• M level QAM 2 x level PAM signals
• PAM Pulse Amplitude Modulation

19
The optimal PAM Detector
For PAM
20
The optimal PAM Detector
21
Optimal rectangular QAM Demodulator

• d spacing of rectangular grid

22
Probability of Error for rectangular M-ary QAM

• Related to error probability of PAM

Accounts for ends
23
Probability of Error for rec. QAM

• Assume Gaussian noise

0
24
Probability of Error for rectangular M-ary QAM

• Error probability of PAM

25
SNR for M-ary QAM

• Related to PAM
• For PAM find average energy in equally
  probable signals

26
SNR for M-ary QAM

• Related to PAM

Find average Power
27
SNR for M-ary QAM

• Related to PAM

Find SNR
(ratio of powers)
Then SNR per bit
28
SNR for M-ary QAM

• Related to PAM

29
SNR for M-ary QAM

• Related to PAM
• Now need to get M-ary QAM from PAM

M½16
M½8
M½4
M½2
30
SNR for M-ary QAM

• Related to PAM

(1- probability of no QAM error)
(Assume ½ power in each PAM)
31
SNR for M-ary QAM

• Related to PAM

M