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
Channel Capacity (Chapter 7) (week 6)

• Discrete Memoryless Channels
• Random Codes
• Block Codes
• Trellis Codes

3
Channel Models

• Discrete Memoryless Channel
• Discrete-discrete
• Binary channel, M-ary channel
• Discrete-continuous
• M-ary channel with soft-decision (analog)
• Continuous-continuous
• Modulated waveform channels (QAM)

4
Discrete Memoryless Channel

• Discrete-discrete
• Binary channel, M-ary channel

Probability transition matrix
5
Discrete Memoryless Channel

• Discrete-continuous
• M-ary channel with soft-decision (analog) output

x0 x1 x2 . . . xq-1
AWGN
y
6
Discrete Memoryless Channel

• Continuous-continuous
• Modulated waveform channels (QAM)
• Assume Band limited waveforms, bandwidth W
• Sampling at Nyquist 2W sample/s
• Then over interval of N 2WT samples use an
  orthogonal function expansion

7
Discrete Memoryless Channel

• Continuous-continuous
• Using orthogonal function expansion

8
Discrete Memoryless Channel

• Continuous-continuous
• Using orthogonal function expansion get an
  equivalent discrete time channel

Gaussian noise
9
Capacity of binary symmetric channel

• BSC

10
Capacity of binary symmetric channel

• Average Mutual Information

11
Capacity of binary symmetric channel

• Channel Capacity is Maximum Information
• earlier showed

12
Capacity of binary symmetric channel

• Channel Capacity
• When p1 bits are inverted but information is
  perfect if invert them back!

13
Capacity of binary symmetric channel

• Effect of SNR on Capacity
• Binary PAM signal (digital signal amplitude 2A)

AGWN
14
Capacity of binary symmetric channel

• Effect of SNR
• Binary PAM signal (digital signal amplitude 2A)

15
Capacity of binary symmetric channel

• Effect of SNR
• Binary PAM signal (digital signal amplitude 2A)

16
Capacity of binary symmetric channel

• Effect of SNR
• Binary PAM signal (digital signal amplitude 2A)

Not sure about this Does it depend on bandwidth?
17
Capacity of binary symmetric channel

• Effect of SNR
• Binary PAM signal (digital signal amplitude 2A)

18
Capacity of binary symmetric channel

• Effect of SNR
• Binary PAM signal (digital signal amplitude 2A)

19
Capacity of binary symmetric channel

• Effect of SNR
• Binary PAM signal (digital signal amplitude 2A)

At capacity SNR 7, so waste lots of SNR to get
low BER!!!
20
Capacity of binary symmetric channel

• Effect of SNRb
• Binary PAM signal (digital signal amplitude 2A)

21
Channel Capacity of Discrete Memoryless Channel

• Discrete-discrete
• Binary channel, M-ary channel

Probability transition matrix
22
Channel Capacity of Discrete Memoryless Channel

• Average Mutual Information

23
Channel Capacity of Discrete Memoryless Channel

• Channel Capacity is Maximum Information
• Occurs for
• only if
• Otherwise must work out max

24
Channel Capacity Discrete Memoryless Channel

• Discrete-continuous
• Channel Capacity

25
Channel Capacity Discrete Memoryless Channel

• Discrete-continuous
• Channel Capacity with AWGN

26
Channel Capacity Discrete Memoryless Channel

• Binary Symmetric PAM-continuous
• Maximum Information when

27
Channel Capacity Discrete Memoryless Channel

• Binary Symmetric PAM-continuous
• Maximum Information when

28
Channel Capacity Discrete Memoryless Channel

• Binary Symmetric PAM-continuous
• Versus Binary Symmetric discrete

29
Discrete Memoryless Channel

• Continuous-continuous
• Modulated waveform channels (QAM)
• Assume Band limited waveforms, bandwidth W
• Sampling at Nyquist 2W sample/s
• Then over interval of N 2WT samples use an
  orthogonal function expansion

30
Discrete Memoryless Channel

• Continuous-continuous
• Using orthogonal function expansion get an
  equivalent discrete time channel

Gaussian noise
31
Discrete Memoryless Channel

• Continuous-continuous
• Capacity is (Shannon)

32
Discrete Memoryless Channel

• Continuous-continuous
• Maximum Information when

Statistically independent zero mean Gaussian
inputs
then
33
Discrete Memoryless Channel

• Continuous-continuous
• Constrain average power in x(t)

34
Discrete Memoryless Channel

• Continuous-continuous
• Thus Capacity is

35
Discrete Memoryless Channel

• Continuous-continuous
• Thus Normalized Capacity is