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)

Digital Communication System: Transmitter Receiver

Receivers (Chapter 5) (week 3 and 4) Optimal Receivers Probability of Error

Optimal Receivers Demodulators Optimum Detection

Demodulators Correlation Demodulator Matched filter

Correlation Demodulator Decomposes the signal into orthonormal basis vector correlation terms These are strongly correlated to the signal vector coefficients s m

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

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

Correlation Demodulator Consider each demodulator output

Correlation Demodulator Noise components {n k } are uncorrelated Gaussian random variables

Correlation Demodulator Correlator outputs Have mean = signal For each of the M codes Number of basis functions (=2 for QAM)

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

Matched filter Demodulator We find that this Demodulator Maximizes the SNR Essentially show that any other function than f 1 () decreases SNR as is not as well correlated to components of r(t)

The optimal Detector Maximum Likelihood (ML):

The optimal Detector Maximum Likelihood (ML):

Optimal Detector Can show that so

Optimal Detector Thus get new type of correlation demodulator using symbols not the basis functions:

Alternate Optimal rectangular QAM Detector M level QAM = 2 x level PAM signals PAM = Pulse Amplitude Modulation

The optimal PAM Detector For PAM

The optimal PAM Detector

Optimal rectangular QAM Demodulator d = spacing of rectangular grid Select s i for which

Probability of Error for rectangular M-ary QAM Related to error probability of PAM Accounts for ends

Probability of Error for rec. QAM Assume Gaussian noise 0

Probability of Error for rectangular M-ary QAM Error probability of PAM

SNR for M-ary QAM Related to PAM For PAM find average energy in equally probable signals

SNR for M-ary QAM Related to PAM Find average Power

SNR for M-ary QAM Related to PAM Find SNR Then SNR per bit (ratio of powers)

SNR for M-ary QAM Related to PAM

SNR for M-ary QAM Related to PAM Now need to get M-ary QAM from PAM M ½ =16 M ½ =8 M ½ =4 M ½ =2

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

SNR for M-ary QAM Related to PAM M=M=