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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)

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Outline l.jpg
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)


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Digital Communication System:

Transmitter

Receiver


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Receivers (Chapter 5) (week 3 and 4)

  • Optimal Receivers

  • Probability of Error


Optimal receivers l.jpg
Optimal Receivers

  • Demodulators

  • Optimum Detection


Demodulators l.jpg
Demodulators

  • Correlation Demodulator

  • Matched filter


Correlation demodulator l.jpg
Correlation Demodulator

  • Decomposes the signal into orthonormal basis vector correlation terms

  • These are strongly correlated to the signal vector coefficients sm


Correlation demodulator7 l.jpg
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 l.jpg
Additive White Gaussian Noise (AWGN)

i.e., the noise is flat in Frequency domain


Correlation demodulator9 l.jpg
Correlation Demodulator

  • Consider each demodulator output


Correlation demodulator10 l.jpg
Correlation Demodulator

  • Noise components

{nk} are uncorrelated Gaussian random variables


Correlation demodulator11 l.jpg
Correlation Demodulator

  • Correlator outputs

Have mean = signal

For each of the M codes

Number of basis functions (=2 for QAM)


Matched filter demodulator l.jpg
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 demodulator13 l.jpg
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)


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The optimal Detector

  • Maximum Likelihood (ML):


The optimal detector15 l.jpg
The optimal Detector

  • Maximum Likelihood (ML):


Optimal detector l.jpg
Optimal Detector

  • Can show that

so


Optimal detector17 l.jpg
Optimal Detector

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


Alternate optimal rectangular qam detector l.jpg
Alternate Optimal rectangular QAM Detector

  • M level QAM = 2 x level PAM signals

  • PAM = Pulse Amplitude Modulation




Optimal rectangular qam demodulator l.jpg

Select si for which

Select si for which

Optimal rectangular QAM Demodulator

  • d = spacing of rectangular grid


Probability of error for rectangular m ary qam l.jpg
Probability of Error for rectangular M-ary QAM

  • Related to error probability of PAM

Accounts for ends


Probability of error for rec qam l.jpg
Probability of Error for rec. QAM

  • Assume Gaussian noise

0



Snr for m ary qam l.jpg
SNR for M-ary QAM

  • Related to PAM

  • For PAM find average energy in equally probable signals


Snr for m ary qam26 l.jpg
SNR for M-ary QAM

  • Related to PAM

Find average Power


Snr for m ary qam27 l.jpg
SNR for M-ary QAM

  • Related to PAM

Find SNR

(ratio of powers)

Then SNR per bit


Snr for m ary qam28 l.jpg
SNR for M-ary QAM

  • Related to PAM


Snr for m ary qam29 l.jpg
SNR for M-ary QAM

  • Related to PAM

  • Now need to get M-ary QAM from PAM

M½=16

M½=8

M½=4

M½=2


Snr for m ary qam30 l.jpg
SNR for M-ary QAM

  • Related to PAM

(1- probability of no QAM error)

(Assume ½ power in each PAM)


Snr for m ary qam31 l.jpg
SNR for M-ary QAM

  • Related to PAM

M=


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