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Matched Filtering and Digital Pulse Amplitude Modulation (PAM)
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Part  I Part  II
Transmitting One Bit ,[object Object],[object Object],receive ‘0’ bit receive‘1’ bit How does the receiver decide which bit was sent?  b t A ‘ 1’ bit Additive Noise Channel input output x ( t ) y ( t )  b - A ‘ 0’ bit t  b - A  b t A
Transmitting One Bit ,[object Object], h t 1 Model channel as LTI system with impulse response   h ( t ) t Assume that  T h  <  T b  b t A ‘ 1’ bit  b - A ‘ 0’ bit Communication Channel input output x ( t ) y ( t ) - A T h receive ‘0’ bit t  h +  b  h t receive‘1’ bit  h +  b  h A T h
Transmitting Two Bits (Interference) ,[object Object],[object Object],[object Object],Assume that  T h  <  T b t  b A ‘ 1’ bit ‘ 0’ bit   b * = - A T h t  b ‘ 1’ bit ‘ 0’ bit  h +  b Intersymbol interference  h t 1
Preventing ISI at Receiver ,[object Object],[object Object],[object Object],[object Object],[object Object], h t 1 Assume that  T h  <  T b * = t  b A ‘ 1’ bit ‘ 0’ bit  h +  b  h t - A T h  b ‘ 1’ bit ‘ 0’ bit  h +  b
Digital 2-level PAM System ,[object Object],[object Object],Transmitter Channel Receiver b i Clock  T b PAM g ( t ) h ( t ) c ( t ) 1 0 a k  {- A , A } s(t) x(t) y(t) y(t i ) AWGN w ( t ) Decision Maker Threshold    Sample at t=iT b bits Clock   T b pulse shaper matched filter
Matched Filter ,[object Object],[object Object],[object Object],Additive white Gaussian noise (AWGN) with zero mean and variance  N 0  /2 T  is the symbol period g ( t ) Pulse signal w ( t ) x ( t ) h ( t ) y ( t ) t  =  T y ( T ) Matched filter
Matched Filter Derivation ,[object Object],[object Object],[object Object],[object Object],T  is the symbol period g ( t ) Pulse signal w ( t ) x ( t ) h ( t ) y ( t ) t  =  T y ( T ) Matched filter
Power Spectra ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],t 1 x ( t ) 0 T s  R x (  ) - T s T s T s
Power Spectra ,[object Object],[object Object],[object Object],[object Object],N = 16384;  % finite no. of samples gaussianNoise = randn(N,1); plot( abs(fft(gaussianNoise)) .^ 2 ); approximate noise floor
Matched Filter Derivation ,[object Object],[object Object],f Noise power spectrum  S W ( f ) AWGN Matched filter T  is the symbol period g ( t ) Pulse signal w ( t ) x ( t ) h ( t ) y ( t ) t  =  T y ( T ) Matched filter
Matched Filter Derivation ,[object Object],[object Object],[object Object],[object Object], a b
Matched Filter Derivation T  is the symbol period
Matched Filter ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Matched Filter for Rectangular Pulse ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],t=nT T Sample and dump h ( t )  = ___
Digital 2-level PAM System ,[object Object],[object Object],Transmitter Channel Receiver b i Clock  T b PAM g ( t ) h ( t ) c ( t ) 1 0 a k  {- A , A } s(t) x(t) y(t) y(t i ) AWGN w ( t ) Decision Maker Threshold    Sample at t=iT b bits Clock   T b pulse shaper matched filter
Digital 2-level PAM System ,[object Object],[object Object],[object Object],[object Object],actual value (note that t i  = i T b ) intersymbol interference (ISI) noise p ( t ) is centered at origin
Eliminating ISI in PAM ,[object Object],[object Object],[object Object],[object Object],[object Object]
Eliminating ISI in PAM ,[object Object],[object Object],[object Object],[object Object],ideal Nyquist channel impulse response dampening adjusted by rolloff factor  
Bit Error Probability for 2-PAM ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],r ( t ) =  h ( t ) *  r ( t ) h ( t ) s ( t ) Sample at t = nT b Matched filter v ( t ) r ( t ) r ( t ) r n
Bit Error Probability for 2-PAM ,[object Object],[object Object],[object Object],[object Object],Probability density function (PDF) See slide 13-16 0 -
Bit Error Probability for 2-PAM  ,[object Object],[object Object],Q function on next slide PDF for  N (0, 1) T b  = 1 0
Q Function ,[object Object],[object Object],[object Object],Erfc[ x ] in Mathematica erfc( x ) in Matlab
Bit Error Probability for 2-PAM ,[object Object],[object Object],[object Object],See slide 8-17 T b  = 1
PAM Symbol Error Probability ,[object Object],[object Object],[object Object],[object Object],[object Object],2-PAM d -d 4-PAM Constellation points with receiver decision boundaries d  d 3 d  3 d
PAM Symbol Error Probability ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],two-sided power spectral density of AWGN channel noise after matched filtering and sampling
Alternate Derivation of Noise Power Noise power T  =  T sym Filtered noise  2    (  1 –  2 ) Optional
PAM Symbol Error Probability ,[object Object],[object Object],M-2 interior points 2 exterior points
Visualizing ISI ,[object Object],[object Object],[object Object],[object Object]
Eye Diagram for 2-PAM ,[object Object],[object Object],M =2 t  -  T sym Sampling instant Interval over which it can be sampled Slope indicates sensitivity to timing error Distortion over zero crossing Margin over noise t + T sym t
Eye Diagram for 4-PAM  Due to startup transients. Fix is to discard first few symbols equal to number of symbol periods in pulse shape. 3d d -d -3d

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Lecture13

  • 1. Matched Filtering and Digital Pulse Amplitude Modulation (PAM)
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  • 14. Matched Filter Derivation T is the symbol period
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  • 28. Alternate Derivation of Noise Power Noise power T = T sym Filtered noise  2  (  1 –  2 ) Optional
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  • 32. Eye Diagram for 4-PAM Due to startup transients. Fix is to discard first few symbols equal to number of symbol periods in pulse shape. 3d d -d -3d