Chapter Three

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Chapter Three

• Baseband Demodulation/Detection

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Signal and Noise

• Error performance degradation in communication
  systems
• Filtering effect at the transmitter, channel, and
  receiver, which causes intersymbol interference
  (ISI)
• Electrical noise and interference produced by a
  variety sources
• Thermal noise (Gaussian distributed, its two-side
  spectral density is N0/2, which is flat for all
  frequencies)
• Demodulation and detection
• Demodulation is recovery of a waveform
• Detection means the decision-making process of
  selecting the digital meaning of the waveform

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Two basic Steps in the Demodulation / Detection
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Receiving Filter and Decision Making

• The goal of the receiving filter is to recover a
  baseband pulse with the best possible
  signal-to-noise ratio (SNR), free of any ISI
• Matched filter or correlator is the optimum
  receiver
• Equalizing filter is only needed for systems
  where channel-induced ISI can distort the signals
• Decision making is regarding the digital meaning
  of the sample
• Assume the input noise is a Gaussian random
  process
• Decision making is performed according to the
  threshold measurement hypothesis H1 is chosen if
  z(T)gtg, and hypothesis H2 is chosen if z(T) ltg

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Conditional Probability Density
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Vectorial Representation of Signal Waveform
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Waveform Representation in Orthonormal Functions
Assume there are M signal waveforms and N
orthonormal basis functions
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Signal and Noise in Vector Space
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Example Orthogonal Representation of Waveforms
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Representing White Noise
AWGN noise can be partitioned into two components
In other words, may be thought of the
noise that is effectively tuned out by
the detector
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Detection of Binary Signal in Gaussian Noise
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Components of the Decision Theory Problem
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Decision Theory

• Likelihood ratio test
• Maximum Likelihood Criterion

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Maximum Likelihood Binary Decision

• Binary decision rule (Assume the binary
  transmitted waveforms are s1(t) and s2(t) )
• where a1 is the signal component of z(T) when
  s1(t) is transmitted, and a2 is the signal
  component of z(T) when s2(t) is transmitted. The
  threshold level g0 is the optimum threshold for
  minimizing the probability of error.
• ( Reference to Appendix B.3 Signal Detection
  Example )

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Error Probability

• According to Fig.3.2, the error probability can
  be derived by
• Where Q(x) is called the complementary error
  function, and

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Matched Filter

• The goal of the matched filter is to provide the
  maximum signal-to-noise power ratio
• Signal-to-noise power ratio
• Transfer function and impulse response of matched
  filter
• Maximum signal-to-noise power ratio

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Correlation Realization of the Matched filter
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Optimizing Error Performance

• Minimize the error probability
  is to maximize
• where (a1-a2) is the difference of the desired
  signal components at the filter output at time
  tT

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Signaling Characteristic in Error Probability

• Error probability function is rewritten by

• Define a time cross-correlation coefficient r as
  a measure of similarity between two signals s1(t)
  and s2(t)