Chapter 6 Bandpass Random Processes

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Chapter 6Bandpass Random Processes

• Bandpass Random Processes
• PSD of bandpass random processes
• BP Filtered White Noise
• Sinusoids in Gaussian Noise

Huseyin Bilgekul EEE 461 Communication Systems
II Department of Electrical and Electronic
Engineering Eastern Mediterranean University
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Homework Assignments

• Return date December 13, 2005.
• Assignments
• Problem 6-38
• Problem 6-41
• Problem 6-45
• Problem 6-48
• Problem 6-50

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Equivalent Representations of Bandpass Signals

• Remind Equivalent representations of a bandpass
  signal

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Bandpass Random Process

• If x(t) and y(t) are jointly WSS processes, the
  real bandpass process
• Will be WSS stationary if and only if

• If v(t) is a Gaussian random process then g(t),
  x(t) and y(t) are Gaussian processes since they
  are linear functions v(t). However R(t) ?(t) are
  NOT Gaussian because they are NONLIEAR functions
  of v(t).

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Bandpass Random Process

• What happens to a signal at a receiver? How does
  the PSD of the signal after a BPF correspond to
  the signal before the BPF?
• Remind Equivalent representations of a bandpass
  signal

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BPF System

• Bandpass random process can be written as
• With the impulse response

cos(wctq)
Baseband x(t) Inphase
2cos(wctq)
Ideal LPF H0(f)
x
x
v(t) BP Process
v(t) BP Process

sin(wctq)
2sin(wctq)
Ideal LPF H0(f)
x
x
Baseband y(t) Quadrature
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Impulse Response

• Impulse Response
• Transfer Function
• So, x(t) and y(t) are low-pass random processes,
  what else can be deduced?
• Assume theta is uniformly distributed phase noise

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PSD of BP Random Processes

• PSD of x(t) and y(t)

Pv(f-fc)
Pv(ffc)
LPF
-2fc
f
2fc
0
Px(f) or Py(f)
f
-2fc
0
2fc
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PSD of BP Random Processes
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PSD of BP Random Processes
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Properties of WSS BP Processes

• If the narrowband noise is Gaussian, then the
  in-phase x(t) and quadrature y(t) components are
  jointly Gaussian
• If the narrow band noise is wide-sense stationary
  (WSS), then the in-phase and quadrature
  components are jointly WSS.
• In-phase and quadrature components have the same
  PSD.
• In-phase and quadrature components of narrowband
  noise are zero-mean
• Noise comes original signal being passed through
  a narrowband linear filter
• Variance of the processes is the same (area
  under PSD same)

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Properties of WSS BP Processes Continued

• Bandpass PSD from baseband PSD.
• PSD of I and Q from bandpass PSD

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BP White Noise Process

• The PSD of a BP white noise process is No/2. What
  is the PSD and variance of the in-phase and
  quadrature components?
• From the SNR calculations, it is clear that the
  variance of the white noise is

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Sinusoids in Gaussian Noise

• Signal is a sinusoid mixed with narrow-band
  additive white Gaussian noise (AWGN)
• Can be written in terms of ENVELOPE and PHASE
  terms as

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Sinusoids in Gaussian Noise

• In-phase and Quadrature terms of noise Gaussian
  with variance s2.
• Similar transformation to that used for
  calculating the dart board example, the joint
  density can be found in polar coordinates.
• Marginal density of the Envelope is Rician type.
• Approaches a Gaussian if A gtgt s