Todays Schedule

1
Todays Schedule

• Reading Lathi 11.5,13.1
• Mini-Lecture 1
• Go over Quiz 2
• Mini-Lecture 2
• Bandpass Random Processes -Equivalent filters
• Mini-Lecture 3
• Optimal Threshold Detection

2
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?

Sx(w)
-wc
w
wc
0
3
BPF System

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

cos(wctq)
2cos(wctq)
xc(t)
Ideal LPF H0(w)
x
x
x(t)
x(t)

sin(wctq)
2sin(wctq)
xs(t)
Ideal LPF H0(w)
x
x
4
Impulse Response

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

H0(w)
1
2pB
4pB
H (w)
1
-wc
wc
0
5
PSD of BP Random Processes

• PSD of xc(t) and xs(t)

Sx(w)
-wc
wc
w
Sx(w-wc) Sx(wwc)
LPF
-2wc
w
2wc
0
Sxc(w) or Sxs(w)
w
-2wc
0
2wc
6
Mean and variance of narrowband noise

• 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)

7
Properties

• If the narrowband noise is Gaussian, then the
  in-phase (nI) and quadrature (nQ) are jointly
  Gaussian.
• If the narrow band noise is wide-sense stationary
  (WSS), then the in-phase and quadrature
  components are jointly WSS.

8
Cross Correlations

• Definition Cross Correlation
• Definition Jointly Stationary

9
Jointly Stationary Properties

• Properties
• Uncorrelated
• Orthogonal
• Independent if x(t1) and y(t2) are independent
  (joint distribution is product of individual
  distributions)

10
Activity

• Working with a partner come up with a list of
  communications systems that will need bandpass
  analysis for performance assessment.
• The PSD of a BP white noise process is N/2. What
  is the PSD and variance of the in-phase and
  quadrature components?

11
Example White noise process

• The PSD of a BP white noise process is N/2. What
  is the PSD and variance of the in-phase and
  quadrature components?

12
Variance of White Noise Process

• From the SNR calculations, it is clear that

13
Sinusoid in Gaussian Noise

• Signal is a sinusoid mixed with narrow-band
  additive white Gaussian noise (AWGN)
• Can be written as

14
Example (continued)

• In-phase and Quadrature terms of noise Gaussian
  with variance s2
• In a similar transformation to that used for
  calculating the dart board example, the joint
  density can be found in polar coordinates

15
Marginal Density of E

• Rician Density
• Approaches a Gaussian if Agtgts

16
Digital Communications Systems in Noise

• Analog Comm Goal is to reproduce the waveform
  accurately
• Figure of Merit output signal to noise ratio
• Digital Communications Goal is to decide which
  waveform was transmitted accurately
• Figure of Merit Probability of error in making
  this decision at the receiver

17
Detection Bipolar Signaling
s(t)

• If 1, send p(t)
• If 0, send p(t)
• Received signal r(t) /- p(t) n(t)
• Optimal threshold?
• Noise is additive, Gaussian noise

Peak Detect and Sample
Detector

n(t)
18
PE Calculations

• P(1/0) prob of detecting a 1 when 0 sent
• P(ngtAp)
• P(0/1) prob of detecting a 0 when 1 sent
• P(nlt-Ap)

19
Bipolar

• Total Error Probability
• If P(1)P(0)0.5

Q(x)
0.5
x
20
Minimize PE

• To minimize PE, must maximize r since Q decreases
  monotonically as r increases
• Ap is the signal amplitude and sn is the rms
  noise.
• Goal filter signal to enhance signal and reduce
  noise power

21
Linear Filtering
Linear Network h(t) H(f)
x(t)/- p(t)n(t) X(f)
y(t)/- po(t)no(t) Y(f)

• Recall

22
Signal Amplitude

• The goal is maximize
• (same as maximizing r)

/- po(t)no(t)
/- p(t)n(t)
ttm
h(t) H(f)
Threshold Detector
p(t)n(t)
p(t)n(t)
p(t)
p(t)
TS
TS
23
Signal and Noise Calculation

• Signal output
• Output noise power or variance
• So, the ratio becomes

24
Next Time

• Reading Lathi 13.1, 13.2
• Mini-Lecture 1
• Optimum Threshold Detection
• Mini-Lecture 2
• Optimum Binary receivers
• Mini-Lecture 3
• Optimum Binary receivers