Todays Schedule

1
Todays Schedule

• Reading Lathi 10.4-10.5
• Mini-Lecture 1
• Central limit theorem
• Mini-Lecture 2
• Correlation
• Activity
• Mini-Lecture 3
• Solving problems

2
Central Limit Theorem

• Under certain conditions, sum of a large number
  independent RVs tends to a Gaussian RV
• Lets look at two RVs
• If z xy, then fz(z) fx(x) fy(y)
• Why?

3
Why?
y
Look at probability that z lies in a small range
zdzxy
x
z xy
Within strip, y is constrained by x y z - x
4
Convolution
y
Note dz dy and the strip has constant width
zdzxy
x
Term in brackets is the pdf of z
z xy
5
Summing RVs is Convolution
fx1

• Same logic holds for sums of more RVs
• With each additional convolution the result
  approaches the normal distribution

x1
f x1 x2
x1 x2
6
Central Limit Theorem

• Sums of independently distributed RVs tend to
  become a Gaussian Distribution
• Holds for most distributions as long as they are
  unimodal (one hump) and tend to zero at /-
  infinity
• One of the main reasons that the Normal
  distribution is used

7
Covariance and Correlation Coefficient

• Covariance
• Correlation Coefficient

8
Correlation

• Correlation is used to determine how closely one
  input resembles another
• Used in matched filter type demodulators to
  decide which waveform was sent

R1
R0
R -1
9
Correlation-Type Demodulators

• Decomposes the received signal and noise into
  N-dimensional vectors
• Received signal
• R(t) si(t) n(t)
• Multiply the signal by a basis function that
  spans the space

x
x
To detector
x
10
Correlation Receivers

• Received signal after demodulation is
• Branch with the highest match is the detected
  output
• Derivation requires some understanding of random
  processes so we will do later

11
Matched Filters

• Can replace the correlators with a linear filter
  of the form
• Output of these filters

To detector
12
Matched Filters

• This is the time autocorrelation function of s(t)
• It peaks at T
• For AWGN (additive white Gaussian noise) the
  filter with the matched impulse response,
  maximizes the output SNR

A
A
0
0
T
T
Signal, s(t)
h (t) s(T-t)
y(T)
0
T
13
Activity-Movies

• You can learn more about densities and
  distributions at
• http//www.public.iastate.edu/dicook/JSS/paper/pa
  per.html
• Calibrating your eyes

14
Bivariate Normal Distribution

• If r0, f(x,y) f(x) f(y)

15
Bivariate Normal
Rho 0 rho 0.5
16
Next Time

• Reading Lathi 10.6-10.7
• Mini-Lecture 1
• Linear estimation
• Mini-Lecture 2
• Optimal receivers
• Activity

17
Example Rayleigh Distribution

• Given two independent, identically distributed
  (IID) Gaussian RVs, x and y
• Find the PDFs of the amplitude and phase of these
  variables (polar coordinates)

18
Example Rayleigh Distribution

• Given two independent, identically distributed
  (IID) Gaussian RVs, x and y
• Find the PDFs of the amplitude and phase of these
  variables (polar coordinates)

19
Expected Value of a Function

• The expected value (or ensemble average) of
  yh(x) is

20
Error in PCM

• Each sample encoded by a group of n binary pulses
• Two sources of error
• Quantization noise
• Channel noise