TELECOMMUNICATIONS

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TELECOMMUNICATIONS

• Dr. Hugh Blanton
• ENTC 4307/ENTC 5307

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POWER SPECTRAL DENSITY
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Summary of Random Variables

• Random variables can be used to form models of a
  communication system
• Discrete random variables can be described using
  probability mass functions
• Gaussian random variables play an important role
  in communications
• Distribution of Gaussian random variables is well
  tabulated using the Q-function
• Central limit theorem implies that many types of
  noise can be modeled as Gaussian

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Random Processes

• A random variable has a single value. However,
  actual signals change with time.
• Random variables model unknown events.
• A random process is just a collection of random
  variables.
• If X(t) is a random process then X(1), X(1.5),
  and X(37.5) are random variables for any specific
  time t.

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Terminology

• A stationary random process has statistical
  properties which do not change at all with time.
• A wide sense stationary (WSS) process has a mean
  and autocorrelation function which do not change
  with time.
• Unless specified, we will assume that all random
  processes are WSS and ergodic.

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Spectral Density
Although Fourier transforms do not exist for
random processes (infinite energy), but does
exist for the autocorrelation and cross
correlation functions which are non-periodic
energy signals. The Fourier transforms of the
correlation is called power spectrum or spectral
density function (SDF).
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Review of Fourier Transforms
Definition A deterministic, non-periodic signal
x(t) is said to be an energy signal if and only if
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The Fourier transform of a non-periodic energy
signal x(t) is The original signal can
be recovered by taking the inverse Fourier
transform
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Remarks and Properties
The Fourier transform is a complex function in w
having amplitude and phase, i.e.
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Example 1
Let x(t) eat u(t), then
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Autocorrelation

• Autocorrelation measures how a random process
  changes with time.
• Intuitively, X(1) and X(1.1) will be more
  strongly related than X(1) and X(100000).
• Definition (for WSS random processes)
• Note that Power RX(0)

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Power Spectral Density

• P(w) tells us how much power is at each frequency
• Wiener-Klinchine Theorem
• Power spectral density and autocorrelation are a
  Fourier Transform pair!

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Properties of Power Spectral Density

• P(w) ? 0
• P(w) P(-w)

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Gaussian Random Processes

• Gaussian Random Processes have several special
  properties
• If a Gaussian random process is wide-sense
  stationary, then it is also stationary.
• Any sample point from a Gaussian random process
  is a Gaussian random variable
• If the input to a linear system is a Gaussian
  random process, then the output is also a
  Gaussian process

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Linear System

• Input x(t)
• Impulse Response h(t)
• Output y(t)

x(t)
h(t)
y(t)
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Computing the Output of Linear Systems

• Deterministic Signals
• Time Domain y(t) h(t)
  x(t)
• Frequency Domain Y(f)Fy(t)X(f)H(f)
• For a random process, we still relate the
  statistical properties of the input and output
  signal
• Time Domain RY(?) RX(?)h(?)
  h(-?)
• Frequency Domain PY(?) PX(?)?H(f)2

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Power Spectrum or Spectral Density Function (PSD)

• For deterministic signals, there are two ways to
  calculate power spectrum.
• Find the Fourier Transform of the signal, find
  magnitude squared and this gives the power
  spectrum, or
• Find the autocorrelation and take its Fourier
  transform
• The results should be the same.
• For random signals, however, the first approach
  can not be used.

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Let X(t) be a random with an autocorrelation of
Rxx(t) (stationary), then and
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• Properties
• SXX(w) is real, and SXX(0) ? 0.
• Since RXX(t) is real, SXX(-w) SXX(w), i.e.,
  symmetrical.
• Sxx(0)

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Special Case
For white noise, Thus,
SXX(w)
sX2
???
t
w
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Example 1
Random process X(t) is wide sense stationary and
has a autocorrelation function given by Find
SXX.
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Example 1
RXX(t)
sX2
t
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Example 2
Let Y(t) X(t) N(t) be a stationary random
process, where X(t) is the actual signal and N(t)
is a zero mean, white gaussian noise with
variance sN2 independent of the signal. Find SYY.
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Correlation in the Continuous Domain

• In the continuous time domain

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• Obtain the cross-correlation R12 (t) between the
  waveform v1 (t) and v2 (t) for the following
  figure.

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• The definitions of the waveforms are
• and

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• We will look at the waveforms in sections.
• The requirement is to obtain an expression for
  R12 (t)
• That is, v2 (t), the rectangular waveform, is to
  be shifted right with respect to v1 (t) .

t
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The situation for is shown in the figure. The
figure show that there are three regions in the
section for which v2(t) has the consecutive
values of -1, 1, and -1, respectively. The
boundaries of the figure are
t
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v(t)
The situation for is shown in the figure. The
figure show that there are three regions in the
section for which v2(t) has the consecutive
values of 1, -1, and 1, respectively. The
boundaries of the figure are
T/2
1.0
t
T
t-T/2
-1.0
t
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0.25
T/2
T
t
-0.25
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• Let X(t) denote a random process. The
  autocorrelation of X is defined as

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Properties of Autocorrelation Functions for
Real-Valued, WSS Random Processes

• 1. Rx(0) EX(t)X(t) Average Power
• 2. Rx(t) Rx(-t). The autocorrelation function
  of a real-valued, WSS process is even.
• 3. Rx(t)? Rx(0). The autocorrelation is
  maximum at the origin.

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Autocorrelation Example
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?
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Correlation Example
t
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t0.012 y(t.3./24.-t./2.2/3) plot(t,y)
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t0.012 y(-t.3./24.t./2.2/3) plot(t,y)
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tint0 tfinal10 tstep.01
ttinttsteptfinal x5((tgt0)(tlt4)) subplot
(3,1,1), plot(t,x) axis(0 10 0
10) h3((tgt0)(tlt2)) subplot(3,1,2),plot(t,h)
axis(0 10 0 10) axis(0 10 0 5) t22tinttstep
2tfinal yconv(x,h)tstep subplot(3,1,3),plot(
t2,y) axis(0 10 0 40)
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Matched Filter
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Matched Filter

• A matched filter is a linear filter designed to
  provide the maximum signal-to-noise power ratio
  at its output for a given transmitted symbol
  waveform.
• Consider that a known signal s(t) plus a AWGN
  n(t) is the input to a linear time-invariant
  (receiving) filter followed by a sampler.

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• At time t T, the sampler output z(t) consists
  of a signal component ai and noise component n0.
  The variance of the output noise (average noise
  power) is denoted by s02, so that the ratio of
  the instantaneous signal power to average noise
  power, (S/N)T, at time t T is

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Random Processes and Linear Systems

• If a random process forms the input to a
  time-invariant linear system, the output will
  also be a random process.
• The input power spectral density GX(f) and the
  output spectral density GY(f) are related as
  follows

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• We wish to find the filter transfer function
  H0(f) that maximizes
• We can express the signal ai(t) at the filter
  output in terms of the filter transfer function
  H(f) and the Fourier transform of the input
  signal, as

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• If the two-sided power spectral density of the
  input noise is N0/2 watts/hertz, then we can
  express the output noise power as
• Thus, (S/N)T is

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• Using Schwarzs inequality,
• and

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• Or
• where

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• The maximum output signal-to-noise ratio depends
  on the input signal energy and the power spectral
  density of the noise.
• The maximum output signal-to-noise ratio only
  holds if the optimum filter transfer function
  H0(f) is employed, such that

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• Since s(t) is a real-valued signal, we can use
  the fact that
• and

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• to show that
• Thus, the impulse response of a filter that
  produces the maximum output signal-to-noise ratio
  is the mirror image of the message signal s(t),
  delayed by the symbol time duration T.

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s(t)
s(-t)
h(t)s(T-t)
t
t
t
T
-T
T
Signal waveform
Mirror image of signal waveform
Impulse response of matched filter
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• The impulse response of the filter is a delayed
  version of the mirror image (rotated on the t 0
  axis) of the signal waveform.
• If the signal waveform is s(t), its mirror image
  is s(-t), and the mirror image delayed by T
  seconds is s(T-t).

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• The output of the matched filter z(t) can be
  described in the time domain as the convolution
  of a received input wavefrom r(t) with the
  impulse response of the filter.

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Substituting ks(T-t) with k chosen to be unity
for h(t) yields.
When T t
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• The integration of the product of the received
  signal r(t) with a replica of the transmitted
  signal s(t) over one symbol interval is known as
  the correlation of r(t) with s(t).

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• The mathematical operation of a matched filter
  (MF) is convolution a signal is convolved with
  the impulse response of a filter.
• The mathematical operation of a correlator is
  correlation a signal is correlated with a
  replica of itself.

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• The term matched filter is often used
  synonymously with correlator.
• How is that possible when their mathematical
  operations are different?

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s1(t)
s0(t)
A
A
Tb
Tb
-A
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h0s1(Tb -t)
h0s0(Tb -t)
A
A
Tb
Tb
-A
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y0(t)
y0(t)
A2Tb
Tb
2Tb
Tb
2Tb