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TELECOMMUNICATIONS. Dr. Hugh Blanton ENTC 4307/ENTC 5307. POWER SPECTRAL DENSITY. 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

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### TELECOMMUNICATIONS

Dr. Hugh Blanton

ENTC 4307/ENTC 5307

### POWER SPECTRAL DENSITY

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 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 whaving 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)=F{y(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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RXX(t)

sX2d(t)

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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v1(t)

1.0

t

T

2T

3T

v2(t)

1.0

t

T

2T

3T

-1.0

• 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) .

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t

v(t)

T/2

1.0

T

-1.0

t+T/2

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

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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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Dr. Blanton - ENTC 4307 - Correlation 38

Properties of Autocorrelation Functions for Real-Valued, WSS Random Processes
• 1. Rx(0) = E[X(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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y(t)

(t+t)/2

t/2

t

t

2

2-t

Autocorrelation Example

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y(t)

(t+t)/2

t/2

t

t

2

0

2-t

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y(t)

1

0

1

6

7

2

3

4

5

-1

Correlation Example

t

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t=0:.01:2;

y=(t.^3./24.-t./2.+2/3);

plot(t,y)

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t=0:.01:2;

y=(-t.^3./24.+t./2.+2/3);

plot(t,y)

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tint=0;

tfinal=10;

tstep=.01;

t=tint:tstep:tfinal;

x=5*((t>=0)&(t<=4));

subplot(3,1,1), plot(t,x)

axis([0 10 0 10])

h=3*((t>=0)&(t<=2));

subplot(3,1,2),plot(t,h)

axis([0 10 0 10])

axis([0 10 0 5])

t2=2*tint:tstep:2*tfinal;

y=conv(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 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)Tis

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Using Schwarz’s 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

Dr. Blanton - ENTC 4307 - Correlation 58

• 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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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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• 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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h0=s1(Tb-t)

h0=s0(Tb-t)

A

A

Tb

Tb

-A

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y0(t)

y0(t)

A2Tb

Tb

2Tb

Tb

2Tb

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