telecommunications l.
Download
Skip this Video
Download Presentation
TELECOMMUNICATIONS

Loading in 2 Seconds...

play fullscreen
1 / 71

TELECOMMUNICATIONS - PowerPoint PPT Presentation


  • 217 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'TELECOMMUNICATIONS' - Patman


Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
telecommunications

TELECOMMUNICATIONS

Dr. Hugh Blanton

ENTC 4307/ENTC 5307

summary of random variables
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

Dr. Blanton - ENTC 4307 - Correlation 3

random processes
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.

Dr. Blanton - ENTC 4307 - Correlation 4

terminology
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.

Dr. Blanton - ENTC 4307 - Correlation 5

spectral density
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).

Dr. Blanton - ENTC 4307 - Correlation 6

review of fourier transforms
Review of Fourier Transforms

Definition: A deterministic, non-periodic signal x(t) is said to be an energy signal if and only if

Dr. Blanton - ENTC 4307 - Correlation 7

slide8

The Fourier transform of a non-periodic energy signal x(t) is

The original signal can be recovered by taking the inverse Fourier transform

Dr. Blanton - ENTC 4307 - Correlation 8

remarks and properties
Remarks and Properties

The Fourier transform is a complex function in whaving amplitude and phase, i.e.

Dr. Blanton - ENTC 4307 - Correlation 9

example 1
Example 1

Let x(t) = eat u(t), then

Dr. Blanton - ENTC 4307 - Correlation 10

autocorrelation
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)

Dr. Blanton - ENTC 4307 - Correlation 11

power spectral density12
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!

Dr. Blanton - ENTC 4307 - Correlation 12

properties of power spectral density
Properties of Power Spectral Density
  • P(w)  0
  • P(w) = P(-w)

Dr. Blanton - ENTC 4307 - Correlation 13

gaussian random processes
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

Dr. Blanton - ENTC 4307 - Correlation 14

linear system
Linear System
  • Input: x(t)
  • Impulse Response: h(t)
  • Output: y(t)

x(t)

h(t)

y(t)

Dr. Blanton - ENTC 4307 - Correlation 15

computing the output of linear systems
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

Dr. Blanton - ENTC 4307 - Correlation 16

power spectrum or spectral density function psd
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.

Dr. Blanton - ENTC 4307 - Correlation 17

slide18

Let X(t) be a random with an autocorrelation of Rxx(t) (stationary), then

and

Dr. Blanton - ENTC 4307 - Correlation 18

slide19

Properties:

  • SXX(w) is real, and SXX(0)  0.
  • Since RXX(t) is real, SXX(-w) = SXX(w), i.e., symmetrical.
  • Sxx(0) =

Dr. Blanton - ENTC 4307 - Correlation 19

special case

RXX(t)

sX2d(t)

Special Case

For white noise,

Thus,

SXX(w)

sX2



t

w

Dr. Blanton - ENTC 4307 - Correlation 20

example 121
Example 1

Random process X(t) is wide sense stationary and has a autocorrelation function given by:

Find SXX.

Dr. Blanton - ENTC 4307 - Correlation 21

example 122
Example 1

RXX(t)

sX2

t

Dr. Blanton - ENTC 4307 - Correlation 22

example 2
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.

Dr. Blanton - ENTC 4307 - Correlation 24

correlation in the continuous domain
Correlation in the Continuous Domain
  • In the continuous time domain

Dr. Blanton - ENTC 4307 - Correlation 25

slide26

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.

Dr. Blanton - ENTC 4307 - Correlation 26

slide27
The definitions of the waveforms are:

and

Dr. Blanton - ENTC 4307 - Correlation 27

slide28
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) .

Dr. Blanton - ENTC 4307 - Correlation 28

t

slide29

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

Dr. Blanton - ENTC 4307 - Correlation 29

slide33

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

Dr. Blanton - ENTC 4307 - Correlation 33

slide37

0.25

T/2

T

t

-0.25

Dr. Blanton - ENTC 4307 - Correlation 37

slide38
Let X(t) denote a random process. The autocorrelation of X is defined as

Dr. Blanton - ENTC 4307 - Correlation 38

properties of autocorrelation functions for real valued wss random processes
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.

Dr. Blanton - ENTC 4307 - Correlation 39

autocorrelation example

y(t)

(t+t)/2

t/2

t

t

2

2-t

Autocorrelation Example

Dr. Blanton - ENTC 4307 - Correlation 40

slide41

y(t)

(t+t)/2

t/2

t

t

2

0

2-t

Dr. Blanton - ENTC 4307 - Correlation 41

slide42

Dr. Blanton - ENTC 4307 - Correlation 42

correlation example

y(t)

1

0

1

6

7

2

3

4

5

-1

Correlation Example

t

Dr. Blanton - ENTC 4307 - Correlation 43

slide44

t=0:.01:2;

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

plot(t,y)

Dr. Blanton - ENTC 4307 - Correlation 44

slide46

t=0:.01:2;

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

plot(t,y)

Dr. Blanton - ENTC 4307 - Correlation 46

slide49

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])

Dr. Blanton - ENTC 4307 - Correlation 49

matched filter
Matched Filter

Dr. Blanton - ENTC 4307 - Correlation 50

matched filter51
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.

Dr. Blanton - ENTC 4307 - Correlation 51

slide52
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

Dr. Blanton - ENTC 4307 - Correlation 52

random processes and linear systems
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:

Dr. Blanton - ENTC 4307 - Correlation 53

slide54
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

Dr. Blanton - ENTC 4307 - Correlation 54

slide55
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

Dr. Blanton - ENTC 4307 - Correlation 55

slide56
Using Schwarz’s inequality,
    • and

Dr. Blanton - ENTC 4307 - Correlation 56

slide57
Or
    • where

Dr. Blanton - ENTC 4307 - Correlation 57

slide58
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

slide60
Since s(t) is a real-valued signal, we can use the fact that
  • and

Dr. Blanton - ENTC 4307 - Correlation 60

slide61
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.

Dr. Blanton - ENTC 4307 - Correlation 61

slide62

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

Dr. Blanton - ENTC 4307 - Correlation 62

slide63
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).

Dr. Blanton - ENTC 4307 - Correlation 63

slide64
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.

Dr. Blanton - ENTC 4307 - Correlation 64

slide65

Substituting ks(T-t) with k chosen to be unity for h(t) yields.

When T = t

Dr. Blanton - ENTC 4307 - Correlation 65

slide66
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).

Dr. Blanton - ENTC 4307 - Correlation 66

slide67
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.

Dr. Blanton - ENTC 4307 - Correlation 67

slide68
The term matched filter is often used synonymously with correlator.
  • How is that possible when their mathematical operations are different?

Dr. Blanton - ENTC 4307 - Correlation 68

slide69

s1(t)

s0(t)

A

A

Tb

Tb

-A

Dr. Blanton - ENTC 4307 - Correlation 69

slide70

h0=s1(Tb-t)

h0=s0(Tb-t)

A

A

Tb

Tb

-A

Dr. Blanton - ENTC 4307 - Correlation 70

slide71

y0(t)

y0(t)

A2Tb

Tb

2Tb

Tb

2Tb

Dr. Blanton - ENTC 4307 - Correlation 71