telecommunications dr. hugh blanton entc 4307/entc 5307
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TELECOMMUNICATIONS
Dr. Hugh Blanton
ENTC 4307/ENTC 5307
POWER SPECTRAL DENSITY
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Summary of Random VariablesSummary 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 ProcessesRandom 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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TerminologyTerminology
• 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 DensitySpectral 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 TransformsReview of Fourier Transforms
Definition: A deterministic, non-periodic signal x(t) is said to be an energy signal if and only if
dttxE )(2
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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
dtetxXtx tj )()()}({
deXtxX tj)()()}({1
)()( Xtx
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Remarks and PropertiesRemarks and Properties
The Fourier transform is a complex function in having amplitude and phase, i.e.
jeXX )()(
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Example 1Example 1
Let x(t) = eat u(t), then
jae
ja
dtedteetx
tja
tjatjat
11
)(
0
00
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AutocorrelationAutocorrelation
• 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)
)()()( tXtXERX
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Power Spectral DensityPower Spectral Density
• P() tells us how much power is at each frequency
• Wiener-Klinchine Theorem:
• Power spectral density and autocorrelation are a Fourier Transform pair!
)()( RP
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Properties of Power Spectral DensityProperties of Power Spectral Density
• P() 0
• P() = P(-)
dPPower )(
Dr. Blanton - ENTC 4307 - Correlation 14
Gaussian Random ProcessesGaussian 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 SystemLinear 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 SystemsComputing 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)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() (stationary), then
and
deRS jXXXX
)()(
deSR jXXXX
)(
2
1)(
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Properties:
(1)SXX() is real, and SXX(0) 0.
(2)Since RXX(t) is real, SXX(-) = SXX(), i.e., symmetrical.
(3)Sxx(0) =
(4)
dSR XXXXX )(2
1)0(2
dRXX
)(
)()( XXXX StR
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Special CaseSpecial Case
For white noise,
Thus,
)()( 2XXXR
22 )()( Xtj
XXX dteS
RXX()
X
SXX()
X
Dr. Blanton - ENTC 4307 - Correlation 21
Example 1Example 1
Random process X(t) is wide sense stationary and has a autocorrelation function given by:
Find SXX.
eR XXX2)(
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Example 1Example 1
eR XXX2)(
RXX()
X
deedee
deedeRS
jX
jX
jjXXXX
0
2
0
2
)()(
Dr. Blanton - ENTC 4307 - Correlation 23
2
2
2
0
202
0
202
0
0
2
0
0
2
1
2
1
1
1
1
11
11
11
X
XjXjX
jXjX
jjX
jjX
jje
je
j
djej
djej
dededeedee
Dr. Blanton - ENTC 4307 - Correlation 24
Example 2Example 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 N
2 independent of the signal.
Find SYY.
2
2
)()(
)()()(
NXXYY
NXXYY
SS
RR
Dr. Blanton - ENTC 4307 - Correlation 25
Correlation in the Continuous DomainCorrelation in the Continuous Domain
• In the continuous time domain
dttxtx
TR )()(
1)( 21
012
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• Obtain the cross-correlation R12 () between the waveform v1 (t) and v2 (t) for the following figure.
T 2T 3T
v1(t)
t
1.0
v2(t)
tT 2T 3T
1.0
-1.0
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• The definitions of the waveforms are:
and
TtforTttv 0/)(1
TtTfor
Ttfortv
2/1
2/01)(2
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• We will look at the waveforms in sections.• The requirement is to obtain an expression
for R12 ()
• That is, v2 (t), the rectangular waveform, is to be shifted right with respect to v1 (t) .
Dr. Blanton - ENTC 4307 - Correlation 29
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:
20 Tt
Tt
Tt
t
2
v(t)
T
1.0
-1.0
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T
T
T
T
T
T
tdtT
tdtT
tdtT
dtT
t
Tdt
T
t
Tdt
T
t
TR
22
2
20
2
2
2
012
111
11
11
11
)(
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42
2
42
2
2
1
2
2
222
2
2
1
2
1
2
1
2
1)(
2222
222
2
22222
2
2
2
2
22
2
0
2
212
TTT
TT
T
TTT
T
t
T
t
T
t
TR
T
T
T
Dr. Blanton - ENTC 4307 - Correlation 32
20
4
1
4
2
2
4
2
1
42
2
42
2
2
1
42
2
42
2
2
1)(
22
2
22
2
2
2222
222
212
Tfor
TT
TT
T
TTT
TT
T
TTT
TT
Tr
Dr. Blanton - ENTC 4307 - Correlation 33
t
v(t)
T
1.0
-1.0
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:
TtT 2
Tt
t
Tt
2
Dr. Blanton - ENTC 4307 - Correlation 34
T
T
T
T
T
T
tdtT
tdtT
tdtT
dtT
t
Tdt
T
t
Tdt
T
t
Tr
22
2
2
02
2
2
012
111
11
11
11
)(
Dr. Blanton - ENTC 4307 - Correlation 35
222
222
22
22222
2
2
2
2
2
2
2
0
2
212
42
2
42
2
2
1
2022
1
2
1
2
1
2
1)(
TTTTT
T
TTTT
t
T
t
T
t
Tr
T
T
T
Dr. Blanton - ENTC 4307 - Correlation 36
TT
forT
TTT
Tr
24
3
4
2
2
4
2
1)( 2
2
212
Dr. Blanton - ENTC 4307 - Correlation 37
t
T/2 T
0.25
-0.25
Dr. Blanton - ENTC 4307 - Correlation 38
• Let X(t) denote a random process. The autocorrelation of X is defined as
Dr. Blanton - ENTC 4307 - Correlation 39
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() = Rx(-). The autocorrelation function of a real-valued, WSS process is even.
• 3. |Rx()| Rx(0). The autocorrelation is maximum at the origin.
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Autocorrelation ExampleAutocorrelation Example
t 2
2-t
y(
t
Dr. Blanton - ENTC 4307 - Correlation 41
t 2
2-t
y(
t
0
dt
Rx 22)(
dtRt
x
2
0
2
4
1)(
dtdRt t
x
2
0
2
0
2
4
1
4
1)(
t
x
tR
2
0
23
234
1)(
2
2
3
2
4
1)(
23 tttRx
Dr. Blanton - ENTC 4307 - Correlation 42
2
44
3
6128
4
1)(
2332 ttttttRx
tt
ttttRx 22
2324
3
8
4
1)( 2
332
3
82
64
1)(
3
tt
Rx
12
8
224)(
3
tt
Rx 3
2
224)(
3
tt
Rx
Dr. Blanton - ENTC 4307 - Correlation 43
Correlation ExampleCorrelation Example
t
y(t
0 1 2 3 4 5 6 7
1
-1
Dr. Blanton - ENTC 4307 - Correlation 44
t=0:.01:2;y=(t.^3./24.-t./2.+2/3);plot(t,y)
Dr. Blanton - ENTC 4307 - Correlation 45
Dr. Blanton - ENTC 4307 - Correlation 46
t=0:.01:2;y=(-t.^3./24.+t./2.+2/3);plot(t,y)
Dr. Blanton - ENTC 4307 - Correlation 47
Dr. Blanton - ENTC 4307 - Correlation 48
Dr. Blanton - ENTC 4307 - Correlation 49
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 FilterMatched Filter
Dr. Blanton - ENTC 4307 - Correlation 51
Matched FilterMatched 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 52
• 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 0
2, so that the ratio of the
instantaneous signal power to average noise power, (S/N)T, at time t = T is
20
2
i
T
a
N
S
Dr. Blanton - ENTC 4307 - Correlation 53
Random Processes and Linear SystemsRandom 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:
2)()()( fHfGfG XY
Dr. Blanton - ENTC 4307 - Correlation 54
• 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
20
2
i
T
a
N
S
dfefSfHa ftj
i2)()(
Dr. Blanton - ENTC 4307 - Correlation 55
• 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
dffH
N 2020 )(
2
dffHN
dfefSfH
N
Sftj
T 20
2
2
)(2
)()(
Dr. Blanton - ENTC 4307 - Correlation 56
• Using Schwarz’s inequality,
• and
dffSdffHdfefSfH ftj 22
2
2 )()()()(
0
2)(2
N
dffS
N
S
T
Dr. Blanton - ENTC 4307 - Correlation 57
• Or
• where0
2max
N
E
N
S
T
dffSE
2
)(
Dr. Blanton - ENTC 4307 - Correlation 58
• 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
tfj
tfj
efkSth
or
efkSfHfH
21
20
)(*)(
)(*)()(
Dr. Blanton - ENTC 4307 - Correlation 59
tfj
tfj
efkSth
or
efkSfHfH
21
20
)(*)(
)(*)()(
Dr. Blanton - ENTC 4307 - Correlation 60
• Since s(t) is a real-valued signal, we can use the fact that
• and
)(*)( fXfX
tfjefXttx 20 )()(
Dr. Blanton - ENTC 4307 - Correlation 61
• 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.
elsewhere
TttTksth
0
0)()(
Dr. Blanton - ENTC 4307 - Correlation 62
Tt
s(t)
-Tt
s(-t) h(t)=s(T-t)
tT
Signal waveform Mirror image of signal waveform
Impulse response of matched filter
Dr. Blanton - ENTC 4307 - Correlation 63
• 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 64
• 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.
t
dthrthtrtz0
)()()()()(
Dr. Blanton - ENTC 4307 - Correlation 65
t
t
dtTsr
dtTsrtz
0
0
)()(
)(()()(
Substituting ks(T-t) with k chosen to be unity for h(t) yields.
When T = t
T
dsrtz0
)()()(
Dr. Blanton - ENTC 4307 - Correlation 66
• 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 67
• 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 68
• 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 69
s0(t) s1(t)
Tb
Tb
A A
-A
Dr. Blanton - ENTC 4307 - Correlation 70
h0=s0(Tb -t) h0=s1(Tb -t)
TbTb
A A
-A
Dr. Blanton - ENTC 4307 - Correlation 71
y0(t)
Tb
A2Tb
2Tb
y0(t)
Tb 2Tb
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