3f4 power and energy spectral density
DESCRIPTION
3F4 Power and Energy Spectral Density. Dr. I. J. Wassell. Power and Energy Spectral Density. The power spectral density (PSD) S x ( w ) for a signal is a measure of its power distribution as a function of frequency - PowerPoint PPT PresentationTRANSCRIPT
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3F4 Power and Energy Spectral Density
Dr. I. J. Wassell
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Power and Energy Spectral Density
• The power spectral density (PSD) Sx() for a signal is a measure of its power distribution as a function of frequency
• It is a useful concept which allows us to determine the bandwidth required of a transmission system
• We will now present some basic results which will be employed later on
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PSD
• Consider a signal x(t) with Fourier Transform (FT) X()
dtetxX tj )()(
• We wish to find the energy and power distribution of x(t) as a function of frequency
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Deterministic Signals• If x(t) is the voltage across a R=1 resistor,
the instantaneous power is,2
2
))(())((
txR
tx
• Thus the total energy in x(t) is,
dttx 2)(Energy
• From Parseval’s Theorem,
dfX2
)(Energy
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Deterministic Signals• So,
dfX2
)(Energy
dffE
dffX
)2(
)2(2
Where E(2f) is termed the Energy Density Spectrum (EDS), since the energy of x(t) in the range fo to fo+fo is,
oo ffE )2(
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Deterministic Signals• For communications signals, the energy is
effectively infinite (the signals are of unlimited duration), so we usually work with Power quantities
• We find the average power by averaging over time
2
2
2))((1lim
power AverageT
T
T dttxTT
Where xT(t) is the same as x(t), but truncated to zero outside the time window -T/2 to T/2
• Using Parseval as before we obtain,
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Deterministic Signals
2
2
2))((1lim
power AverageT
T
T dttxTT
dffS
dfT
fX
T
dffXTT
x
T
T
)2(
)2(lim
)2(1lim
2
2
Where Sx() is the Power Spectral Density (PSD)
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PSD
T
X
TS T
x
2)(lim
)(
The power dissipated in the range fo to fo+fo is,
oox ffS )2(
And Sx(.) has units Watts/Hz
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Wiener-Khintchine Theorem• It can be shown that the PSD is also given by
the FT of the autocorrelation function (ACF), rxx(),
derS jxxx )()(
Where,
2
2
)()(1lim
)(
T
Txx dttxtx
TTr
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Random Signals
• The previous results apply to deterministic signals
• In general, we deal with random signals, eg the transmitted PAM signal is random because the symbols (ak) take values at random
• Fortunately, our earlier results can be extended to cover random signals by the inclusion of an extra averaging or expectation step, over all possible values of the random signal x(t)
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PSD, random signals
• The W-K result holds for random signals, choosing for x(t) any randomly selected realisation of the signal
T
XE
TS T
x
])([lim)(
2
Where E[.] is the expectation operator
Note: Only applies for ergodic signals where the time averages are the same as the corresponding ensemble averages
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Linear Systems and Power Spectra• Passing xT(t) through a linear filter H() gives
the output spectrum,)()()( TT XHY
• Hence, the output PSD is,
T
XHE
TT
YE
TS TT
y
])()([lim])([lim)(
22
T
XE
THS T
y
])([lim)()(
22
)()()(2 xy SHS