Unformatted text preview: Correlation Operations Do not distribute these notes 214 Prof. R.T. M’Closkey, UCLA Correlation Functions Let u1 and u2 be two signals for which the Fourier transforms exist. We deﬁne two new functions of time as follows
∞ “Autocorrelation": Ru1 u1 (t) = −∞
∞ = −∞ and,
“Crosscorrelation": Ru2 u1 (t) = = u1 (t + τ )u1 (τ )dτ
(82) u1 (τ )u1 (τ − t)dτ ∞
−∞
∞
−∞ u2 (t + τ )u1 (τ )dτ
(83) u2 (τ )u1 (τ − t)dτ Note that the energy in u1 and u2 are given as the t = 0 value of their respective autocorrelation functions,
Energy in u1 =
Energy in u2 = ∞
−∞
∞
−∞ u1 (τ )2 dτ = Ru1 u1 (0)
u2 (τ )2 dτ = Ru2 u2 (0) It can be shown that the auto and crosscorrelation functions also posses Fourier transforms. Another fact is that
the autocorrelation function is symmetric about t = 0: Ruu (−t) =
= ∞
−∞
∞ u(−t + τ )u(τ )dτ
σ u(σ )u(t + σ )dσ −∞ = Ruu (t). Do not distribute these notes 215 Prof. R.T. M’Closkey, UCLA Now consider conducting an experiment in which an energy signal, u, is the input to an asymptotically stable linear
system with impulse response, h, and in which the output, y , is produced. Remarkably, if the system’s input is now
speciﬁed to be Ruu , then the system’s output is Ryu . Pictorially, u Ruu y (t) = Timeinvariant
Linear System ∞ −∞ Timeinvariant
Linear System h(t − τ )u(τ )dτ Ryu The bottom block diagram can be viewed as conducting a “virtual" experiment. This is proven as follows. If Ruu is
the input to a system with impulse response h then the output is
∞
−∞ h(t − τ )Ruu (τ )dτ =
=
=
= ∞
−∞
∞ h(t − τ )
∞ ∞ u(τ + σ )u(σ )dσ dτ −∞ h(t − (s − σ ))u(s)u(σ )dsdσ −∞
∞ −∞
∞ −∞
∞ −∞ h(t + σ − s)u(s)ds u(σ )dσ (84) y (t + σ )u(σ )dσ −∞ = Ryu (t).
Applying the Fourier transform to (84) yields ˆ
ˆ
ˆ
Ryu (ω ) = h(ω )Ruu (ω ). Do not distribute these notes 216 (85) Prof. R.T. M’Closkey, UCLA The Fourier transform of Ruu is =
= ∞ ∞ −∞
∞ −∞
∞ −∞
∞ −∞
∞ −∞ ˆ
F (Ruu ) = Ruu (ω ) = −∞
∞ u(t + τ )e−jω(t+τ ) u(τ )ejωτ dtdτ
u(t + τ )e−jω(t+τ ) dt u(τ )ejωτ dτ u(τ )ejωτ dτ = u(ω )
ˆ
= u(ω )
ˆ u(t + τ )u(τ )dτ e−jωt dt −∞
∞
∗ u(τ )e −jωτ dτ ∗ , (u is assumed to be realvalued) −∞ = u(ω )ˆ (ω )
ˆu
= u(ω )2
ˆ In other words, the Fourier transform of the autocorrelation of u is just the energy spectral density function of u.
The Fourier transform of the crosscorrelation of y and u is called the crossspectrum and is computed to be ˆ
F (Ryu ) = Ryu (ω ) = y (ω )ˆ∗ (ω ).
ˆu
Note that (85) is no surprise because ˆ
ˆu
y (ω ) = h(ω )ˆ(ω ) =⇒ y (ω )ˆ∗ (ω ) = h(ω ) u(ω )ˆ∗ (ω ) .
ˆ
ˆu
ˆu
ˆ
Ryu ˆ
Ruu Example. Consider the system given by the differential equation y + 3y + 100y = 100u.
¨
˙
Let the input be u(t) = e−t µ(t).
The response to this input, and the auto and crosscorrelation and their relation to the system are shown below.
Do not distribute these notes 217 Prof. R.T. M’Closkey, UCLA Input, u
2 1.5 Output, y
2 u 1.5 1 y 1 Timeinvariant
Linear System 0.5 0.5 0 0 0.5 0.5 Impulse response, h
10
1
8 6 4 2 0
seconds 2 4 6 8 1
8 8 6 4 2 h 6 0
seconds 2 4 6 8 2 4 6 8 4
2
0
2
4 Input, R Output, R uu yu 1 6
8 Ruu 1 6 4 2 0
seconds 2 4 0.5 6 8 Ryu
0.5 Timeinvariant
Linear System
0 0.5
8 0 6 4 2 0
seconds Do not distribute these notes 2 4 6 0.5
8 8 218 6 4 2 0
seconds Prof. R.T. M’Closkey, UCLA Correlation Applications • searching for periodic features in a “noisy" signal (autocorrelation)
• searching for relative delays between two signals (crosscorrelation)
• estimating the impulse response from test data in which the input is not impulsive  elaborate on this Do not distribute these notes 219 Prof. R.T. M’Closkey, UCLA ...
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 Spring '06
 TSAO
 Autocorrelation, LTI system theory, Prof. R.T. M’Closkey

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