Transcript Linear Systems (SISO)

Geology 6600/7600
Signal Analysis
21 Sep 2015
Last time:
• The Cross-Power Spectrum relating two random
~
~
processes x and y is given by:
Sxy (w) =
¥
ò
Rxy (t )e-iwt dt
-¥
and has the property Sxy (w) = Syx (-w). Generally the
cross-power spectrum will be complex-valued (i.e., it will
contain phase as well as amplitude information).
• The Coherence Function is defined as:
2
Sxy (w)
and ranges from zero (uncorrelated
2
g xy (w) =
Sxx (w) Syy (w) processes) to one.
• The spectrum of the sum of two random processes is
Szz = Sxx + Syx + Sxy + Syy
© A.R. Lowry 2015
Linear Systems with Stochastic Inputs:
Signal analysis has a long list of applications in systems
engineering, electrical engineering, etc. (with signals
including sound, images, telecommunications).
Consider for example a deterministic signal (i.e., a
signal f for which f(t) is fixed and can be determined at
every t from some mathematical expression) that has
¥
finite energy:
2
E=
ò f (t)dt < ¥
-¥
For example, power dissipated in a  resistor is P = I 2R = I 2.
Energy relates to power as
Ett12 =
t2
ò
t1
P( t)dt =
t2
ò
t1
I 2 ( t)dt
Consider the energy in the frequency domain ():
¥
ò f (t)e
2
-iwt
dt =
-¥
¥
ò f (t) f (t)e
-iwt
-¥
¥
ò
1
dt =
F ( l ) F (w - l )dl
2p -¥
Notation convention: we will denote the Fourier
transform of a (lower-case letter) function or process by
an (upper-case letter) amplitude.
If we let  so that
¥
ò
-¥
¥
ò
1
f ( t) dt =
F (l ) F (-l )dl
2 p -¥
2
¥
and if f(t) is real F() = F*(–) (exer: Use F (l ) = ò f (t)e-ilt dt)
-¥
then:
¥
ò
¥
ò
¥
ò
2
1
1
1
F (l ) F (-l )dl =
F ( l ) F * (l ) dl =
F (l ) dl
2p -¥
2p -¥
2p -¥
This gives us Parseval’s Relation:
E=
¥
ò
¥
ò
2
1
f ( t) dt =
F ( l ) dl
2 p -¥
2
-¥
This is the Energy Density for . More generally, for
2
a signal x:
¥
2
Exx (w) = X (w) =
ò
-¥
x ( t)e-iwt dt
is the Energy Density Spectrum.
The integral of energy in a particular frequency band [1,2]
Eww12
-w
ò
1 1
1
=
Exx (w) dw +
2p -w
2p
2
w2
ò E (w)dw
xx
w1
is the Power Spectral Density over that frequency
band.
A Linear System is any system for which an output
signal y(t) represents a convolution of the system’s
“impulse response” (or “system function”) h(t)
with an input x(t).
x(t)
h(t)
y(t)
Then if the input and system functions are known, the
output can be found by:
1) y(t) = h(t)  x(t)
2) Y() = H()X()
From 2), we can get |Y()|2 = |H()X()|2 = |H()|2|X()|2
and the energy density spectrum is:
2
Eyy (w) = H (w) Exx (w)
Autocorrelation for a (deterministic) finite-energy signal:
fxx (t ) =
has properties:
¥
ò x(t) x(t + t )dt
-¥
fxx (t ) = fxx (-t )
fxx (0) ³ fxx (t )
(similar to those for random processes).
Cross-correlation:
¥
fxy (t ) =
Correlation Theorem:
ò x(t) y(t + t )dt
-¥
Note the primary difference from random processes is that
we can integrate over ±∞ for expectation operator E{•}!
From these relations we have two ways to find an energy
density spectrum:
Exx (w) = X (w) X * (w)
Exercise: find the energy spectrum of a pulse function
x(t) = APT(t) in two different ways:
x
–T/2
T/2
t
Random Signals:
(Finite Power; Infinite Energy)
~
Consider a linear system with a random process input x(t)
and impulse response h(t):
x˜ ( t)
h( t)
y˜ ( t)
~
In this instance, the output y(t) will also be a random process
corresponding to the convolution:
y˜ ( t) = h (t) Ä x˜ ( t) º
¥
ò h(a ) x˜ (t - a )da
-¥
If this is a (time-domain) causal system, that’s equivalent
to integrating from t = 0 to ∞, as h(t) = 0 for t < 0.
The cross-correlation of the input and output from a SISO
linear system is now:
E{ y˜ ( t + t ) x˜ (t)} =
So:
Ryx (t ) =
¥
ò h(a )E{ x˜(t + t - a ) x˜ (t)}da
-¥
¥
ò h(a ) R (t - a )da
xx
-¥
is the convolution of the autocorrelation of the input with the
impulse response,
Ryx (t ) = Rxx (t ) Ä h(t )
The autocorrelation of the output signal is:
E{ y˜ ( t + t ) y˜ (t)} =
Ryy (t ) =
¥
ò h(a )E{ x˜(t + t - a ) y˜ (t)}da
-¥
¥
ò h(a ) R (t - a )da
xy
-¥
Ryy (t ) = Rxy (t ) Ä h(t )