Geology 5600/6600
Signal Analysis
Last time:
16 Sep 2015
• A process is ergodic if time averages equal ensemble
averages. Properties of weakly WSS ergodic processes:
and hence, statistical properties can be derived by averages
(We don’t need the pdf a priori to evaluate expectation!)
• The convolution of two functions
f and g is defined as:
¥
[ f Ä g](t ) º ò f (t)g(t - t)dt
so we can also write:
-¥
The convolution theorem: convolution in the time
domain is equivalent to multiplication in the freq domain.
© A.R. Lowry 2015
Geology 6600/7600
Signal Analysis
16 Sep 2015
Last time (Continued):
• The Auto-Power Spectrum of a random variable is
the Fourier transform of the autocorrelation function,
given by the Wiener-Khinchin relation:
Sxx (w) =
¥
ò R (t )e
xx
-¥
-iwt
dt
¥
ò
1
Rxx (t ) =
Sxx (w)e +iwt dw
2p -¥
which we can also write as:
© A.R. Lowry 2015
Grokking the Fourier Transform:
Power spectra and the Fourier Transform to the frequency
domain are fundamental to signal analysis, so you should
spend a little time familiarizing yourself with them. For the
following functions, I’d like you to first evaluate the integral
by hand, & then calculate and plot the Fourier transform
using Matlab.
(Send me by class-time Monday Sep 28).
1) Autocorrelation of a zero-mean, WSS white noise
process (use  2 = 3)
2) A constant (use a = 3)
3) A cosine function (use amplitude 1; )
4) A sine function (as above)
5) A box function (0 on [–,–/2] & [/2,]; 1 on [–/2,/2])
Use the continuous Fourier transform to do calculations by
hand; use the DFT for the matlab exercises
Matlab functions you’ll need to learn:
fft
fftshift
Be sure to turn in matlab scripts so if you did something
wrong I can figure out what happened!
Important: Defns of forward/inverse Fourier transform
Useful: Euler’s eqn, function product relations
e-ia = cos(a ) - i sin(a )
1
1
cos a sin b = sin(a + b ) - sin(a - b )
2
2
1
1
sin a sin b = cos(a - b ) - cos(a + b )
2
2
1
1
cos a cos b = cos(a - b ) + cos(a + b )
2
2
Some key things to recognize:
• Integral of a sinusoid on [–,] (hence [–∞,∞]) is always
zero unless the sinusoid is cos(0).
• Ergo, integral of a product of sin’s/cos’s is nonzero only when
. In the case of FT, the Fourier transform Acos(t) and
Asin(t) is thus nonzero only at 0.
• By definition this means that sinusoids are a class of
orthogonal functions. (Their inner product
f ,g º ò f * (t)g(t)dt is zero unless f = g!)
• This makes them a very important class of functions…
As a shorthand for the forward and inverse Fourier
transform, we will use e.g.:
Some properties of the Fourier transform: Recalling Euler’s
relation, e–it = cos(t) – isin(t), the FT of an even
function will always be even (and real), and the FT of an
odd function will always be odd and imaginary.
Hence, because the autocorrelation function Rxx is real
and even, the autopower spectrum Sxx will always be
real and even as well!
Note however this also implies that the power spectrum
does not contain any phase information about the
signal…
Some practical applications of cross-correlation
in geophysics:
If we assume ergodicity of WSS random processes that
are sampled at N regular intervals in time (latter is commonly
the case for geophysical data), the cross-correlation Rxy[l]
at lag l can be estimated as:
Rxy [ l] =
N -l
N -l
N -l
i=1
i=1
i=1
( N - l )å x [ i + l ] y [ i ] - å x [ i + l ] å y [ i ]
ì
N -l
N -l
æ N -l
ö2 üïìï
æ N -l
ö2 üï
ï
í( N - l) x 2 [ i + l] - çç x [ i + l] ÷÷ ýí( N - l) y 2 [ i ] - çç y [ i ] ÷÷ ý
ïî
è i=1
ø ïþïî
è i=1
ø ïþ
i=1
i=1
å
å
å
å
Note that for zero-mean signals, this simplifies to:
N -l
Rxy [ l] =
å x [ i + l] y [i ]
i=1
N -l
N -l
å x [ i + l] å y [ i ]
2
i=1
2
i=1
=
Cxy [ l]
Cxx [ l] Cyy [ l]
Example: The change in reflection travel-time with offset is
given by the Normal MoveOut (NMO) equation:
TNMO
x2 2
= t ( x ) - t0 = 2
+ t0 - t 0
2
V
NMO correction cross-correlates signals shifted using
a range of assumed velocities to identify the “stacking
velocity” that maximizes the cross-correlation.
Example: Pleistocene Lake Bonneville
shorelines exhibit erosional and
depositional features that are easily
recognized in plots of elevation
slope and curvature….
Cross-correlation of slope & curvature
allows accurate estimates of -height!
Generally, because both the autocorrelation function and
the autopower spectrum are real and even, we can
write:
¥
¥
Sxx (w) =
ò
Rxx (t )e-iwt dt = 2
-¥
Rxx (t ) =
1
p
ò R (t ) cos(wt )dt
xx
0
¥
ò S (w) cos(wt )dw
xx
0
The average power of x˜ ( t) is simply E{ x˜ 2 ( t)} = R(0)
We can derive this by recalling that
Rxx (t ) = E{ x˜(t) x˜ (t + t )}
and plugging zero in for  in the Wiener-Khinchin relation:
¥
ò
1
R(0) =
Sxx (w) dw
2 p -¥
~
Hence the average power is the mean square of x.
The Cross-Power Spectrum relating two random
processes x~ and y~ is given by:
Sxy (w) =
¥
ò R (t )e
xy
-iwt
dt
-¥
Recall that Rxy() = Ryx(–): If we substitute
¥
Sxy (w) =
and let –,
ò
Ryx (-t )e-iwt dt
-¥
Sxy (w) =
However
Syx (w) =
so consequently
¥
ò
Ryx (a )e iwa da
-¥
¥
ò R (a )e
yx
-iwa
-¥
Sxy (w) = Syx (-w)
da