Notes on Generalized Transforms- Sandra Chapman (MPAGS: Time series analysis)
Some Generalizations of Fourier Theory
see also Dudok de Wit review
Higher Order Spectra (summary)
Consider the nonlinear system:
!u(x,t)
= f (u(x,t))
!x
decompose "
!u(x,t)
= $ g(# 1 )u(x,t " # 1 )d# 1
!x
+ $$ g(# 1 , # 2 )u(x,t " # 1 )u(x,t " # 2 )d# 1 d# 2
+ $$$ g(# 1 , # 2 , # 3 )u(x,t " # 1 )u(x,t " # 2 )u(x,t " # 3 )d# 1 d# 2 d# 3
+...
Taking the DFT we obtain the Volterra series:
!u p
= " pu p + # " kl uk ul$ k +l, p + # " klm uk ul um$ k +l + m, p + ...
!x
k,l
k,l, m
with
u p = u(x, % p )
The leading term is a linear (cf Fourier) decomposition. The rest are mode coupling. Ensemble
average over x and consider a homogeneous medium:
setting
!
=0
!x
" p u *pu p +
#"
k +l = p
kl
uk ul uk* +l +
#
k +l + m = p
" klm uk ul um uk* +l + m + .. = 0
Now recall convolution:
N "1
g k ! hk = # gu hk"u
u=0
The DFT is Gm H m where Gm is the DFT of g k etc.,
N !1
This relates to cross correlation: C! = " g k hk+ !
DFT is Gm* H m
k =0
N !1
auto correlation:
R! = " xk xk +!
DFT is S m* S m (the power spectrum)
k =0
1
Notes on Generalized Transforms- Sandra Chapman (MPAGS: Time series analysis)
generalise these to:
bispectrum
Bkl = S k Sl S k*+l
trispectrum
Tklm = S k Sl S m S k*+l +m
There are normalized versions, eg:
bicoherence
=
Bkl
S k Sl
2
2
S k +l
2
= bkl
One can obtain averaged bispectra in the same way as averaged power spectra – average over M
consecutive intervals.
N !1
Alternatively, use the 2nd order autocorrelation: M !1!2 = " xk xk +!1 xk +!2
k =0
The bispectrum is the twice applied DFT on M (cf convolution theorem to prove).
Physical meaning (see also Dudok de Wit review)
m
" !m
N!t
Thus : S k ! S ( ! k ), Sl ! S ( !l ), S k +l ! S ( ! k + !l )
Recall frequency f m =
This tests for coherence between beating oscillations
Envelope ! k + ! L
Difference ! k ! ! L
where !k ,l = !k + !l , for strongest mode we see strong bicoherence.
2
Notes on Generalized Transforms- Sandra Chapman (MPAGS: Time series analysis)
For wave- wave coupling we need to satisfy physical constraints in both space and time:
Principal domain of bicoherence- due to symmetries:
3
Notes on Generalized Transforms- Sandra Chapman (MPAGS: Time series analysis)
Example- water waves: (courtesy T. Dudok De Wit)
Real part and imaginary part of bicoherence
asymmetry wrt time reversal- imaginary bispectra
4
Notes on Generalized Transforms- Sandra Chapman (MPAGS: Time series analysis)
up- down asymmetry- real bispectra
so wave is not steepening- simply have a ‘wobble’ on amplitude.
Linear Time Invariant (LTI) Filters
A way to generalize the Fourier world to other transforms.
So far – everything flowed from the idea:
"
#
x( t ) =
S m e2!ifm t
fm =
m
T
fm =
m
N!t
m=!"
Sm =
with
1
T
T /2
" x( t )e
!2!if m t
dt
!T / 2
Discrete version:
xk =
1
N!t
N "1
#S
m
e2!imk / N
m=0
N "1
S m = !t # xk e"2!ikm/ N
tk = k!t
k =0
and orthogonality property of the Fourier kernel ! = e 2 !ift
A framework to consider other ! Write filter as: L [ x ( t ) ] = y ( t )
where L is a linear operator, with the properties:
1.
2.
3.
L [ ax ] = aL [ x ]
scale preserving
distributative (superposition)
time invariant
L [ x1 + x2 ] = L [ x1 ] + L [ x2 ]
L[ x ( t )] = y ( t )
⇒
Then in general:
L[ x ( t + ! )] = y ( t + ! )
N
" N
%
L $ ! ! p x p ( t ) ' = ! ! p L "# x p ( t ) %&
$ p=1
' p=1
#
&
Now consider for input x to the filter, the Fourier kernel:
! f ( t ) = e 2 !ift
f = const
y f = L[! f ] = y f ( t )
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Notes on Generalized Transforms- Sandra Chapman (MPAGS: Time series analysis)
then: y f ( t + " ) = L [ ! f ( t + " ) ] = L [ e 2 !if " ! f ( t ) ] = e 2 !if " y f ( t )
This is just the "shift theorem" from Fourier theory.
Now let t = 0 .
y f ( " ) = e 2 !if " y f ( 0 )
true for any ! so let ! …t
y f ( t ) = e 2 !ift y f ( 0 )
so y f ( 0 ) is a constant – there is one value for each f.
We can consider a spectrum of values of y f ( 0 )
y f ( 0 ) = G f = A f ei ! f
Then
generally:
y f ( t ) = G f ! " f ( t ) = L #$ " f ( t ) %& = G f e2!ift
! f " eigenvalues %#%%
$ of L
G f " eigenvectors %%
%&
We can think of any transform in this way.
We need to identify an appropriate ! f ( t ) . Two methods:
1.
Choose ! f ( t ) as basis on which we expand, ie: y ( t ) = " y f ( t ) = " G f ! f ( t )
f
f
! f may be orthogonal – chosen for "appropriate" properties.
#
This is equivalent to the transform: y ( t ) =
$ G ( f )!( f ,t ) d f
"#
2.
again, ! ( f , t ) = e 2 !ift for the Fourier transform.
Perform an SVD (single value decomposition, or principle component analysis) on
L, so that the ! f are generated by the data (beyond the scope of this course).
6
Notes on Generalized Transforms- Sandra Chapman (MPAGS: Time series analysis)
Application of LTI filters – coloured noises
x( t ) =
Consider some
"
#
S m e2!ifm t
m=!"
The discrete version of this is
xk =
1 N "1
S m e2!imk / N
#
N!t m=0
fm =
m
N!t
We now have:
y ( t ) = L !" x ( t ) #$ =
&
'G
m
S m e2!ifm t
m=%&
N "1
yk =
1
# G S e2!imk / N .
N!t m=0 m m
Now for a stochastic process we have
1 N "1
xk =
# D e2!imk / N
N!t m=0 m
where
Dm = S m ei!m
Dm ,!m are random iid processes (these are stationary)
then as above
y f (t ) = Gf xf (t )
yk ,m = Gm xk ,m
or
where xk ,m = S m e 2 !imk / N
The G f are just constants.
Then for process xk stochastic, there will be a process yk also stochastic, with spectral components
Gf Df .
The xk , yk should share the statistical properties .
More formally, since the R-S integral gives, for stochastic dz x (see notes on stationarity)
1/ 2
x( t ) = "
!1/ 2
e2!ift dz x ( f
)
then
1/ 2
y( t ) = "
!1/ 2
e2!ift G ( f ) dz x ( f
="
1/ 2
2
so this filter generates "coloured" noises if G ~
!1/ 2
ie
dz y ( f )
2
= G 2 ( f ) dz x ( f )
)
e2!ift dz y ( f
7
)
1
f!