sampling decimation seismometer amplifier aaa filter daa ...€¦ · removing the acausal response...
TRANSCRIPT
Digital LTI systemAnalog LTI system
DecimationSampling
DAA filterAAA filterAmplifierSeismometer
Filtering (Digital Systems)
x n[ ]
X~ k[ ]
X z[ ]
y n[ ] = h[n] *x[n]
Y~ k[ ] T~ k[ ]X~ k[ ]=
Y z[ ] T z[ ]X z[ ]=
T~ k[ ]
T z[ ]
h n[ ]
input outputfilter
Convolution of Sequences
Discrete Fourier Transform (DFT)
z-Transform
The z-Transform
Z x n[ ]{ } x n[ ]z n–
n ∞–=
∞∑ X z( )= =
T z( ) Z y n[ ]{ }Z x n[ ]{ }--------------------- Y z( )
X z( )-----------= =
Transfer Function
Properties
(shifting theorem) x n n0–[ ] zn
0–
X z( )⇔
x n–[ ] X 1 z⁄( )⇔
x1 n[ ]*x2 n[ ] x1 m[ ]x2 n m–[ ] X1 z( ) X2 z( )⋅⇔
m ∞–=
∞∑=
(convolution theorem)
T z( ) Y z( )X z( )-----------
blz l–l 0=
M
∑
akz k–k 0=
N
∑------------------------= =
Rational Transfer Function
akk 0=
N
∑ y n k–[ ] blx n l–[ ]l 0=
M
∑=
Linear Difference Equation⇔
⇔
akk 0=
N
∑ y n k–[ ] blx n l–[ ]
l 0=
M
∑=
non-recursive
FIR Filtersa0 1= ak 0= k 1≥and for( )
a0y n[ ] akk 1=
N
∑ y n k–[ ]+ blx n l–[ ]
l 0=
M
∑=
Recursive/Non-Recursive Filters
y n[ ]aka0------
k 1=
N
∑ y n k–[ ]bla0------x n l–[ ]
l 0=
M
∑+–=
y n[ ] blx n l–[ ]
l 0=
M
∑=
recursive
( IIR Filters )
T z( ) Y z( )X z( )-----------
blz l–l 0=
M
∑
akz k–k 0=
N
∑------------------------= =
Rational Transfer Function
FIR filter
y n[ ] blx n l–[ ]l 0=
M
∑ x n[ ]*b[l]= =
a0 1= ak 0= k 1≥and for( )
T z( ) blz l–l 0=
M
∑ b0 1 clz 1––( )l 1=
M
∏= =
z M– b0 z cl–( )l 1=
M
∏⋅=
Special Case:
Linear Difference Equation
• FIR filters : + Always stable. - Steep filters need many coefficients.+ Both causal and noncausal filters can be implemented. + Filters with given specifications are easy to implement!
• IIR filters : - Potentially unstable and subject to quantization errors. + Steep filters can easily be implemented with a few coefficients. Speed. - Filters with given specifications are in general, difficult, if not impossible, to implement exactly(!).
-t1 -t2
akk 0=
N
∑ y n k–[ ] blx n l–[ ]
l 0=
M
∑=
non-recursive
FIR Filtersa0 1= ak 0= k 1≥and for( )
a0y n[ ] akk 1=
N
∑ y n k–[ ]+ blx n l–[ ]
l 0=
M
∑=
Recursive/Non-Recursive Filters
y n[ ]aka0------
k 1=
N
∑ y n k–[ ]bla0------x n l–[ ]
l 0=
M
∑+–=
y n[ ] blx n l–[ ]
l 0=
M
∑=
recursive
( IIR Filters )
How can we remove these effects?
Back to Transfer function:
Frequency
Waveform Properties and Root Positions
minimum delay/phase
maximum delay/phase
mixed delay/phase(zero phase)
mixed delay/phase
mixed delay/phase
flip
time
axis
Problem: Two-Sided IRCure: Change IR into Minimum Phase
Zero Phase FIR Filter
Methods:
1) Add phase of Minimum Phase Filter to trace spectrum
2) Recursive Filtering of time inverted trace
Removing the acausal response of a Zero Phase FIR Filter
Linear PhaseFIR filter
zlp
X z( ) Y z( )
F z( ) time delay lp
00
Linear PhaseFIR filter
X z( )Y z( )
F z( )
lp00 0
linear phaseshift correction
Linear Phase and Zero Phase Filter:
Linear-phase FIR Filter:
Zero-phase FIR Filter:
= Time delay correction by samples
F z( )
F z( ) F z( ) zlp⋅=
zlp lp
General rule:Any filter can be expressed by convolution of its minimum phase and maximum phase component.
roots within UC roots outside UC
=> Linear phase FIR Filter: F z( ) Fmax z( ) Fmin z( )⋅=
maximum phase component -> left-sided (acausal) componentminimum phase component -> right-sided (causal) component
Removal of acausal response:
Replace maximum phase component Fmax(z) by its minimum phase equivalent MinPhase{Fmax(z)}.
MinPh ase Fmax z( ){ } =
Answer: flip maximum phase component in time. In terms of z-transform:
=> MinPhase Fmax z( ){ } Fmax 1 z⁄( )=
minimum phase
maximum phase
z-transform representation of digital seismogram
Y~ z( ) F z( ) zlp X~ z( )⋅ ⋅=
digital seismogram before FIR filtering
digital seismogramafter FIR filtering but before decimation
LP FIR
0P FIR
LP correction
x~ n[ ] = unfiltered digital seismogram~X z( ) = z-transform of the input signal x~ n[ ]
F z( ) = z-transform of the linear phase FIR filter
y~ n[ ] = filtered digital seismic trace before decimation1
1 This is a fictitious signal since decimation is commonly done while filtering.
= z-transform of Y~ z( ) y~ n[ ]
F z( ) zlp⋅corresponds to a zero phase filter in which the linear phase component of F(z) is corrected.
In practice: Treat time shift zlp separately from F(z)
Removing the maximum phase component of a FIR filter
Principle:
Fmax z( ) -> MinPh ase Fmax z( ){ }
‘Corrected’ seismogram Y(z): Y z( ) 1Fmax z( )-------------------- Fmax 1 z⁄( ) Y
~z( )⋅ ⋅=
Problem: Since Fmax(z) has only zeros outside the unit circle, 1/ Fmax(z) will have poles outside the unit circle.
Solution: Flip time axis.
Y 1 z⁄( ) 1Fmax 1 z⁄( )--------------------------- Fmax z( ) Y~ 1 z⁄( )⋅ ⋅=
Impulse response corresponding to 1/ Fmax(z) becomes a stable causal sequence in nominal time and the deconvolution of the maximum phase component Fmax(z) poses no stability problems.
The difference equation
Fmax 1 z⁄( ) Y 1 z⁄( )⋅ Fmax z( ) Y~ 1 z⁄( )⋅=
Rewrite to
A' z( ) Y ' z( )⋅ B' z( ) X ' z( )⋅=
Written as convolution sum:
A' z( )
<=>
Fmax 1 z⁄( ) Y' z( ) Y 1 z⁄( )
B' z( )
<=>
Fmax z( ) X' z( ) Y~ 1 z⁄( )
<=>
<=>
a' k[ ] y' i k–[ ]⋅k ∞–=
∞
∑ b' l[ ] x' i l–[ ]⋅l ∞–=
∞
∑=
Assumption: F(z) contains mx zeros outside the unit circle=> wavelets a’[k] and b’[k] will be of length mx + 1.
a' k[ ] y' i k–[ ]⋅
k 0=
mx
∑ b' l[ ] x' i l–[ ]⋅
l 0=
mx
∑=
Rearrange to
y' i[ ] a' 0[ ]⋅ a' k[ ] y' i k–[ ]⋅
k 1=
mx
∑+ b' l[ ] x' i l–[ ]⋅
l 0=
mx
∑=
which is equivalent to
y' i[ ] a' k[ ]a' 0[ ]------------ y' i k–[ ]⋅
k 1=
mx
∑– b' l[ ]a' 0[ ]------------ x' i l–[ ]⋅
l 0=
mx
∑+=
This is
for k = 1 to mxa k[ ]a' k[ ]a' 0[ ]------------–
fmax mx k–[ ]
fmax mx[ ]--------------------------------= =
and
for l = 0 to mxb l[ ] b' l[ ]a' 0[ ]------------
fmax l[ ]fmax mx[ ]-----------------------= =
The convolution sum
y' i[ ] a k[ ] y' i k–[ ]⋅k 1=
mx
∑ b l[ ] x' i l–[ ]⋅l 0=
mx
∑+=
y' i[ ] = the time reversed ‘corrected’ sequence.
To obtain y[i] flip y’[i] back in time.
‘Corrected’ seismogram: Y z( ) F~
z( ) Y~
z( )⋅=
‘Correction’Y~z( ) Y z( )
F~z( )
lp0
filter
lp0
F~ z( ) = changes the linear phase filter F(z) into a minimum phase filter
=> onset of output signal is advanced by lp samples.
To account for this time shift in the corrected seismogram:
a) change time tag of ‘corrected’ trace
or
b) delay ‘corrected’ trace by lp samples.
What is needed for ‘correction’?
mx + 1 coefficients of the maximum phase portion of the a linear phase FIR filter
How to get maximum phase component?
F z( ) b lz l–
l 0=
m
∑ b0 1 c lz l––( )l 1=
m
∏= =
mn zeros inside UC (cimin) <=> Fmin(z)
mx zeros outside UC (cimax) <=> Fmax(z)
Fmax z( ) fimaxz i–
i 0=
mx
∑ b0 1 cimaxz i––( )
i 1=
mx
∏= =
From zeros ouside UC -> polynomial Fmax(z). fmax[l]for l = 0 to mx: coefficients of polynomial Fmax(z).
Correction Procedure
‘decimation while filtering’
x n[ ]y n[ ]
digitalsignal
Fs :f1decimateddigitalsignal
Stage 1f1:f2
Stage 2fn 1– :fn
Stage n...
Fs
f1 f2
fnintermediate data streams
initial samplingfrequency
final sampling frequency
Full Correction (not always necessary):
interpolation -> fn 1– -> correction for FIR filter stage n
interpolation -> fn 2– -> correction for FIR filter stage n-1
interpolation -> f1 -> correction for FIR filter stage 2interpolation -> Fs -> correction for FIR filter stage 1
finally: decimation back to fn
. . .
.
Decimation stages
a) b)
a) b)
1 423
13
515
14
16 17
12
7 8109
11
6 1 2
34
5
1514
17
16
12
611
78
109
13
Conclusions
FIR filter generated precursory artefacts:
• can become impossible to be identified visually
• can have similar scaling properties as nucleation phases
Zero - phase FIR filters in general
• affect the determination of all onset properties (onset times, onset polarities)
For the interpretation of onset properties (onset times, onset polarities, nucleation phases, etc.) the acausal response of thezero-phase FIR filter has to be removed
but not
for waveform analysis.
Consequence