error function filter robert l. coldwell university of florida lsc august 2005
DESCRIPTION
Error Function Filter Robert L. Coldwell University of Florida LSC August 2005. LIGO-G050352-00-Z. Reducing the number of data points. Introduction. Discrete Fourier Transform Definitions The Nyquist Theorem Ideal Filter/Error Function Filter Pulsar numbers - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/1.jpg)
Error Function FilterRobert L. Coldwell
University of Florida
LSC August 2005
Reducing the number of data points
LIGO-G050352-00-Z
![Page 2: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/2.jpg)
Introduction
• Discrete Fourier Transform Definitions
• The Nyquist Theorem
• Ideal Filter/Error Function Filter
• Pulsar numbers
• Dramatically reducing the number of needed data points
![Page 3: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/3.jpg)
/ 2 1
/ 2
exp 2N
m i m ii N
TD f d t j f t
N
/ 2 1
/ 2
1exp 2
N
i m m im N
d t D f j f tT
/i tt iT N i /mf m T
/ 2
/ 2
exp 2T
m
T
d t j f t dt
1/ 2
1/ 2
exp 2t
t
iD f j ft df
DFT definitions
1j
![Page 4: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/4.jpg)
/ 2 1 / 2 1
/ 2 / 2
1exp 2
N N
n m n t i i m in N i N
m
X f S f f x t s t j f tT
XS f
/ 2 1 / 2 1
/ 2 / 2
1exp 2
N N
t n m n i i m in N i N
m
x t s t t X f S f j f tT
xs t
ConvolutionFrequency
Time
![Page 5: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/5.jpg)
Nyquist Theorem1
N MN
tT N N
Define
2 2m
N Nmf m
T
1 Loosely based on Alan V. Oppenheimer, Ronald W. Schafer with John R. Buck, Discrete-time Signal Processing – Prentice Hall Signal Processing Series – Alan V. Oppenheimer, editor, Second edition 1999 – first 1989
![Page 6: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/6.jpg)
Data versus time
![Page 7: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/7.jpg)
/ 2 1
,/ 2
N
k k Mii N
s t M
/ 2 1/ 2 1
,/ 2 / 2
/ 2 1
/ 2
/ 2 1
/ 2
exp 2
exp 2
exp 2
NN
m k Mi m ki N k N
N
ti N
N
ti N
S f M j f t
m Tj Mi
T N
mj i
N
The function s(t)
Note upper limit of N/2-1
This sum is N for m=kN, zero otherwise
![Page 8: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/8.jpg)
/ 2 1 / 2 1
, ,/ 2 / 2
M M
m m kN m kNk M k M
TS f N T
N
samp k cont k kd t d t s t
/ 2 1
/ 2
1 N
samp m cont n m nn N
D f D f S f fT
This is set up for convolution
![Page 9: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/9.jpg)
/ 2 1/ 2 1
,/ 2 / 2
1 NM
samp m cont n n m kNk M n N
D f T D fT
kNt
kN kf
T
Nyquist theorem in frequency
/ 2 1/ 2
,/ 2 / 2
NM
cont n n m kNk M n N
D f
/ 2 1
/ 2
M
cont m kNk M
D f
Inserting S(f)
Dsamp is periodic by construction
![Page 10: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/10.jpg)
Back Transform
• The back transform over all N points is not wanted since it will produce the spiky function transformed forward.
• Define an ideal filter as
1/ 2 1/ 2
1 1/ 2 1/ 2
0 1/ 2
t
t t
t
f
H f f
f
![Page 11: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/11.jpg)
/ 2 1
/ 2
1 M
samp m m cont m kN mk M
D f H f D f H fT
With appropriate restrictions
cont m samp m mD f D f H f
This form is a setup for convolution
The sum is over M as in N=MN
But in any case define
H m samp m mD f D f H f
![Page 12: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/12.jpg)
/ 2 1
/ 2
/ 2 1
/ 2
/ 2 1
/ 2
N
H m samp i m ii N
N
cont i ik k i ii N
N
t cont k m k k tk N
d t m d t h t t
M d t h t t t i
d t h t t t k
Using the convolution theorem
The Nyquist Theorem in time
The time tm is any time, the time tk is for a data point.
This is where the dramatic reduction in data points needed takes place.
![Page 13: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/13.jpg)
Ideal Filter/Error Function Filter
The ideal f0 to f1 filter
H is not required to extend from -1/2t to 1/2t
![Page 14: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/14.jpg)
Details of the filter near the two ends
![Page 15: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/15.jpg)
/ 2 1
0 1 0 1/ 2
1; , ; , exp 2
N
i m m ii N
h t f f H f f f j f tT
1
0
1
0 1
0 1
1; , exp 2
1exp 2 exp 2
2
m m
im m
h t f f j ftT
m mj t j t
T T T
Ideal filter transformation
A few steps are skipped involving 1/(1-exp(-j21/T)). All steps are rigorous for the sums
The fact that this sum is to N includes the extra point at N/2
The extra term ↑Subrtacting ½ the first term ↑
![Page 16: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/16.jpg)
Ideal h(t,f0,f1)
0 1
1 0
exp
sin cos
sin
h t j f f t
tf f t
Tt
TT
The second term as T 1 0sin f f t
t
![Page 17: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/17.jpg)
Error function filter2
00
1( ; , ) exp
f x fAiGauss f f w dx
ww
Let x=(f-f0)/w, then for f<f0
0( ; , ) .5 1AiGauss f f w erf x And for f > f0
0( ; , ) .5 1AiGauss f f w erf x
Equivalent erfs allow overlapping regions to exactly sum to 1
![Page 18: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/18.jpg)
Herrf(f,f0,f1)
0 1
0 1
; , ,
( ; , ) ( ; , )
errfH f f f w
AiGauss f f w AiGauss f f w
The f0 = 59 Hz, f1=61 Hz w = 0.125 Hz.
![Page 19: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/19.jpg)
Error function filter/ ideal filter
f = 1/7 sec. w=f
![Page 20: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/20.jpg)
herrf(t,f0,f1,w)
2 2 20 1
1 0
0 1
; , , exp
sinexp
errfIh t f f w w t
f f tj f f t
t
Approximation of the integral result requires integration by parts
This now differs from the ideal filter only by the exponential factor.
In a reversal of the Nyquist theorem, the correctly periodic version is
0 1 0 1; , , ; , ,errf errfIk
h t f f w h t kT f f w
The exp(-(wt)2) term makes the sum rapidly convergent
![Page 21: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/21.jpg)
Limiting the convolution rangeFor |t-tk| > 6/(w), the exponential part of
h(t,f0,f1,w) is less then
2
6exp exp exp 36 2.3e-16f w
w
This leads to a definition kmin(t)=(t-6/(w))/t such that
max
min
( )k t
H t cont k k k tk t
d t d t h t t t k Even for infinite T, this sum is finite. For t such that kmin(t) > -N/2 and kmax(t)<N/2 dH(t) does not depend on T
![Page 22: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/22.jpg)
|h(t,f0,f1,w)|
Time in seconds.
![Page 23: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/23.jpg)
h(t,f0,f1,w)
Small region of time showing the oscillations in real and imaginary h(t)
The oscillations can be used to shift the frequency. Note that h can be calculated once, then shifted and re-used over and over, the sine and cosines need not be recalculated. Convolution reduces to a single set of multiplications and sums for each output data point.
![Page 24: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/24.jpg)
Splitting the Space
![Page 25: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/25.jpg)
Second region
The convolution with h(t,f1,f2,w) produces complex data. The imaginary part is shown above.
![Page 26: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/26.jpg)
Second region in frequency
Real part of transform of convoluted data between 32/Time and 50/time using 50-32+10 data points
![Page 27: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/27.jpg)
Second region in frequency
Real part of transform of convoluted data between 32/Time and 50/time using 50-32+10 data points
Ignoring the 10, the data reduction factor is 2 1
/
f f
F N T
![Page 28: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/28.jpg)
Pulsar numbersThe size of F was found by Cornish and Larson to be ~ 0.01 Hz[i], Thus there needs to be an output point every 10 seconds to follow the Doppler motion of B0531+21 which has a quadrupole frequency of 59.620.01 Hz.
[i] Neil J. Cornish and Shane L. Larson, “LISA data analysis: Doppler demodulation”, Class. Quantum Grav. 20 (2003) S163-S170 – online at stacks.iop.org/CQG/20/S163
Data reduction factor ~ 16384/0.02 = 819200
![Page 29: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/29.jpg)
|h(t)| for 0.02 width signal
The time range on this plot is from –500 seconds to + 500 seconds.
![Page 30: Error Function Filter Robert L. Coldwell University of Florida LSC August 2005](https://reader036.vdocuments.net/reader036/viewer/2022062519/56814e43550346895dbbb730/html5/thumbnails/30.jpg)
Omissions
If the convolution went straight from the input data, noise in the region would in principle rise as T1/2 while the signal would rise as T.
The noise is systematic and has many properties that identify it, an intermediate step in which known sources of frequencies that overlap the pulsar frequency are examined and removed will be investigated.
The phase will need to be monitored, if it drifts the signal will cancel to zero. – possibly the violin modes will help.