Error Function FilterRobert L. Coldwell

University of Florida

LSC August 2005

Reducing the number of data points

LIGO-G050352-00-Z

Introduction

• Discrete Fourier Transform Definitions

• The Nyquist Theorem

• Ideal Filter/Error Function Filter

• Pulsar numbers

• Dramatically reducing the number of needed data points

/ 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

/ 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

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

Data versus time

/ 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

/ 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

/ 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

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

/ 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

/ 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.

Ideal Filter/Error Function Filter

The ideal f0 to f1 filter

H is not required to extend from -1/2t to 1/2t

Details of the filter near the two ends

/ 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 ↑

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

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

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.

Error function filter/ ideal filter

f = 1/7 sec. w=f

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

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

|h(t,f0,f1,w)|

Time in seconds.

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.

Splitting the Space

Second region

The convolution with h(t,f1,f2,w) produces complex data. The imaginary part is shown above.

Second region in frequency

Real part of transform of convoluted data between 32/Time and 50/time using 50-32+10 data points

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

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

|h(t)| for 0.02 width signal

The time range on this plot is from –500 seconds to + 500 seconds.

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.