DT5: Design of whitening filters for DT5 part2
National Astronomical Observatory, TAMA project
Koji Arai
March 11, 2001

1Introduction

In the ﬁfth Data Taking (DT5, from 2001/3/2∼8), two boxes of the WTF have used. They
were used to whiten the feedback signals of L− before (HDAQ 0ch) and after (HDAQ 1ch) a
summing point where a sinusoidal calibration peak at 625Hz was applied. These whitening
ﬁlters are hereafter called WTF#1 and WTF#2, respectively.
The design priciples and the performances of the prepared whitening ﬁlters have beendescribed in the earlier document [1]. After that document was written, the circuits weremodiﬁed just before the data taking in order to increase immunity of the circuit against
nonstationary noise.
In this document, the circuit of the WTF, measured transfer functions, whitened output
taken with the actual signal, and noise performance are described, concentrating on themodiﬁcation from the previous version of the circuits.

2Circuit

The WTF, depicted in Figure. 1, is formed by four stages of analog ﬁlters. Although theprevious version of the ﬁlter worked ﬁne, it was pointed out that the 2nd stage was about
being saturated in the daytime operation. As a quick remedy, the second stage and the laststage was swapped.

3Transfer function

After the modiﬁcation of the ﬁlters, the gain matching at 625Hz were perfomed again. Also,
the transfer funcitons of the ﬁlters were measured again.
In order to simplify the calibration process, the gains of the two whitening ﬁlters areadjusted to be same as far as possible. The diﬀerence of the transfer functions measured
by an FFT spectrum analyzer are given by|HWTF#2(625Hz)/H WTF#1(625Hz)| = −0.00066 ± 0.001 [dB](1)

Arg[HWTF#2(625Hz)/H WTF#1(625Hz)] = −4.0108 ± 0.005 [degree] ,(2)where HWTF#j (f ) is a transfer function of WTF#j. The diﬀerence of the phase wasintroduced by small deviation of cut-oﬀ frequencies from the designed values.

1Figure 1: Circuit of the whitening ﬁlter actually used in DT52

In order to characterize the WTFs, the measured transfer functions were ﬁtted using
LISO [2]. This time, the polynomial functions were used to represent ﬁve zeros at the origin
in the Laplace plane. The source code for the ﬁtting are shown below.

a0 0
a1 0
a2 0
a3 0
a4 0
a5 1 # a0-a5 composes a polynomial which represents (s^5)
zero 3.5238299
pole 77.759692
pole 34.336386
pole 359.91245 697.81246m
pole 101.68389 1.276314
pole 99.681401 571.4866m
pole 71.110443k
factor -59.041861nparam zero0:f 1 10
param pole0:f 20 1500
param pole1:f 20 1500
param pole2:f 200 1000
param pole2:q 0.1 10
param pole3:f 10 200
param pole3:q 0.1 10
param pole4:f 10 200
param pole4:q 0.1 10
param pole5:f 1e3 1e6param factor 1e-15 1e-6
fit llm_WTF1_tf.bod absdeg semi
rewrite samebetter#gnuterm cps
tfoutput abs:deg
freq log 1 1e5 400

Overall transfer functions were parametrized as the following:H(f ) = A1(if )2(if )2
1
h(f, f0 ) if,h(f, f1 ) h(f, f2 ) h(f, f3 , q3 ) h(f, f4 , q4 ) h(f, f5 , q5 ) h(f, f6 )(3)where a single pole (or zero) on the imaginary axis and a double pole pair are respectively3A
f0 (Hz)f1 (Hz)f2 (Hz)f3 (Hz)q3
f4 (Hz)q4
f5 (Hz)q5
f6 (Hz)

WTF#1

WTF#2

−59.041861n
3.5238299
34.336386
77.759692
359.91245
0.69781246
101.68389
1.276314
99681401
0.5714866
71.110443k−75.224937n
1.8831484
27.543661
33.801999
358.86281
0.69343594
99.362801
1.2476081
111.72927
0.62278847
221.2854k

Table 1: Identiﬁed poles and zeros from the measurements of the transfer functions.

expressed byh(f, f0 ) = 1 + ih(f, f0 , q0 ) = 1 + i

ff0f−f0 q0

(4)
ff0

2.

(5)

Note that 1/h(f, f6 ) was added so as to represent phase delay caused by the opamps.
The result of the ﬁtting are shown in Table. 1. Figures. 2 and 3 show the calculated
and measured transfer functions of the whitening ﬁlters. Figure. 4 shows the errors in theﬁtting. The ﬁtting has errors smaller than ±1% and ±1 degree in magnitude and phasefrom 30Hz to 2kHz.

4Whitened output

The performance of the ﬁlters has been tested with the actual interferometer signals.
In order to conﬁrm that saturation is not present in the circuits with the typical interferometer signal, the outputs of the circuits were observed with an oscilloscope. Table 2
shows the observed typical outputs. No saturation has been observed in the outputs andmonitor points of the ﬁlters during the normal state of the interferometer.
Typical peak values for the calibration at 625Hz were measured. During the measurement the calibration peak at 625Hz has been added to the L− loop. The amplitudeof the injected signal was 1.22 × 10−4Vrms . The peak amplitudes of 3.26310 −1Vrms and2.19110 −1Vrms were observed at the outputs of WTF#1 and WTF#2. From the ratio of4

410

Magnitude [V/V]

210010101010-2-4-6Phase [Degree]

10010

1102

103104510180120600-60-120-180
100101102103104510Figure 2: Measured (solid) and ﬁtted (dashed) transfer functions of WTF#1. Values from
10Hz to 10kHz were used for the ﬁtting.

410Magnitude [V/V]

2100101010

Phase [Degree]

10-2-4-610180120
60
0-60-120-180
1000

1010

11101022101033101044510

510

Frequency [Hz]Figure 3: Measured (solid) and ﬁtted (dashed) transfer functions of WTF#2. Values from
10Hz to 10kHz were used for the ﬁtting.

5

Phase difference (Meas.-Fit.) Ratio (Meas./Fit.)[Degree]

1.04
1.02
1.00
0.98

WTF#1
WTF#2

0.96
010110210103410105543210-1-2-3-4-5010110210103410105Frequency [Hz]Figure 4: Comparison between the measured transfer functions and the ﬁtted ones. Upperplots shows the magnitude ratio of the measured one and the calculated one. Lower plotsshow the diﬀerence of their phases.

OUTPUT

Observed Voltage

InS1 Mon
S2 MonS3 Mon
S4 MonOut< 4Vpp< 4Vpp< 3Vpp< 1Vpp< 300mV pp< 3Vpp

Table 2: Observed output at each output port.

6these amplitudes the open-loop transfer function of 1.489 at 625Hz was deduced. Thiscorresponds to the unity gain frequency of 942Hz.
The whitened output spectra are shown in Figs. 5 and 6. The obtained spectra were√√typically about 10−3 V/ Hz, which is larger than the required value of 2.5 × 10−5 V/ Hz.
During normal operations, the calibration peak at 625Hz is a dominant component in thewhitening ﬁlter outputs, though excited violin modes of the suspension wires are dominant
just after the lock acquisition. Even with the excited violin modes, it was conﬁrmed thecircuits have no saturation.
The output noise of the ﬁlters was also measured by grounding the circuit inputs. Thenoise level were plotted in Figs. 5 and 6 together with the whitened outputs. It was conﬁrmed that the noise of the circuits does not disturbe the target signal.

5Summary

The whitening ﬁlters for DT5 were built and tested. The circuit was designed and conﬁrmed
to have suﬃcient low noise for the L− feedback signals. The ﬁlters have enough gain
to record the output signals. Also, absence of saturation during normal operation wasconﬁrmed.
The transfer functions of the ﬁlters were measured and characterized by pole-zero models
with accuracy of 1% in amplitude and 1 Degree in phase.

References[1] K. Arai, “DT5: Design of whitening ﬁlters for DT5”,
http://tamago.mtk.nao.ac.jp/
tama/recom/general lib/circuits/010217 DT5 WTF/whitening.pdf[2] G. Heinzel, “LISO - Program for Linear Simulation and Optimization of analog electronic citcuits”.

7010-1

1/2

Filter Output [V/Hz ]

10-2

10-3

10-410-510-610110210103410

Frequency [Hz]Figure 5: Whitened output of WTF#1 and the output noise level.

010-1

1/2

Filter Output [V/Hz ]10-2

10-310-410-510-61011021010

3410

Frequency [Hz]Figure 6: Whitened output of WTF#2 and the output noise level.

8-2

1/2Input referred voltage [V/Hz ]10-3

10-410-510-610-710-810-9101234 5 6102234 5 610103234 5 6410

Frequency [Hz]Figure 7: Input-referred noise of WTF#1 and reconstructed feedback signal for L− .

-2

1/2Input referred voltage [V/Hz ]10-310-410-510-610-710-810-9

10

1102

34 5 62234 5 610103234 5 6

410Frequency [Hz]Figure 8: Input-referred noise of WTF#2 and reconstructed feedback signal for L− .

9