DT5: Design of whitening filters for DT5
National Astronomical Observatory, TAMA project
Koji AraiFebruary 21, 2001

1Introduction

Analog-to-digital converters (ADCs) are indispensable to measure and record analog signals
on computers. Since ADCs have limited resolution and dynamic range, whitening ﬁlters(WTFs) are commonly used before sampling of the signals by ADCs. WTFs amplify thesignals in order to exploit full resolution of the ADCs. In addition, to compensate the limited
dynamic range of the ADCs, WTFs ampliﬁes the signals especially in the frequency region
where the power of the signals are small compared with the other frequency component.
As a result, the output of the WTFs has an approximately ﬂat spectrum.
The high-speed data acquisition system of TAMA300 (HDAQ) requires the following
features for the WTF of the L− feedback signal [1].
• The whitened output must be larger than half 1 of the noise level of the HDAQ system.
√Thus, the output must be 2.5 × 10−6V/ Hz from 50Hz to 5kHz.
• The input-referred noise of the WTF must be smaller than the spectrum of the L−feedback signal from 50Hz to 5kHz (Fig. 2).
• The output of the WTF must be within ±5V 2 . Also, during the operation of theinterferometer, any saturation in the WTF should be avoided as far as possible 3 .
For the forthcoming ﬁfth Data Taking (DT5, from 2001/3/2∼8), two boxes of the WTF
are built. They are 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 is applied.
They are hereafter called WTF#1 and WTF#2, respectively.
In this document, the circuit of the WTF, measured transfer functions, whitened output
taken with the actual signal, and noise performance are described.

2Circuit

The WTF, depicted in Figure. 1, is formed by four stages of analog ﬁlters.

1

The diﬀerential driver used before HDAQ has a gain of 2 [1].
HDAQ itself has a range of ±10V [1].
3Though, it is diﬃcult to avoid saturation caused by sudden large disturbance to the interferometer.
21Figure 1: Circuit for the whitening ﬁlter for DT52101/2Output Voltage [V/Hz ]10101010101010-1-2-3-4-5-6-7-8100101102103104510Frequency [Hz]Figure 2: Feedback signal of TAMA300, taken on January 24th, 2001.

1st stage
This stage ampliﬁes only above 36Hz by a factor of 10.
In order to increase the signal in the observation band above the noise level of thefollowing stages, the input stage must have some amount of gain. The feedback signalof L− has a large amplitude below 10Hz, as seen in Figure. 2. Therefore, low frequency
components should not be ampliﬁed in order to avoid saturation.
The non-inverting ampliﬁer with gain boost above 36Hz is used to satisfy the issue of
having the gain without saturation. This stage does not amplify the low frequency signal,
on the other hand the frequency components in the observation band are ampliﬁed. Notethat a simple highpass ﬁlter was not applied as inverting ampliﬁers with low input noise
inevitably involve the small value of a resistor in the input. It is safer to have a high input
impedance (10kΩ) to preserve precision of the circuit.
The operational ampliﬁer of AD797 [2] is used in this stage. The input-referred noise ofthe circuits are set to be dominated by the noise of this stage. The simulation by LISO [3]√showed that the input-referred noise is to be 2nV/ Hz caused by the noise of the opamp
and the feedback resistor of 910Ω.
A capacitance of 22pF was used in order to prevent from oscillation of the circuit.

32nd stage
The second stage is used to cut the low frequency component and to amplify the signalfurthermore.
In order to have a suﬃcient gain without saturation, a highpass ﬁler with an inverting
ampliﬁer using an OP-27 [4] is used. The gain in the transmission band is −51. Here, theinput resistance of 1kΩ could safely be used because the signal was already ampliﬁed by
the ﬁrst stage.

3rd stage
The third stage is a 2nd-order butterworth lowpass ﬁlter with the cut-oﬀ frequency ofabout 350Hz. This stage is used to whiten the high frequency component which has anapproximate slope of +12dB/Oct.

4th stage
The fourth stage is a 4th-order butterworth highpass ﬁlter with the cut-oﬀ frequency of100Hz. This stage is used to further whiten the low frequency component.
A SR Resistor Tunable ﬁlter by NF circuit block design corporation is used for the ﬁlter.
This ﬁlter allows us to make a 4th-order ﬁlter by just putting four equivalent resistors. On√
the other hand, this ﬁlter has an output noise of about 10−7 V/ Hz. In spite of its poor
noise performance, this output noise does not degrade the performance of our WTF due tothe gain of the previous stages.

5th stage
The ﬁfth stage is an inverting ampliﬁer with gain of 0 to −10. This stage ampliﬁes thesignal so that the resulting output has suﬃcient amplitude for the ADC range of HDAQ.

Miscellaneous
The circuit has not only the connectors for the input and the output of the 5th stage,
but also the connectors for monitoring the output of the other stages, labeled “S1 MON”,“S2 MON”, “S3 MON”, and “S4 MON”. They are used to conﬁrm that each stage has nosaturation.

3Transfer function

In order to simplify the calibration process, the gains of the two whitening ﬁlters are adjusted
to be same as far as possible. The diﬀerence of the transfer functions measured by an FFT4

A
f0 (Hz)f1 (Hz)f2 (Hz)f3 (Hz)q3
f4 (Hz)q4
f5 (Hz)q5
f6 (Hz)

WTF#1

WTF#2

−6.3296461 × 10−8

−1.1138385 × 10−7
3.0998213
29.983979
41.599029
345.90772
0.68685034
100.83677
1.3192324
104.45711
0.53522177
68.083472k

3.8316963
38.236954
67.846192
348.34772
0.68275571
101.02979
1.308833
106.51815
0.53343746
57.720477k

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

spectrum analyzer are given by|HWTF#2(625Hz)/H WTF#1(625Hz)| = −0.0008 ± 0.0005 [dB](1)

Arg[HWTF#2(625Hz)/H WTF#1(625Hz)] = −4.123 ± 0.003 [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.
In order to characterize the WTFs, the measured transfer functions were ﬁtted using
LISO 4 . Overall transfer functions were parametrized as the following:H(f ) = A1(if )2(if )21h(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 respectively
expressed byh(f, f0 ) = 1 + iff0f−h(f, f0 , q0 ) = 1 + if0 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. 3 and 4 show the calculated
and measured transfer functions of the whitening ﬁlters. Figure. 5 shows the errors in theﬁtting. The ﬁtting has errors smaller than ±1% and ±1 degree in magnitude and phasefrom 30Hz to 2kHz.
4In LISO, I could not ﬁnd the directive to put zeros on the origin in the fourier plane. Therefore beforethe ﬁttings have been performed, (if )5 was multiplied to the transfer functions in advance

5Magnitude [V/V]

101010420-210-410-6100

Phase [Degree]

10180120
600-60-120-180

100110101231010102103410104105510

Frequency [Hz]Figure 3: Measured (solid) and ﬁtted (dashed) transfer functions of WTF#1. Values from
10Hz to 10kHz were used for the ﬁtting.

Magnitude [V/V]

101010420-210-4

10-6100

Phase [Degree]

10110231010410105180120600-60-120-180
100101102103104510

Frequency [Hz]Figure 4: Measured (solid) and ﬁtted (dashed) transfer functions of WTF#2. Values from
10Hz to 10kHz were used for the ﬁtting.

6

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

1.04
1.02
1.00
0.98

WTF#1
WTF#2

0.96
0

10

10

12310

10

4

105105432
10-1-2-3-4-50

1101021010341010

5

Frequency [Hz]Figure 5: Comparison between the measured transfer functions and the ﬁtted ones. Upper
plots shows the magnitude ratio of the measured one and the calculated one. Lower plotsshow the diﬀerence of their phases.

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 except for occasional large nonstationary burst.
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 amplitude of
the injected signal was 1.2145 × 10−4 Vrms . The peak amplitudes of 3.03110 −1Vrms andOUTPUT

Observed Voltage

In
S1 MonS2 MonS3 Mon
S4 MonOut< 1Vpp< 2Vpp< 10Vpp< 10Vpp< 0.5V pp< 5Vpp

Table 2: Observed output at each output port.

71.71610 −1Vrms were observed at the outputs of WTF#1 and WTF#2. From the ratio ofthese amplitudes the open-loop transfer function of 1.766 at 625Hz was deduced. Thiscorresponds to the unity gain frequency of 1.11kHz.
The whitened output spectra are shown in Figs. 6 and 7. 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. The
noise level were plotted in Figs. 6 and 7 together with the whitened outputs. It was conﬁrmed that the noise of the circuits does not disturbe the target signal. The input-referrednoise of the circuits are estimated by dividing the output noises by the transfer functions√(Fig. 8 and Fig. 9). The input-referred noises of 2 ∼ 3nV/ Hz were achieved.

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. These parameters will be used inreconstruction of the signals from the restored HDAQ signals.

References[1] D. Tatsumi, private communications.
[2] Analog Devices Inc., “Ultralow Distortion, Ultralow Noise Op Amp AD797”
http://www.analog.com/pdf/ad797.pdf[3] G. Heinzel, “LISO - Program for Linear Simulation and Optimization of analog electronic citcuits”.
[4] Analog Devices Inc., “Low Noise, Precision Operational Ampliﬁer OP27”
http://www.analog.com/pdf/op27.pdf

8HDAQ0whitened outnoise010

1/2Voltage Noise [m/Hz ]10101010101010-1-2-3-4-5-6-710

1102103410Figure 6: Whitened output of WTF#1 and the output noise level.

HDAQ1
whitened outnoise0101/2

Voltage Noise [m/Hz ]10101010101010-1-2-3-4-5-6-7

101102103410

Frequency [Hz]Figure 7: Whitened output of WTF#2 and the output noise level.

9101/2

Voltage Noise [m/Hz ]10101010101010

HDAQ0input signal
input noise-2-3-4-5-6-7-8-910

1102103410

Frequency [Hz]Figure 8: Input-referred noise of WTF#1 and reconstructed feedback signal for L− .

10

1/2

Voltage Noise [m/Hz ]10101010101010

HDAQ1
input signal
input noise-2-3-4-5-6-7-8-910

1102103410

Frequency [Hz]Figure 9: Input-referred noise of WTF#2 and reconstructed feedback signal for L− .

10