1
Introduction
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 been described in the earlier document [1]. After that document was written, the circuits were modified 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 the modification from the previous version of the circuits.
Circuit
2
The WTF, depicted in Figure. 1, is formed by four stages of analog filters. Although the previous 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 last stage was swapped.
3
Transfer 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 are adjusted 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)/Hwtf#1(625Hz)| = −0.00066 ± 0.001 [dB] Arg[Hwtf#2(625Hz)/Hwtf#1(625Hz)] = −4.0108 ± 0.005 [degree] , (2) where HwTF# i (f) is a transfer function of WTF# j .  The difference of the phase was introduced 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 [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 -59.041861n param 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 1e6 param 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: a(f,f1)h(f,f2)h(f,f3,q3)h(f,f4,q4)h(f,f5,q5)h(f,f6)
where a single pole (or zero) on the imaginary axis and a double pole pair are respectively
expressed by
Note that 1/h(f, f 6 ) 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 the fitting. The fitting has errors smaller than ±1% and ±1 degree in magnitude and phase from 30Hz to 2kHz.
4
Whitened 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 inter- ferometer 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 and monitor points of the filters during the normal state of the interferometer. Typical peak values for the calibration at 625Hz were measured.  During the mea- surement the calibration peak at 625Hz has been added to the L_ loop. The amplitude of the injected signal was 1.22 × 10− 4 V rms.  The peak amplitudes of 3.26310− 1 V rms and 2.19110-1Vrms were observed at the outputs of WTF#1 and WTF#2. From the ratio of
these amplitudes the open-loop transfer function of 1.489 at 625Hz was deduced.  This corresponds to the unity gain frequency of 942Hz. The whitened output spectra are shown in Figs. 5 and 6. The obtained spectra were typically about 10-3V/√Hz, which is larger than the required value of 2.5 × 10-5V/√Hz. During normal operations, the calibration peak at 625Hz is a dominant component in the whitening 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 the circuits have no saturation. The output noise of the filters was also measured by grounding the circuit inputs. The noise level were plotted in Figs. 5 and 6 together with the whitened outputs. It was con- firmed that the noise of the circuits does not disturbe the target signal.
5
Summary
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 was confirmed. 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.