Function: normalize_gw()

0 void normalize_gw(FILE *fpss,int npoint,float srate,float *response)

This routine generates an array of complex numbers $R(f)$ from the information in the swept sine file and an overall calibration constant. Multiplying this array of complex numbers by (the FFT of) channel.0 yields the (FFT of the) differential displacement of the interferometer arms $\Delta l$, in meters: $\widetilde{\Delta l}(f) = R(f) \widetilde{C_0}(f)$. The units of $R(f)$ are meters/ADC-count.

The arguments are:

fpss: Input. Pointer to the file in which the swept sine normalization data can be found.

npoint: Input. The number of points $N$ of channel.0 which will be used to calculate an FFT for normalization. Must be an integer power of 2.

srate: Input. The sample rate in Hz of channel.0.

response: Output. Pointer to an array response[0..s] with $s=N+1$ in which $R(f)$ will be returned. By convention, $R(0)=0$ so that response[0]=response[1]=0. Array elements response[$2 i$] and response[$2 i + 1$] contain the real and imaginary parts of $R(f)$ at frequency $f= i{\tt srate}/N$. The response at the Nyquist frequency response[N]=0 and response[N+1]=0 by convention.

The absolute normalization of the interferometer can be obtained from the information in the swept sine file, and one other normalization constant which we denote by $Q$. It is easy to understand how this works. In the calibration process, one of the interferometer end mirrors of mass $m$ is driven by a magnetic coil. The equation of motion of the driven end mass is

$$m {d^2 \over dt^2} {\Delta l} = F(t)$$ (3.12.9)

where $F(t)$ is the driving force and $\Delta l$ is the differential length of the two interferometer arms, in meters. Since the driving force $F(t)$ is proportional to the coil current and thus to the coil voltage, in frequency space this equation becomes

$$(- 2 \pi i f)^2 \widetilde{\Delta l} = {\rm constant} \times... ...} = {\rm constant} \times {{\tilde V}_{\rm IFO} \over S^*(f)}.$$ (3.12.10)

We have substituted in equation () which relates ${\tilde V}_{\rm IFO}$ and ${\tilde V}_{\rm coil}$. The IFO voltage is directly proportional to the quantity recorded in channel.0: $V_{\rm IFO} = {\rm ADC} \times C_0$, with the constant ${\rm ADC}$ being the ratio of the analog-to-digital converter's input voltage to output count.

Putting together these factors, the properly normalized value of $\Delta l$, in meters, may be obtained from the information in channel.0, the swept sine file, and the quantities given in Table  by

$$\widetilde{\Delta l} = R(f) \times \widetilde{C_0 } \qquad {... ...} \quad R(f) = {Q \times {\rm ADC} \over -4 \pi^2 f^2 S^*(f)},$$ (3.12.11)

where the $\tilde {}$ denotes Fourier transform, and $f$ denotes frequency in Hz. (Note that, apart from the complex conjugate on $S$, the conventions used in the Fourier transform drop out of this equation, provided that identical conventions (,) are applied to both $\Delta l$ and to $C_0$).

Table: Quantities entering into normalization of the IFO output.

Description	Name	Value	Units
Gravity-wave signal (channel.0)	$C_0$	varies	ADC counts
A$\rightarrow$D converter sensitivity	ADC	10/2048	$\rm V_{\rm IFO} \left({\rm ADC\ counts}\right)^{-1}$
Swept sine calibration	S(f)	from file	$\rm V_{\rm IFO} \left( V_{\rm coil}\right) ^{-1}$
Calibration constant	$Q$	$1.428\times 10^{-4}$	$\rm meter\ Hz^2 \left( V_{\rm coil} \right)^{-1}$

The constant quantity $Q$ indicated in the above equations has been calculated and documented in a series of calibration experiments carried out by Robert Spero. In these calibration experiments, the interferometer's servo was left open-loop, and the end mass was driven at a single frequency, hard enough to move the end mass one-half wavelength and shift the interference fringe's pattern over by one fringe. In this way, the coil voltage required to bring about a given length motion at a particular frequency was established, and from this information, the value of $Q$ may be inferred. During the November 1994 runs the value of $Q$ was given by

$$Q = {\sqrt{9.35 \; \rm Hz} \over k} = 1.428 \times 10^{-4} {... ... where\ } k=21399 {\rm\ V_{\rm coil} \over meter \; Hz^{3/2}}.$$ (3.12.12)

Author: Bruce Allen, ballen@dirac.phys.uwm.edu

Comments: See comment for calibrate().