To/Mail Code: Albert Lazzarini/51-33
From/Mail Code: Bruce Allen/51-33
Phone/Fax 818-395-3677/818-304-9834
Refer to: LIGO-T970128-02-E
Date: September 10, 1997
Subject: LIGO ADC digitization/quantization accuracy
Quantization Noise in Ligo Interferometers
Bruce Allen
Department of Physics
University of Wisconsin - Milwaukee
Patrick Brady
California Institute of Technology
Pasadena CA 91125
The quantization of the Interferometer Dierential Output (IFO) voltage by the Analog-to-
Digital Converter (ADC) reduces the Signal-to-Noise Ratio (SNR) obtained by optimal ltering
for an astrophysical signal. We calculate the fractional loss of SNR, and show that it is largely
determined by the design of the whitening lter used at the IFO. We show that if three simple
conditions on the whitened IFO output are satised, then the SNR loss is small. If the IFO voltage
is perfectly white (in the signal band) then a 6 bit ADC (1) gives SNR loss of less than 0
.4% and (2)
has enough headroom (dynamic range) so that the IFO output can be up to eight times the root-
mean-square voltage
V
rms
before clipping occurs. Our analysis applies to arbitrary astrophysical
searches. As a concrete example, we analyze the SNR loss for a binary inspiral search in the the
November 1994 conguration of the 40-meter prototype. We also give bounds on the timing jitter
of the sample-and-hold clock of the ADC, which ensure that the loss of SNR will be small.
I. INTRODUCTION
In converting analog signals to digital form one must consider eects which arise due to sampling the signal. The
Nyquist sampling theorem proves that a bandwidth limited signal can be reconstructed from samples taken at twice
the maximum frequency of interest. An additional issue arises because the samples are stored as nite precision
numbers. While the requirement that one may reconstruct the continuous input signal drives the choice of sampling
rate, potential loss in signal to noise for detection purposes is an important factor in determining the accuracy (number
of bits) with which to record the sampled signal.
For the gravitational wave detectors which are currently being built, the following two issues arise:
1. The whitening lters must work well enough to reduce the dynamic range of the IFO voltage so that the there
is very small probability that it exceeds the maximum input level of the ADC. In other words, the signal should
not clip, or should clip very infrequently.
2. The number of bits should be chosen so that the expected signal to noise from a gravitational wave signal is not
signicantly reduced by the quantization process.
In section III we present a simple theoretical framework in which to address these issues. Our conclusions may be
summarized by the following three conditions, which may be regarded as requirements for the design of the IFO
whitening lters:
Loss of SNR
To ensure that the fractional loss
of SNR is small (for example
< 0.01) the quantization step size in volts
must be less than 2
< 24 f
N
min
ff
sig
|S
v
(f )| .
(1.1)
Here f
N
is the Nyquist frequency (half the sample rate) and S
v
(f ) is the voltage power spectrum (volts/Hz
2
) of
the whitened IFO. The value of f is that value which minimizes the right-hand-side in the astrophysical signal
band, typically 100 Hz 2000 Hz.
1 Dynamic Range
To ensure sucient headroom (dynamic range) in the ADC process (i.e. to prevent clipping) there must be
enough bits b so that
(2
b1
)
2
N
2
f
N
0
S
v
(f )df .
(1.2)
Here N is the safety factor: the ratio of ADC input clipping voltage to the ADC rms input voltage. One would
typically like to have N > 32 to enable careful inspection of transient glitches in the IFO voltage, i.e. for
diagnostic purposes.
Dithering
The nal condition that must be satised by the whitening lter is that there is enough power at high frequencies
to adequately dither the ADC. This requires that
f
N
0
df [1 cos(2f
min
)] S
v
(f ) 2
/2 .
(1.3)
Here
min
is (less than or equal to) the period of the highest frequency waves of astrophysical interest. It may
safely be taken to be the sample time. In this case, the integral above gives no weight to the IFO voltage output
spectrum at DC, and maximum weight to the spectrum at the Nyquist frequency.
Provided that the IFO whitening lters are designed to satisfy these three conditions, the loss of SNR from the
digitization/quantization process will be small.
Note that there is an additional restriction, arising from the accuracy of the clock that drives the ADC sample-
and-hold circuit. Timing jitter in this clock gives rise to an eective noise source. Detailed analysis shows that the
eects of this timing jitter decrease the SNR by less than 1% if the largest timing jitter is less than about 1/22nd the
width of the clock signal. For a 16kHz sample rate, this corresponds to less than 3�sec of timing jiter.
II. THE 40-METER PROTOTYPE
It is illustrative to examine these three conditions, which we will derive in the following sections, for the November
1994 conguration of the Caltech 40-meter prototype interferometer. For this system, the sample rate was approxi-
mately 9868 Hz, corresponding to a Nyquist frequency f
N
= 4934 Hz. Fig. 1 shows a graph of the power spectrum
S
v
(f ) of the IFO during a period of quiet operation, as well as a line showing the spectrum of quantization noise,
with amplitude S
q
(f ) =
2
/12f
N
= 1.69 � 10
5 2
/Hz.
2 10
100
1000
10000
f (Hz)
10
6
10
4
10
2
10
0
10
2
ADC counts
2
/Hz
IFO ADC Spectrum
effects of quantization noise
quantization noise
adc spectrum
determines
loss of SNR
FIG. 1. The IFO power spectrum in (ADC counts)
2
/Hz is shown in red. This may be compared to the expected spectrum
of quantization error
2
/12f
N
shown in green. The expected level of quantization noise was also simulated by replacing the
IFO output with uniformly distributed random numbers on the interval (
/2, /2), shown in blue. The ratio of the powers
at the indicated point determines our upper bound on the fractional loss in signal to noise ratio.
During this (typical) period of interferometer operation, the RMS output value was about 23 (i.e.,
�23 ADC
output counts). The ADC itself had b = 12 bits and an output range from 2048 to +2047. Examining each of
the above three conditions in turn, we nd:
Loss of SNR
An upper bound on the fractional loss
in signal to noise ratio (SNR) for a gravitational wave signal is provided
by
max
f
sig
S
q
(f )
2S
v
(f ) = max
f
sig 2
24f
N
S
v
(f ) ,
(2.1)
In Fig. 1, the minimum value of the ratio
in the signal band from 120 Hz to 2000 Hz is 9
� 10
3
. Hence, for
this particular stretch of data, no more than 0.9% of the SNR is lost due to quantization error.
Dynamic Range
The safety factor N is simply the ratio of the rms output voltage to the peak (clipping level) input of the ADC,
which is
�2
b1
. For this 40-meter data, one nds that:
N = 2
b1 V
rms
=
2
b1 f
N
0
df S
v
(f )
1/2
= 89.
(2.2)
3 Thus, the IFO can exceed 89 times its rms value without overloading or clipping. In practice, clipping is very
infrequent.
Dithering
As we will show later, the dithering condition is set by requiring that the output of the ADC changes by more
than a single count over the timescale of interest. It is easy to show that for the spectrum we have shown, that
f
N
0
df [1 cos(f /f
N
)] S
v
(f ) 100
2
/2 ,
(2.3)
so this condition is easily satised, even when the timescale of interest is the sample time,
min
= t = 1/2f
N
.
Later, we will return to the November 1994 conguration of the 40-meter prototype, and show some additional details
concerning SNR loss from digitization error.
III. QUANTIZATION PROCESS
In this section we derive some the results which have just been outlined. In particular, we consider the properties
of the IFO voltage and the eects of the analog-to-digital conversion on the recorded signal. During normal operation
the partially whitened IFO voltage v(t) will be a stationary random process. This voltage is passed through an ADC
which determines an output voltage v = Q(v) for the given input voltage v. The function Q(v) is represented in Fig. 2
and maps the real numbers into signed integer multiples of by rounding; has units of volts. The number of output
levels available is 2
b
where b is the number of bits used by the ADC, thus the dynamic range is [2
b1
, (2
b1
1)].
The error introduced by the quantization process is given by
W (v) = Q(v) v
(3.1)
and lies in the range (
/2, /2] as shown in Fig 2. 3/2
/2 W(v)
v
(a)
(b)
/2
v
Q(v)
FIG. 2. (a) The quantization function
Q(v) which maps the analog signal into integer multiples of , the quantization level,
by rounding. (b) The error function which is dened as
W (v) = Q(v) v. The raw voltage is electronically ltered to reduce the dynamic range. This process removes the dominant second-order
correlations from IFO, however the ltered voltage is not completely white.
4 Since we have no knowledge of the precise input voltage which produced a given ADC output, it is tempting to
think of the quantization process as the addition of noise to the IFO voltage in such a way that
v(t) = Q[v(t)] = v(t) + W [v(t)] = v(t) + n
q
(t) ,
(3.2)
where n
q
(t) = W [v(t)]. To make progress we must make several assumptions about this noise process which will be
justied a posteriori in the case of the 40m prototype interferometer:
1. n
q
(t) is a stationary, white random process.
2. n
q
(t) is uncorrelated with the voltage v(t).
3. n
q
(t) is uniformly distributed from (/2, /2].
Now, if the sampling rate is 2f
N
, where f
N
is the Nyquist frequency measured in Hz, the one-sided power spectral
density of the noise n
q
(t) can be determined from
f
N
0
S
q