An autocorrelator directly measures the autocorrelation function
of signal x(t) with itself for a number of evenly-spaced delays or
``lags'' 
, n=0,1..N. The Fourier transform of the
autocorrelation function then yields the power spectrum
. The bandwidth and frequency resolution in the spectrum
are inversely proportional to the delay spacing 
and maximum
delay 
, respectively. In radio astronomy applications signal
x(t) is usually assumed to be a real stationary Gaussian variable, so
that averaging in (
) can be done by integrating over
time. When digital processing is employed the signal of bandwidth B
is sampled at (at least) the Nyquist frequency 
and, with
samples, () becomes
Here 
is the sampled signal. The number of accumulated
samples 
(T is total accumulation time) has to be very large
since the uncertainty of 
decreases only as 
. In
radio astronomy, typical integration periods are 10 milliseconds
or more [30] and for a sampling rate of about
we have 
.
The digital computation of () is greatly simplified by
coarse quantization of the signal x. The quantization characteristic
for two-level (one bit) sampling is shown in Fig. .
Figure: Characteristic curve for two-level quantization.
With 1-bit sampling () becomes
The gain here is obvious: the multiplication of 1-bit numbers 
is a much simpler operation than the multiplication of
multibit numbers 
. The relation between the true correlation
function and the correlation function of the 1-bit quantized
signal r(n) is less trivial and is given by the Van Vleck
relationship [31]:
A disadvantage of the coarse quantization is the degradation of the
signal-to-noise ratio: the variance in an -sample estimate of the
correlation function is increased over what it would be for multibit
correlation. In a one-bit correlator the number of samples should be
increased by a factor of 2.46 to achieve the same accuracy as in the
multibit method [28]. It is not necessary, however, to
increase the integration time by the same factor. Oversampling, that
is, sampling at a frequency higher than the Nyquist frequency can
provide additional information about the signal because the bandwidth
of the quantized signal is larger than that of the original signal due
to introduced discontinuities. The advantage of oversampling can be
significant: in one-bit correlation the relative integration time
drops from 2.46 to 1.81 at twice the Nyquist rate (double
oversampling). A further increase in sampling speed does not give any
significant advantage: if sampled continuously (at infinite speed),
one-bit quantization would require a minimum relative integration time
of 1.65 to achieve the same accuracy as multi-bit multiplication.
Many other quantization schemes are, of course, possible (three-level,
two-bit, etc.) and usually the best trade-off between sensitivity and
complexity of hardware is chosen. For RSFQ implementation one-bit
quantization appears to be the most attractive since it requires only
the simplest hardware. It is easy to see that the multiplication in
() can be performed using a simple XOR function. Let 
be the quantized signal and 
its binary coding. Then
the identity
maps a=0 into s=1 and a=1 into s=-1. Next, we have the following chain of identities:
Here the definition of XOR operator ( 
) as modulo 2 addition
operator was used: 
and 
.
To summarize, autocorrelation function () can be directly
measured in a digital autocorrelator characterized by
For RSFQ implementation, a 1-bit quantization scheme with double oversampling seems to offer the best trade-off between sensitivity and complexity of hardware.
