# Frequency Domain Variant of Pearson's Sample Correlation Coefficient?

Suppose that you need to find (i.e. acquire) $x_1$, a short data packet with a known bit sequence preamble (i.e. a PRBS BPSK), in $x_2$, quantized samples recorded from the output of an ADC. You don't know where the PRBS starts in the collected data. The preamble has an unknown phase but no carrier frequency offset (CFO). You need to set a threshold at the output of the matched filter to declare if the preamble is present. It seems to me that a detection threshold based on the the correlation coefficient would be more robust to narrow-band interference (NBI) than a fixed signal-to-noise ratio (SNR) threshold:

$\frac{S}{N}=\frac{signal_{peak}}{noise_{avg}}=\frac{max(fft(x_1)*fft(x_2)'))}{median(fft(x_1)*fft(x_2)')}$

As mentioned in a previous post and on Wikipedia, the sample correlation coefficient can be calculated as:

$r_{x_1x_2} = \frac{max(abs(ifft(fft(x_1)*fft(x_2)')))}{sqrt(max(abs(ifft(fft(x_1)*fft(x_1)'))))*sqrt(max(abs(ifft(fft(x_2)*fft(x_2)'))))}$

Why do you need the ifft's? What's wrong with leaving the numerator and denominator in the frequency domain like:

## $r_{x_1x_2}' = \frac{max(abs(fft(x_1)*fft(x_2)'))}{sqrt(max(abs(fft(x_1)*fft(x_1)')))*sqrt(max(abs(fft(x_2)*fft(x_2)')))}$

In an effort to save the unnecessary floating-point operations (FLOPS), would the metric $r_{x_1x_2}'$ still be a measure of the degree of the relative likeness or similarity between $x_1$ and $x_2$ (and therefore a useful metric to base a detection threshold on)? If so, does $r_{x_1x_2}'$ have an official name from statistics or detection theory?

[I understand that $r_{x_1x_2}$ is a cross-correlation of $x_1$ and $x_2$ divided by the product of the auto-correlation of $x_1$ and the auto-correlation of $x_2$ with each correlation implemented in the frequency domain using fft's by the OP.]