# Coherent FSK Modulation correlation coefficient derivation

FSK has a large bandwidth requirement and its minimum bandwidth is around $4 B_T$, where $B_T$ is the bit rate.

In a coherent FSK system the signals $s_1(t)$ and $s_2(t)$ representing bit 1 and bit 0 respectively are given by:

$s_1(t) = A_C . \cos(2\pi (f_c + \frac{\Delta f}{2})t)$

$s_1(t) = A_C . \cos(2\pi (f_c - \frac{\Delta f}{2})t)$

Assuming $f_c > \Delta f$

Show that the correlation coefficient between $s_1(t)$ and $s_2(t)$ is given by:

$\rho = \frac{\int_0^{T_B}{s_1(t)s_2(t)dt}}{\int_0^{T_B}{s_1^2(t)dt}}$

is approximately given by:

$\rho \approx sinc(2\Delta fT_B)$

How do I approach this question? I can do the integral but I don't know how the integral is derived, the squared looks like a matched filter or power calculation $(V^2/R)$?

• Do these 4Xdatarate apply to MFSK? – analogsystemsrf Apr 21 '18 at 2:11
• @analogsystemsrf That was just my comment. It happens when the two spectra can overlap at 2/T. So to your question no, that equation applies for only a binary signal. – Lewis Apr 21 '18 at 2:16
• Did U learn Correlation functions yet? – Sunnyskyguy EE75 Apr 21 '18 at 6:24
• @TonyStewartEEsince1975 autocorrelation, yes! Should I use that? – Lewis Apr 21 '18 at 15:20