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This is a basic figure that explains how Non-Coherent FSK actually works. The incoming signal

$ r (t)=\sqrt{2 P}{Cos}(2 \pi t (\text{fc}+\text{fi})+\text{n(t)}$

where P denotes the signal power in Watts, $T$ denotes the symbol time in seconds, $f$, $i$ $s$ the carrier frequency in Hz, Also, n(t) denotes the AWGN with single-sided power spectral density $No. T.

Now this incoming signal matches to integrator output

$ Zck =\int _ 0^T r(t) \sqrt{2 P}{Cos}(2 \pi t (\text{fc}+\text{fi})+\text{n(t)}$ equation- 1

Similarly, the quadrature integrator output $Zs,k$

$ Zsk =\int _ 0^T r(t) \sqrt{2 P}{Sin}(2 \pi t (\text{fc}+\text{fi})+\text{n(t)}$ $ equation- 2

Eventually in $fc$ terms are canceled out by low -pass-filters and we are left with other terms.

I tried to solve it by putting r(t) into equation-1 and integrated it. The end result is this enter image description here

Similarly to the quadrature integrator output $Zs,k$ the end result is enter image description here

In the end, I added them and got $1T$ and Noise.

My question is what happens to other integrators where the miss-matching occurs, how we get 0 from it. As this is Non-coherent FSK and spacing is $1/T$ so should we put 1/T instead of fi-fi in equation-1 and 2 to get zero from these orthogonal functions? like the same equations, 1 and 2 now become?

enter image description here

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