# Non-coherent FSK and Orthogonality and square-law detectors

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

Similarly to the quadrature integrator output $$Zs,k$$ the end result is

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?