# Orthogonality of FSK modulation for non-coherent detection

For an FSK modulation with coherent detection, in order to ensure orthogonality, the frequency separation between symbols is $\Delta f=f_2 - f_1 =1/2T$. However, in case of non-coherent demodulation, which must be this value $1/2T$ or $1/T$?

For continuous-phase FSK, frequencies $f_1$ and $f_2$ and phases can be chosen so that a frequency spacing of $\frac{1}{2T}$ gives orthogonality. Note that it is not sufficient to have just $\Delta f = \frac{1}{2T}$; both $f_1$ and $f_2$ have to be integer multiples of $\frac{1}{2T}$ or $\frac{1}{4T}$ etc. Look for Sunde's FSK and minimum-shift keying (MSK) regarded as coherent FSK. If the phases are not carefully specified (as they indeed are for Sunde's FSK and MSK), orthogonality requires $\Delta f$ to be an integer multiple of $\frac{1}{T}$ and so of course the minimum spacing is $\frac{1}{T}$.
The theory of noncoherent detection assumes that the phase of the received signal is arbitrary rather than carefully controlled, and it is guaranteed to give the specified error-rate vs SNR performance regardless of signal phase as long as the frequency-spacing is $\frac{1}{T}$. The noncoherent receiver will work and it will provide the specified error-rate vs SNR performance if its input is a continuous-phase FSK signal with minimum frequency spacing $\frac{1}{2T}$ etc. However, it is worth keeping in mind that if the received signal is such that coherent detection is possible, then use of noncoherent detection in lieu of coherent detection is a suboptimal choice as far as BER vs SNR is concerned, though it might be preferable for other reasons such as ease of implementation, cost, weight, power consumption, etc of the receiver.