Frequency-Shift Keying (FSK)

In a coherent FSK signalling scheme the signals $s_1(t)$ and $s_2(t)$ are defined to be

$$\begin{array}{ccl} s_1(t) & = & A \cos(2 \pi ( f_c + \frac{\... ... & = & A \cos(2 \pi ( f_c - \frac{\Delta\!f}{2})t) \end{array}$$

These signals are used to transmit equally-likely bits over a channel which adds white Gaussian noise of spectral height $\frac{N_0}{2}$ to the signal.

1. Assuming $f_c \gg \Delta\!f$, show that the ``correlation coefficient'' $\rho$ of the signals $s_1(t)$ and $s_2(t)$, defined as
   
   $$\rho = \frac{(s_1(t),s_2(t))}{\vert\vert s_1(t)\vert\vert \cdot \vert\vert s_2(t)\vert\vert}$$
   
   
   is given approximately by
   
   $$\rho = \frac{\sin(2\pi\Delta\!fT)}{2\pi\Delta\!fT} = \mbox{sinc}(2\pi\Delta\!fT),$$
   
   
   where $T$ is the duration of a bit interval.
2. What is the minimum value of the frequency shift $\Delta\!f$ for which the signals are orthogonal?
3. Draw a block diagram of the receiver which attains the minimum probability of error.
4. Compute the minimum probability of error as a function of $\rho$.
5. What is the value of $\Delta\!f$ that minimizes the probability of error?

Hint: You may need these equations:

$$\cos(x+y) \cdot \cos(x-y) = \cos^2(x) - \sin^2(y)$$

$$\int \cos^2(ax)\,dx = \frac{x}{2} + \frac{\sin(2ax)}{4a}$$

$$\int \sin^2(ax)\,dx = \frac{x}{2} - \frac{\sin(2ax)}{4a}$$