BFSK generation, non-coherent detection and signal space

Binary FSK transmits symbol 1 as a tone of frequency $f_1$ and symbol 0 as a tone of frequency $f_2$, each lasting $T_b$ and having constant amplitude:

$$s_i(t)=\sqrt{\frac{2E_b}{T_b}}\cos 2\pi f_it,\qquad f_i=\frac{n_c+i}{T_b},\ i=1,2,\ 0\le t\le T_b$$

The choice $f_i=(n_c+i)/T_b$ makes the tones orthogonal over a bit.

Generation

 data b(t) (on-off) ------->(X)<-- sqrt(2Eb/Tb) cos w1 t
     |                       |
     |                      (+)--------> BFSK
     |                       |
     +-->[Inverter]-------->(X)<-- sqrt(2Eb/Tb) cos w2 t

1. The binary data is in on-off (unipolar) form: 1 is "on", 0 is "off".
2. For bit 1 the upper product modulator passes the $f_1$ oscillator; the inverter blocks the lower one.
3. For bit 0 the inverter output is 1, so the $f_2$ oscillator passes.
4. The adder combines both branches; only one tone appears in each bit.

If the two oscillators are synchronised, phase is continuous at bit boundaries (CPFSK). A single VCO driven by polar data gives the same result.

Non-coherent detection

The phase of the received tone is not needed. Two filter-envelope branches compare energy at $f_1$ and $f_2$.

        +->[BPF f1]->[Env. det]--l1--+
        |                            |
 x(t) --+                      [Comparator]--> data
        |                            |   l1>l2: 1
        +->[BPF f2]->[Env. det]--l2--+   l1<l2: 0

1. Each band-pass filter (or filter matched to $\cos 2\pi f_it$) passes only its own tone.
2. The envelope detector outputs the amplitude of that tone at the end of the bit (sampled at $t=T_b$).
3. The comparator decides 1 if $l_1>l_2$, else 0.

An equivalent form uses quadrature correlators (cos and sin) in each branch and squares and adds their outputs. Error probability: $P_e=\tfrac12 e^{-E_b/2N_0}$, slightly worse than coherent BFSK but with a much simpler receiver (no carrier recovery).

Signal space diagram

Orthonormal basis: $\phi_i(t)=\sqrt{2/T_b}\cos 2\pi f_it$, $i=1,2$. Then

$$\mathbf{s}_1=(\sqrt{E_b},\,0),\qquad \mathbf{s}_2=(0,\,\sqrt{E_b})$$

  phi2
   |
   * s2 (0, sqrt Eb)     / boundary x1 = x2
   |    Z2           /
   |             /
   |         /      Z1
   |     /
   +---------------*--------> phi1
  0             s1 (sqrt Eb, 0)

• Two-dimensional, two orthogonal message points.
• Euclidean distance $d=\sqrt{2E_b}$.
• Decision boundary: the bisector $x_1=x_2$.
• Coherent $P_e=\tfrac12\text{erfc}\sqrt{E_b/2N_0}$, i.e. 3 dB worse than BPSK because of the smaller distance.