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Carrier Frequency Modulation

Module by: Tuan Do-Hong

Frequency Shift Keying (FSK)

The data is impressed upon the carrier frequency. Therefore, the M M different signals are
s m t=A P T tcos2π f c t+2πm-1Δft+ θ m s m t A P T t 2 f c t 2 m 1 Δ f t θ m (1)
for m∈12…M m 1 2 … M
The M M different signals have M M different carrier frequencies with possibly different phase angles since the generators of these carrier signals may be different. The carriers are
f 1 = f c f 1 f c (2)
f 2 = f c +Δf f 2 f c Δ f f M = f c +M-1Δf f M f c M 1 Δ f Thus, the MM signals may be designed to be orthogonal to each other.
< s m , s n >=∫0TA2cos2π f c t+2πm-1Δft+ θ m cos2π f c t+2πn-1Δft+ θ n dt=A22∫0Tcos4π f c t+2πn+m-2Δft+ θ m + θ n dt+A22∫0Tcos2πm-nΔft+ θ m - θ n dt=A22sin4π f c T+2πn+m-2ΔfT+ θ m + θ n -sin θ m + θ n 4π f c +2πn+m-2Δf+A22sin2πm-nΔfT+ θ m - θ n 2πm-nΔf-sin θ m - θ n 2πm-nΔf s m s n t 0 T A 2 2 f c t 2 m 1 Δ f t θ m 2 f c t 2 n 1 Δ f t θ n A 2 2 t 0 T 4 f c t 2 n m 2 Δ f t θ m θ n A 2 2 t 0 T 2 m n Δ f t θ m θ n A 2 2 4 f c T 2 n m 2 Δ f T θ m θ n θ m θ n 4 f c 2 n m 2 Δ f A 2 2 2 m n Δ f T θ m θ n 2 m n Δ f θ m θ n 2 m n Δ f (3)
If 2 f c T+n+m-2ΔfT 2 f c T n m 2 Δ f T is an integer, and if m-nΔfT m n Δ f T is also an integer, then < S m , S n >=0 S m S n 0 if ΔfT Δ f T is an integer, then < s m , s n >≈0 s m s n 0 when f c f c is much larger than 1T 1 T .
In case ∀m, θ m =0 m θ m 0
< s m , s n >≈A2T2sinc2m-nΔfT s m s n A 2 T 2 sinc 2 m n Δ f T (4)
Therefore, the frequency spacing could be as small as Δf=12T Δ f 1 2 T since sincx=0 sinc x 0 if x=±1 x ± 1 or ±2 ± 2 .
If the signals are designed to be orthogonal then the average probability of error for binary FSK with optimum receiver is
P ‐ e =Q E s N 0 P ‐ e Q E s N 0 (5)
in AWGN.
Note that sincx sinc x takes its minimum value not at x=±1 x ± 1 but at ±1.4 ± 1.4 and the minimum value is -0.216-0.216. Therefore if Δf=0.7T Δ f 0.7 T then
P ‐ e =Q1.216 E s N 0 P ‐ e Q 1.216 E s N 0 (6)
which is a gain of 10log1.216≈0.85dθ 10 1.216 0.85 d θ over orthogonal FSK.

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