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Carrier Frequency Modulation

Module by: Behnaam Aazhang

Summary: A method of carrier modulation in which signal information is carried in the carrier frequency of the signal.

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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