Diversity Analysis of Peaky FSK Signaling in Fading Channels

Error performance of noncoherent detection of on-off frequency shift keying (OOFSK) modulation over fading channels is analyzed when the receiver is equipped with multiple antennas. The analysis is conducted for two cases: 1) the case in which the receiver has the channel distribution knowledge only; and 2) the case in which the receiver perfectly knows the fading magnitudes. For both cases, the maximum a posteriori probability (MAP) detection rule is derived and analytical probability of error expressions are obtained. Numerical and simulation results indicate that for sufficiently low duty cycle values, lower error probabilities with respect to FSK signaling are achieved. Equivalently, when compared to FSK modulation, OOFSK with low duty cycle requires less energy to achieve the same probability of error, which renders this modulation a more energy efficient transmission technique. Also, through numerical results, the impact of number of antennas, antenna correlation, duty cycle values, and unknown channel fading on the performance are investigated.

💡 Research Summary

The paper investigates the non‑coherent detection performance of on‑off frequency‑shift keying (OOFSK) modulation when the receiver employs multiple antennas. Two distinct channel‑knowledge scenarios are considered: (i) the receiver knows only the statistical distribution of the fading coefficients (Rician with mean dℓ and variance σ²) and (ii) the receiver has perfect knowledge of the instantaneous fading magnitudes. In both cases the authors derive the maximum‑a‑posteriori (MAP) decision rule and obtain closed‑form expressions for the symbol error probability.

System model: An M‑ary OOFSK transmitter sends one of M orthogonal sinusoidal tones with probability v/M each, or remains silent (zero signal) with probability 1 − v, where v∈(0,1] is the duty cycle. The average power is P and the peak power is Pv = P/v, so the peak‑to‑average ratio grows as v decreases. The receiver has L antennas; each antenna feeds M correlators matched to the orthogonal frequencies. The correlator outputs Yℓ,m are complex Gaussian with mean A dℓ e^{jθk} (when tone k is transmitted) and variance A²σ²+1, where A = √(P Ts/(v N₀)). The energy per branch Rℓ,m = |Yℓ,m|² follows a non‑central chi‑square distribution. By equal‑gain combining across antennas, the total energy for tone m is R_m = Σ_{ℓ=1}^L Rℓ,m, which is chi‑square with 2L degrees of freedom and a non‑centrality parameter ξ = A² Σ_{ℓ=1}^L |dℓ|².

MAP detection: The joint pdf of the vector R =