Multi-Level Signaling With Noncoherent SIMO Over a Rayleigh Channel

Multi-level signaling for a single-input multiple-output system over a Rayleigh fading channel with noncoherent detection is considered. The decision rule for the optimum noncoherent receiver, that has features of both noncoherent amplitude-shift keying (ASK) and noncoherent frequency-shift keying (FSK) receivers, is derived. Using a characteristic function approach, a closed form expression for the union bound on the symbol error probability (SEP) is obtained; this serves as an approximation for the SEP under the condition of high signal-to-noise ratio (SNR). A new M-ary signaling scheme, termed amplitude-frequency keying (AFK), which uses <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M_{\!a}$ </tex-math></inline-formula> amplitude levels and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\!f}$ </tex-math></inline-formula> orthogonal tones such that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M=M_{\!a}^{N_{\!f}}$ </tex-math></inline-formula>, is presented, and its diversity order as well as a closed form high SNR approximation for its SEP is obtained. The advantages of AFK over ASK and FSK schemes are shown through numerical results.