Performance of Coherent M-ary Signaling ENSC 428 – Spring 2007

Digital Communication System

1. M-ary PSK T sin

cont …

Integration over IQ plane

cont …

2. M-ary Orthogonal Signaling

cont … 3-ary orthogonal signal space

cont … assume equally likely M-ary symbols a priori

cont … assume equally likely symbols a priori

cont … union bound

cont …

 Performance improves as M increases (??)  In the limit (M  ∞), error probability can be made arbitrarily small as long as E b /N 0 > ln2 (-1.59 dB). Proof in Gallager Lecture 19 section 4.3 In fact, Information Theory also proves that we cannot achieve error probability arbitrarily small if E b /N 0 < ln2.  Most practical systems use non-coherent FSK rather than coherent FSK. We will discuss non-coherent FSK soon.

Biorthogonal Signaling

cont … 6-ary biorthogonal signal constellation

Simplex Signaling

cont … 3-ary simplex signal constellation

cont …

3. M-ary QAM 16-ary QAM

cont …

Symbol Error Rate (SER) or SEP

Bit Error Rate (BER) or BEP (cf SEP, symbol error probability)

Orthogonal signaling BEP k bits, M=2 k, each bit error pattern corresponds to a unique symbol, which is not the transmitted. e e e

Example: 8-ary PSK

Gray Coding

Gray-coded MPSK

cont …

Gray code (Reflected binary code by Frank Gray) generation  Can be generated recursively by reflecting the bits (i.e. listing them in reverse order and concatenating the reverse list onto the original list), prefixing the original bits with a binary 0 and then prefixing the reflected bits with a binary 1.

Gray code generation: another view