# Consider a baseband M-ary system using M discrete amplitude

Consider a baseband M-ary system using M discrete amplitude levels. The receiver model is as shown in Figure, the operation of which is governed by the following assumptions;
(a) The signal component in the received wave is m (t) = S n a n sin (t/T – n) where if 1/T is the signaling rate in bauds.
(b) The amplitude levels are an, +A/2, ±3A/2… ± (M — l) A/2 if M is even, and an, = 0, ±A,…, ±(M -1)A/2 if M is odd
(c) The M levels are equiprobable, and the symbols transmitted in adjacent rime slots are statistically independent.
d) The channel noise w (t) is white and Gaussian with zero mean and power spectral density N0/2.
(e) The low-pass filter is ideal with bandwidth B = 1/2T.
(f) The threshold levels used in the decision device are 0, ±A… ± (M — 2) A/2 if M is even and tA/2, ±3A/2… ± (M — 2) A/2 if M is odd, the average probability of symbol error in this system is defined by Pe = (1 – 1/M) erfc (A/2v2s, where s is the standard deviation of the noise at the input of the decision device. Demonstrate the validity of this general formula by determining Pe for the following three cases: M — 2, 3, 4.