Question

(15 pts) Given the 5-point signal space diagram shown below, can you find an alter- native 2-dimensional signal set with the same number of signals, lower average energy and a lower probability of symbol error at high SNR? Assume signals are equally likely. Draw and label your signals very neatly. Justify your answer 3 3 -21 -1 -2 -3

Answer 1

x5 x2 3 -21 -1 -2 ● x3 X4 -3

The five points are labelled. The magnitude and angle can be determined for each point.

Magnitude and angle for the 5 points

Point	Magnitude	Angle
x1	V10	0.33 * П
x2	3	0
x3	5	一0.3 * 11
x4	ИЗ	-0.21 * П
x5	V8	0.75 * П

Net magnitude,

r=V10 + 9 + 5 + 13 + 8 = V45-6.7

Net angle, $\Phi =(0.33-0.3-0.21+0.75)*\Pi =0.57*\Pi$

Therefore,

Ψ 1(t) = 6.7cos (2n * t/T + 0.57 * π)

and

2 (t) = 6.10.0 s (2π * t/T-0.57 * π)

Average energy, $\varepsilon =\sum_{i=0}^{4}\left \| xi \right \|^{2}*p=45*1/5=9J$ (Since signals are equally likely)

Hope it helps. All the best!