Problem Set I: Fourier Transforms and Constellations

ECE300 Communication Theory
Problem Set I: Fourier Transforms and Constellations
August 30, 2021
1. Consider 16-QAM versus 8-PSK. For transmission through an AWGN channel with
a given noise power, the probability of symbol error in the two constellations will be
roughly the same if the minimum Euclidean distance dmin between signal points are
the same. For purposes of comparison, assume the 16-QAM constellation is comprised
of points Ik + jQk where Ik; Qk 2 f1; 3g, while the 8-PSK constellation consists of
points on a circle of radius A. In this problem, assume equiprobable transmission.
(a) Find the average energy per symbol for each constellation, and then the average
energy per bit. For 8-PSK, these answers depend on A, of course.
(b) Find the amplitude A for 8-PSK so that the two constellations have the same
dmin.
(c) Let  16QAM be the SNR per bit required for 16-QAM, and  8P SK be the SNR
per bit required for 8-PSK, both in decibels, assuming the same dmin found above.
Which requires more SNR per bit? Find the di§erence between these two parameters.
(d) As suggested in the notes, let us deÖne the spectral e¢ ciency  as the number of
bits per symbol, divided by the number of (real) dimensions. Compute 16QAM
and 8P SK, and identify the constellation that is more spectrally e¢ cient.
2. Continue the previous problem. Now consider 8-ary orthogonal signalling, where each
symbol has equal energy Eo.
(a) Express the dmin for this constellation in terms of Eo.
(b) If dmin is now set to be the same as for the constellations considered in the previous
problem, Önd  8orth, the energy per bit for this orthogonal signalling scheme.
(c) Compute 8orth for the 8-orthogonal signalling scheme.
(d) Finally, list the 3 constellations in order of power e¢ ciency (most power e¢ cient
Örst), and spectral e¢ ciency (most spectrally e¢ cient Örs