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

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# Problem Set I: Fourier Transforms and Constellations

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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