Page 381 - Applied Numerical Methods Using MATLAB

370 OPTIMIZATION
N 0 and the bit error rate (probability of bit error) P e , Seung-hee, a commu-
nication system expert, formulated the following constrained minimization
problem:
a
4 N 0 −1 2 i
x i
Min f(x) = (2 − 1) 2(erfc (P e /2)) (P7.11.1)
i=1 3 x i
subject to
4
a i
g(x) = − N = 0 (P7.11.2)
i=1 x i
with N = 128, and a i : the data rate of each user
−1
where erfc (x) is the inverse function of the complementary error function
defined by Eq. (P4.9.3) and is installed as the MATLAB built-in function
‘erfcinv()’. He defined the objective function and the constraint func-
tion as below and save them in the M-files named “fp_bits1.m”and
“fp_bits_c.m”.

function y = fp_bits1(x,a,N,Pe)
N0 = 1; y = sum((2.^x-1)*N0/3*2*erfcinv(Pe/2).^2.*a./x);
function [C,Ceq] = fp_bits_c(x,a,N,Pe)
C = []; Ceq = sum(a./x) - N;

Compose a program that solves the above constrained minimization problem
−4
(with N 0 = 1and P e = 10 ) to get the modulation order x i of each user
for five different sets of data rates
a = [32 323232], [64 323232], [128 32 32 32], [256 32 32 32], and [512 32 32 32]

and plots a 1 /x 1 (the number of subchannels assigned to user 1) versus a 1
(the data rate of user 1). If you feel uneasy about the results obtained with
your initial guesses, try with the initial guesses as follows for each set of
data rates, respectively:

x 0 = [0.50.50.50.5], [1 111], [111 1], [2 222], and[4 444]