Random Signals for Engineers using MATLAB and Mathcad
Copyright  1999 Springer Verlag NY
Example 2.9 Uniform Random Number Generators
In this example we how we may use the linear congruential formula to generate random
numbers. We will select parameters that produce acceptable and unacceptable random number
sequences and show how we may detect the differences.
Let us start with a set of parameters for a, c and m that produce a small periodic sequence.
a=7; c=3; m=10;
Since the sequence is between 0 and m-1 we expect to have or 1 to m sequences in Matlab
Let us generate a two dimensional array with the row index corresponding to starting seeds of 0
to 9
z=[0:9]';
The generator formula is given below
for j=1:9
for i=1:10
z(i,j+1) = mod(a*z(i,j) +c,m);
end
end
z
z =
0
1
2
3
4
5
6
7
8
9
3
0
7
4
1
8
5
2
9
6
4
3
2
1
0
9
8
7
6
5
1
4
7
0
3
6
9
2
5
8
0
1
2
3
4
5
6
7
8
9
3
0
7
4
1
8
5
2
9
6
4
3
2
1
0
9
8
7
6
5
1
4
7
0
3
6
9
2
5
8
0
1
2
3
4
5
6
7
8
9
3
0
7
4
1
8
5
2
9
6
There a three cyclic sequences, 2 of length 4 - {1 0 3 4} and {5 8 9 6} and one of length 2 - {2 7}
with different starting seeds. We clearly desire a sequence where the period is long and
approaching m-1 and should not exhibit periodic sequences for certain starting seeds.
Let us examine a set of uniform numbers generated by the parameters
a=137; c=0; m=256;
x(1)=87;
for j=1:256
x(j+1) = mod(a*x(j) +c,m);
end
Normalizing the number to the 0 - 1 interval we have
u=x/256;
We first draw a histogram of the number sequence and verify that it indeed conforms to a uniform
density distribution. We select 5 bins of uniform width, and use the Matlab built in histogram
function
[nw x]= hist(u,5); bar(x,nw,.3)
60
50
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
The curve does indeed look uniform since all of the bins contain about 50 samples. We must
perform other tests to verify that indeed we do not have a periodic sequence. If we plot the values
ui vs. i we discover that indeed a periodic sequence exists and its period is approximately 40.
plot(u)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
200
250
300
Another informative plot can be made which shows the interval between adjacent numbers in the
sequence. We plot ui+1 vs. ui as
plot(u(2:256),u(1:255))
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
This sequence of plots can be made for different N and the length of the sequence can be
observed by increasing N until the plot does not change. This indicates that the sequences are
overlaying one another as N is increased and contain repeating sequences.
If we use the parameters suggested in Table 2.7-1 we find
a=314159269; c=453806245; m=2^31-1;
x(1)=87;
for j=1:256
x(j+1) = mod(a*x(j)+c,m);
end
u=x/m;
plot(u(2:255),u(1:254))
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
As we increase N we see that the curve fills the space with no repeats indicating the absence of
periodicity in the sequence.
