Random Processes - Random Signals and Processes Primer with MATLAB - page 423

X 1

2

1

0

-1 -0.5 0 0.5 1

X2

2

1

0

-1 -0.5 0 0.5 1

X3

2

1

0

-1 -0.5 0 0.5 1

Fig. 6.24 Estimated PDFs in Exercise M.6.4

Choose a realization of the process N 1 ¼ 547.
The amplitudes of the realization are random variable Y 1 .

M 1 ¼ 1 ; 203 ; N 1 ¼ 547 : (6.252)

(a) Plot the process at the point M 1 .
Plot the N 1 th realization.
(b) Estimate the mean values of X 1 , and Y 1 .
(c) Estimate the variances of variables of X 1 , and Y 1 .
(d) Estimate the PDFs of variables X 1 , and Y 1 .
(e) Is this process ergodic?

Solution The process is generated using the MATLAB code X ¼ (R2 R1)*rand
(M,N) + R1 ; where R1 ¼ 1 and R2 ¼ 5.

(a) The random variables are shown in Fig. 6.25 for the first n ¼ 1,000 samples.
(b) The estimated mean values are approximately equal:

m X ¼ 3 : 002 ; m Y ¼ 3 : 019 : (6.253)

(c) The estimated variances are approximately equal:

s 2 s 2

X ¼ 1

: 3608 ;

Y ¼ 1

: 3237 : (6.254)