Question

Please help me in this question using MATLAB and Calculations please by hand

Problem 2 Consider the causal non-linear discrete-time system characterized b difference equation: y the following n of amplitude P (i.e If we use as input x[n] to this system (algorithim) a step functio rge after several iterations to the square root of P t implements the above recursion to compute the square n)-P uIn), then yIn] will conver roots of 25, 9, 3, and 2. How many iterations does it take to converge to th e true value starting at y-1]-0.2?

Answer 1

NOTE : The solution to the problem is very straight and has no calculation parts. It only has MATLAB coding and result display parts.

(i) P = 25

MATLAB CODE:

clear
clear all
y(1) = 0.2; %% You can take any initial value of y
P = 25;
x = P; %% Since, starting time of our iteration is n = 0
for k=2:50 %% when n = 0, k = 2. This is to ensure integral array number
y(k) = 0.5*y(k-1) + 0.5*x/y(k-1);
end
y %% To display the values of y

RESULT:

y =

Columns 1 through 14

0.2000 62.6000 31.4997 16.1467 8.8475 5.8366 5.0600 5.0004 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

Columns 15 through 25

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

From the above results, we can see that iterations taken to converge to the true value = 9 ==> ANSWER

(ii) P = 9

MATLAB CODE:

clear
clear all
y(1) = 0.2; %% You can take any initial value of y
P = 9;
x = P; %% Since, starting time of our iteration is n = 0
for k=2:25 %% when n = 0, k = 2. This is to ensure integral array number
y(k) = 0.5*y(k-1) + 0.5*x/y(k-1);
end
y %% To display the values of y

RESULT:

y =

Columns 1 through 14

0.2000 22.6000 11.4991 6.1409 3.8032 3.0848 3.0012 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000

Columns 15 through 25

3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000

From the above results, we can see that iterations taken to converge to the true value = 8 ==> ANSWER

(iii) P = 3

MATLAB CODE:

clear
clear all
y(1) = 0.2; %% You can take any initial value of y
P = 3;
x = P; %% Since, starting time of our iteration is n = 0
for k=2:25 %% when n = 0, k = 2. This is to ensure integral array number
y(k) = 0.5*y(k-1) + 0.5*x/y(k-1);
end
y %% To display the values of y

RESULT:

y =

Columns 1 through 14

0.2000 7.6000 3.9974 2.3739 1.8188 1.7341 1.7321 1.7321 1.7321 1.7321 1.7321 1.7321 1.7321 1.7321

Columns 15 through 25

1.7321 1.7321 1.7321 1.7321 1.7321 1.7321 1.7321 1.7321 1.7321 1.7321 1.7321

From the above results, we can see that iterations taken to converge to the true value = 7   ==> ANSWER

(iv) P = 2

MATLAB CODE:

clear
clear all
y(1) = 0.2; %% You can take any initial value of y
P = 2;
x = P; %% Since, starting time of our iteration is n = 0
for k=2:25 %% when n = 0, k = 2. This is to ensure integral array number
y(k) = 0.5*y(k-1) + 0.5*x/y(k-1);
end
y %% To display the values of y

RESULT:

y =

Columns 1 through 14

0.2000 5.1000 2.7461 1.7372 1.4442 1.4145 1.4142 1.4142 1.4142 1.4142 1.4142 1.4142 1.4142 1.4142

Columns 15 through 25

1.4142 1.4142 1.4142 1.4142 1.4142 1.4142 1.4142 1.4142 1.4142 1.4142 1.4142

From the above results, we can see that iterations taken to converge to the true value = 7   ==> ANSWER

For any help, please write in the comments.