1. Using only the z-transforms table, determine the z-transform of each of the f
ID: 2266612 • Letter: 1
Question
1. Using only the z-transforms table, determine the z-transform of each of the following signals: a) 2"u[n (Use Matlab to find the answers) b) cos£n)uln] (Use Matlab to find the answers) Hint: use the 'ztrans' command 2. Find the inverse z-transforms of the following: (Use Matlab to find the answers) z(-5z+22) (z+1)z-2)2 b) 28837+134) (z-2)(z2-5z+25) Hint: use the 'iztrans' command Solve y|n + 1] + 2y[n-x(n + 1] when y[0] = 1 and x[n] = e-(n-1)u[n] and find the zero-input and zero-state components of the response. Is the system stable? 3. 4. Solve y(n + 2]-3y[n + 1] + 2y[n] = x[n] when y[0-y[1] = 0 and x[n] = [n] and find the zero-input and zero-state components of the response. Is the system stable? 5. Find the zero-state response of an LTID system with transfer function 22+3 H[2] = and the input x[n] = e(n+1)u[n] Using Matlab, define the transfer function above using command with sampling time of 0.1 and find (plot) the step and impulse responses using 'step' and 'impulse' commands. (z-2)0-3)Explanation / Answer
1.
(a)
Use the following MATLAB Command to get z-transformation:
>> syms n
>> F=2^(-n)*heaviside(n);
>> ztrans(F)
MATLAB should produce the following output:
ans =
1/(2*z - 1) + 1/2
(b)
Use the following MATLAB Command to get z-transformation:
>> clearvars
>> syms n
>> F=cos(pi*n/3)*heaviside(n);
>> ztrans(F)
MATLAB should produce the following output:
ans =
ztrans(cos((pi*(n + 1))/3), n, z)/z + 1/2
(c)
Use the following MATLAB Command to get z-transformation:
>> clearvars
>> syms n gamma
>> F=gamma^(n-2)*heaviside(n-2);
>> ztrans(F)
MATLAB should produce the following output:
ans =
-(gamma/(gamma - z) - 1/2)/z^2
(d)
Use the following MATLAB Command to get z-transformation:
>> clearvars
>> syms n
>> F=2^(-n)*cos((pi*n)/3)*heaviside(n-1);
>> ztrans(F)
MATLAB should produce the following output:
ans =
1/(8*z) + ztrans(cos((pi*(n + 2))/3), n, 2*z)/(4*z^2)
2.
Use the following MATLAB Command to get inverse z-transformation
>> syms z
>> F=(-5*z^2+22*z)/((z-1)*(z-2)^2);
>> iztrans (F)
MATLAB should produce the following output:
ans =
6*2^n*(n - 1) - 11*2^n + 17
(b)
Use the following MATLAB Command to get inverse z-transformation:
>>clear all
>>syms z
>> F=(3.83*z^2+11.34*z)/((z-2)*(z^2-5*z+34));
>> iztrans (F)
(a)MATLAB should produce the following output:
ans =
(19*2^n)/28 + ((-1)^n*111^(1/2)*(- (111^(1/2)*1i)/2 - 5/2)^(n - 1)*99229i)/194250 - ((-1)^n*111^(1/2)*((111^(1/2)*1i)/2 - 5/2)^(n - 1)*99229i)/194250 + (1731*(-1)^n*34^n*cos(n*(pi - acos((5*34^(1/2))/68))))/(1750*(34^(1/2))^n)
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