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134 Chapter 4 THE z-TRANSFORM
MATLAB script:
>> n = [0:7]; x = cos(pi*n/3); y = filter(b,a,x,xic)
y =
Columns 1 through 4
1.80750000000000 4.35545833333333 2.83975000000000 -1.56637197916667
Columns 5 through 8
-4.71759442187500 -3.40139732291667 1.35963484230469 5.02808085078841
% Matlab Verification
>> A=real(2*R(1)); B=imag(2*R(1)); C=real(2*R(3)); D=imag(2*R(4));
>> y=A*cos(pi*n/3)+B*sin(pi*n/3)+((0.95).^n).*(C*cos(pi*n/3)+D*sin(pi*n/3))
y =
Columns 1 through 4
1.80750000000048 4.35545833333359 2.83974999999978 -1.56637197916714
Columns 5 through 8
-4.71759442187528 -3.40139732291648 1.35963484230515 5.02808085078871 □
4.6 PROBLEMS
P4.1 Determine the z-transform of the following sequences using the definition (4.1). Indicate the
region of convergence for each sequence and verify the z-transform expression using
MATLAB.
1. x(n) ={3, 2, 1, −2, −3}.
↑
2. x(n) =(0.8) n u(n − 2). Verify the z-transform expression using MATLAB.
3. x(n) =[(0.5) n +(−0.8) n ]u(n). Verify the z-transform expression using MATLAB.
4. x(n) =2 n cos(0.4πn)u(−n).
5. x(n) =(n + 1)(3) n u(n). Verify the z-transform expression using MATLAB.
P4.2 Consider the sequence x(n) =(0.9) n cos(πn/4)u(n). Let
{
x(n/2), n=0, ±2, ±4, ···;
y(n) =
0, otherwise.
1. Show that the z-transform Y (z) ofy(n) can be expressed in terms of the z-transform
X(z) ofx(n) asY (z) =X(z 2 ).
2. Determine Y (z).
3. Using MATLAB, verify that the sequence y(n) has the z-transform Y (z).
P4.3 Determine the z-transform of the following sequences using the z-transform table and the
z-transform properties. Express X(z) asarational function in z −1 .Verify your results using
MATLAB. Indicate the region of convergence in each case, and provide a pole-zero plot.
1. x(n) =2δ(n − 2)+3u(n − 3)
2. x(n) =3(0.75) n cos(0.3πn)u(n)+4(0.75) n sin(0.3πn)u(n)
3. x(n) =n sin( πn 3 )u(n)+(0.9)n u(n − 2)
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