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Problems 135
4. x(n) =n 2 (2/3) n−2 u(n − 1)
5. x(n) =(n − 3)( 1 4 )n−2 cos{ π (n − 1)}u(n)
2
P4.4 Let x(n) beacomplex-valued sequence with the real part x R(n) and the imaginary part
x I(n).
1. Prove the following z-transform relations:
X R(z) △ = Z [x R(n)] = X(z)+X∗ (z ∗ )
2
2. Verify these relations for x(n) =exp {(−1+j0.2π)n} u(n).
and X I(z) △ = Z [x I(n)] = X(z) − X∗ (z ∗ )
2
P4.5 The z-transform of x(n) isX(z) =1/(1 + 0.5z −1 ), |z| ≥0.5. Determine the z-transforms of
the following sequences and indicate their region of convergence.
1. x 1(n) =x(3 − n)+x(n − 3)
2. x 2(n) =(1+n + n 2 )x(n)
3. x 3(n) =( 1 2 )n x(n − 2)
4. x 4(n) =x(n +2)∗ x(n − 2)
5. x 5(n) =cos(πn/2)x ∗ (n)
P4.6 Repeat Problem P4.5 if
X(z) =
1+z −1
1+ 5 6 z−1 + 1 6 z−2 ; |z| > 1 2
P4.7 The inverse z-transform of X(z) isx(n) =(1/2) n u(n). Using the z-transform properties,
determine the sequences in each of the following cases.
1. X 1(z) = z−1 X(z) z
2. X 2(z) =zX(z −1 )
3. X 3(z) =2X(3z)+3X(z/3)
4. X 4(z) =X(z)X(z −1 )
5. X 5(z) =z 2 dX(z)
dz
P4.8 If sequences x 1(n), x 2(n), and x 3(n) are related by x 3(n) =x 1(n) ∗ x 2(n), then
(
∞∑
∞
)(
∑
∞
)
∑
x 3(n) = x 1(n) x 2(n)
n=−∞
n=−∞
n=−∞
1. Prove this result by substituting the definition of convolution in the left-hand side.
2. Prove this result using the convolution property.
3. Verify this result using MATLABand choosing any two random sequences x 1(n), and
x 2(n).
P4.9 Determine the results of the following polynomial operations using MATLAB.
1. X 1(z) =(1− 2z −1 +3z −2 − 4z −3 )(4 + 3z −1 − 2z −2 + z −3 )
2. X 2(z) =(z 2 − 2z +3+2z −1 + z −2 )(z 3 − z −3 )
3. X 3(z) =(1+z −1 + z −2 ) 3
4. X 4(z) =X 1(z)X 2(z)+X 3(z)
5. X 5(z) =(z −1 − 3z −3 +2z −5 +5z −7 − z −9 )(z +3z 2 +2z 3 +4z 4 )
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