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Problems 137
2. Using (4.32), determine the signal parameters A c, A s, r, and v 0 in terms of the rational
function parameters b 0, b 1, a 1, and a 2.
3. Using your results in part b above, design a MATLABfunction, invCCPP, that computes
signal parameters using the rational function parameters. The format of this function
should be:
function [As,Ac,r,v0] = invCCPP(b0,b1,a1,a2)
P4.13 Suppose X(z) isgiven as follows:
2+3z −1
X(z) =
, |z| > 0.9
1 − z −1 +0.81z−2 1. Using the MATLABfunction invCCPP given in Problem P4.12, determine x(n) inaform
that contains no complex numbers.
2. Using MATLAB, compute the first 20 samples of x(n), and compare them with your
answer in the above part.
P4.14 The z-transform of a causal sequence is given as
−2+5.65z −1 − 2.88z −2
X(z) =
(4.33)
1 − 0.1z −1 +0.09z −2 +0.648z −3
which contains a complex-conjugate pole pair as well as a real-valued pole.
1. Using the residuez function express (4.33) as
( )+( )z −1
X(z) =
1+( )z −1 +( )z + ( )
(4.34)
−2 1+( )z −1
Note that you will have to use the residuez function in both directions.
2. Now using your function invCCPP and the inverse of the real-valued pole factor,
determine the causal sequence x(n) from the X(z) in(4.34) so that it contains no
complex numbers.
P4.15 For the linear and time-invariant systems described by the following impulse responses,
determine (i) the system function representation, (ii) the difference equation representation,
(iii) the pole-zero plot, and (iv) the output y(n) ifthe input is x(n) = ( 1 n
4) u(n).
1. h(n) =5(1/4) n u(n)
2. h(n) =n(1/3) n u(n)+(−1/4) n u(n)
3. h(n) =3(0.9) n cos(πn/4+π/3)u(n +1)
4. h(n) = (0.5)n sin[(n +1)π/3]
u(n)
sin(π/3)
5. h(n) =[2− sin(πn)]u(n)
P4.16 Consider the system shown below.
1. Using the z-transform approach, show that the impulse response, h(n), of the overall
system is given by
h(n) =δ(n) − 1 δ(n − 1)
2
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