5.6
๎
-transform MATLAB Laboratory Experiment
Purpose: This experiment presents the frequency domain analysis of discrete-time
systems using MATLAB. The impulse, step, sinusoidal, and exponential responses of
discrete-time systems will be examined using the system transfer function method based
on the
๎
-transform. In addition, MATLAB will be used to perform the partial fraction
expansion and to find the inverse
๎
-transform. Note that MATLAB uses only the integral
representation of linear discrete-time systems.
Part 1. Consider the linear discrete-time system represented by the system transfer
function
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Using MATLAB, find and plot:
(a) The impulse response of the system.
(b) The step response of the system.
(c) The zero-state response due to the input signal
!๎" #๎$
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" +๎#๎$-,." #๎$
.
(d) The zero-state response due to the input signal
!."๎#๎$
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+
๎(1
,."๎#๎$
.
Part 2. Consider the systen transfer function
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+
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+
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๎56๎
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๎78๎๎4๎๎
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๎9๎
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+
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๎:4๎๎
๎๎<;
(a) Find the factored form of the transfer function using the MATLAB function
[z,p,k]=tf2zp(num,den).
(b) The partial fraction expansion of rational functions can be performed using
the MATLAB function residue. Find the inverse
๎
-transform of the given transfer
function using residue, that is, find analytically the system impulse response. Plot the
system impulse response.
(c) Repeat appropriately the steps outlined in (b) such that the system step response
is obtained.
Part 3. Find and plot the zero-input response of the system whose transfer function
is given in Part 2, and whose initial conditions are given by
=
"
๎?>
$
๎@;๎A
=
" BC$
๎&๎DA
B
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AH๎?4DAH๎
+
AH๎M;
(Hint: Find
NPO
Q
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and
R
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as defined in formulas (5.45) and (5.67) respectively.
Then, use the MATLAB function dimpulse with the coefficients of
NPO
Q
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๎๎
and
R
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representing, respectively, the corresponding numerator and denominator coefficients.)
Submit four plots for Part 1, one plot for Part 2, and one plot for Part 3, and present
analytical results obtained in Parts 2โ3.
Comment: Instructors can design additional MATLAB laboratory experiments or
additional parts of laboratory experiments by selecting from the problem section (Section
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