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Experiments on Convolution - Linear Dynamic Systems and Signals - Lab Handout, Exercises of Electronic Circuits Analysis

Jamia Millia IslamiaElectronic Circuits Analysis

Main points are: Experiments on Convolution, Convolution of Signals, Arbitrary Input, Discrete-Time Convolutions, Distributivity Properties, Associativity Properties, Finite Sum of Form, Convolution Integral

Typology: Exercises

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6.6 MATLAB Laboratory Experiments on Convolution
Purpose: In this section we design two experiments dealing with continuous- and
discrete-time convolutions and their applications to linear continuous- and discrete-time
dynamic systems. The purpose of the first experiment is to present the convolution
operator, and to demonstrate some of its properties in both continuous- and discrete-time
domains. By writing and modifying the corresponding MATLAB programs, students will
master every step of the convolution process. In the second experiment, the convolution
method will be used to determine the zero-state responses of both continuous- and
discrete-time linear dynamic systems by using the famous formula that states that the
response of a linear system at rest due to an arbitrary input is the convolution of that
input with the system impulse response.
6.6.1. Convolution of Signals
In this experiment, students must verify the commutativity and distributivity properties
of continuous- and discrete-time convolutions.
Part 1. Continuous-Time Signals
The convolution of two signals is defined by
๎˜€๎˜‚๎˜๎˜„๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡๎˜Š๎˜‰๎˜‹๎˜€๎˜๎˜Œ๎˜Ž๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡๎˜๎˜ ๎˜‘
๎˜’
๎˜“
๎˜‘
๎˜€๎˜‚๎˜๎˜Ž๎˜ƒ๎˜•๎˜”๎˜–๎˜‡๎˜—๎˜€๎˜˜๎˜Œ๎˜˜๎˜ƒ๎˜†๎˜…๎˜š๎˜™๎˜›๎˜”๎˜–๎˜‡๎˜—๎˜œ๎˜๎˜”๎˜Ÿ๎˜ž ๎˜™! #"๎˜›๎˜…๎˜‹"$
(6.30)
Consider the continuous-time signals
๎˜€๎˜‚๎˜๎˜Ž๎˜ƒ%๎˜…๎˜ˆ๎˜‡๎˜๎˜'&(๎˜Œ๎˜Ž๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡)๎˜ž*๎˜€๎˜˜๎˜Œ+๎˜ƒ%๎˜…๎˜ˆ๎˜‡,๎˜.-/๎˜Œ๎˜Ž๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡)๎˜ž*๎˜€+0๎˜˜๎˜ƒ1๎˜…๎˜ˆ๎˜‡2๎˜43657๎˜ƒ8๎˜…๎˜ˆ๎˜‡2๎˜™957๎˜ƒ%๎˜…2๎˜™๎˜›:๎˜๎˜‡๎˜•;
Write a MATLAB program to perform and verify the continuous-time convolution
commutativity and associativity properties:
๎˜€๎˜๎˜ƒ1๎˜…๎˜ˆ๎˜‡๎˜Š๎˜‰๎˜‹๎˜€ ๎˜Œ๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡๎˜๎˜$๎˜€ ๎˜Œ๎˜ƒ1๎˜…๎˜ˆ๎˜‡๎˜Š๎˜‰๎˜‹๎˜€ ๎˜๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡
๎˜€๎˜๎˜ƒ%๎˜…๎˜ˆ๎˜‡๎˜Š๎˜‰<๎˜ƒ=๎˜€ ๎˜Œ๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡?>'๎˜€ 0๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡@๎˜‡๎˜๎˜A๎˜€ ๎˜๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡๎˜Š๎˜‰๎˜‹๎˜€ ๎˜Œ๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡?>'๎˜€ ๎˜๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡?๎˜‰๎˜‹๎˜€ 0๎˜ƒ%๎˜…๎˜ˆ๎˜‡
While writing the program you must discretize the convolution integral given in (6.30);
that is the integral in (6.30) must be approximated by a finite sum of the form
๎˜€๎˜๎˜ƒ=B๎˜ŸC!๎˜‡?๎˜‰๎˜‹๎˜€ ๎˜Œ๎˜ƒDBEC!๎˜‡๎˜šFGC H
I
J%K
H)L
๎˜€๎˜๎˜ƒ๎˜•M๎˜†C!๎˜‡๎˜ˆ๎˜€ ๎˜Œ36๎˜ƒ=BN๎˜™OM1๎˜‡=CP;
(6.31)
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Download Experiments on Convolution - Linear Dynamic Systems and Signals - Lab Handout and more Exercises Electronic Circuits Analysis in PDF only on Docsity!

6.6 MATLAB Laboratory Experiments on Convolution

Purpose: In this section we design two experiments dealing with continuous- and discrete-time convolutions and their applications to linear continuous- and discrete-time dynamic systems. The purpose of the first experiment is to present the convolution operator, and to demonstrate some of its properties in both continuous- and discrete-time domains. By writing and modifying the corresponding MATLAB programs, students will master every step of the convolution process. In the second experiment, the convolution method will be used to determine the zero-state responses of both continuous- and discrete-time linear dynamic systems by using the famous formula that states that the response of a linear system at rest due to an arbitrary input is the convolution of that input with the system impulse response.

6.6.1. Convolution of Signals

In this experiment, students must verify the commutativity and distributivity properties of continuous- and discrete-time convolutions. Part 1. Continuous-Time Signals The convolution of two signals is defined by

    

Consider the continuous-time signals

%'&( ) +%,.-/ )+0124365782957%2: ;

Write a MATLAB program to perform and verify the continuous-time convolution commutativity and associativity properties:

1 $ 1  % <= ?>' 0 @A  ?>' ? 0 %

While writing the program you must discretize the convolution integral given in (6.30); that is the integral in (6.30) must be approximated by a finite sum of the form

=BC!? DBEC!FGC IH J %K H )L

MC! 3 6=BNOM1=CP; (6.31)

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Take QSRUTWVYX , plot the results obtained, and comment on the durations of the signals obtained through the convolution procedure. Observe that the commutativity and distribu- tivity properties hold for the approximate continuous-time convolution, and conclude that when Q.Z[T these properties also hold for the convolution integral (6.30). Part 2. Discrete-Time Signals In the discrete-time domain, the convolution of two signals is defined by

\]^ _a \b ^_ (^) Rdcfehgi

cfekjlg

\]^ mno\b^6_qprmst p!u#vw_sv$u (^) (6.32)

Consider the discrete-time signals

]^(^) RGx@y8z({ ^_o|~}h^6ย€ยย‚p9}h^ _nprย1ยƒ \b^_(^) Rย„X pย† b^6_t (^) TNย‡ _ ย‡ ย \ยˆ^ _(^) R }?^6_p}h^ _Npยย‰`

Write a MATLAB program to perform the operations defined in (6.17โ€“18). Plot the results obtained and comment on the convolution signal durations. Verify the results by using the MATLAB function conv. Hint: Use the MATLAB program given in Section 6.5 and write its modifications. The program can be used for calculating both the continuous-time and discrete-time convolutions. Note that you must plot the convolution results obtained in the proper time intervals. Use plot(t,fapr) with t=t1+t2:T:T1+T2, where ^ยŠ@ย‹t^ Qย‹ยŒ`Dtย^ RdX t@ยŽWt define the duration intervals of the signals being convolved.

SUPPLEMENT:

  1. Commutativity
    ] ^ 6_a \b^6_ (^) R \b^_a \]^_ (^) (6.17)

  2. Distributivity \]^ a<|~\b^ _\ ยˆ^oยƒ (^) R \]^6_EaP\b^ E\]^6_a \ยˆ^` (^) (6.18)

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