## Exercises in Continuous-Time Convolution

Conference

2017 ASEE Annual Conference & Exposition

Location

Columbus, Ohio

Publication Date

June 24, 2017

Start Date

June 24, 2017

End Date

June 28, 2017

Conference Session

Electrical and Computer Division Technical Session 8

Tagged Division

Electrical and Computer

Page Count

14

DOI

10.18260/1-2--28321

Permanent URL

https://peer.asee.org/28321

11035

#### Anthony M. Richardson University of Evansville

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Anthony Richardson has been teaching engineering for over 20 years. He joined the electrical engineering department at the Univerisity of Evansville in 2000. His interests are in signal processing and communications.

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#### Abstract

Continuous-time convolution is one of the more difficult topics that is taught in a Signals and Systems course. This paper presents several analytical and MATLAB based assignments that help students develop a better understanding of continuous-time convolution.

In the first assignment students are introduced to a set of basis functions that can be used to compose any piece-wise polynomial function. (Such functions include step and ramp functions as well as rectangular, triangular and trapezoidal pulses.) Each basis function can be represented by a triplet of numbers (amplitude, delay, and order) and any piece-wise polynomial function can be represented as an n x 3 matrix (with each row in the matrix corresponding to a single basis function). MATLAB routines accept the matrix representation and an array of time values as input arguments and return an array of function values.

Convolution between members of this particular set of basis functions can be performed exactly using only simple addition and multiplication operations (no integration is required). Piece-wise polynomial functions can be convolved using corresponding operations on their matrix representations. These operations can be easily performed analytically. MATLAB routines can also be used to compute the exact continuous-time convolution of any pair of piece-wise polynomial functions from their basis function representation. In a second assignment, students analytically convolve several interesting piece-wise polynomial functions using conventional convolution (integration) methods and compare their results to those obtained using the basis function representation and the corresponding convolution operations. The basis function results can be found from pencil-and-paper calculations or computed using MATLAB.

MATLAB can also be used to approximately compute the continuous time convolution of a wider class of functions than the piece-wise polynomial class. In a third assignment students compare the results of analytical convolution to those obtained approximately using MATLAB and also to those obtained from circuit simulation.

Richardson, A. M. (2017, June), Exercises in Continuous-Time Convolution Paper presented at 2017 ASEE Annual Conference & Exposition, Columbus, Ohio. 10.18260/1-2--28321

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