EE3TP4_9_DTConvolution_2_v2_Lecture 11

# EE3TP4_9_DTConvolution_2_v2_Lecture 11 - 2.2 Computing D-T...

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2.2 “Computing” D-T convolution We know about the impulse response h [ n ]. We found out that h [ n ] interacts with x [ n ] through convolution to give the zero-state response: y [ n ]= i =−∞ x [ i ] h [ n i ] How do we compute this? Two cases, depending on the form of x [ n ]: 1. x [ n ] is known analytically 2. x [ n ] is known numerically or graphically Analytical Convolution (used for “by-hand” analysis): Pretty straightforward conceptually: - put functions into convolution summation - exploit math properties to evaluate/simplify

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Example: x [ n ]= a n u [ n ] h [ n ]= b n u [ n ] Recall this form from 1 st - order difference equation example b n u [ n ] a n u [ n ] y [ n ]= ? y [ n ]= i =−∞ x [ i ] h [ n i ] = i =−∞ a i u [ i ] b n i u [ n i ] a function of n i gets “summed away” u [ i ]= { 1, i 0 0, i 0 = i = 0 a i b n i u [ n i ] Now use: u [ n i ]= { 1, i n 0, i n = i = 0 n a i b n i = b n i = 0 n a b i Now use: You should be able to go here directly
y [ n ]= { n 1, a = b [ 1 a b n 1 1 a b ] , a b “Geometric Sum” y [ n ]= b n i = 0 n a b i If a = b you are adding ( n + 1) 1’s and that gives n + 1 So now we simplify this summation… If a b , then a standard math relationship gives: i = 0 N 1 r i = 1 r N 1 r , r 1

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Commutativity Property of Convolution A simple change of variables shows that y [ n ]= i =−∞ x [ i ] h [ n i ] x [ n ]∗ h [ n ] = i =−∞ h [ i ] x [ n i ] h [ n ]∗ x [ n ] So … we can use which ever of these is easier!
Step 1 : Write both as functions of i : x [ i ] & h [ i ] Step 2 : Flip h [ i ] to get h [ -i ] (The book calls this “fold ”) Step 3 : For each output index n value of interest, shift by n to get h [ n - i ] (Note: positive n gives right shift!!!!) Step 4 : Form product x [ i ] h [n – i ] and sum its elements to get the number y [ n ] Repeat for each n Graphical Convolution Steps Can do convolution this way when signals are know numerically or by equation - Convolution involves the sum of a product of two signals: x [ i ] h [ n i ] - At each output index n , the product changes

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