1.Input-Output Relationship 1.Input-Output Relationship 1.Input-Output Relationship Let h(n)denote the impulse response of an .Since h(n)is the response of input 6(n)and Taking advantage of the property of linear, LTI discrete-time system the system is time invariant,we have we have x(n) h(n) m 6(n-)→n-k) Likewise,as the system is linear -)十2a- Recall that x(n)can be written as x(6(n-)→xk)hn-) Eventually,the I-O relationship of an LTI system can be written as follows x(m)=x(k)6(n-k) Note that.x(A)is considered as a constant in this case ym)=∑xkhn-k)=∑)xn-k) 二 20 2 1.Input-Output Relationship 1.Input-Output Relationship 1.Input-Output Relationship Solution This operation is called a linear convolution Schematic Representation sum and can be represented compactly as Time-reverse h(k)to form h) h(n-k) y(n)=x(n)*h(n) Shift /n-k)to the right by n sampling periods "⑧ y(n) if n>0 or shifth(k)to the left by n sampling The linear convolution sum has periods if n<0 to form h(n) n Commutative property .Form the product vx()h(n-k) If the lengths of x(n)and h(n)are N and N Associative property Sum all samples of v)to develop the n-th respectively,the length of the convolution Distributive property sample ofy(n)of the convolution sum sum y(n)will be N +N2-1 2219 1. Input-Output Relationship Let h(n) denote the impulse response of an LTI discrete-time system Recall that x(n) can be written as h(n) x(n) y(n) () ()( ) n x n xk n k     20 1. Input-Output Relationship Since h(n) is the response of input ¥(n) and the system is time invariant, we have ¥(nˉk) ėh(nˉk) Likewise, as the system is linear x(k)¥(nˉk) ė x(k)h(nˉk) Note that, x(k) is considered as a constant in this case 21 1. Input-Output Relationship Taking advantage of the property of linear, we have Eventually, the I-O relationship of an LTI system can be written as follows () ()( ) ()( ) k k y n xkhn k hkxn k        ()( ) ()( ) k k x k n k xkhn k       22 1. Input-Output Relationship This operation is called a linear convolution linear convolution sum and can be represented compactly as The linear convolution sum has Commutative property Associative property Distributive property yn xn hn () () ()  23 1. Input-Output Relationship Solution Time-reverse reverse h(k) to form h(ˉk) Shift h(ˉk) to the right by n sampling periods if n > 0 or shift h(ˉk) to the left by n sampling periods if n <0 to form h(nˉk) Form the product product v(k)=x(k)h(nˉk) Sum all samples of v(k) to develop the n-th sample of y(n) of the convolution sum 24 1. Input-Output Relationship Schematic Representation If the lengths of x(n) and h(n) are N1 and N2 respectively, the length of the convolution sum y(n) will be N1+N2ˉ1 z h(ˉk) n h h(nˉk) k  y(n) x(n)