ece306-6 - 1&& ¢£¤¥¤ ¦ ECE 306 Discrete-Time...

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Unformatted text preview: 1 &¡& ¢£¤¥¤ ¦ ECE 306 Discrete-Time Signals and Systems &¡ ¢£¤¥¦§¤¨¤©ª£« ¬ £­¨®¯ ¤¨¦£ ¦°±²©³´«®­¯¬°ª¤°­­¯ ¤°ª µ­´¦¯®³­°® ²¦£ ¶©£¥ ¶©³©°¦ &¡& ¢£¤ ¥¤ &¡& ¢£¤¥¤ § Analysis of Linear Time Invariant Systems (LTI) Intoduction Two basic methods for analyzing the response of LTI: ¨ The direct solution of the input-output equation for the linear system [ ] ( ) ( 1), ( 2),..., ( ), ( ), ( 1),..., ( ) y n F y n y n y n N x n x n x n M =----- In general form of the input-output relationship is (called a difference equation ), which is given 1 ( ) ( ) ( ) N M k k k k y n a y n k b x n k = = = -- +- & & where a k and b k are constant parameters 2 &¡& ¢£¤¥¤ ¢ Analysis of Linear Time Invariant Systems (LTI) The second methods is to resolve the input signal into sum of elementary signals. Then determine the response of each input elementary signals. We can use linearity property to find total response Let ¦ s have arbitrary discrete-time signal . We can represent in the following form ( ) ( ) ( ) k x n x k n k δ ∞ =-∞ =- & Example: A finite-duration sequence ( ) {2,4,0,3} x n ↑ = ( ) 2 ( 1) 4 ( ) 3 ( 2) x n n n n δ δ δ = + + +- Using the previous equation, can be written as ( ) x n &¡& ¢£¤¥¤ § Analysis of Linear Time Invariant Systems (LTI) The convolution Sum The response of the system to x(n) is The response of the LTI system to the unit sample sequence is denoted as h(n), and it is called the impulse response of a linear time invariant system [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) k k y n x n x k n k x k n k δ δ τ τ τ ∞ ∞ =-∞ =-∞ ¡...
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This note was uploaded on 03/11/2008 for the course ECE 306 taught by Professor Aliyazicioglu during the Winter '08 term at Cal Poly Pomona.

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ece306-6 - 1&& ¢£¤¥¤ ¦ ECE 306 Discrete-Time...

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