lect2 - SOME USEFUL Z-TRANSFORMS We found earlier that a z...

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Unformatted text preview: SOME USEFUL Z-TRANSFORMS We found earlier that a z z n U a n- ↔ ) ( . Equivalently, a z a z z z a n n n- = ∑ ∞ =- ; . (2.1) Differentiating both sides with respect to a yields ( 29 a z a z z z na n n n- = ∑ ∞ =-- ; 2 1 . (2.2) Now, the sum on the left is the z-transform of the signal ) ( n U na n . Thus, we have the new transform pair ( 29 2 1 ) ( a z z n U na n- ↔- with ROC a z (2.3) Differentiating (2.2) now yields ( 29 a z a z z z a n n n n n- =- ∑ ∞ =-- ; 2 ) 1 ( 3 2 , which gives us the transform pair ( 29 3 2 2 ) ( ) 1 ( a z z n U a n n n- ↔-- with ROC a z (2.4) Differentiating again with respect to a yields the pair ( 29 4 3 ) 3 )( 2 ( ) ( ) 2 )( 1 ( a z z n U a n n n n- ↔--- with ROC a z (2.4) Continuing this process, we find the general pair: ( 29 1 ! ) ( ) 1 ( ) 1 ( +-- ↔ +- ⋅ ⋅ ⋅- m m n a z z m n U a m n n n which can be re-written as ( 29 1 ) ( +-- ↔       m m n a z z n U a m n with ROC a z (2.5) If we repeat the above process starting with the anticausal pair a z z n U a n- ↔--- ) 1 ( , we arrive at the general transform pair ( 29 1 ) 1 ( +-- ↔--      - m m n a z z n U a m n with ROC a z < (2.6) Example 2.1 a) With m = 1 (2.5) and (2.6) yield ( 29 2 1 ) ( a z z n U na n- ↔- ( 29 2 1 ) 1 ( a z z n U na n- ↔---- . b) With m = 3, ( 29 4 3 ) ( 6 ) 2 )( 1 ( a z z n U a n n n n- ↔--- ( 29 4 3 ) 1 ( 6 ) 2 )( 1 ( a z z n U a n n n n- ↔------ . SOME Z-TRANSFORM THEOREMS 1. The time shifting theorem ) ( ) ( z F z k n f k- ↔- Proof: ∑ ∑ ∑ ∞-∞ =-- ∞-∞ = +- ∞-∞ =- = =- ↔- n n k n k n n n z n f z z n f z k n f k n f ) ( ) ( ) ( ) ( ) ( . Q.E.D. 2. The convolution theorem ) ( ) ( ) ( ) ( z G z F n g n f ↔ . Proof: ∑ ∞-∞ =- = k k n g k f n g n f ) ( ) ( ) ( ) ( ∑ ∑ ∑ ∑ ∞-∞ =- ∞-∞ = ∞-∞ =- ∞-∞ =      - =      - ↔ k n n n n k z k n g k f z k n g k f ) ( ) ( ) ( ) ( ∑ ∞-∞ =- = k k z z G k f ) ( ) ( (by the time shifting theorem) ) ( ) ( ) ( ) ( z F z G z k f z G k k = = ∑ ∞-∞ =- . Q.E.D. 3. A conjugate sequence property       ↔- z F n f 1 ) ( . Proof: ∑ ∞-∞ =- = n n z n f z F ) ( ) ( ∑ ∞-∞ = =       n n z n f z F ) ( ) ( 1 ∑ ∑ ∞-∞ =- ∞-∞ =- = =       n n n n z n f z n f z F ) ( ) ( ) ( 1 . Q.E.D. 4. Parseval’s formula ∫ ∑- ∞-∞ = = p p w w w p d e G e F n g n f j j n ) ( ) ( 2 1 ) ( ) ( . Proof: Let ) ( ) ( z F n f ↔ and ) ( ) ( z G n g ↔ . Then by the conjugate sequences property above,       ↔- z G n g 1 ) ( , and by the convolution theorem       ↔- z G z F n g n f 1 ) ( ) ( ) ( ....
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This note was uploaded on 10/18/2009 for the course EE ee332 taught by Professor Ee during the Spring '07 term at Istanbul Technical University.

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lect2 - SOME USEFUL Z-TRANSFORMS We found earlier that a z...

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