ece306-5 - ECE 306 Discrete-Time Signals and Systems...

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1 °±° ²³´µ¶ · ECE 306 Discrete-Time Signals and Systems °± ²³´µ¶·´¸´¹º³» ¼ ³½¸¾¿ ´¸¶³ ¶ÀÁ¹ÃÄ»¾½¿¼Àº´À½½¿ ´Àº ŽĶ¿¾Ã½À¾ ¶³ ƹ³µ ƹùÀ¶ °±° ²³´ µ¶ °±° ²³´µ¶ ¸ Addition, Multiplication, and Scaling of Sequences Amplitude Scaling: (A Constant Multiplier) ( ) ( ), y n Ax n n = - ∞ < < ∞ x(n) A y(n)=Ax(n) Addition of two signals (An Adder) 1 2 ( ) ( ) ( ), y n x n x n n = + - ∞ < < ∞ + x 1 (n) y(n)=x 1 (n)+ x 2 (n) x 2 (n)
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2 °±° ²³´µ¶ ² Addition, Multiplication, and Scaling of Sequences The product of two signals (A signal Multiplier) A unit delay element A unit advance element 1 2 ( ) ( ) ( ), y n x n x n n = - ∞ < < ∞ x x 1 (n) x 2 (n) y(n)=x 1 (n) x 2 (n) z -1 x(n) y(n)=x(n-1) z x(n) y(n)=x(n+1) °±° ²³´µ¶ ¹ Input-Output Description of Systems The relation between the input and output signals are known input- output relationship Mathematical representation of the transformation is Discrete- time System x(n) y(n) Output signal or response input signal or excitation τ denotes the transformation. In general input-output relationship can be also shown as [ ] ( ) ( ) y n x n τ = ( ) ( ) x n y n τ
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3 °±° ²³´µ¶ ¶ Input-Output Description of Systems The input signal is Example: 3 3 ( ) 0 otherwise n n x n ° - = ± ² ( ) ( ) y n x n = ( ) ( 1) y n x n = - ( ) ( 1) y n x n = + [ ] 1 ( ) ( 1) ( ) ( 1) 3 y n x n x n x n = + + + - [ ] ( ) max ( 1), ( ), ( 1) y n x n x n x n = + - ( ) ( ) ( ) ( 1) ( 2) ... n k y n x k x n x n x n =-∞ = = + - + - + ³ a b c d e f Solution { } ( ) ...,0,3,2,1,0,1,2,3,0,... x n = ( ) ( ) y n x n = { } ( ) ...,0,3,2,1,0,1,2,3,0,... y n = a ( ) ( 1) y n x n = - { } ( ) ...,0,3,2,1,0,1,2,3,0,... y n = b °±° ²³´µ¶ ´ Input-Output Description of Systems Solution (cont) ( ) ( 1) y n x n = + { } ( ) ...,0,3,2,1,0,1,2,3,0,... y n = c d f [ ] 1 ( ) ( 1) ( ) ( 1) 3 y n x n x n x n = + + + - [ ] [ ] 1 1 2 (0) (1) (0) ( 1) 1 0 1 3 3 3 y x x x = + + - = + + = [ ] [ ] 1 1 ( 1) (0) ( 1) ( 2) 0 1 2 1 3 3 y x x x - = + - + - = + + = 5 2 5 ( ) ...,0,1, ,2,1, ,1,2, ,1,0,...
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