Slide 48 CUHKSZ EIE 3001 Spring 201819 Interaction of Multiple Systems cont a d

# Slide 48 cuhksz eie 3001 spring 201819 interaction of

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Slide 48 CUHK(SZ) EIE 3001, Spring 2018/19 Interaction of Multiple Systems (cont.) a (d) Feedback #1 #2 R ³ f ² t dt t i C t v ' ) ' ( 1 ) ( 1 C v ( t ) R t V i ) ( 2 Ex. i(t) i 2 (t) i 1 (t) + v(t) _ ³ f ² t C dt t i C t v ' ) ' ( 1 ) ( 1 R i t V R 2 ) ( Note: - i ( t ) i 1 ( t )

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Slide 49 CUHK(SZ) EIE 3001, Spring 2018/19 Properties of System a Memory/Memoryless a Invertibility and Inverse System a Causality a Stability a Time-invariance a Linearity
Slide 50 CUHK(SZ) EIE 3001, Spring 2018/19 (1). Memory and Memoryless a A system is memoryless if the output at a given time is dependent only on the input at the same time Ex. (i) Resister is a memoryless component. y(t)=R x(t) where y(t) : voltage x(t) : current R: resistance ¦ f ²f k k x n y ] [ ] [ (ii) With memory : e.g. Capacitor is also with memory.

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Slide 51 CUHK(SZ) EIE 3001, Spring 2018/19 Memory and Memoryless
Slide 52 CUHK(SZ) EIE 3001, Spring 2018/19 Memory and Memoryless

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Slide 53 CUHK(SZ) EIE 3001, Spring 2018/19 (2). Invertibility and Inverse System a A system is invertible if distinct inputs lead to distinct outputs. x(t) y(t) z(t) System #1 System #2 If z(t)=x(t), then system #2 is the inverse system of system #1. E.g. y(t)=2 x(t) then z(t) =0.5 y(t) ] 1 [ ] [ ] [ ] [ ] [ ² ¦ ²f n -y n y n z then k x n y n k
Slide 54 CUHK(SZ) EIE 3001, Spring 2018/19 (3). Causality a A system is causal if the output at anytime only depends on the input at the present time and before. E.g. y[n]=x[n]-x[n-1] : causal y(t)=x(t+1) : non-causal a Causal property is more important for real-time processing. a But for some applications, such as image-processing, no need to process the data causally. E.g. ¦ ² ² µ M M k k n x M n y ] [ 1 2 1 ] [ Q: Memoryless l Causal ?

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Slide 55 CUHK(SZ) EIE 3001, Spring 2018/19 (4). Stability a A system is stable if bounded input gives bounded output. --- BIBO stable a Eg. x(t) : the horizontal force; y(t) : vertical displacement x(t) y(t) Stable System x(t) y(t) Unstable System
Slide 56 CUHK(SZ) EIE 3001, Spring 2018/19 a E.g. ] [ ) 1 ( ] [ ] [ n u n k u n y n k µ ¦ ²f ] [ ) 1 ( n u n µ Input: u[n], bounded Output: (n+1)u[n], non-bounded as f o f o ] [ lim n y n (4). Stability (cont.)

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Slide 57 CUHK(SZ) EIE 3001, Spring 2018/19 (5). Time-invariance a A system is time-invariant if a time shift in the input only causes a time shift in the output. ` i.e. If x[n] o y[n], then x[n-n 0 ] o y[n-n 0 ] a Ex.1 y(t)=sin(x(t)) Let y 1 (t)= sin(x 1 (t)), x 2 (t)=x 1 (t-t 0 ) Then y 2 (t)=sin(x 2 (t))=sin(x 1 (t-t 0 )) = y 1 (t- t 0 ) ? time-invariant (T.I.)
Slide 58 CUHK(SZ) EIE 3001, Spring 2018/19 (5). Time-invariance (cont.) a Ex.2 y[n]=n x[n] Let y 1 [n]= n x 1 [n] & x 2 [n]=x 1 [n-n 0 ] Then y 2 [n]=n x 2 [n]=n x 1 [n-n 0 ] However y 1 [n-n 0 ]=(n-n 0 ) x 1 [n-n 0 ] z y 2 [n] ? not time-invariant (T.I.)

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Slide 59 CUHK(SZ) EIE 3001, Spring 2018/19 (6). Linearity a A system is linear if 1. the response to x 1 (t)+x 2 (t) is y 1 (t) +y 2 (t) --- additivity 2. the response to a·x 1 (t) is a·y 1 (t), where a is any complex constant. --- scaling a Combine the above two properties, we can conclude a·x 1 (t)+ b·x 2 (t) a·y 1 (t) + b·y 2 (t) --- superposition property For discrete-time : a·x 1 [n]+ b·x 2 [n] a·y 1 [n] + b·y 2 [n]
Slide 60 CUHK(SZ) EIE 3001, Spring 2018/19 a If linear, zero input gives zero output.

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