W08-102-Week1-Recap - EE 102 WEEK 1 RECAP NHAN LEVAN 1...

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EE 102 WEEK 1 RECAP NHAN LEVAN 1. Systems: IPOP Description Let S be a system. S is said to have an IPOP Description if it can be described by: (1.1) x ( t ) −→ [ S ] −→ y ( t ) , t R and (1.2) y ( t ) := T [ x ( t )] , t R where T [ · ] is a transformation taking IP x ( · ) to OP y ( · ). Thus in this course whenever we say a “System” we mean a “ System with an IPOP Description ”. Note that there also exist “Systems with State Space Descrip- tion” (take EE 142 in Sp). 2. Basic Properties of Systems with IPOP Description LINEAR (L) and NON-LINEAR (NL) . For any scalars a and b , and any x ( · ) , x 1 ( · ) , and x 2 ( · ): S is L T [ ax 1 + bx 2 ] = aT [ x 1 ] + bT [ x 2 ] , (2.1) equivalently S is L T [ ax ] = aT [ x ] and T [ x 1 + x 2 ] = T [ x 1 ] + T [ x 2 ] (2.2) If S is not L it is called NL. One can use either (2.1) or (2.2) to test whether S is L or not. If S is not L it is called NL. Note : any = all = every. TIME-INVARIANT (TI) and TIME-VARYING (TV). S is TI x ( t ) −→ [ S ] −→ y ( t ) , t R (2.3) and x ( t A ) −→ [ S ] −→ y ( t A ) , t R (2.4) where A is any real constant.
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