102_1_Week1and2_notes

102_1_Week1and2_notes - WINTER 2008 EE 102 WEEK 1 RECAP...

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WINTER 2008 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” (e.g., 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. TI TEST: x ( t ) -→ [ S ] -→ y ( t ) , t R (2.5) x ( t - A ) -→ [ S ] -→ z ( t ) , (2.6) If z ( t ) = y ( t - A ) TI (2.7) Date : Winter 2008. 1
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2 NHAN LEVAN If S is not TI it is called TV (Time-Varying) . CAUSAL (C) and NON-CAUSAL (NC) A system S is called C if its OP at any time t , i.e., y ( t ), depends only on the values of the input x ( · ) for all · t. If one takes t to be NOW. Then causality means “present” value of output, i.e., value at “now”, depends only on “past” and “now” values of input, but NOT “future” values of input. In other words, y ( t ) depends only on values of IP x ( · ) from the beginning of time up to and including NOW:= t — but not on the future values, i.e., values at times beyond t . If S is not C it is called NC. SYSTEMS with MEMORY. Heuristically speaking a causal system is said to have memory if it can “remember” past values of its inputs. If the OP at anytime t depends only on value of the IP at the same time t then the system is called Instantaneous, or has Zero memory. 3. UNIT STEP FUNCTION U ( · ) and DELTA FUNCTION δ ( · ) U ( t ) := 1 , t 0 , ( or t > 0) (3.1) := 0 , t < 0 , ( or t 0) (3.2) δ ( t ) := 0 , t = 0 , (3.3) := , t = 0 , (3.4) -∞ δ ( t ) dt = 1 (3.5) More to come next week!
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3 4. Typical Problems of The Week Consider the DE: dy ( t ) dt + y ( t ) = dx ( t ) dt , t 0 , given y (0) = x (0) = 0 (i) Solve for y ( · ) in terms of x ( · ) —————————– We have: d dt { e t y ( t ) } = e t dx ( t ) dt , e t y ( t ) = e t dx ( t ) dt dt + K, e t y ( t ) = t 0 e τ dx ( τ ) d τ d τ , t 0 .
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