精品課程《大學(xué)物理》-信號(hào)與系統(tǒng)(文件)
【正文】 o m o g en ei t ofp r o p er t y t h )()(),()(21212121112211tbytaytbftaftytytftftaytaftytftytf?????????：)(1 tf )(2 tf)(1 tf )(2 tfExample: system S， input is f(t), output is y(t), )()( 2 tfty ?Solution ： )()()( t h e n),()(),()( 2132211tbftaftftytftytfif????2212333 )]()([)()()( tbftaftftytfso ????)()(2)()()()(2)()(21221221222212tytyabtybtyatfta b ftfbtfa??????)()( 21 tbytay ??∴ This system is nonlinear. Linear and nonlinear systems Example: Consider a system S whose input f(t) and output y(t) are related by )()( txtty ?? To determine whether or not S is linear, we consider two arbitrary inputs f1(t) and f2(t): )()( 11 tytx ?)]()([)( 213 tbftaftty ???)(1 tft ?? )()( 22 tytf ? )(2 tft ??)()()( 213 tbftaftf ??)()( 21 tbytay ??Linear system Linear and nonlinear systems Example: Consider the system 3)(2)( ?? tftyWe can see: 3)(2)()(111 ??? tftytf3)(2)()( 222 ??? tftytfIf the input signal is )()()(213 tbftaftf ??then the output signal is 3)}()({2)( 213 ??? tbftafty )()( 21 tbytay ??So that this system is nonlinear. Linear and nonlinear systems Timevarying and timeinvariant systems A timevarying system is one in which the system operator changes with time. That is, the rule used to pute the outputs depends on time. For a singleinput singleoutput system , we would write y=T [ f(t)] A timeinvariant system is one in which the system operator does not change with time. We will study timeinvariant systems almost exclusively. timeinvariant system )(tf)(ty timeinvariant system )( ??tf )( ??tyDefinition : When T [ f(t)] = yf (t), if the system satisfy T [ f(t?t0)] = yf (t?t0), we call it timeinvariant systems. t)( tf21 t)(ty f21t)(0ttf ?10t02 t?t )( 0tty f ?10t 02 t?timeinvariant system )(tf )(tyTimevarying and timeinvariant systems Example : Consider the continuoustime system defined by )](s in [)( tfty ?Let x1(t) be an arbitrary input signal, the corresponding output signal is: )](s in [)( 11 tfty ?If we shift x1(t) by t0 seconds, then the corresponding output signal is: )](s i n [)( 0101 ttftty ??? Timeinvariant Timevarying and timeinvariant systems Example : Consider the system y(t)=f(2t) Assume f(t) is a rectangular pulse: y(t)=T[f(t)]= f(2t) T[f(t?t0)] = f(2t?t0) y(t?t0) = f(2t ?2t0) T[f(t?t0)] ? y(t?t0) t)( tf10 2t10 1)]([ tfTt)1( ?tf10 31 t)]1([ ?tfT1Timevarying and timeinvariant systems The system is timevariant Example: Consider the discretetime system ][][ nnfny ?Let x1[n] be an arbitrary input signal, the corresponding output signal is: ][][ 11 nnfny ?If we shift x1[n] by n0 points, then the corresponding output signal is: ][][ 01 nnnfny ??][)(][ 01001 nnfnnnny ??????Timevariant Timevarying and timeinvariant systems 例：已知某線性系統(tǒng)有兩個(gè)初始狀態(tài) x1(0)與 x2(0) 當(dāng) x1(0) =4, x2(0) =2,f(t)=0時(shí) ,零輸入響應(yīng) y1(t)=6e2t+4e3t,t0； 當(dāng) x1(0) =2, x2(0) =6,f(t)=0時(shí) ,零輸入響應(yīng) y2(t)=2e2t+8e3t , t0 ； 當(dāng) x1(0) =1, x2(0) =2,輸入為 f(t)時(shí) ,完全響應(yīng) y3(t)=4e2t+2e3t+3et, t0 (1) 當(dāng) x1(0) =1, x2(0) =0, f(t)=0時(shí) ,零輸入響應(yīng) =? ?????????????????????010T ????????????????????????????????620101240103T )()( 21 tyty ?(2) 當(dāng) x1(0) =0, x2(0) =0,輸入為 f(t)時(shí) ,零狀態(tài)響應(yīng) =? ?????????????????????00)( tfT???????????????????????????????????????????21)
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