【正文】 ystem 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)有兩個初始狀態(tài) x1(0)與 x2(0) 當(dāng) x1(0) =4, x2(0) =2,f(t)=0時 ,零輸入響應(yīng) y1(t)=6e2t+4e3t,t0； 當(dāng) x1(0) =2, x2(0) =6,f(t)=0時 ,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 ,完全響應(yīng) y3(t)=4e2t+2e3t+3et, t0 (1) 當(dāng) x1(0) =1, x2(0) =0, f(t)=0時 ,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 ,零狀態(tài)響應(yīng) =? ?????????????????????00)( tfT???????????????????????????????????????????21)(620240tfT)()()( 321 tytyty ????Conclusions Systems are typically described by an arrangement of subsystems each of which is defined by a functional relation. Many different physical systems are defined by the same mathematical model so that understanding one system leads to an understanding of others. Systems are classified according to such properties as: memory, causality, stability, linearity, and timeinvariance. Linear, timeinvariant systems are special systems for which a rich and powerful description is available. We will focus on such systems. Actually, we will focus on the systems which have the properties of dynamic, causal, linear, timeinvariant, deterministic models. 。 若設(shè)電感中電流 iL(t)為電路響應(yīng) ， 則由基爾霍夫定律列出節(jié)點 a的支路電流方程為 )(1)]()([1)(1)(1)( 39。 零輸入響應(yīng)：若沒有輸入信號加入，則系統(tǒng)的響應(yīng)由其狀態(tài) 唯一確定。 若 a1， 則 f(at)是 f(t)的壓縮。 21 /TT 為無理數(shù)時，則是非周期函數(shù)。Signals amp。抽樣頻率 =22050Hz 0 0 Classification of signals and Multidimensional signal 2. Continuous Time（ CT） and discrete time（ DT） Signals (analogue signals and digital signals) 3. Pe