【正文】 (7) 3 Where T{x1(n)}=y1(n) , T{x2(n)}=y2(n), and a and b are any scalar constants. Time invariance of a system is defined as Time invariance T{x(nn0)}=y(nn0) (8) Where y(n)=T{x(n)} and 0n is a integer linearity and time inva riance are independent properties, ,a system may have one but not the other property ,both or neither . For a linear and time invariant (LTI) system ,the system response y(n) is given by y(n)= ?????? ??k nhnxknhkx )(*)()()( (9) where x(n) is the input and h(n) is the response of the system when the input is δ(n).Eq(9) is the convolution sum . As with continuous time convolution ,the convolution operator in Eq(9) is mutative and associative and distributes over addition: Commutative : x(n)*y(n)= y(n)* x(n) (10) Associative: [x(n)*y(n)]*w(n)= x(n)*[ y(n)*w(n)] (11) Distributive: x(n)*[y(n)+w(n)]=x(n)*y(n)+x(n)*w(n) (12) In continuous time systems, convolution is primarily an analytical tool. For discrete time system ,the convolution sum. In addition to being important in the analysis of LTI systems, namely those for which the impulse response if of finite length (FIR systems). Two additional system properties that are referred to frequently are the properties of stability and causality .A system is considered stable in the bounded inputbounder output(BIBO)sense if and only if a bounded input always leads to a bounded output. A necessary and sufficient condition for an LTI system to be stable is that unit sample response h(n) be absolutely summable For an LTI system, 4 Stability ???????n nh )( (13) Because of Eq.(13),an absolutely summable sequence is often referred to as a stable sequence. A system is referred to as causal if and only if ,for each value of n, say n, y(n) does not depend on values of the input for n necessary and sufficient condition for an LTI system to be causal is that its unit sample response h(n) be zero for n an LTI system. Causality： h(n)=0 for n 0 (14) Because of sequence that is zero for n0 is often referred to as a causal sequence. representation of signals In this section, we summarize the representation of sequences as linear binations of plex exponentials, first for periodic sequence using the discretetime Fourier series, next for stable sequences using the discretetime Fourier transform, then through a generalization of discretetime Fourier transform, namely, the ztransform, and finally for finiteextent sequence using the discrete Fourier transform. In section review the use of these representation in charactering LIT systems. Discretetime Fourier series Any periodic sequence x(n) with period N can be represented through the discrete time series(DFS) pair in Eqs.(15)and (16) Synthesis equation : )(~nx = ???10~ )/2()(1 NnnkNjekXN ? (15) Analysis equation: )(~kX = e nkNjNn nxN)/2(10~ )(1 ????? (16) The synthesis equation expresses the periodic sequence as a linear bination of 5 harmonically related plex exponentials. The choice of i nterpreting the DFS coefficients X(k) either as zero outside the range 0≦ k≦ (N1) or as periodically accepted convention , however ,to interpret X(k) as periodic to maintain a duality between the analysis and synthesis equations. Time Fourier Transform Any stable sequence x(n) (. one that is absolutely summable ) can be represented as a linear bination of plex exponentials. For a periodic stable sequences, the synthesis equation takes the form of Eq.(17),and the analysis equation takes the form of Eq.(18) synthesis equation: x(n)= ??? ?? ? dX e nj?? )(21 (17) analysis equation: X(ω)= ??????rnjenx ?)( (18) To relate the discrete time Fourier Transform and the discrete time Fourier Transform series, consider a stable sequence x(n) and the periodic signal x1(n) formed by time aliasing x(n), ?)(1nx ????? ?r rNnx )( (19) Then the DFS coefficients of x1(n) are proportional to samples spaced by 2π/N of the Fourier Transform x(n). Specifically, X1(k0=1/N X(ω) )/2( Nk?? ? (20) Among other things ,this implies that the DFS coefficients of a peri