【正文】 ?nA (3) Play a role in discrete time signal processing similar to the role played by exponential functions in continuous time signal processing .Specifically, they are eigenfunctions of 2 discrete time linear system and for that reason form the basis for transform analysis techniques. When ︳ α ︳ =1, x(n) takes the form x(n)= Aenj? (4) Because the variable n is an integer ,plex exponential sequences separated by integer multiples of 2π in ω(frequency) are identical sequences ,I .e: ee njkj ??? ?? )2( (5) This fact forms the core of many of the important differences between the representation of discrete time signals and systems . A general sinusoidal sequence can be expressed as x(n)=Acos( 0w n +Φ) (6) where A is the amplitude , 0w the frequency, and Φ the phase . In contrast with continuous time sinusoids, a discrete time sinusoidal signal is not necessarily periodic and if it is the periodic and if it is ,the period is 2π/ω0 is an integer . In both continuous time and discrete time ,the importance of sinusoidal signals lies in the facts that a broad class of signals and that the response of linear time invariant systems to a sinusoidal signal is sinusoidal with the same frequency and with a change in only the amplitude and phase . Systems:In general, a system maps an input signal x(n) to an output signal y(n) through a system transformation T{.}.The definition of a system is very broad . without some restrictions ,the characterization of a system requires a plete inputoutput relationship knowing the output of a system to a certain set of inputs dose not allow us to determine the output of a system to other sets of inputs . Two types of restrictions that greatly simplify the characterization and analysis of a system are linearity and time invariance, alternatively referred as shift invariance . Fortunately, many system can often be approximated by a linear and time invariant system . The linearity of a system is defined through the principle of superposition: T{ax1(n)+bx2(n)}=ay1(n)+by2(n) (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 u