1、1 外 文 资料 Signals and System Signals are scalar-valued functions of one or more independent variables. Often for convenience, when the signals are one-dimensional, the independent variable is referred to as time The independent variable may be continues or discrete. Signals that are continuous in bot
2、h amplitude and time (often referred to as continuous -time or analog signals) are the most commonly encountered in signal processing contexts. Discrete-time signals are typically associated with sampling of continuous-time signals. In a digital implementation of signal processing system, quantizati
3、on of signal amplitude is also required . Although not precisely Correct in every context, discrete-time signal processing is often referred to as digital signal processing. Discrete-time signals, also referred to as sequences, are denoted by functions whose arguments are integers. For example , x(n
4、) represents a sequence that is defined for integer values of n and undefined for non-integer value of n . The notation x(n) refers to the discrete time function x or to the value of function x at a specific value of n .The distinction between these two will be obvious from the contest . Some sequen
5、ces and classes of sequences play a particularly important role in discrete-time signal processing .These are summarized below. The unit sample sequence, denoted by (n)=1 ,n=0 ， (n)=0,otherwise (1) The sequence (n) play a role similar to an impulse function in analog analysis . The unit step sequenc
6、e ,denoted by u(n), is defined as U(n)=1 , n 0 u(n)=0 ,otherwise (2) Exponential sequences of the form X(n)= 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 disc
7、rete time linear system and for that reason form the basis for transform analysis techniques. When =1, x(n) takes the form x(n)= Aenj (4) Because the variable n is an integer ,complex exponential sequences separated by integer multiples of 2 in (frequency) are identical sequences ,I .e: ee njkj )2(
8、(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 +) (6) where A is the amplitude , 0w the frequency, and the phase . In contrast with continuous time s
9、inusoids, a discrete time sinusoidal signal is not necessarily periodic and if it is the periodic and if it is ,the period is 2/0 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 l
10、inear 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 . w
11、ithout some restrictions ,the characterization of a system requires a complete input-output 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 chara
12、cterization 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: Tax1(n)+bx2(n)=
13、ay1(n)+by2(n) (7) 3 Where Tx1(n)=y1(n) , Tx2(n)=y2(n), and a and b are any scalar constants. Time invariance of a system is defined as Time invariance Tx(n-n0)=y(n-n0) (8) Where y(n)=Tx(n) and 0n is a integer linearity and time inva riance are independent properties, i.e ,a system may have one but n
14、ot the other property ,both or neither . For a linear and time invariant (LTI) system ,the system response y(n) is given by y(n)= knhnxknhkx )(*)()()( (9) where x(n) is the input and h(n) is the response of the system when the input is (n).Eq(9) is the convolution sum . As with continuous time convo
15、lution ,the convolution operator in Eq(9) is commutative 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 primaril
16、y 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 st
17、ability and causality .A system is considered stable in the bounded input-bounder 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,
