數(shù)字信號處理教學(xué)課件：Chapter2 Discrete-Time Signals and Systems in the Time-Domain

1、Chapter 2Discrete-Time Signals and Systems in the Time-Domain1Three Concepts:Analog signal: continuous in time, continuous in amplitude.Discrete time signal: discrete in time, continuous in amplitude.Digital signal: discrete in time, quantization in amplitude. 22.1 Discrete-Time SignalsHow to get a

2、Discrete-Time Signal?The signal is defined at equally spaced discrete values of time, with the signal amplitude at these discrete times being continuous.The signal that is got by periodic sampling the continuous signal.3Discrete time signal is got by periodic sampling the continuous signal，sampling

3、interval being T：Discrete Time Signal-Sequence0txa(t)0 xa(nT)tT2T4Main contents:Representation: a sequence of numbers.Operation: time-shift, time-reversal, adder, product(modulator), differential, time-scaling, convolution, Several common sequences: n ,un, cos(n), ejn 52.1 Time-Domain Representation

4、 We use sequences to represent discrete-time signals.There are two ways to represent a sequence:(1) Descriptive way:(2) Graphic way: Example: look at books P4243.62.1.1 Length of a Discrete-Time SignalLeft-sided sequence. i.e.(a left-sided sequence is called an anticausal sequence.) Finite-length Se

5、quence: Infinite-length Sequence:Right-sided sequence. i.e. (a right-sided sequence is called a causal sequence.)7Finite-length SequenceA finite-length discrete-time signal can be represented as a finite-length sequence in a transform-domain.If we denote the vector of time-domain samples of length N

6、 defined for 0 n N-1 as x = x0 x1 xN-1t, the vector of the transform-domain samples of length N defined for 0 k N-1, X = X0 X1 XN-1t.So, X is obtained from x by multiplying by an NN invertible matrix D: X=Dx.82.1.2 Strength of a Discrete-Time SignalThe Lp-norm of a sequencexn is defined byWhere p is

7、 a positive integer and its value are 1 or 2 or . The norm provides an estimate of the size of a signal.92.2 Operations on Sequences(46) (1) Time-shifting: xnxn-m: if m is positive, xn-m is xn right-shifting m; if m is negative, xn-m is xn left-shifting m. 10(2) Time-reversal: xnx-n (3) Addition: xn

8、,ynxn+ynx1(n)n0 x2(n)n0 x1(n) +x2(n)n011(4) Multiplication: xnAxn, where A is a constant. (5)Progressive addition:xn (6)Differential: xn=xn+1-xnx(n)=xn-xn-112(7) Decimation/Interpolation:(Sampling Rate Alteration)xnxmn or xn/mUp-sampling (L=3)13Down-sampling (M=3)14(8) Convolution:(9) Combination of

9、 elementary operations15Example:convolutiongn=1 2 0 1, hn=2 2 1 1.16Convolution of two finite-lengh sequences:Note：1. Each point multiply and add respectively, and dont carry cross point; 2. The starting index equals to the sum of the two sequences index.172.3 Operation on Finite-Length Sequences Fo

10、r any arbitrary integer n0 , x1n = xn n0is no longer defined for the range 0nN-1We thus need to define another type of operation that will always keep the operated sequence defined for the range 0nN-1.18 2.3.1 Circular Time-reversal OperationModulo operationInteger mInteger r:0,N-1residueExample Thu

11、s,Thus,19circular time-reversal operation20n0 1 2 3 4 5Example 2.7 Illustration of Circular Time-Reversal 0 1 2 3 4 5nN=621For n00 (right circular shift), the above equation impliesthe circular shift, is defined using a modulo operation:2.3.2 Circular Shift of a Sequence22Illustration of the concept

12、 of a circular shiftRead Figure 2.17 by yourself. Alternate illustration of circular shiftFigure 2.16232.3.3 The Classifications of the Discrete-Time SignalsClass standard:Based on sequence lengthBased on symmetryBased on periodicityBased on energy and powerOther types of classification24SymmetryCon

13、jugate-symmetric sequence:A real conjugate-symmetric sequence is called an even sequence.Conjugate-antisymmetric sequence:A real conjugate-antisymmetric sequence is called an odd sequence.25Any complex sequence xn can be expressed as:Where, xcsn is its conjugate-symmetric part, and xcan is its conju

14、gate-antisymmetric part, we can get:Please look at P52 example2.5.26For any real sequence xn, it can be expressed as:Special condition: for a length-N sequence defined for , we will introduce the concept about periodic(circular) conjugate-symmetric part and antisymmetric part.Where:27Likewise, for a

15、 real sequence xn , the conjugate-symmetric part is denoted by xpen , and the other part is denoted by xpon. If a length-N sequence satisfy: xn=x*N-nWe call it periodic conjugate-symmetric.28PeriodicityPeriodic signalsIt is denoted by . Non-periodic signalsIt is denoted by xn.29Energy and PowerThe e

16、nergy of sequencesIt equals to the sum of the squares of all samples of the sequence.The average power of an aperiodic sequence xn is defined by30The average power of a periodic sequence Power signal :a infinite energy signal with finite average power.Energy signal :a finite energy signal with zero

17、average power.31Other TypesBounded sequence: Absolutely-summable: Square-summable:322.4 Typical Sequences1.Unit Sample SequenceThe unit sample sequence shifted by k samples is thus given by: 012.-1-233Unit Sample Sequence in MATLAB342. Unit Step SequenceThe unit step sequence shifted by k samples is

18、 thus given by:-10 123 135Unit Step Sequence in MATLAB363.Rectangle Sequence-10 123 1N-1NThe relationship among is :37Rectangle Sequence in MATLABN11384.Real Exponential Sequences 1 1 0 2 3 4 1 0 2 4 3 1Where a is in real value.39Real Exponential Sequence in MATLAB1.20.7405.Sinusoidal and Exponentia

19、l SequencesA real sinusoidal sequence(P57): Where A , 0 , and are real numbers.A exponential sequence(P59):Where A and are real or complex numbers.410=0 xn=cos0*n0 =0.80 =0.10 =42By expressing We can get:If we write xn=Then43The relationship between sin&exp sequence:44Complex Exponential Sequence Ge

20、neration Using MATLAB% Program 2_1 % Generation of complex exponential sequence a = input(Type in real exponent = ); b = input(Type in imaginary exponent = ); c = a + b*i; K = input(Type in the gain constant = ); N = input (Type in length of sequence = ); n = 1:N; x = K*exp(c*n); stem(n,real(x); xla

21、bel(Time index n);ylabel(Amplitude); title(Real part); disp(PRESS RETURN for imaginary part); pause stem(n,imag(x); xlabel(Time index n);ylabel(Amplitude); title(Imaginary part);45Input ParametersWe will generate a complex exponential sequence. 46Real Part & Imaginary Part47Real Sequence Generation

22、Using MATLAB% Program 2_2 % Generation of real exponential sequence % a = input(Type in exponent = ); K = input(Type in the gain constant = ); N = input (Type in length of sequence = );n = 0:N; x = K*a.n; stem(n,x); xlabel(Time index n);ylabel(Amplitude); title(alpha = ,num2str(a);48Input Parameters

23、We will generate two real exponential sequences.4950The Periodicity of Sinusoidal and Exponential SequencesIf for all n ,there is the least positive integer N existing, which satisfies xn=xn+N. We call xn is a periodic sequence, and N is its period. Lets see the sinusoidal sequence:51If:When k is a

24、integer, thenThat isThe sinusoidal sequence is a periodic sequence, its period is (N,k must be integers).52(1)When is a integer, if k=1, is the least integer, and is the period.(2) is not a integer, but a rational number, then: where, k and N are prime with each other, then is the least positive int

25、eger, and is the period.(3)When is irrational ,any k can not let N be positive integer ,so the sin sequence is not periodic.There is several conditions to discus:53Note:The sequence ej(0+) has the same period condition as the sin sequence.Whether the sin and the complex exp sequence are periodic seq

26、uence or not , 0 is their frequency. If the period of basic frequency is N, then its frequency is , so the basic frequency of sin sequences above is .54Fundamental and Harmonic ComponentsThe sinusoidal sequence with the lower frequency is called the fundamental component. The sinusoidal sequences whose angular frequencies are integer multiples(such as k times) of a sinusoidal sequence of lower angular frequency are called the harmonics(k-th harmonic).552.5 The Sampling Process:If a sin sequence is got by sampling a continuous sin signal, then what relationship between sample int