数字信号处理英文版课件：Chap 2 Discrete-Time Signals in T-Domain

1、 Chap 2 Discrete-Time Signals in T-Domain Discrete-Time Signals in the Time Domain Representation of sequences Operations on sequences; Classification of Sequences Typical SequencesDSP group 2007-chap2-ed12 2.1 Time-Domain Representation DT signals represented as sequences of numbers called samples.

2、 xn (n +, n is integer)Example A-1 x1= 0.2, x0 = 2.1, x1=1.1, Eg. DSP group 2007-chap2-ed13 2.1 Time-Domain Representation Graphical representation Sequence can be obtained by samplingP34xn= xa(t)|t=nT = xa(nT), n=,2, 1, 0, 1, 2, DSP group 2007-chap2-ed14 2.1 Time-Domain Representation sampling peri

3、od (Sampling interval) : T (sec) sampling frequency : FT = 1/T (Hz) (2.3) nth sample : xn= xa(nT) DSP group 2007-chap2-ed15 2.1.1 Length of a DT signal Finite-length sequence (Finite-duration sequence): xn=0, for nN2 . length: N = N2 N1 + 1 N-point sequence right-sided sequence: xn=0, for nN1 causal

4、 sequence: xn=0, for nN2 anticausal sequence: xn=0, for nN2 and N2 0 two-sided sequence: xn for n be positive and negative valuesDSP group 2007-chap2-ed16 2.1.1 Length of a DT signalP 37XXXDSP group 2007-chap2-ed17 2.1.2 Strength of a DT signal Lp -norm of a sequence xn: p= DSP group 2007-chap2-ed18

5、 2.1 Time-domain representation Examples Real sequence: if the nth sample xn is real for all values of n. Complex sequence: Complex conjugate sequence of xn : Sequence xn represented in simple form by ignoring the brace.DSP group 2007-chap2-ed19 2.1 Time-domain representation ExamplesExample A-2 rea

6、l sequence complex sequenceWhere Complex conjugate sequence DSP group 2007-chap2-ed110 2.1 Time-domain representation Examples Sampled-data signals: samples are continuous-valued . Digital signals: samples are discrete-valuedIn practical cases:Digital signals are obtained by quantizing the sample va

7、lues either by rounding or by truncation.Digital systems only deals with digital signals. DSP group 2007-chap2-ed111 ExamplesExample A-4 Finite-length sequence 8-point sequenceInfinite-length sequence two-sided sequence 12-point sequence zero-paddingDSP group 2007-chap2-ed112 ExamplesExample 2Right-

8、sided sequenceLeft-sided sequence0: causal sequence0: anticausal sequenceDSP group 2007-chap2-ed113 2.2 Operations on sequences 2.2.1 Elementary Operations:Discrete-Time SystemxnynInput sequenceOutput sequence Product (Modulation) operation: w1n= xn yn Modulatorxnynw1nwindowingDSP group 2007-chap2-e

9、d114 2.2 Operations on sequences Scalar multiplication operation : w2n= Axn Multiplier Addition operation: w3n= xn+ yn Adderxnynw3n Time-shifting operation: w4n= xn m Delayerz1xnw4n= xn 1xnAw2nDSP group 2007-chap2-ed115 2.2 Operations on sequences Time-reversal / folding : w6n= xn Branching: Used to

10、 provide multiple copies of a sequencexnxnxnDSP group 2007-chap2-ed116 2.2 Operations on sequences ExamplesP41 Example 2.2Two 5-point sequences: (0n4)Solution:DSP group 2007-chap2-ed117 2.2 Operations on sequences ExamplesExample 2.3Two sequences:Solution:Zero paddingP39 Example 2.1Ensemble Averagin

11、gDSP group 2007-chap2-ed118 2.2.3 Convolution Sum Convolution sum: Definition Commutative operation DSP group 2007-chap2-ed119 2.2.3 Convolution Sum distributive operation Interpretation of convolution sum1) time-reverse hk to hk associative operation DSP group 2007-chap2-ed120 2.2.3 Convolution Sum

12、2) time-shift hk to hnk3) product vk = xk hnk4) sum Schematic representation-nDSP group 2007-chap2-ed121 2.2.3 Convolution SumExample:Find DSP group 2007-chap2-ed122 Figures representation of ConvolutionDSP group 2007-chap2-ed123 2.2.3 Convolution Sum The result sequence of convolution of two finite

13、 -length sequences is also a finite sequence. If sequence yn is the convolution of N-point sequence xn and M-point sequence hn, then the length of yn is N+M1.DSP group 2007-chap2-ed124 2.2.3 Convolution Sum Generally, if then hkxkkkDSP group 2007-chap2-ed125 2.2.3 Convolution SumEx1 Suppl.：hn=an un,

14、 0a1, xn=un-un-N, determine yn=xn*hnN-1DSP group 2007-chap2-ed126 2.2.3 Convolution SumEx1 Suppl.：1) When n yn=0nkhn-kN-112DSP group 2007-chap2-ed127 2.2.3 Convolution SumEx1 Suppl.：2)When 3）when DSP group 2007-chap2-ed128 2.2.3 Convolution Sum Convolution computed by Multiplication -2 1 3 4 2 -4 2

15、6 8 3 -6 3 9 12 4 -8 4 12 16DSP group 2007-chap2-ed129 2.2.3 Convolution Sum Convolution computed by Multiplication Example: one 3-point and one 4-point sequenceSolution:8 12 166 9 122 3 44 6 8 4 4 1 21 24 16 2 3 4) 2 1 3 4DSP group 2007-chap2-ed130 2.2.4 Sampling Rate Alteration Sampling rate alter

16、ation ratio is R = FT / FT If R 1, the process called interpolation, or called upsampling; If R 1, the process called decimation. or called downsampling L Mxn=xa(nT)FT=1/TP46 Eq. 2.23, 2.24P47 Fig 2.14.DSP group 2007 chap5-ed1312.3.1 Circular shift of a sequence1. Consider length-N sequences defined

17、 for 0nN1 2. i.e. xn=0, for n 0 (right circular shift), the above equation implies 2.3.1 Circular shift of a sequenceWhere: if r = m +l N, and 0m N1, then moduloDSP group 2007 chap5-ed1337. Illustration of the concept of a circular shift 2.3.1 Circular shift of a sequence exampleDSP group 2007 chap5

18、-ed1348. As can be seen from the previous figure, a right circular shift by n0 is equivalent to a left circular shift by N n0 sample periods;9. A circular shift by an integer number greater than N is equivalent to a circular shift by n0 N. 2.3.1 Circular shift of a sequence comments on the example10

19、. Circular shift can be realized by extend the sequence periodically as xn+mN first, and then shift, or vice versa, and at last get the sample for 0nN1.DSP group 2007 chap5-ed135xn-1xnx4 2.3.1 Circular shift of a sequence example1 xn+4m,DSP group 2007 chap5-ed136 2.3.1 Circular shift of a sequence s

20、tepsStep1: time-shifting;Step2: N-point period extension; Step3: get the principle; DSP group 2007 chap5-ed137 2.3.2 Circular time-reversal of a sequenceP48 Figure. 2.15DSP group 2007-chap2-ed138 2.3.3 Classification of SequencesClassification based on symmetry for real sequence: even sequence ( xev

21、n) Conjugate-symmetric sequence xcsn : xn= x*nFig. 2.18 (a) An even sequenceDSP group 2007-chap2-ed139 2.3.3 Classification of Sequences for real sequence: odd sequence ( xodn) Conjugate-antisymmetric sequence xcan : xn= x*n xn= xcsn+ xcan (2.27) Fig. 2.18 (b) An odd sequenceDSP group 2007-chap2-ed1

22、40 2.3.3 Classification of SequencesExample 2.8Solution:DSP group 2007-chap2-ed141 2.3.3 Classification of Sequences even and odd sequence xn= xevn+ xodn (2.29) DSP group 2007-chap2-ed142 2.3.3 Classification of Sequences based on periodicity Periodic sequence : fundamental Period : NN minimum posit

23、ive integerPeriod: N=7Aperiodic sequenceDSP group 2007-chap2-ed143 2.3.3 Classification of Sequences average power : energy of sequence : Example: 2.10 Not energy signal Power signalDSP group 2007-chap2-ed144 2.3.3 Classification of Sequences bounded sequence : N-periodic extension : DSP group 2007-

24、chap2-ed145 2.4 Typical Sequences and Sequence Representation2.4.1 Some basic sequences Unit sample(impulse) sequence: Unit step sequence: x=zeros(1,N);x(1)=1Plot(x)ones(1,N)46 2.4.1 Some basic sequences Sinusoidal sequence: Angular frequency: 0 Amplitude: A Phase: DSP group 2007-chap2-ed147 2.4.1 Some basic sequences Exponential sequence: A and are real or complex numbers LetReal partImaginary partDSP group 2007-chap2-ed148 2.4.1 Some basic sequence ExampleExample: Fig.2.23 DSP group 2007-chap2-ed149Real exponential sequence xn=A n, n where A and are real