數(shù)字信號處理英文版課件：Chapter2 Discrete-Time Signals In the Time-Domain第四版

1、Discrete-time signals xn-represented as sequences with argument n being an integer in the range - n The nth sample value of sequence xn is also denoted as xnvalues of argument n-defined only for integer and undefined for noninteger values of nDiscrete-time signal represented by xn2 Discrete-Time Sig

2、nals In the Time-Domain2.1 Time-Domain RepresentationDiscrete-time signal may also be written as a sequence of numbers inside braces: xn=,-0.2,2.2,1.1,0.2,-3.7,2.9, The arrow is placed under the sample at time index n = 0In the above, x-1= -0.2, x0=2.2, x1=1.1, etc. 2 Discrete-Time Signals In the Ti

3、me-DomainGraphical representation of a discrete-time signal with real-valued samples is as shown below:2 Discrete-Time Signals In the Time-DomainIn some applications, a discrete-time sequence xn may be generated by periodically sampling a continuous-time signal xa(t) at uniform intervals of time2 Di

4、screte-Time Signals In the Time-DomainHere, n-th sample is given by xn=xa(t) |t=nT=xa(nT), n=,-2,-1,0,1, The spacing T between two consecutive samples is called the sampling interval or sampling periodReciprocal of sampling interval T, denoted as FT , is called the sampling frequency: FT=1/T2 Discre

5、te-Time Signals In the Time-DomainUnit of sampling frequency is cycles per second, or hertz (Hz), if T is in secondsWhether or not the sequence xn has been obtained by sampling, the quantity xn is called the n-th sample of the sequence xn is a real sequence, if the n-th sample xn is real for all val

6、ues of nOtherwise, xn is a complex sequence2 Discrete-Time Signals In the Time-DomainA complex sequence xn can be written as xn=xren+jximn (2.4) where xren and ximn are the real and imaginary parts of xnThe complex conjugate sequence of xn is given by x*n=xren - jxim n Often the braces are ignored t

7、o denote a sequence if there is no ambiguity2 Discrete-Time Signals In the Time-DomainExample - xn=cos0.25n is a real sequence yn=ej0.3n is a complex sequenceWe can write yn=cos0.3n + jsin0.3n = cos0.3n + jsin0.3n where yren=cos0.3n yimn=sin0.3n2 Discrete-Time Signals In the Time-DomainTwo types of

8、discrete-time signals: - Sampled-data signals in which samples are continuous-valued- Digital signals in which samples are discrete-valuedSignals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding or truncation2 Discrete-Ti

9、me Signals In the Time-DomainExampleAmplitudeDigital signalAmplitudeBoxedcar signalTime,tTime,t2 Discrete-Time Signals In the Time-DomainA discrete-time signal may be a finite-length or an infinite-length sequenceFinite-length (also called finite-duration or finite-extent) sequence is defined only f

10、or a finite time interval:N1 n N2where - N1 and N2 with N1 N2Length or duration of the above finite-length sequence is N= N2 - N1+ 1Zero-paddingappend with zero-valued samples.2.1.1 Length of a discrete-time signal Vector form xn=x0 x1 xN-1t2 Discrete-Time Signals In the Time-DomainA right-sided seq

11、uence xn has zero-valued samples for n N2If N2 0, a left-sided sequence is called a anticausal sequence.A left-sided sequence2 Discrete-Time Signals In the Time-Domain2.1.2 Strength of a Discrete-Time Signalp is a positive integer.In practice, the value of p used is typically 1 or 2 or .(2.9)The str

12、ength of a DT Signal usually is given by its norm. of xn is defied by 2 Discrete-Time Signals In the Time-Domain(2.10)Peak absolute value For a length-N seqence, is the root-mean-squared(rms) value of xn,and is the mean absolute value of xn.It can be shown thatnorm(x,1)norm(x,2)norm(x,inf) M-file in

13、 MATLAB2 Discrete-Time Signals In the Time-Domain(2.11)For example,a length-N seqence yn, another xn,One application of the norm is in estimatingthe error in approximation of a DT signal byanother DT signal in some sense.0 n N-1mean-squared error(MSE)relative error(2.12)2 Discrete-Time Signals In th

14、e Time-Domain2.2 The Basic Operations On Sequences 2.2.1 Elementary Operations modulator(product or windowing) 2 Discrete-Time Signals and Systemsnxny1nynxnw=modulatornx2nAxnw=multiplerpick-off node2 Discrete-Time Signals and Systemsaddition(subtraction) nxny3nynxnw+=adder1-Znx14-=nxnwunit delayZnx1

15、5+=nxnwunit advance time-reversal (folding operation)x-nExample 2.1 Ensemble average(Application of addition operation)2 Discrete-Time Signals and Systemsoriginal uncorrupted data NoiseEnsemble average2 Discrete-Time Signals and SystemsVector form s=s0 s1 sN-1tThe i-th measurement of data vector2.2.

16、2 Combination of Elementary Operations 2 Discrete-Time Signals and Systems2.2.3 Convolution Sumyn = xn hn*is called the convolution sum of the sequences xn and hn and represented compactly as2 Discrete-Time Signals and SystemsExample Develop the sequence yn generated by the convolution of the sequen

17、ces xn and hn : xn = hn = n + n-1 + n-2*2 Discrete-Time Signals and Systemsxkh-kxkh1-kxkh2-kxkxkxn = hn = n + n-1 + n-2Xkh-k=1Xkh-kkkXkh1-kXkh1-k=2kXkh2-kkXkh3-kkXkh4-kh4-kh3-kXkh2-k=3Xkh3-k=2Xkh4-k=1xkhk112 Discrete-Time Signals and SystemsIn general, if the lengths of the two sequences being convo

18、lved are M and N, then the sequence generated by the convolution sum is of length M+N-1Read example 2.5 and 2.6 by yourself.yn=n+2n-1+3n-2+2n-3+n-4 2 Discrete-Time Signals and Systemsyn n11023422312.2.4 Sampling rate alterationUp-samplingDown-sampling2 Discrete-Time Signals and Systems2.2.4 Sampling

19、 rate alterationUp-samplingDown-sampling2 Discrete-Time Signals and SystemsUp-sampling (L=3)2 Discrete-Time Signals and SystemsDown-sampling (M=3)2 Discrete-Time Signals and Systems5 Finite-Length DiscreteTransforms 2.3 Operations on Finite-length SequencesConsider length-N sequences xn defined for

20、0nN-1Sample values of such sequences are equal to zero for values of n 0 (right circular shift), the above equation implies5 Finite-Length DiscreteTransforms the circular shift, is defined using a modulo operation:2.3.2 Circular Shift of a SequenceIllustration of the concept of a circular shift5 Fin

21、ite-Length DiscreteTransforms Read Figure 2.17 by yourself. Alternate illustration of circular shiftFigure 2.16xn is a real sequence, if the n-th sample xn is real for all values of nOtherwise, xn is a complex sequenceA complex sequence xn can be written as xn=xren+jximn where xre and xim are the re

22、al and imaginary parts of xnThe complex conjugate sequence of xn is given by x*n=xren - jxim n2 Discrete-Time Signals and Systems2.3.3 Classification of Sequences1. Classification based on Symmetry Conjugate-symmetry sequence x*n=x-nConjugate-antisymmetry sequence x*n=-x-nIf xn is a real sequence, w

23、hat is xn called?2 Discrete-Time Signals and Systemscomplex sequence xn Evenxn=x-n Oddxn=-x-n real xnConjugate-symmetry part Conjugate-antisymmetry part xn is complex sequence, 2 Discrete-Time Signals and Systemsgn=0, 1-j4, -2-j3, 4+j2, -5+j6, j2, 3 g-n=3, j2, -5+j6, 4+j2, -2-j3, 1-j4, 0 2 Discrete-

24、Time Signals and Systemsgn= 0, 1+j4, -2+j3, 4-j2, -5-j6, -j2, 3 Example 2.8 A length-7 sequence(n:-3,3):gcs n=1.5,0.5+j3,-3.5+j4.5,4,-3.5-j4.5,0.5-j3,1.5 gca -n=-1.5,0.5+j,1.5-j1.5,-j2,-1.5-j1.5,-0.5+j,1.5 2.Classification of Sequences based on periodicity Example 2 Discrete-Time Signals and Systems

25、A sequence satisfying the periodicity condition is called an periodic sequence,for all nAdding two periodic sequences with different fundamental periodsThe fundamental periodGCD( )-greatest common divider2 Discrete-Time Signals and SystemsLCM( )-least common multiplefundamental periodsExampleIts fun

26、damental period is 48 ,because the GCD of 16 and 12 is 4.2 Discrete-Time Signals and Systems3. Energy and Power SignalsTotal energy of a sequence xn is defined by2 Discrete-Time Signals and SystemsThe average power of a sequence is defined byExample - Consider the causal sequence defined by Note: xn

27、 has infinite energy Its average power is given by2 Discrete-Time Signals and Systems4. Other Types of ClassificayionA sequence xn is said to be bounded if Example - The sequence xn=cos(0.3n) is a bounded sequence as2 Discrete-Time Signals and SystemsA sequence xn is said to be absolutely summable i

28、f Example - The sequenceis an absolutely summable sequence as2 Discrete-Time Signals and SystemsA sequence xn is said to be square-summable if Example - The sequenceis square-summable but not absolutely summable2 Discrete-Time Signals and SystemsIt is neither absolutely summable nor square-summable

29、.2 Discrete-Time Signals and Systems Example - The sequence obtained by adding an absolutely summable sequence with its replicas shifted by integer multiples of N,is called an N-periodic extension of .Unit sample sequence Unit step sequence 2.4 Typical Sequences and Representation2.4.1 Basic Sequenc

30、es2 Discrete-Time Signals and Systems(2.4.3 )Representation of an Arbitrary Sequence by impulsesAn arbitrary sequence can be represented in the time-domain as a weighted sum of some basic sequence and its delayed (advanced) versions2 Discrete-Time Signals and SystemsExample2 Discrete-Time Signals an

31、d Systems2 Discrete-Time Signals and SystemsRectangular Window SequenceBox-car sequence(2.56)(2.57)Windowing2 Discrete-Time Signals and SystemsReal sinusoidal sequence xn=Acos(0n+) (2.48)where A is the amplitude, 0 is the normalized angular frequency, and is the phase of xn.Sinusoidal and Exponentia

32、l sequenceThe unit of 0 and is radians/sampleOften in practice, 0 is expressed as 0=2f0 (2.54) Where f0 is normalized frequency in cycles/sample.2 Discrete-Time Signals and Systemsxin=Acos()cos(0n),xqn=-Asin()sin(0n) xn= xin+xqn (2.49)where xin+xqn are, respectively,the in-phase and the quadrature c

33、omponents of xn,and are given bywhen A and are real or complex numbers.By expressing2 Discrete-Time Signals and Systems(2.51)General form of Exponential Sequencewe can rewriteAnd Then,2 Discrete-Time Signals and SystemsThe Complex Exponential Sequence(2.52a)(2.52b)2 Discrete-Time Signals and Systems

34、complex sinusoidal sequencewhen A and are real numbers,its a real exponential sequence.=1.2=0.92 Discrete-Time Signals and SystemsReal Exponential Sequence Why? How to determine the fundamental period?To verify the above fact, consider x1 n= Acos(0n+) x2 n= Acos(0( n+N)+)2 Discrete-Time Signals and

35、SystemsIf 2/0 is an rational number, the sequence xn=Acos(0n+) is periodic.Otherwise,its aperiodic.The periodicity of sinusoidal sequence x2 n= cos(0( n+N) + ) = cos(0n + )cos 0N - sin(0n + )sin0N which will be equal to cos(0n+ )=x1n only if sin 0N= 0 and cos 0N = 1.These two conditions are met if and only if 0N= 2r (2.53a) or 2/ 0 = N/r . (2.53b)where N and r are positive integers.Smallest value of N satisfying 0N=2ris the fundamental period of the sequence.