1、数字信号处理数字信号处理(英文版英文版)全册配套精品完整课件全册配套精品完整课件2 2Main Contents:1 Discrete-Time Signals2 Typical Sequences and Sequences Representations3 Discrete-Time Systems4 Linear Time Invariant(LTI) Systems5 Classification of LTI Discrete-Time SystemsChap 2 DT-Signals & DT Systems3 31. Discrete Time SignalsA. Time-Do
2、main Representation Signals represented as sequences of numbers,called samples and their values , denoted asSequence ,(). ,0.95, 0.2,2.17,1.1,s:0.2, 3.67,2.9,x nnx n 4 4TSamples:sampling int (ervalsampling fr), 2, 1,e0,1,2,T, que. F1cy/n, at nTx nx tnT1. Discrete Time Signals ,if 0. .: cosReal sequ(
3、0.25 )ence.s:reimimx nxnjxnxne gx nn0.3 Complex sequ .ence.: .s:reimjnx nxnjxne gx ne5 5B. Operations on SequencesDiscrete-time system Input sequencexn Output sequencey n(a) Product(modulation) (b) Scaling (amplification, attenuation) y nw n x ny nax n x n wn y na xn yn6 612(c) Sum and difference (d
4、) Ideal delay(Time-shifting) - y nx n x ny nx n d 1 x n2 x n y n xn ynZ-dB. Operations on Sequences7 71234Combination of elementary operation(e) 123sy nx nx nx nx nB. Operations on Sequences8 8(f) :Ensemble averaging11 , where . KiaveiiiixxxsdKsddat a， noi s e Let actual uncorrupted data, denote the
5、 noise vector corrupting the i-th measurement.isd A very simple application of addition operation in improvingquality of measured data corrupted with additive noise. B. Operations on Sequences9 9B. Operations on Sequences1010(g) Sampling rate alte ration: x n uxn L / , 0, 2 , 0, .ux n LnLLx notherwi
6、se if R=L1, inserting L-1 zero-valued samples between two consequence samplUp-es samplinofg: . x ninterpolati If 1,called . 1,called ondecimatio . nRR Sampling rate alteration ratio is R=/.TTFF Employed to generate a new sequence with a sampling rate higer or lower than that of the sampling rate of
7、a given sequence TTy nFFx n.B. Operations on Sequences1111Up-samp g linB. Operations on Sequences1212 1 if R=1, keeping every M-th sample,M and removing M-1 samples in-between samDownples-sampling of : .x n x n y n MB. Operations on Sequences1313l On Symmetry: Conjugate-symmetric or Conjugateantisym
8、metric sequencesC. Classification of Sequences Any real sequence xn can be divided into even part and odd part , where 2 2eoeox nx n x n, x nx nx -n/ ;x nx nx -n/ . (a) the real even sequence: , for all (b) the real odd sequence: , for alleeoox nx -n nx n-x -n n1414(c) the conjugate symmetric sequen
9、ce: , for all(b) the conjugate anti-symmetric sequence: , for allcscscaca xnx-n nxn-x-n nAny complex sequence can be divided into the conjugatesymmetric and the conjugate anti-symmetric parts: 2 cscacscax n x nxn xn ,where xnx nx -n/ ; xnx nx2-n/ .C. Classification of Sequences1515l On Periodicity P
10、eriodic or aperiodic sequencesl On Energy or Power Bounded or unbounded sequences for all x nx Nn n2 ,bounded sequenceIf , absolutely-summable.If , square-summable.xnnx nBx nx nx nx nC. Classification of Sequences1616sin , - Square-summable but not absolutely-sum. .m bl:a e. caengx nnn sin, - Neithe
11、r square-summable nor absolutely-summable. bcx nnn C. Classification of Sequences17172. Typical Sequences0 1,01) Unit sample: 0,1,02) UnitA. Some Basic Se step : quences0,3) Sinusoidal sequence : sin:x nnnotherwisenu notherwise nnunnx 18184) Exponential sequence: na5) Rect. sequence NRnu nu nN：N-1n
12、NR n2. Typical Sequences1919-Analysis of signal: B. Representation of an Arbi trary Sequencekx nx kn-k. .: 0.5 2 1.5 120.75 6e gx nnnnn2. Typical Sequences2020 3. Discrete Time SystemsA. Definition and examples nyTnx It is a transformation or operator that maps an inputsequence into an output sequen
13、ce. y nT x n21211. .: 1 Accumulato.rnnlle gy nx lx lx ny nx n3. Discrete Time Systems0or, = -1 where -1 is the initial condition of a causal systennly nyx ly2222101Moving average filter: Mly nx nlMTime index n Time index nSuppose , where 2 (0.9) nx ns nd ns nnd ns i gnal ，noi s e.3. Discrete Time Sy
14、stems2323 where 2 (0.8) ns nnd ns i gnal ，noi s e. s n s nd n 3M 10M 24241 F n+(n-1+ n+1 ),2actor of 2 interpolatoruuuy nxxx xn ux n L DT yn Linear interpolator 12 n+(n-1+ n+2 )+(n-2+ n+1 )33Factor of 3 interpolatoruuuuuy nxxxxx3. Discrete Time Systems2525Factor of 2 interpolator3. Discrete Time Sys
15、tems2626Factor of 4 interpolatorLi near i nt er pol at i on3. Discrete Time Systems272712122(a) : e.g. Moving average system, but not Linear sy stemT ax nbx naT x nbT x nT x nx nB. Classification of Discrete-Time Systems(b) : e.g. Moving average system, but ninotShif dowvariant nsampling system syst
16、e t-mT x ny nT x ndy ndy nx Mn3. Discrete Time Systems2828(c) : only deponds on values , Causal or, 0, 0systemy nx k knh nfor nk -(d) : . () B i.e.Stable syste mI BOx ny nh k (e) :Linear Time-InvarianLTI Syt Sysstemtem 3. Discrete Time Systems2929the response to a unit sample sequence .Determination
17、 of theImpulse resp impulse resonse:ponse.nC. Impulse and Step Response123Consider an LTI DT system with an input-output relation 12.y nx nx nx n123123Let , then . : 12., ,02.x nny nh nieh nnnnorh nn the impulse response of the accumulator SStep respo . nse:nlnh l3. Discrete Time Systems30304. Time-
18、Domain Characterization of LTI DTS A. Input-Output Relationship kx nx knksignalanalysis by linearity by timeinvariancekkky nT x nTx knkx k Tnkx k h nk- sum theimpulse response of the LTIsystconvolutioenm.y nx nh nh n- -3131 Calculation of the Convolution S: For eam:uchky nx k h nkn(a) reverse h khk(
19、b) delay bysamples,. ., ()hkniehknh nk(c) multiply the sequences and sum the results.x kwith h nk4. Time-Domain Characterization of LTI DTS 32324. Time-Domain Characterization of LTI DTS 33334. Time-Domain Characterization of LTI DTS 34344. Time-Domain Characterization of LTI DTS 35354. Time-Domain
20、Characterization of LTI DTS 36364. Time-Domain Characterization of LTI DTS 37374. Time-Domain Characterization of LTI DTS 38384. Time-Domain Characterization of LTI DTS 39394. Time-Domain Characterization of LTI DTS 40404. Time-Domain Characterization of LTI DTS 41414. Time-Domain Characterization o
21、f LTI DTS 42424. Time-Domain Characterization of LTI DTS 4343i.e.: .nSh n B. Stability condition in terms of the impulse response absolutely-summable,iff h n .kkxxky nh k x nkh kx nkBh kB S where, . So the system is BIBO stablexx nB 4. Time-Domain Characterization of LTI DTS 4444000 the output sampl
22、e depends only on input sa Causal symples st for em:.nthy nx nnnC. Cusality condition in terms of the impulse response1212 Let and be the responses of a causal DT system to the inputs and , respectively.y ny nx nx n1212 Then = for implies also that = for .x nx nnNy ny nnN In a causal system, changes
23、 in output samples do not precede the changes in the input samples.4. Time-Domain Characterization of LTI DTS 4545D. Linear constant coefficient difference equationThe representation of the LTI systems in the time domainthe -th-order linear constant coefficient difference equation:N00 can be determi
24、ned based on initial conditions; if initial rest, corresponding system is LTI and causal.Initial rest: If 0,for ,then 0, for .y nNx nnny nnn0001 or NMkmkmMNmkmkd y nkp x nmy np x nmd y nk，4. Time-Domain Characterization of LTI DTS 46465. Classification of LTI DT SystemsA. Based on impulse response l
25、engthl Finite impulse response (FIR): 1212 0 for and with .the system output is h nnNnNNNl Infinite impulse response (IIR):0For a causal IIR system, its output is .nky nh k x nk21 .Nk Ny nh k x nk4747B. Based on the Output Calculation Processl Non-recursive DT System: the output depend only on the p
26、resent and the past input samples.l Recursive DT System: the output depend not only on the present and the past input samples, but also the past output samples.0 Mkky np x nk01 MNkkkky np x nkd y nkC. Based on the Impulse Response Coefficients Real-valued System and Complex-valued System.5. Classifi
27、cation of LTI DT Systems4848l The Continuous-time Fourier Transform (CTFT)l The Discrete-time Fourier Transform (DTFT)l DTFT TheoremslThe Frequency Response of LTI DT SystemlPhase and Group DelayChap 3 Discrete-Time Fourier Transform 4949Definition -The CTFT Xa(j) of a continuous-time signalxa(t) is
28、 given by j(j )( )ed (3.1)taaXx tt The CTFT often is referred to as the Fourier spectrum, or simply the spectrum of the CT signal3.1 Continuous-Time Fourier Transform5050 Inverse Continuous-Time Fourier Transform j1( )(j )ed (3.2)2taax tXDefinition -The Inverse CTFT of a Fourier transformXa(j) is gi
29、ven by The ICTFT often is referred to as the Fourier integral. CTFT pair-CTFT ( )(j ) aax tXt 5151 Magnitude spectrum and phase spectrum Magnitude spectrum - Phase spectrum -Polar formj()(j ) |(j )|e, ( )arg(j )aaaaaXXX |Xa(j)|a() Total Energy E E x of a finite-energy CT complex signal221|( )| d|(j
30、)| d (3.9)2xaax ttXE Parsevals relation5252 Energy Density Spectrum Sxx( )Definition -2( ) |(j )|xxaSX Energy E E x ,r over a specified range of frequencies a b of the signal xa(t) is computed by,1( )d2bax rxxSE5353 Band-Limited CT Signals Ideal Band-limited signal has a spectrum that is zero outsid
31、e a finite frequency a | | b :(j ), | | (j )0, otherwise aabaXX An ideal band-limited signal cannot be generated in practice Lowpass CT signal:0, | |(j )(j ), 0| |paapXX Bandwidth : p5454 Band-Limited CT Signals Highpass CT signal:0, 0| |(j )(j ), | |paapXX Bandpass CT signal:0, 0| |, | |(j )(j ), |
32、 | LHaaLHXX Bandwidth : H L5555 3.2 Discrete-Time Fourier TransformDefinition -The DTFT X(e j) of a sequence xn is given by jj(e ) e (3.12)nnXx n In general, X(e j) is a complex function of the real variable and can be written asjjjreim(e )(e )j(e ) (3.19)XXX5656 Magnitude function and phase functio
33、n Magnitude function : Phase function :Polar formjjj ()j(e ) |(e )|e (3.21)( )arg(e ) (3.22)XXX j|(e )|X( ) Likewise, j|(e )|Xand ( ) are called the magnitude spectrum and phase spectrum.5757 Examples of DTFTExample 3.5 Find the DTFT of unit sample sequence n. Example 3.6 Find the DTFT of causal seq
34、uence xn=anun, |a|1. as | a e j | = |a| 1jj(e ) e = 01nnnSolution:jjjj01(e ) e =e1ennnnnXx naaSolution:5858jj|(e )| |(e)|XX( )() / magnitude / Phase in radians The magnitude and phase function of sequence 0.5nun. Examples of DTFT5959 Characteristics of DTFT The DTFT X(e j) of a sequence xn is a cont
35、inuous function of ; The DTFT X(e j) of a sequence xn is also a periodic function of with a period 2 .j(2)jj(e)(e ) e,knnXXx ni.e. for all integer values of k6060 The Inverse DTFT jj1 (e ) ed (3.16)2nx nXDefinition -The DTFT pairj (e ) (3.17)x nX F F6161 Commonly used DTFT pairs table 3.3SequenceDTF
36、Tn11unanun, |a|1 F F0jen2(2)kk 02(2)kk j1(1e)aj1(1 e)(2)kk 6262 Basic Propertiesjjj ()(e ) |(e )|eXX jj () 2|(e )|ekX The phase function ()of DTFT cannot be uniquely specified for all values of . Principal value( ) 6363 Symmetry Relations (I) table 3.1Sequence the DTFT j(e )X x nj(e)Xxn*j(e )X*xnjj*
37、j1cs2(e )(e )(e)XXXRe x njj*j1ca2(e )(e )(e)XXXjIm x njre(e )Xcs xnjimj(e )Xca xn6464 Symmetry Relations (II) table 3.2Real Sequence the DTFT jjjreim(e )(e )j(e )XXX x nj*j(e )(e)XXjre(e )Xev xnjimj(e )Xod xnSymmetryrelationsjjrere(e )(e)XXjjimim(e )(e)XX jj|(e )| |(e)|XXjjarg(e )arg(e)XX 6565 3.3 D
38、TFT Theorems table 3.4 Theorem Sequence DTFT j(e)G g nj(e)H h njj(e )(e )GH g nh nLinearityj(e)GgnTime-reversalTime-shifting0jje(e )nG0g nnFrequency-shifting0j()(e)G 0je ng nConvolutionjj(e )(e )GH g nh nModulation+()1()()d2jjG eH e g n h nParsevals Relation+*1 ()()d2jjng n h nG eHeDifferentiation-i
39、n frequencyjd(e)jdG ng n6666 Example of DTFT theorems (I) Example 3.13 Determine the DTFT of yn. (1) , | 1ny nnu nSolution:Let , | 1nx nu nthen y nnx nx njj1(e )1eXandthereforejjDTFTj2d (e )e j=d(1e)Xnx n 6767 Example of DTFT theorem (II) According to the linear theorem:jDTFTj2jj2 e1 (1e)1e1 (1e)y n
40、nx nx n 6868 Example of DTFT theorem (IV) Example 3.11 Determine the DTFT V(e j) of vn. 0101 1 1d v nd v npnpnSolution:Using time-shifting and linearity theorem of DTFTjjjj0101(e )e(e )ed VdVppthereforejj01j01e(e )eppVdd6969 Example of DTFT theorem (V) ExampleLet X(ej) denote the DTFT of a length-9
41、sequence xn given by. 2,3, 1,0, 4,3,1,2,426x nn Evaluate the follow of X(ej) without computing transform itself. 022( )()( )()( )()()( )|()|( )|jjjjja X eb X ecX eddX edX ededd7070 Example of DTFT theorem (VI) Solution:7171 Total Energy E Ex of DT Signal Total Energy E Ex of a finite-energy DT compl
42、ex signal xn2j21| |(e )| d2xnx nXE Definition of Energy Density Spectrum Sxx( )j2( ) |(e )|xxSX The area under this curve in the range divided by 2 is the energy of the sequence7272 Band-limited Discrete-Time signal Full-Band Signal Since the spectrum of a DT signal is a periodic functionof with a p
43、eriod 2, a full-band signal has a spectrumoccupying the frequency range . Ideal Band-limited signal has a spectrum that is zero outside a finite frequency 0 a b :jj0, 0| |,| |,(e )(e ), otherwise abXX 7373 Band-Limited DT Signals An ideal band-limited signal cannot be generated in practice Lowpass D
44、T signal:jj(e ), 0| |(e )0, | |ppXXBandwidth : p Highpass DT signal:jj0, 0| |(e )(e ), | |ppXXBandwidth : p7474 Band-Limited DT Signals Bandpass DT signal: Bandwidth : H Ljj0, 0| |, | |(e )(e ), | | LHLHXX7575 3.4 DTFT Computation Using MATLAB Function: Freqz( ) To compute the values of the DTFT of
45、a sequence, described as a rational function in the form:jjj01jj01ee(e )eeMMNNpppXddd Example: H= Freqz (num,den,w)7676 Example of Using MATLABExample 3.14jj2j3j4jjj2j3j40.008 0.033e0.05e0.033e0.008e(e )1 2.37e2.7e1.6e0.41eXNum=0.008, 0.033, 0.05, 0.033, 0.008;Den=1, 2.37, 2.7, 1.6, 0.41;W=0:0.01*pi
46、:pi;H=freqz(Num,Den,w);subplot(2,2,1);plot(w/pi,real(H);grid;subplot(2,2,2);plot(w/pi,imag(H);grid;subplot(2,2,3);plot(w/pi,abs(H);gridsubplot(2,2,4);plot(w/pi,angle(H);grid; The phase spectrum has discontinuity of 2 at =0.72.7979 3.5 The Unwrapped Phase Function Unwrapping the phase: Process the di
47、s- continuity removal. c() Unwrapped phase functioncc0( )( )ddd with the constraintc(0)08080 3.6 The Frequency Response of LTI DT System Eigenfunction : e j n Let xn = ej n, LTI system with impulse response hn, the output of the LTI system isj ()jj ee( e) (3.77)kn knkkky nh nx nh k x n kh kh kOr rew
48、ritten as jj (e )e (3.78)ny nH8181 Property of Frequency Responsejj(e) e (3.79)nnHh n Frequency response H(e j ) : is the DTFT of the impulse response hn; is a continuous function of ; is a periodic function of with a period 2 ; is a complex function of real variable .jjjjj ( )reim(e )(e )j(e ) |(e
49、)|eHHHH 8282 Gain and Attenuationj10( )20log|(e )| (dB) (3.81)GH Gain function : Attenuation (loss) function :( )( )AG H(ej ) provides a frequency-domain description of the systemjjj(e )(e )(e ) (3.82)YHX8383 Frequency-Domain Characterization of the LTI DT Systemjjj(e )(e ) (3.83)(e )YHXExample 3.15
50、Input sequence xn = anun, |a|1,LTI system with impulse response:hn = bnun, |b|1. Find the output sequence yn .Solution: y nx nh njjj(e )(e )(e )YXHjj1(e )1eXajjjjj1(e )(e )(e )(1e)(1e)YXHabjj1(e )1eHbjj()()1e1eabababab11 nnnnababy na u nb u nu nababab0 nkn kka bu n8585 Example of Frequency ResponseE
