自动化南理工数字信号处理英课件dsp mitra ql

1、1Main 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 Systems21. Discrete Time SignalsA. Time-Domain Representation Signals represented

2、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 3TSamples:sampling int (ervalsampling fr), 2, 1,e0,1,2,T, que. F1cy/n, at nTx nx tnT1. Discrete Time Signals ,if 0. .: cosReal sequ(0.25 )ence.s:reimimx nxnjxnxne gx nn0.3 Co

3、mplex sequ .ence.: .s:reimjnx nxnjxne gx ne4B. 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 yn512(c) Sum and difference (d) Ideal delay(Time-shifting) - y nx n x ny nx

4、n d 1 x n2 x n y n xn ynZ-dB. Operations on Sequences61234Combination of elementary operation(e) 123sy nx nx nx nx nB. Operations on Sequences7(f) :Ensemble averaging11 , where . KiaveiiiixxxsdKsddat a， noi s e Let actual uncorrupted data, denote the noise vector corrupting the i-th measurement.isd

5、A very simple application of addition operation in improvingquality of measured data corrupted with additive noise. B. Operations on Sequences8B. Operations on Sequences9(g) Sampling rate alte ration: x n uxn L / , 0, 2 , 0, .ux n LnLLx notherwise if R=L1, inserting L-1 zero-valued samples between t

6、wo 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 a given sequence TTy nFFx n.B. Operations on Sequences1

7、0Up-samp g linB. Operations on Sequences11 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 Sequences12l On Symmetry: Conjugate-symmetric or Conjugateantisymmetric sequencesC. Classification of Sequences Any real seque

8、nce 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 n13(c) the conjugate symmetric sequence: , for all(b) the conjugate anti-symmetric sequence: , for a

9、llcscscaca 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 Sequences14l On Periodicity Periodic or aperiodic sequencesl On Energy or Power Bounded or unb

10、ounded sequences for all x nx Nn n2 ,bounded sequenceIf , absolutely-summable.If , square-summable.xnnx nBx nx nx nx nC. Classification of Sequences15sin , - Square-summable but not absolutely-sum. .m bl:a e. caengx nnn sin, - Neither square-summable nor absolutely-summable. bcx nnn C. Classificatio

11、n of Sequences162. Typical Sequences0 1,01) Unit sample: 0,1,02) UnitA. Some Basic Se step : quences0,3) Sinusoidal sequence : sin:x nnnotherwisenu notherwise nnunnx 174) Exponential sequence: na5) Rect. sequence NRnu nu nN：N-1n NR n2. Typical Sequences18-Analysis of signal: B. Representation of an

12、Arbi trary Sequencekx nx kn-k. .: 0.5 2 1.5 120.75 6e gx nnnnn2. Typical Sequences19 3. Discrete Time SystemsA. Definition and examples nyTnx It is a transformation or operator that maps an inputsequence into an output sequence. y nT x n201. .: 1 Accumulato.rnnlle gy nx lx lx ny nx n3. Discrete Time

13、 Systems0or, = -1 where -1 is the initial condition of a causal systennly nyx ly21101Moving 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 Systems22 where 2 (0.8) ns nnd ns i gnal ，noi s e. s n s nd n 3M 10M 231 F n+(n-1

14、+ 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 Systems24Factor of 2 interpolator3. Discrete Time Systems25Factor of 4 interpolatorLi near i nt er pol at i on3. Discrete Time Systems2612

15、122(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 syste t-mT x ny nT x ndy ndy nx Mn3. Discrete Time Systems27(c) : only deponds on values , Ca

16、usal 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 Systems28the response to a unit sample sequence .Determination of theImpulse resp impulse resonse:ponse.nC. Impulse and Step Response123Consider an LTI DT

17、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 Systems294. Time-Domain Characterization of LTI DTS A. Input-Output Relationship kx nx knksignalanalysis by line

18、arity by timeinvariancekkky nT x nTx knkx k Tnkx k h nk- sum theimpulse response of the LTIsystconvolutioenm.y nx nh nh n- -30 Calculation of the Convolution S: For eam:uchky nx k h nkn(a) reverse h khk(b) delay bysamples,. ., ()hkniehknh nk(c) multiply the sequences and sum the results.x kwith h nk

19、4. Time-Domain Characterization of LTI DTS 314. Time-Domain Characterization of LTI DTS 324. Time-Domain Characterization of LTI DTS 334. Time-Domain Characterization of LTI DTS 344. Time-Domain Characterization of LTI DTS 354. Time-Domain Characterization of LTI DTS 364. Time-Domain Characterizatio

20、n of LTI DTS 374. Time-Domain Characterization of LTI DTS 384. Time-Domain Characterization of LTI DTS 394. Time-Domain Characterization of LTI DTS 404. Time-Domain Characterization of LTI DTS 414. Time-Domain Characterization of LTI DTS 42i.e.: .nSh n B. Stability condition in terms of the impulse

21、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 43000 the output sample depends only on input sa Causal symples st for em:.nthy nx nnnC. Cusality condition in terms of the impulse response1212

22、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 in output samples do not precede the changes in the input samples.4. Time-Domain Characterization of LTI DTS 44D. Linear c

23、onstant coefficient difference equationThe representation of the LTI systems in the time domainthe -th-order linear constant coefficient difference equation:N00 can be determined 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 NMk