數(shù)字信號處理英文影印版課件4-1_圖文

1、數(shù)字信號處理Digital Signal Processing 電子信息工程系韓建峰KeywordsSectionsSampling sinusoidsSampling theoremDiscrete-to-Continuous Conversion Summary LECTURE 1Reading assignments This lecture Chapter 4Section 4-1 KeywordsPart AContinuous-to-Discrete Conversion Sampling & ReconstructionAliasing & Folding LEC

2、TURE OBJECTIVES SAMPLING can cause ALIASING Spectrum for digital signals,x nNormalized Frequency22+=ss f f T ALIASING Review SignalsSampling Reconstruction Continuous-time SignalBut the key point is that any computer represent ation is discrete.So, do sampling!And, how?(cos(x t A t=+ Sample a contin

3、uous-time signal at equally spaced time instants.Take a “snapshot” every Ts.Speech, audio andso on. Or, compute the values of a discrete-time signal directly from a formula.2=-+53x n n n SAMPLING x(tSAMPLING PROCESSConvert x(t to numbers xn“n” is an integer; xn is a sequence ofvaluesThink of “n” as

4、the storage address inmemoryUNIFORM SAMPLING at t = nTsIDEAL: xn = x(nTs SAMPLING RATE, f s SAMPLING RATE (fsf=1/T ssNUMBER of SAMPLES PER SECOND T= 125 microsec f s= 8000 samples/secsUNITS ARE HERTZ: 8000 Hz UNIFORM SAMPLING at t = nT= n/f ssIDEAL: xn = x(nT=x(n/f ss Examples of continuous-time sig

5、nals exist in the “real-world” outside the computer.Simple mathematical formula.More general continuous-time signals can be represented as sum of sinusoids.So, we will use sinusoidal signal as the basis for our study of sampling. sf s T n A n x =+=cos(cos(cos(+=+=s s nT A nT x n x t A t x Change x(t

6、 into xn DERIV ATIONcos(+=n T A n x s DEFINE DIGITAL FREQUENCYDigital Frequency V ARIES from 0to 2, as f varies from 0 to the sampling frequencyUNITS are radians, not rad/secDIGITAL FREQUENCY is NORMALIZEDss f f T 2= Sample RateHow to select theT sSample Theorem A interesting phenomenonExercise 4.1I

7、s this the only possible answer?Hz 1000at sampled 2400cos(2=s f t t x 21000cos(2400cos(2.4cos(0.42cos(0.4nx n n n n n =+=(cos(400x t t =Aliasingcos(0.4x n n = Illustration of aliasingDifferent frequency, but same values at n=0,1,2,32.4is an alias of 0.4Exercise 4.2 AliasingHow does aliasing arise in

8、 a mathematical treat ment of discrete-time signal?The last example:12cos(0.4cos(2.4x n n x n n =2cos(0.42cos(0.4x n n n n =+=Periodic function with period 2Aliasing Derivation-1and we substitute: t n f sIf x (t =A cos(2(f + f s t +then: x n =A cos(2(f + f s n f s +or, x n =A cos(2f f s n +2 n + Ali

9、asing Derivation-22s sf T f = +2 2(22then: s s s s s f f f f f f f +=+ and we want: cos(x n A n =+If x (t =A cos(2(f + f s t +t n f s Folded Aliasx (t =A cos(2(-f + f s t -SAME DIGITAL SIGNAL cos(x n A n =+x n =A cos(2f T s n -2 n +x n =A cos(-2fT s n +(2 f s T s n - x n =x (nT s =A cos(2(-f + f s n

10、T s - Aliasing2s s f T f = +2 22s s f T f =-+ Folded Alias AlisingPrincipal Aliasing, 2, 2 integer l l l +-=General Formula Spectrum of a Discrete-Time SignalPLOT versus NORMALIZED FREQUENCY INCLUDE ALL SPECTRUM LINES ALIASESADD MULTIPLES of 2SUBTRACT MULTIPLES of 2FOLDED ALIASESALIASES of NEGATIVE

11、FREQS SPECTRUM (Aliasing Case12X *0.512X 1.512X 0.5 2.52.512X 12X *12X *1.580/(100(2cos(+=n A n x 80s f Hz =sf f 2= SPECTRUM (Folding Case2sf f =f s =125Hz 12X *0.412X 0.4 1.61.612X 12X *125/(100(2cos(+=n A n x DEMO: Strobe Movies 12What is the meaning of this DEMO?Can you give us more examples in the real world? f Camera: 30 Frames/s Human Eyessf'f Summary2s s f T f