測(cè)試信號(hào)英文版課件：Chapter3 Discrete-Time Signals in the Transform Domain_Lec1

1、Chapter 3 Discrete-Time Signals in the Transform DomainTransform-Domain Representation of Discrete-Time SignalsThree useful representations of discrete-time sequences in the transform domain:Discrete-time Fourier Transform (DTFT)Discrete Fourier Transform (DFT)z-Transform2Discrete-time Fourier Trans

2、form (DTFT)Definition Convergence ConditionDTFT Properties Energy Density SpectrumDTFT Computation Using MATLABLinear Convolution Using DTFT3Discrete-Time Fourier TransformDefinition - The discrete-time Fourier transform (DTFT) of a sequence xn is given byIn general, is a complex function of the rea

3、l variable w and can be written as4w-數(shù)字頻率（單位：弧度 (radian) 范圍：p）Discrete-Time Fourier Transform (DTFT) and are, respectively, the real and imaginary parts of , and are real functions of w can alternately be expressed aswhere 5Discrete-Time Fourier Transform is called the magnitude function is called t

4、he phase functionBoth quantities are again real functions of wIn many applications, the DTFT is called the Fourier spectrumLikewise, and are called the magnitude and phase spectra 6Discrete-Time Fourier TransformFor a real sequence xn, and are even functions of w, whereas and are odd functions of w7

5、Discrete-Time Fourier TransformExample - The DTFT of the unit sample sequence dn is given byExample - Consider the causal sequence8Discrete-Time Fourier TransformIts DTFT is given byas9Discrete-Time Fourier TransformThe magnitude and phase of the DTFT are shown below10Discrete-Time Fourier Transform

6、The DTFT of a sequence xn is a continuous function of wIt is also a periodic function of w with a period 2p:1112Discrete-Time Fourier TransformThe DTFT of a sequence xn is a continuous function of wIt is a periodic function of w with a period 2pIt is a complex function13FT vs. DTFT14Time Domain Freq

7、uency DomainsignalsystemInput-outputDiscrete-Time Fourier TransformInverse discrete-time Fourier transform:15Discrete-Time Fourier TransformDTFT: analysis equationIDTFT: synthesis equationtime domainfrequency domain16Discrete-Time Fourier TransformConvergence Condition - An infinite series of the fo

8、rmmay or may not convergeLet17Discrete-Time Fourier TransformThen for uniform convergence of , Now, if xn is an absolutely summable sequence, i.e., 18Discrete-Time Fourier TransformThen for all values of wThus, the absolute summability of xn is a sufficient condition for the existence of the DTFT 19

9、If(absolutely summable) Discrete-Time Fourier TransformExample - The sequence for is absolutely summable asand its DTFT therefore converges to uniformly20Discrete-Time Fourier TransformSincean absolutely summable sequence has always a finite energyHowever, a finite-energy sequence is not necessarily

10、 absolutely summable21Discrete-Time Fourier TransformExample - The sequence has a finite energy equal toBut, xn is not absolutely summableE22Discrete-Time Fourier TransformTo represent a finite energy sequence xn that is not absolutely summable by a DTFT , it is necessary to consider a mean-square c

11、onvergence of :where23Discrete-Time Fourier TransformHere, the total energy of the errormust approach zero at each value of w as K goes to In such a case, the absolute value of the error may not go to zero as K goes to24Discrete-Time Fourier TransformExample - Consider the DTFTshown below25Discrete-

12、Time Fourier TransformThe inverse DTFT of is given byThe energy of is given byTherefore, is a finite-energy sequence, but it is not absolutely summable 26Discrete-Time Fourier TransformAs a resultdoes not uniformly converge to for all values of w, but converges to in the mean-square sense27Discrete-

13、Time Fourier TransformThe DTFT can also be defined for a certain class of sequences which are neither absolutely summable nor square summableExamples of such sequences are the unit step sequence mn, the sinusoidal sequence and the exponential sequenceFor this type of sequences, a DTFT representation

14、 is possible using the Dirac delta function d(w)28Discrete-Time Fourier TransformA Dirac delta function d(w) is a function of w with infinite height, zero width, and unit area29Discrete-Time Fourier TransformIt is the limiting form of a unit area pulse function as D goes to zero satisfyingThe sampli

15、ng property of the delta function:w30Discrete-Time Fourier TransformExample - Consider the complex exponential sequenceIts DTFT is given bywhere is an impulse function of w31Discrete-Time Fourier TransformThe functionis a periodic function of w with a period 2p and is called a periodic impulse train

16、To verify that given above is indeed the DTFT of we compute the inverse DTFT of 32Discrete-Time Fourier TransformThuswhere we have used the sampling (or sifting) property of the impulse function 33How to Interpret the Frequency of DTFTAnalog frequency（模擬頻率）f：每秒經(jīng)歷多少個(gè)周期，單位Hz，即1/s；Analog angular freque

17、ncy（模擬角頻率）：每秒經(jīng)歷多少弧度，單位rad/s；Digital frequency（數(shù)字頻率） w ：每個(gè)頻率采樣點(diǎn)間隔之間的弧度，單位rad。Relations: 34Relationship among f, and w35How to Interpret the Frequency of DTFTExample: Sampling frequency fs=8000Hz, analog frequency of a sinusoidal signal f=1000Hz, then the digital frequency of the DTFT for the correspo

18、nding discrete-time sinusoidal signal36Commonly Used DTFT Pairs Sequence DTFT37Commonly Used DTFT Pairs38DTFT PropertiesThere are a number of important properties of the DTFT that are useful in signal processing applicationsWe will illustrate the applications of some of the DTFT properties39Table 3.

19、4:General Properties of DTFT40Table 3.1: DTFT Properties: Symmetry Relationsxn: A complex sequence41Table 3.2: DTFT Properties: Symmetry Relationsxn: A real sequence42DTFT PropertiesExample - Determine the DTFT ofLetWe can therefore writeFrom previous examples, the DTFT of xn is given by43DTFT Prope

20、rtiesUsing the differentiation property of the DTFT given in Table 3.4, we observe that the DTFT of is given byNext using the linearity property of the DTFT given in Table 3.4 we arrive at44DTFT PropertiesExample - Determine the DTFT of the sequence vn defined byFrom previous part, the DTFT of is 1U

21、sing the time-shifting property of the DTFT given in Table 3.4 we observe that the DTFT of is and the DTFT of is45DTFT PropertiesUsing the linearity property of Table 3.4 we then obtain the frequency-domain representation ofasSolving the above equation we get46Energy Density SpectrumThe total energy

22、 of a finite-energy sequence gn is given byFrom Parsevals relation given in Table 3.4 we observe thatEE47Energy Density SpectrumThe quantityis called the energy density spectrumThe area under this curve in the range divided by 2p is the energy of the sequence48Energy Density SpectrumExample - Comput

23、e the energy of the sequenceHerewhere49Energy Density SpectrumThereforeHence, is a finite-energy sequence50DTFT Computation Using MATLABThe function freqz can be used to compute the values of the DTFT of a sequence, described as a rational function in in the form ofat a prescribed set of discrete frequency points51DTFT Computation Using MATLABFor example, the statementH = freqz(num