《数字信号处理英文版课件：Chapter3 Discrete-Time Signals in the Frequency Domain第四版》由会员分享，可在线阅读，更多相关《数字信号处理英文版课件：Chapter3 Discrete-Time Signals in the Frequency Domain第四版（112页珍藏版）》请在金锄头文库上搜索。
1、3 Discrete-Time Signals in the Frequency Domain3.1 The Continuous Time Fourier Translation )3.1.1 Definition of CTFTNormally, is a complex function.(3.1)(3.2)In polar form:In rectangular form:-magnitude spectrum-phase spectrum3 Discrete-Time Signals in the Frequency Domain(a) Finite number of finite
2、 discontinities and a finite number of maxima and minima in any finite interval.(b) Absolutely integrable.Dirichlet conditions: exists if a milder condition of (3.3)(3.3)(3.6)Note: A singularity function has its CTFT existed. 3 Discrete-Time Signals in the Frequency Domain1/|Xa(j )| nExample3.1xa(t)
3、t13 Discrete-Time Signals in the Frequency DomainUnit impulse (t)(t)0tt(j )10 The sampling property of the delta functionExample 3.23 Discrete-Time Signals in the Frequency DomainThe shifted impulse (t-t0) The sampling property of the delta functionExample 3.3xa(t) =(t-t0)0tt03 Discrete-Time Signals
4、 in the Frequency DomainThe total energy of xa(t) By using (3.2) and conjungating operation3.1.2 Energy Density SpectrumInterchanging the order of integrations 3 Discrete-Time Signals in the Frequency DomainParsevals Theorem (3.9)Read example 3.4 By yourself. 3 Discrete-Time Signals in the Frequency
5、 DomainEnergy Density Spectrum The energy over a specified range of frequency3 Discrete-Time Signals in the Frequency Domain3.1.3 Band-Limited Continuous-Time SignalsnAn ideal band-limited signal has a spectrum that is zero outside a frequency range 0 a| | b, that is3 Discrete-Time Signals in the Fr
6、equency DomainBand-limited signals are classified according to the frequency range where most of the signals energy is concentrated.A lowpass continuous-time signal has a spectrum occupying the frequency range 0| |p ,where p is called the bandwidth of the signal.3 Discrete-Time Signals in the Freque
7、ncy DomainA highpass contnuous-time signal has a spectrum occupying the frequency range 0 p| | and has a bandwidth from p to . .A bandpass contnuous-time signal has a spectrum occupying the frequency range 0 L| | H and has a bandwidth of H- L3 Discrete-Time Signals in the Frequency DomainHowever,an
8、ideal band-limited signal can not be generated in practice, and, for practical purposes, it is sufficient to ensure that the signal energy is sufficiently small outside the specified frequency range.A precise definition of the bandwidth depends on applications.3 Discrete-Time Signals in the Frequenc
9、y DomainAs can be seen from Figure3.2(a),the signal in example 3.1 is a lowpass signal. It can be shown that 80% energy of this signal is contained in the frequency range 0| |0.4898Hence, we can define the 80% bandwidth to be 0.4898 3 Discrete-Time Signals in the Frequency DomainType of Property Sig
10、nal CTFT Differentiation dxa(t)/dt j Xa(j )Frequency-shifting e-j 0txa(t) Xa(j - 0)Time-shifting xa(t-t0) e-j t0Xa(j )Linearity axa(t)+bha(t) aXa(j )+bHa(j )ha(t) Ha(j )xa(t) Xa(j )3.1.4 Properties of CTFT (Appended)3 Discrete-Time Signals in the Frequency Domain3 Discrete-Time Signals in the Freque
11、ncy DomainType of Property Sequence CTFT Modulation xa(t)Ha(t)Convolution xa(t)* ha(t) Xa(j ) Ha(j )Conjugation x*a(t) X*a(-j ) Conjugate Symmetry real xa(t)Xa(-j )=X*a(j )Parsevals Theorem 3.2 Discrete Time Fourier Translation(3.10) 3.2.1 Definition nExample 3.5- The DTFT of the unit sample sequenc
12、e n is given by(3.11) 3 Discrete-Time Signals in the Frequency Domain Example 3.6- Consider the causal sequencenIts DTFT is given byas(3.12) (3.13) 3 Discrete-Time Signals in the Frequency DomainThe DTFT X(ej ) of a sequence xn is a periodic continuous function of with a period 2For all values of in
13、teger k.3 Discrete-Time Signals in the Frequency Domain(3.14) Inverse DTFT Equation (3.10)and (3.14) are denoted asA DTFT pair.(3.15) 3 Discrete-Time Signals in the Frequency DomainProof:3 Discrete-Time Signals in the Frequency DomainNowHence(3.16) 3 Discrete-Time Signals in the Frequency Domain X(e
14、j ) = Xre(ej ) + j Xim(ej )In general, X(ej ) is a complex function of the real variable and can be written in rectangular form asXre(ej ) and Xim(ej ) are, respectively, the real and imaginary parts of X(ej ) , and are real functions of 3.2.2 Basic Properties(3.17) 3 Discrete-Time Signals in the Fr
15、equency Domain| X(ej ) | is called the magnitude function ( ) is called the phase functionnX(ej ) can alternately be expressed in polar form as X(ej ) = | X(ej ) |ej ( ) (3.19)where ( ) = argX(ej ) (3.20)3 Discrete-Time Signals in the Frequency DomainThe relations between the rectangular and polar f
16、orms of X(ej) are given by: Xre(ej) = | X(ej)|cos( ) Xim(ej) = | X(ej)|sin( ) |X(ej)|2=X(ej) X* (ej)=X2re(ej)+X2im(ej) tan( ) = Xim(ej) / Xre(ej) 3 Discrete-Time Signals in the Frequency DomainNote: X(ej ) = | X(ej ) |ej ( )+2 k = | X(ej ) |ej ( )for any integer kThe phase function () cannot be uniq
17、uely specified for any DTFT Unless otherwise stated, we shall assume that the phase function () is restricted to the following range of values: - () the principal value (called as wrapped phase)Wrapped Phase and Unwrapped Phase3 Discrete-Time Signals in the Frequency Domain ()= - 4Example -3 Discret
18、e-Time Signals in the Frequency Domain3.2.3 Symmetry Relation3 Discrete-Time Signals in the Frequency DomainSymmetry relations of the DTFT of a real sequence3 Discrete-Time Signals in the Frequency DomainExample3.7 The magnitude and phase of the DTFT X(ej ) = 1/(1 0.5e-j ) are shown below|X(ej)|= |X
19、(e-j)|()=-(-)3 Discrete-Time Signals in the Frequency Domain LetConsider the behavior of the error X(ej)- Xk(ej) as K goes to .3.2.4 Convergence Conditionis said to exist if it converges in some sense.3 Discrete-Time Signals in the Frequency Domainuniform convergence of X(ej ) 1. xn is absolute summ
20、able.for all values of as K goes to sufficient condition :3 Discrete-Time Signals in the Frequency DomainIf xn is an absolutely summable sequence, i.e., ifThenfor all values of 3 Discrete-Time Signals in the Frequency Domain2. xn is a finite-energy sequence.mean-square convergence of X(ej): the tota
21、l energy of the error X(ej)- Xk(ej) must approach zero at each value of as K goes to 3 Discrete-Time Signals in the Frequency DomainExample 3.8 Consider the DTFTShown below(3.27) 3 Discrete-Time Signals in the Frequency Domaindoes not converge uniformly to (3.27) for all values of , but converge to
22、(3.27) in the mean-square sense.(3.30) So,Examining 3 Discrete-Time Signals in the Frequency DomainK=10K=20K=30K=40Gibbs phenomenon (discussed in Section 10.2.3)3 Discrete-Time Signals in the Frequency DomainThe type of sequences that a DTFT representation is possible using the Dirac Delta function
23、() nneither absolutely summable nor square summablen,cos(0n+)3. The DTFT of xn is Dirac Delta function 3 Discrete-Time Signals in the Frequency Domainw Dirac Delta function ( ) is the limiting form of a unit area pulse function p ( ) as goes to zero satisfyingThe sampling property of the Dirac delta
24、 function3 Discrete-Time Signals in the Frequency DomainExample 3.9 Consider the complex exponential sequencewhere ( ) is an impulse function of andIts DTFT is given by3 Discrete-Time Signals in the Frequency Domainis a periodic function of w with a period 2 and is called a periodic impulse train. T
25、o verify that X(ej ) given above is indeed the DTFT of xn=ej 0n we compute the inverse DTFT of X(ej ) The function3 Discrete-Time Signals in the Frequency DomainWhere we have used the sampling property of the Dirac delta function3 Discrete-Time Signals in the Frequency DomainTable 3.3 Commonly Used
26、DTFT Pairs Sequence DTFT3 Discrete-Time Signals in the Frequency Domain3.2.5 Strength of a DTFTp is a positive integer.In practice, the value of p used is typically 1 or 2 or .(3.35)The strength of a DT FT is given by its norm. of X(ej) is defied by 3 Discrete-Time Signals in the Frequency Domain(3.
27、36)Peak absolute value is root-mean-squared(rms) value of X(ej) is the mean absolute value of X(ej) 3 Discrete-Time Signals in the Frequency Domain3.3 DTFT TheoremsThere are a number of important properties of the DTFT that are useful in signal processing applicationsThese are listed here without pr
28、oofTheir proofs are quite straightforwardWe illustrate the applications of some of the DTFT properties3 Discrete-Time Signals in the Frequency DomainType of Property Sequence DTFT Differentiation ngn jdG(ej )/d Frequency-shifting e-j 0ngn G(ej( - 0)Time-shifting gn-n0 e-j n0G(ej )Linearity agn+bhn a
29、G(ej )+bH(ej )hn H(ej )gn G(ej )Table 3.43 Discrete-Time Signals in the Frequency DomainType of Property Sequence DTFT Parsevals relationModulation gnhnConvolution gn*hn G(ej )H(ej )Table 3.4(The convolution is caculated in a periodic interval.)periodical convolutionNote:3 Discrete-Time Signals in t
30、he Frequency DomainExample 3.11 Determine the DTFT V(ej ) of the sequence vn defined by d0vn+d1vn-1 = p0 n + p1 n-1 Using the time-shifting and linearity property of the DTFT we then obtain the frequency-domain representation3 Discrete-Time Signals in the Frequency DomainExample 3.13 Determine the D
31、TFT Y(ej ) of yn=(n+1) n n, | |1Let xn= n n, | |1We can therefore write yn=nxn + xnThe DTFT of xn is given by3 Discrete-Time Signals in the Frequency DomainSupposing We can show that:Example -Note: If , then 3 Discrete-Time Signals in the Frequency DomainUsing the differentiation property of the DTF
32、T, the DTFT of nxn is given by Next using the linearity property of the DTFT we arrive at3 Discrete-Time Signals in the Frequency DomainExample 3.14 If it is known that ,Where,3 Discrete-Time Signals in the Frequency DomainFrom the convolution in time domainorSo , I= 3 Discrete-Time Signals in the F
33、requency Domain3.4 Energy Density Spectrum of a Discrete-Time SequenceThe total energy of a finite-energy sequence gn is given byEE From Parsevals relation we observe that3 Discrete-Time Signals in the Frequency Domainis called the energy density spectrumThe area under this curve in the range - divi
34、ded by 2 is the energy of the sequenceThe quantity3 Discrete-Time Signals in the Frequency DomainExample 3.15 Compute the energy of the sequence hLPn=sincn/n, -nHerewhere3 Discrete-Time Signals in the Frequency DomainThereforeHence, hLPn is a finite-energy lowpass sequence3 Discrete-Time Signals in
35、the Frequency DomainExample 3.16FromAnd Parsevals relation3 Discrete-Time Signals in the Frequency Domain3.5 Band-Limited Discrete-Time SignalsAn ideal band-limited signal has a spectrum that is zero outside a frequency range 0a|b, that is3 Discrete-Time Signals in the Frequency DomainA classificati
36、on of a band-limited discrete-time signal is based on the frequency range where most of the signals energy is concentratedA lowpass discrete-time real signal has a spectrum occupying the frequency range 0|p and has a bandwidth of p3 Discrete-Time Signals in the Frequency DomainA highpass discrete-ti
37、me real signal has a spectrum occupying the frequency range 0p| and has a bandwidth of - p A bandpass discrete-time real signal has a spectrum occupying the frequency range 0L|H and has a bandwidth of H-L3 Discrete-Time Signals in the Frequency DomainExample -Consider the sequence xn=(0.5)nnIts DTFT
38、 is given below on the left along with its magnitude spectrum shown below on the right3 Discrete-Time Signals in the Frequency DomainIt can be shown that 80% of the energy of this lowpass signal is contained in the frequency range 0|0.5081Hence, we can define the 80% bandwidth to be 0.50813 Discrete
39、-Time Signals in the Frequency Domain3.6 DTFT Computation Using MATLABThe function freqz can be used to compute the values of the DTFT of a sequence, described as a rational function in the form ofat a prescribed set of discrete frequency points = l3 Discrete-Time Signals in the Frequency DomainFor
40、example, the statement H = freqz(num, den, w)returns the frequency response values as a vector H of a DTFT defined in terms of the vectors num and den containing the coefficients pi and di, respectively at a prescribed set of frequencies between 0 and 2 given by the vector w 3 Discrete-Time Signals
41、in the Frequency DomainThere are several other forms of the function freqzThe Program 3_1.m in the text can be used to compute the values of the DTFT of a real sequenceIt computes the real and imaginary parts, and the magnitude and phase of the DTFT3 Discrete-Time Signals in the Frequency DomainExam
42、ple 3.17 Plots of the real and imaginary parts, and the magnitude and phase of the DTFT are shown on the next slide3 Discrete-Time Signals in the Frequency Domainwrapped phase3 Discrete-Time Signals in the Frequency DomainIn numerical computation, when the computed phase function is outside the rang
43、e -, , the phase is computed modulo 2, to bring the computed value to this rangeThus, the phase functions of some sequences exhibit discontinuities of 2 radians in the plot.3.7 The unwrapped phase functionnFor example3.17, there is a discontinuity of 2 at =0.72 in the phase function.3 Discrete-Time
44、Signals in the Frequency DomainIn such cases, often an alternate type of phase function that is continuous function of is derived from the original phase function by removing the discontinuities of 2Process of discontinuity removal is called unwrapping the phaseThe unwrapped phase function will be d
45、enoted as c(),which indicates a continuous function of .3 Discrete-Time Signals in the Frequency DomainNote: This discontinuity can be removed using the function unwrap in MATLAB.Wrapped Phase Unwrapped Phase3 Discrete-Time Signals in the Frequency DomainnThe conditions under which the phase functio
46、n will be continuous function of .X(ej ) = | X(ej ) |ej ( ) (3.19)where ( ) = argX(ej ) (3.20)3 Discrete-Time Signals in the Frequency Domain(3.55)IFExists, ( ) can be defined uneqivocally byWith constraint3 Discrete-Time Signals in the Frequency Domain(3.59)(3.60)will be continuous function of Unwr
47、apped phaseMoreover, if will be odd function of 3 .8 Digital Processing of Continuous-Time Signals3 Discrete-Time Signals in the Frequency DomainIn the Time-Domain3.8.1 Effect of Sampling in the Frequency-Domain3 Discrete-Time Signals in the Frequency Domaingn = ga(nT), - n m Then ga(t) is uniquely
48、determined by its samples ga(nT) , -n if T 2 mwhere T=2 /T.The condition T 2 m is often referred to as the Nyquist conditionSampling theorem3 Discrete-Time Signals in the Frequency DomainThe frequency T/2 is usually called the folding frequency orNyquist frequency .Since the minimum sampling frequen
49、cy T =2 m, must be used to fully recover ga(t) from its sampled version,the frequency 2 m is called the Nyquist rate.3 Discrete-Time Signals in the Frequency DomainIf T2 m , ga(t) can be recovered exactly from gp(t) by passing it through an ideal lowpass filter Hr(j ) with a gain T and a cutoff freq
50、uency c greater than m and less than T - m as shown below3 Discrete-Time Signals in the Frequency DomainThe spectra of the filter and pertinent signals are shown below3 Discrete-Time Signals in the Frequency DomainOversampling - The sampling frequency is higher than the Nyquist rateUndersampling - T
51、he sampling frequency is lower than the Nyquist rateCritical sampling - The sampling frequency is equal to the Nyquist rateNote: A pure sinusoid may not be recoverable from its critically sampled version3 Discrete-Time Signals in the Frequency Domain(3.74a)(3.74b)or(3.63)(3.66)we get:Comparing (3.63
52、) and (3.66),The relation between Ga(j ) and G(ej )3 Discrete-Time Signals in the Frequency DomainFrom (3.74) and (3.70),we get: (3.75a)(3.75b)(3.76)3 Discrete-Time Signals in the Frequency DomainRelationship between CTFT and DTFT3 Discrete-Time Signals in the Frequency DomainExample 3.19 Sampling o
53、f a continuous-time signals at two different rate.3 Discrete-Time Signals in the Frequency DomainFig. 3.19 Sampling at 2 HzFig. 3.20 Sampling at 2/3 Hz3 Discrete-Time Signals in the Frequency DomainEffect of aliasing in frequency domainSpeech with aliasingOriginal SpeechOriginal musicMusic with alia
54、sing3 Discrete-Time Signals in the Frequency DomainOriginal musicMusic with aliasingEffect of aliasing in frequency domain3 Discrete-Time Signals in the Frequency Domain3.8.2 Recovery of the Analog Signal(3.78)3 Discrete-Time Signals in the Frequency DomainThe impulse response hr(t) of the lowpass r
55、econstruction filter is obtained by taking the inverse CTFT of Hr(j ) (3.79)3 Discrete-Time Signals in the Frequency Domain The input to the lowpass filter is gp(t): Therefore, the output of the ideal lowpass filter is given by:(3.80)(3.81)3 Discrete-Time Signals in the Frequency DomainAssuming for
56、simplicity c= T/2= /T , we getwhich is called Interpolation formula.(3.82)3 Discrete-Time Signals in the Frequency Domain3.8.3 Implications of the Sampling ProcessExample 3.18 Consider three continuous-time sinusoidal signals: Their corresponding CTFTs are:These three transforms are plotted below3 D
57、iscrete-Time Signals in the Frequency Domain3 Discrete-Time Signals in the Frequency DomainThese continuous-time signals sampled at a rate of T = 0.1 sec, i.e., with a sampling frequency T =20 rad/sec.Plots of the 3 CTFTs are shown belowThe sampling process generates the continuous-time impulse trai
58、ns, g1p(t), g2p(t) , and g3p(t).Their corresponding CTFTs are given by 3 Discrete-Time Signals in the Frequency DomainIn fact, the three discrete-time sinusoidal signals:And 3 Discrete-Time Signals in the Frequency DomainFor all integer k3 Discrete-Time Signals in the Frequency DomainIf reconstructi
59、on filterIf reconstruction filter3 Discrete-Time Signals in the Frequency Domain3.9 Sampling of bandpass SignalsThere are applications where the continuous-time signal is bandpass signal to a higher frequency range L | | H with L 0Of course, to prevent aliasing a bandpass signal can be sampled at a
60、rate greater than twice the highest frequency, i.e. by ensuring T 2 H3 Discrete-Time Signals in the Frequency DomainHowever, due to the bandpass spectrum of the continuous-time signal, the spectrum of the discrete-time signal obtained by sampling will have spectral gaps with no signal components pre
61、sent in these gaps03 Discrete-Time Signals in the Frequency DomainAssume first that H is an integer multiple of the bandwidth, i.e., H = M()A more practical approach is to use under-samplingDefine the bandwidth of the bandpass signal = H - L If H is not an integer multiple of the bandwidth, we can a
62、rtificially extend either to right or left to satisfy above condition.3 Discrete-Time Signals in the Frequency Domain This leads toFigure below illustrate the idea behindChoose the sampling frequency T = 2() = 2 H/M, T 2 H (Nyquist rate)3 Discrete-Time Signals in the Frequency Domain0no aliasing0(M=
63、3)3 Discrete-Time Signals in the Frequency Domain00no aliasing(M=4)3 Discrete-Time Signals in the Frequency DomainAs can be seen, ga(t) can be recovered from gp(t) by passing it through an ideal bandpass filter with a passband given by L | | H and a gain of TRecovery of the Analog Signal3 Discrete-T
64、ime Signals in the Frequency DomainNote: Any of the replicas in the lower frequency bands can be retained by passing through bandpass filters with passbands L- k() | | H - k() , 1 k M-1 providing a translation to lower frequency ranges3 Discrete-Time Signals in the Frequency DomainNote: Effect of Sampling by S/H (Section 3.10). Read by yourself!
