数字信号处理英文版课件：Chap 3 Discrete-Time Signals in the Frequency domain

1、Chapter 3 Discrete-Time Signals in the Frequency domain The Continuous-time Fourier Transform (CTFT) The Discrete-time Fourier Transform (DTFT) DTFT Theorems Sampling of the CT signalsDSP group chap3-ed12 3.1.1 The Definition Fourier spectrum, or simply the spectrum.3.1 The Continuous Time Fourier T

2、ransform (CTFT) Inverse Fourier transform/ Fourier integral CTFT pairDSP group chap3-ed13 3.1 The CTFTPolar form Magnitude spectrum |Xa(j)| phase spectrum a() Total Energy E x of a finite-energy CT complex signal Parsevals relationDSP group chap3-ed14 3.1.2 Energy Density Spectrum Sxx()Definition En

3、ergy E x ,r over a specified range of frequencies a b of the signal xa(t) is computed byDSP group chap3-ed15 3.1.3 Band-Limited CT Signals Ideal Band-limited signal has a spectrum that is zero outside a finite frequency a | b : An ideal band-limited signal cannot be generated in practice Lowpass CT

4、signal:Bandwidth : pppXa( j )10DSP group chap3-ed16 3.1.3 Band-Limited CT Signals Highpass CT signal: Bandpass CT signal: Bandwidth : H LppXa( j )10LLXa( j )10HHDSP group chap3-ed17 3.2 The Discrete-Time Fourier Transform (DTFT) 3.2.1 Definition In general, X(e j) is a complex function of the real v

5、ariable and can be written asThe Inverse DTFT DSP group chap3-ed18 3.2.1 DTFT examples Example 3.5Find the DTFT of unit sample sequence n. Solution:Example 3.6Find the DTFT of causal sequence xn=anun, |a|1. Solution:| a e j| = |a| 1DSP group chap3-ed19 3.2.1 DTFT Examples of DTFT/magnitude/Phase rad

6、ian The magnitude and phase function of sequence 0.5nun.DSP group chap3-ed110 3.2.2 Basic PropertiesPolar form Magnitude function phase function - Likewise, and are called the magnitude and phase spectrum.DSP group chap3-ed111 is a continuous function of ; is also a periodic function of with a perio

7、d 2.for all integer values of k 3.2.2 Basic PropertiesThe phase function ()of DTFT cannot be uniquely specified for all values of . Principal valueDSP group chap3-ed112 3.2.3 Symmetry Relations (I) table 3.2Sequence the DTFT Conjugate-symmetricConjugate-antisymmetricDSP group chap3-ed113 3.2.3 Symme

8、try Relations (II) table 3.1Real Sequence the DTFT Symmetryrelations14 3.2.4 Convergence Conditionabsolutely summableuniform convergenceSince only square summable mean-square convergenceNone of above using Dirac delta function15 3.2.4 Convergence Conditiontable 3.3anun, |a|1un11nDTFTSequence Norm of

9、 DTFT P96 3.3 DTFT Theorems table 3.4 P100 Theorem Sequence the DTFT LinearityTime-reversalTime-shiftingFrequency-shiftingConvolutionModulationParsevals RelationDifferentiation-in frequencyEx3.12 P99DSP group chap3-ed117 3.3 DTFT Theorems linearity & differentiation in frequencyExample 3.13Determine

10、 the DTFT of yn. Solution:Letthen andthereforeDSP group chap3-ed118 3.3 DTFT TheoremsAccording to the linear theorem:DSP group chap3-ed119 3.3 DTFT Theorems time shiftingExample 3.11Determine the DTFT V(e j) of vn. Solution:Using time-shifting and linearity theorem of DTFTthereforeDSP group chap3-ed

11、120 3.3 DTFT Theorems convolutionhnxnyn= hnxnH(e j)X(e j)Y(e j)= H(e j)X(e j)X(e j)DTFTDTFTInverse DTFTH(e j)xnhnynDSP group chap3-ed121 3.4 DTFT Theorems Total Energy of DT Signal Ex Total Energy Ex of a finite-energy DT complex signal xn Definition of Energy Density Spectrum Sxx() The area under t

12、his curve in the range divided by 2 is the energy of the sequenceDSP group chap3-ed122 3.5 Band-limited Discrete-Time signal Full-Band Signal Since the spectrum of a DT signal is a periodic functionof with a period 2, a full-band signal has a spectrumoccupying the frequency range . Ideal Band-limite

13、d signal has a spectrum that is zero outside a finite frequency 0 a | b 2 m For band-limited CT signal, there is two cases:Case 2: s 2 m DSP group chap4-ed140 3.8.1 Effect of sampling in the Frequency-Domain Illustration of the frequency-domain effectsNo overlapOverlapDSP group chap4-ed141 3.8.1 Eff

14、ect of sampling in the Frequency-Domain Baseband signal: the term for k=0 is called baseband portion of Gp(j). Baseband / Nyquist band: frequency range s /2 s /2DSP group chap4-ed142 If s 2m , ga(t) can be recovered exactly from gp(t) by passing if through an ideal lowpass filter Hr(j) with gain T a

15、nd a cutoff frequency c greater than m and less than s m . 3.8.1 Effect of sampling in the Frequency-Domain If s 2m , due to the overlap of the shifted replicas of Ga(j), the spectrum Gp(j) cannot be separated by filtering to recover Ga(j) because of the distortion caused by a part of replicas immed

16、iately outside the baseband being folded Back or aliased into the baseband.DSP group chap4-ed143 3.8.1 Effect of sampling in the Frequency-DomainDSP group chap4-ed144 3.8.1 Effect of sampling in the Frequency-Domain Sampling TheoremSuppose that ga(t) be a band-limited signal withThen ga(t) is unique

17、ly determined by its samples gn= ga(nT), n = 0,1, 2, if Nyquist conditions: Folding frequency: Nyquist Frequency: m Nyquist rate: 2mDSP group chap4-ed145 3.8.1 Effect of sampling in the Frequency-Domain Several Sampling Oversampling:The sampling frequency is higher than the Nyquist rate Undersamplin

18、g:The sampling frequency is lower than the Nyquist rate Critical sampling:The sampling frequency is equal to the Nyquist rate Note: A pure sinusoid may not be recoverable from its critically sampled version. DSP group chap4-ed146 3.8.1 Effect of sampling in the Frequency-Domain Application of Sampli

19、ng In digital telephony, a 3.4 kHz signal bandwidth is adequate for telephone conversation; Hence, a sampling rate of 8 kHz, which is greater than twice the signal bandwidth, is used. In high-quality analog music signal processing, a bandwidth of 20 kHz is used for fidelity; Hence, in CD music syste

20、ms, a sampling rate of 44.1 kHz, which is slightly higher than twice the signal bandwidth, is used.DSP group chap4-ed147Or G(ej) is obtained from Gp(j) simply by scaling according to the relation 3.8.1 Effect of sampling in the Frequency-Domain Relation between G(e j) and Ga(j)48And then passing it

21、through an ideal lowpass filter Hr(j) with a gain T and a cutoff frequency c satisfying Given gn , we can recover exactly ga(t) by generating an impulse train:, 3.8.2 Recovery of the Analog Signal H r (j )gngp(t)ga(t)Convert formSequence toImpulse trainDSP group chap4-ed149 3.8.2 Recovery of the Ana

22、log Signal The lowpass reconstruction filter Hr(j): The impulse response hr(t) of Hr(j):DSP group chap4-ed150 3.8.2 Recovery of the Analog Signal The input to Hr(j) is impulse train gp(t); ga(t) The output of Hr(j) is given by ga(t)(c = s /2) DSP group chap4-ed151With assuming: c = s /2 = /T. 3.8.2

23、Recovery of the Analog SignalDSP group chap4-ed152 Output of ideal D/A convertergz(t)hz(t)gp(t)gr(t) Zero-order hold operationthz(t)T01gr(t)-3T-2T-T0T3Ttyp(t) 3.8.2 Recovery of the Analog SignalDSP group chap4-ed153 zero-order hold frequency response; Reconstruction filterwhere 3.8.2 Recovery of the

24、 Analog SignalDSP group chap4-ed154 Effect of sampling in the Frequency-Domain Examples of SamplingExample 3.17Consider 3 CT sinusoidal signals: The corresponding CTFTs are: They are sampled at a rate of T=0.1 sec, or sampling frequency s =20 rad/sec. The CTFT of the three signals:DSP group chap4-ed

25、156 Comments on example 3.17 In the case of g1(t), the sampling rate satisfies the Nyquist condition and there is no aliasing; The reconstructed output is precisely the original CT signal g1(t); In the other two cases, the sampling rate does not satisfy the Nyquist condition, resulting in aliasing,

26、and outputs are all equal to the aliased signal g1(t) = cos(6t);DSP group chap4-ed157 Comments on example 3.17 In the figure of G2p(j), the impulse appearing at =6 in the positive frequency passband of the lowpass filter results from the aliasing of the impulse in G2(j) at =14; In the figure of G3p(j), the impulse appearing at =6 in the positive frequency passband of the lowpass filter results from the aliasing of the impulse in G3(j) at =26; DSP group chap4-ed158 Ex