1、1l The Continuous-time Fourier Transform(CTFT)l The Discrete-time Fourier Transform(DTFT)l DTFT TheoremslThe Frequency Response of LTI DT SystemlPhase and Group DelayChap 3 Discrete-Time Fourier Transform 2Definition-The CTFT Xa(j)of a continuous-time signalxa(t)is given by j(j)()ed (3.1)taaXx tt Th
2、e CTFT often is referred to as the Fourier spectrum,or simply the spectrum of the CT signal3.1 Continuous-Time Fourier Transform3 Inverse Continuous-Time Fourier Transform j1()(j)ed (3.2)2taax tXDefinition-The Inverse CTFT of a Fourier transformXa(j)is given by The ICTFT often is referred to as the
3、Fourier integral.CTFT pair-CTFT ()(j)aax tXt 4 Magnitude spectrum and phase spectrum Magnitude spectrum-Phase spectrum-Polar formj()(j)|(j)|e,()arg(j)aaaaaXXX|Xa(j)|a()Total Energy E E x of a finite-energy CT complex signal221|()|d|(j)|d (3.9)2xaax ttXE Parsevals relation5 Energy Density Spectrum Sx
4、x()Definition-2()|(j)|xxaSX Energy E E x,r over a specified range of frequencies a b of the signal xa(t)is computed by,1()d2bax rxxSE6 Band-Limited CT Signals Ideal Band-limited signal has a spectrum that is zero outside a finite frequency a|b:(j),|(j)0,otherwise aabaXX An ideal band-limited signal
5、cannot be generated in practice Lowpass CT signal:0,|(j)(j),0|paapXX Bandwidth:p7 Band-Limited CT Signals Highpass CT signal:0,0|(j)(j),|paapXX Bandpass CT signal:0,0|,|(j)(j),|LHaaLHXX Bandwidth:H L8 3.2 Discrete-Time Fourier TransformDefinition-The DTFT X(e j)of a sequence xn is given by jj(e)e (3
6、.12)nnXx n In general,X(e j)is a complex function of the real variable and can be written asjjjreim(e)(e)j(e)(3.19)XXX9 Magnitude function and phase function Magnitude function:Phase function:Polar formjjj()j(e)|(e)|e (3.21)()arg(e)(3.22)XXX j|(e)|X()Likewise,j|(e)|Xand()are called the magnitude spe
7、ctrum and phase spectrum.10 Examples of DTFTExample 3.5 Find the DTFT of unit sample sequence n.Example 3.6 Find the DTFT of causal sequence xn=anun,|a|1.as|a e j|=|a|1jj(e)e=01nnnSolution:jjjj01(e)e=e1ennnnnXx naaSolution:11jj|(e)|(e)|XX()()/magnitude/Phase in radians The magnitude and phase functi
8、on of sequence 0.5nun.Examples of DTFT12 Characteristics of DTFT The DTFT X(e j)of a sequence xn is a continuous function of ;The DTFT X(e j)of a sequence xn is also a periodic function of with a period 2.j(2)jj(e)(e)e,knnXXx ni.e.for all integer values of k13 The Inverse DTFT jj1(e)ed (3.16)2nx nXD
9、efinition-The DTFT pairj(e)(3.17)x nX F F14 Commonly used DTFT pairs table 3.3SequenceDTFTn11unanun,|a|1 F F0jen2(2)kk 02(2)kk j1(1e)aj1(1 e)(2)kk 15 Basic Propertiesjjj()(e)|(e)|eXX jj()2|(e)|ekX The phase function()of DTFT cannot be uniquely specified for all values of.Principal value()16 Symmetry
10、 Relations(I)table 3.1Sequence the DTFT j(e)X x nj(e)Xxn*j(e)X*xnjj*j1cs2(e)(e)(e)XXXRe x njj*j1ca2(e)(e)(e)XXXjIm x njre(e)Xcs xnjimj(e)Xca xn17 Symmetry Relations(II)table 3.2Real Sequence the DTFT jjjreim(e)(e)j(e)XXX x nj*j(e)(e)XXjre(e)Xev xnjimj(e)Xod xnSymmetryrelationsjjrere(e)(e)XXjjimim(e)
11、(e)XX jj|(e)|(e)|XXjjarg(e)arg(e)XX 18 3.3 DTFT Theorems table 3.4 Theorem Sequence DTFT j(e)G g nj(e)H h njj(e)(e)GH g nh nLinearityj(e)GgnTime-reversalTime-shifting0jje(e)nG0g nnFrequency-shifting0j()(e)G 0je ng nConvolutionjj(e)(e)GH g nh nModulation+()1()()d2jjG eH e g n h nParsevals Relation+*1
12、 ()()d2jjng n h nG eHeDifferentiation-in frequencyjd(e)jdG ng n19 Example of DTFT theorems(I)Example 3.13 Determine the DTFT of yn.(1),|1ny nnu nSolution:Let ,|1nx nu nthen y nnx nx njj1(e)1eXandthereforejjDTFTj2d(e)e j=d(1e)Xnx n 20 Example of DTFT theorem(II)According to the linear theorem:jDTFTj2
13、jj2 e1 (1e)1e1 (1e)y nnx nx n 21 Example of DTFT theorem(IV)Example 3.11 Determine the DTFT V(e j)of vn.0101 1 1d v nd v npnpnSolution:Using time-shifting and linearity theorem of DTFTjjjj0101(e)e(e)ed VdVppthereforejj01j01e(e)eppVdd22 Example of DTFT theorem(V)ExampleLet X(ej)denote the DTFT of a l
14、ength-9 sequence xn given by.2,3,1,0,4,3,1,2,426x nn Evaluate the follow of X(ej)without computing transform itself.022()()()()()()()()|()|()|jjjjja X eb X ecX eddX edX ededd23 Example of DTFT theorem(VI)Solution:24 Total Energy E Ex of DT Signal Total Energy E Ex of a finite-energy DT complex signa
15、l xn2j21|(e)|d2xnx nXE Definition of Energy Density Spectrum Sxx()j2()|(e)|xxSX The area under this curve in the range divided by 2 is the energy of the sequence25 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-ban
16、d signal has a spectrumoccupying the frequency range.Ideal Band-limited signal has a spectrum that is zero outside a finite frequency 0 a b :jj0,0|,|,(e)(e),otherwise abXX 26 Band-Limited DT Signals An ideal band-limited signal cannot be generated in practice Lowpass DT signal:jj(e),0|(e)0,|ppXXBand
17、width:p Highpass DT signal:jj0,0|(e)(e),|ppXXBandwidth:p27 Band-Limited DT Signals Bandpass DT signal:Bandwidth:H Ljj0,0|,|(e)(e),|LHLHXX28 3.4 DTFT Computation Using MATLAB Function:Freqz()To compute the values of the DTFT of a sequence,described as a rational function in the form:jjj01jj01ee(e)eeM
18、MNNpppXddd Example:H=Freqz(num,den,w)29 Example of Using MATLABExample 3.14jj2j3j4jjj2j3j40.008 0.033e0.05e0.033e0.008e(e)1 2.37e2.7e1.6e0.41eXNum=0.008,0.033,0.05,0.033,0.008;Den=1,2.37,2.7,1.6,0.41;W=0:0.01*pi:pi;H=freqz(Num,Den,w);subplot(2,2,1);plot(w/pi,real(H);grid;subplot(2,2,2);plot(w/pi,ima
19、g(H);grid;subplot(2,2,3);plot(w/pi,abs(H);gridsubplot(2,2,4);plot(w/pi,angle(H);grid;The phase spectrum has discontinuity of 2 at =0.72.32 3.5 The Unwrapped Phase Function Unwrapping the phase:Process the dis-continuity removal.c()Unwrapped phase functioncc0()()ddd with the constraintc(0)033 3.6 The
20、 Frequency Response of LTI DT System Eigenfunction:e j n Let xn=ej n,LTI system with impulse response hn,the output of the LTI system isj()jj ee(e)(3.77)kn knkkky nh nx nh k x n kh kh kOr rewritten as jj(e)e (3.78)ny nH34 Property of Frequency Responsejj(e)e (3.79)nnHh n Frequency response H(e j):is
21、 the DTFT of the impulse response hn;is a continuous function of ;is a periodic function of with a period 2;is a complex function of real variable .jjjjj()reim(e)(e)j(e)|(e)|eHHHH 35 Gain and Attenuationj10()20log|(e)|(dB)(3.81)GH Gain function:Attenuation(loss)function:()()AG H(ej)provides a freque
22、ncy-domain description of the systemjjj(e)(e)(e)(3.82)YHX36 Frequency-Domain Characterization of the LTI DT Systemjjj(e)(e)(3.83)(e)YHXExample 3.15Input sequence xn=anun,|a|1,LTI system with impulse response:hn=bnun,|b|1.Find the output sequence yn.Solution:y nx nh njjj(e)(e)(e)YXHjj1(e)1eXajjjjj1(e
23、)(e)(e)(1e)(1e)YXHabjj1(e)1eHbjj()()1e1eabababab11 nnnnababy na u nb u nu nababab0 nkn kka bu n38 Example of Frequency ResponseExample 3.16 Determine the frequency response of moving-average filter:1,01 (3.91)0,otherwise nMMh nSolution:j1jjj0j(1)/21 1 e1(e)e1 e1 sin(/2)e (3.92)sin(/2)MMnnMHMMMM39 Ma
24、gnitude response and phase response curveH=freqz(h,1,w);40 Frequency Response of LTI DT Systems LTI FIR DT Systems:2112 *,Nk Ny nx nh nh k x nkNN21jjj(e)e(e)Nkk NYh kX21jj(e)e (3.87)Nkk NHh k4100=NMkkkkd y nkp x nkjjjj00e(e)e(e)NMkkkkkkdYpXjjj0jj0e(e)(e)(3.90)(e)eMkkkNkkkpYHXd LTI IIR DT Systems:Fre
25、quency Response of LTI DT Systems42 The Concept of Filtering One application of an LTI DT system is to pass certain frequency components in an input sequence,without any distortion(if possible)and to block other frequency components.Such systems are called digital filters and are one of the main sub
26、jects of discussion in this course.The filtering process is:jjj1(e)(e)ed 2ny nXH43 The Concept of Filtering44 The Concept of Filtering45 By appropriately choosing the values of magnitude function of the LTI digital filter at frequencies corresponding to the frequencies of the sinusoidal components o
27、f the input,some of these sinusoidal sequences can be selectively heavily attenuated or filtered with respect to the others.jjj1(e)(e)ed2ny nXH The Concept of Filtering46 Lowpass digital filterj1,0|(e)|(3.95)0,|ccH Real coefficient LTI DT system characterized by a magnitude function:input sequence:1
28、212 coscos,0 (3.108)aA We assume that in the frequency range:00|,cc the frequency response of the CT system has a constant magnitude and a linear phase:|(j)|(j)|aacHH d()()()()|dcaaacc ()()()(3.109)cpccgc the CTFT of input signal xa(t)is:1(j)(j)(j)2aacacXAA the output response ya(t)is:00|,cc ()()cos
29、()agccpcy ta tt Because of the constraint imposed by Eq.(3.108),Xa(j)=0 outside the frequency range:With assuming|(j)|1acH the group delay g()is precisely the delay of the envelop a(t)of the input signal xa(t),whereas the phase delay p()is the delay of the carrier signal cos ct.61 Group delay and Phase delay of CT System62 Summaryl CTFT:Xa(j)=|Xa(j)|ej()l DTFT:X(e j)=|X(e j)|ej()l DTFT Theoremsl The Frequency Response H(e j)of LTI DT Systeml Phase Delay p()and Group Delay g()63 ExercisesPage 115:3.2 (a)(b);3.9Page 116:3.20(*);3.37;3.47
