TABLE 14.3 Some Basic DTFT pairs Fourier Transform 2ro+21 4.ll(< 1aa10 6(++9-m7 广sm04+<-20m+r ≤兀 0≤n≤M o(M otherwise in(o/2) 2π60-0.+2π The spectrum S(ej) is periodic in o' with period 2T. The fundamental period in the range -I <O sT, sometimes referred to as the baseband, is the useful frequency range of the DT system because frequency components in this range can be represented unambiguously in sampled form(without aliasing error). In much of the signal-processing literature the explicit primed notation is omitted from the frequency variable. However, the explicit primed notation will be used throughout this section because there is a potential for confusion when so many related Fourier concepts are discussed within the same framework. By comparing Eqs. (14.3)and(1412a), and noting that o'=oT, we see that s())=DTET([n] (14.13) where s[n]=s(r)l=nT. This demonstrates that the spectrum of s(t)as calculated by the CT Fourier transform is identical to the spectrum of s[n] as calculated by the DTFT. Therefore, although sa(t) and s[n] are quite different sampling models, they are equivalent in the sense that they have the same Fourier domain represen tation.A list of common dtFT pairs is presented in Table 14.3. Just as the CT Fourier transform is useful in CT signal system analysis and design, the dTFt is equally useful for DT system analysis and design e 2000 by CRC Press LLC
© 2000 by CRC Press LLC The spectrum S(ejw¢) is periodic in w¢ with period 2p. The fundamental period in the range –p < w¢ £ p, sometimes referred to as the baseband, is the useful frequency range of the DT system because frequency components in this range can be represented unambiguously in sampled form (without aliasing error). In much of the signal-processing literature the explicit primed notation is omitted from the frequency variable. However, the explicit primed notation will be used throughout this section because there is a potential for confusion when so many related Fourier concepts are discussed within the same framework. By comparing Eqs. (14.3) and (14.12a), and noting that w¢ = wT, we see that (14.13) where s[n] = s(t)|t = nT . This demonstrates that the spectrum of sa(t) as calculated by the CT Fourier transform is identical to the spectrum of s[n] as calculated by the DTFT. Therefore, although sa(t) and s[n] are quite different sampling models, they are equivalent in the sense that they have the same Fourier domain representation. A list of common DTFT pairs is presented in Table 14.3. Just as the CT Fourier transform is useful in CT signal system analysis and design, the DTFT is equally useful for DT system analysis and design. TABLE 14.3 Some Basic DTFT Pairs Sequence Fourier Transform 1. 1 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. d[ ] n d n – n [ ]0 e - wj n0 1 ( ) -• < n < • 2pd( ) w + 2p =-• •  k k a u n a n [ ] ( ) < 1 1 1 - - ae jw u n[ ] 1 1 2 - + + ( ) - =-• • e  k j k w pd w p n a u n a n ( + 1 1 ) [ ] ( ) < 1 1 2 ( ) - - ae jw r n u n r n p p sin sin w w ( ) + [ ] ( ) < 1 1 1 1 2 2 2 - + - - r e r e p j j cos w w w sin w p c n n X ej c c w w w w w p ( ) = < < £ Ï Ì Ô Ó Ô 1 0 , , , x n n M [ ] = Ï £ £ Ì Ó 1 0 0 , , otherwise sin sin w w w M e j M ( ) + [ ] ( ) - 1 2 2 2 ejw n0 2 2 0 pd( ) w - + w p =-• •  k k cos w f 0 ( ) n + p d w w p d w w p f f e k e k j j k ( ) - + + ( ) + + [ ] - =-• •  0 0 2 2 F s t { } a ( ) = DTFT{ } s[n]
In the same way that the CT Fourier transform was found to be a special case of the complex Fourier transform(or bilateral Laplace transform), the DTFT is a special case of the bilateral z-transform with z= eje The more general bilateral z-transform is given by S()=∑ (14.14a) 4=(12)52-d (14.14b) where C is a counterclockwise contour of integration which is a closed path completely contained within the ROC of S(z). Recall that the DtFT was obtained by taking the CT Fourier transform of the CT sampling model s(t). Similarly, the bilateral z-transform results by taking the bilateral Laplace transform of s(t). If the lower e summation of Eq. (14.14a)is taken to be n=0, then Eqs. (14.14a)and(14 14b)become the one- ided z-transform, which is the DT equivalent of the one-sided Laplace transform for CT signals Properties of the DTFT Since the DTFt is a close relative of the classical CT Fourier transform, it should come as no surprise that many properties of the DTFT are similar to those of the CT Fourier transform. In fact, for many of the properties presented earlier there is an analogous property for the dtFt. The following list parallels the list that was presented in the previous section for the CT Fourier transform, to the extent that the same property exists. a more complete list of dtft pairs is given in Table 14.4: 1. Linearity(superposition) DTFTlafi[n+ bf2[n]=aDTFTUI[n)+ bDTFTU2[nJI (a and b, complex constants) 2. Index Shifting DTFTUIn-nol=e-jomoDTFTIf[nJ) 3. Frequency Shifting: ejuomf [n]=DTFT-IF((O-O)) 4. Time-Domain Convolutic DTFTUiIn]*f,[n]= DTFTU[n] DTFTIf2[nJ) 5. Frequency-Domain Convolution: DTFTUIIn] f[n)=(1/2I)DTFTUIIn]* DTFTU2InI) nf [n]= DtFT-dF(jo)/dol Note that the time-differentiation and time-integration properties of the CT Fourier transform do not have analogous counterparts in the dtft because time-domain differentiation and integration are not defined for DT signals. When working with DT systems practitioners must often manipulate difference equations in the frequency domain. For this purpose Property 1(linearity)and Property 2(index shifting) are important. As with the CT Fourier transform, Property 4(time-domain convolution) is very important for DT systems because it allows engineers to work with the frequency response of the system in order to achieve proper shaping of ne input spectrum, or to achieve frequency selective filtering for noise reduction or signal detection. Also Property 3(frequency shifting) is useful for the analysis of modulation and filtering common in both anal and digital communication systems. Relationship between the CT and DT Spectra Since DT signals often originate by sampling a CT signal, it is important to develop the relationship between ne original spectrum of the CT signal and the spectrum of the DT signal that results. First, the CT Fourier transform is applied to the CT sampling model, and the properties are used to produce the following result 14.15) c 2000 by CRC Press LLC
© 2000 by CRC Press LLC In the same way that the CT Fourier transform was found to be a special case of the complex Fourier transform (or bilateral Laplace transform), the DTFT is a special case of the bilateral z-transform with z = ejw¢t . The more general bilateral z-transform is given by (14.14a) (14.14b) where C is a counterclockwise contour of integration which is a closed path completely contained within the ROC of S(z). Recall that the DTFT was obtained by taking the CT Fourier transform of the CT sampling model sa(t). Similarly, the bilateral z-transform results by taking the bilateral Laplace transform of sa(t). If the lower limit on the summation of Eq. (14.14a) is taken to be n = 0, then Eqs. (14.14a) and (14.14b) become the onesided z-transform, which is the DT equivalent of the one-sided Laplace transform for CT signals. Properties of the DTFT Since the DTFT is a close relative of the classical CT Fourier transform, it should come as no surprise that many properties of the DTFT are similar to those of the CT Fourier transform. In fact, for many of the properties presented earlier there is an analogous property for the DTFT. The following list parallels the list that was presented in the previous section for the CT Fourier transform, to the extent that the same property exists. A more complete list of DTFT pairs is given in Table 14.4: 1. Linearity (superposition): DTFT{af 1[n] + bf 2[n]} = aDTFT{f 1[n]} + bDTFT{f 2[n]} (a and b, complex constants) 2. Index Shifting: DTFT{f[n – no]} = e –jwnoDTFT{f [n]} 3. Frequency Shifting: ejwonf [n] = DTFT–1{F(j(w – wo))} 4. Time-Domain Convolution: DTFT{f 1[n] * f2[n]} = DTFT{f 1[n]} DTFT{f 2[n]} 5. Frequency-Domain Convolution: DTFT{f 1[n] f2[n]} = (1/2p)DTFT{f 1[n]} * DTFT{f 2[n]} 6. Frequency Differentiation: nf [n] = DTFT–1{dF(jw)/dw} Note that the time-differentiation and time-integration properties of the CT Fourier transform do not have analogous counterparts in the DTFT because time-domain differentiation and integration are not defined for DT signals. When working with DT systems practitioners must often manipulate difference equations in the frequency domain. For this purpose Property 1 (linearity) and Property 2 (index shifting) are important. As with the CT Fourier transform, Property 4 (time-domain convolution) is very important for DT systems because it allows engineers to work with the frequency response of the system in order to achieve proper shaping of the input spectrum, or to achieve frequency selective filtering for noise reduction or signal detection. Also, Property 3 (frequency shifting) is useful for the analysis of modulation and filtering common in both analog and digital communication systems. Relationship between the CT and DT Spectra Since DT signals often originate by sampling a CT signal, it is important to develop the relationship between the original spectrum of the CT signal and the spectrum of the DT signal that results. First, the CT Fourier transform is applied to the CT sampling model, and the properties are used to produce the following result: (14.15) S z s n z n n ( ) = [ ] - = -• • Â s n j S z z dz n C [ ] = ( ) ( ) - Ú 1 2 1 p F sa a t F s t t nT S j F t nT n n { } ( ) = ( ) ( - ) Ï Ì Ô Ó Ô ¸ ˝ Ô ˛ Ô = ( ) ( ) ( - ) Ï Ì Ô Ó Ô ¸ ˝ Ô ˛ = -• Ô • = -• • Â Â d 1 2p w d
Table 14.4 Properties of the DTFt Fourier Transform xin X(er) x(e-in)if x["]real x(elmy x(e r( o) ∑『=a ∑-axy+) In this section it is important to distinguish between o and o, so the explicit primed notation is used in the following discussion where needed for clarification. Since the sampling function(summation of shifted pulses)on the right-hand side of the above equation is periodic with period Tit can be replaced with a Ct Fourier series expansion as follows: 4-)(=(2 Applying the frequency-domain convolution property of the CT Fourier transform yields -)=(2∑0)(xm)N-(2xm)=0m)∑o-m0416) where O, =(2I/T)is the sampling frequency (rad/s). An alternate form for the expression of Eq (1416a)is s-)=(∑so-n27)) (14.16b) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC In this section it is important to distinguish between w and w¢, so the explicit primed notation is used in the following discussion where needed for clarification. Since the sampling function (summation of shifted impulses) on the right-hand side of the above equation is periodic with period T it can be replaced with a CT Fourier series expansion as follows: Applying the frequency-domain convolution property of the CT Fourier transform yields (14.16a) where ws = (2p/T) is the sampling frequency (rad/s). An alternate form for the expression of Eq. (14.16a) is (14.16b) Table 14.4 Properties of the DTFT Sequence Fourier Transform x[n] X(ejw) y[n] Y(ejw) 1. 2. 3. 4. 5. 6. 7. Parseval’s Theorem 8. 9. ax[ ] n + by[ ] n aX e bY e j w wj ( ) + ( ) x n n n d d [ - ] ( ) an integer e X e j n j d - ( ) w w e x n jw n0 [ ] X eÊ j( ) w-w Ë ˆ ¯ 0 x[ ] -n X e x n -j ( ) [ ] w if real X ej *( ) w nx[ ] n j dX e d jw w ( ) x[ ] n *y[ ] n X e Y e jw wj ( ) ( ) x[n]y[ ] n 1 2p q q p p w q X e Y e d j j ( ) Ê Ë ˆ ¯ - ( ) - Ú x n X e d n j [ ] = ( ) =-• • Â Ú- 2 2 1 2p w w p p x n y n X e Y e d n j j [ ] *[ ] = ( ) *( ) = • • Â Ú 1 2p w w p p w – – S e F s t S j F T e j T a j T nt n w p ( ) = { } ( ) = ( p w ) ( ) ( ) Ï Ì Ô Ó Ô ¸ ˝ Ô ˛ Ô ( ) = -• • 1 2 Â 1 2 S e S j T T n T S j n j T n s n w ( ) = ( p) ( w) * ( p )d(w - ( p ) ) = ( ) ( [w - w ]) = -• • = -• • 1 2 Â Â 2 2 1 S e T S j n T j n ¢ = -• • ( ) = ( )Â ( [( ¢ - ) ]) w 1 2 w p
FIGURE 14.7 Relationship between the Ct and dt where a'=OT is the normalized DT frequency axis expressed in radians. Note that S(ejt)=S(er)consists of an infinite number of replicas of the CT spectrum S(jo), positioned at intervals of (2T/T)on the o axis(or at intervals of 2t on the o axis), as illustrated in Fig. 14.7. If S(jo) is band limited with a bandwidth @, and if T is chosen sufficiently small so that @, 20 then the DT spectrum is a copy of S(jo)(scaled by 1/T)in the baseband. The limiting case of o,=2@ is called the Nyquist sampling frequency. Whenever a CT signal is sampled at or above the Nyquist rate, no aliasing distortion occurs (i.e, the baseband spectrum does not overlap with the higher-order replicas)and the CT signal can be exactly recovered from its samples by extracting the baseband spectrum of S(eje) with an ideal low-pass filter that recovers the original CT spectrum by removing all spectral replicas outside the baseband and scaling the baseband by a factor of T. Discrete fourier transform To obtain the dFT the continuous-frequency domain of the dtFt is sampled at n points uniformly spaced around the unit circle in the z-plane, i. e, at the points O=(2TA/N),k=0, 1,.,N-1. The result is the DFT transform pair defined by Eqs. (1417a)and (14 17b). The signal s[n]is either a finite-length sequence of length Nor it is a periodic sequence with period N. =∑ =(N)∑ n=0,l,,N-1 (1417b) Regardless of whether s[n] is a finite-length or a periodic sequence, the DFT treats the N samples of s[n] as though they characterize one period of a periodic sequence. This is an important feature of the DFT, and one that must be handled properly in signal processing to prevent the introduction of artifacts. Important properties of the dFt are summarized in Table 14.5. The notation ()N denotes k modulo N, and RNIn] is a rectangular window such that RNn]=I for n=0,., N-1, and RNIn=0 for n<0 and n2 N The transform relationship given by Eqs. (1417a)and (14 17b)is also valid when s(n] and s[k] are periodic sequences, each of period N. In this case, n and k are permitted to range over the complete set of real integers, and s[k] is referred to as the discrete Fourier series(DFS). The DFS is developed by some authors as a distinct transform pair in its own right [Oppenheim and Schafer, 1975. Whether or not the dFt and the DFS are considered identical or distinct is not very important in this discussion. The important point to be emphasized here is that the DFT treats s[n] ough it were a single period of a periodic sequence, and all signal processing done with the DFt inherit the consequences of this assumed periodicity Properties of the DFT Most of the properties listed in Table 14.5 for the DFT are similar to those of the z-transform and the DTFt, although there are some important differences. For example, Property 5(time-shifting property ) holds for circular shifts of the finite-length sequence s[n], which is consistent with the notion that the DFT treats s[n] as one period of a periodic sequence. Also, the multiplication of two DFTs results in the circular convolution c 2000 by CRC Press LLC
© 2000 by CRC Press LLC where w¢ = wT is the normalized DT frequency axis expressed in radians. Note that S(ejwT) = S(ejw¢) consists of an infinite number of replicas of the CT spectrum S(jw), positioned at intervals of (2p/T) on the w axis (or at intervals of 2p on the w¢ axis), as illustrated in Fig. 14.7. If S(jw) is band limited with a bandwidth wc and if T is chosen sufficiently small so that ws > 2wc, then the DT spectrum is a copy of S(jw) (scaled by 1/T) in the baseband. The limiting case of ws = 2wc is called the Nyquist sampling frequency. Whenever a CT signal is sampled at or above the Nyquist rate, no aliasing distortion occurs (i.e., the baseband spectrum does not overlap with the higher-order replicas) and the CT signal can be exactly recovered from its samples by extracting the baseband spectrum of S(ejw¢) with an ideal low-pass filter that recovers the original CT spectrum by removing all spectral replicas outside the baseband and scaling the baseband by a factor of T. Discrete Fourier Transform To obtain the DFT the continuous-frequency domain of the DTFT is sampled at N points uniformly spaced around the unit circle in the z-plane, i.e., at the points wk = (2pk/N), k = 0, 1, …, N – 1. The result is the DFT transform pair defined by Eqs. (14.17a) and (14.17b). The signal s[n] is either a finite-length sequence of length N or it is a periodic sequence with period N. Regardless of whether s[n] is a finite-length or a periodic sequence, the DFT treats the N samples of s[n] as though they characterize one period of a periodic sequence. This is an important feature of the DFT, and one that must be handled properly in signal processing to prevent the introduction of artifacts. Important properties of the DFT are summarized in Table 14.5. The notation (k)N denotes k modulo N, and RN[n] is a rectangular window such that RN[n] = 1 for n = 0, …, N – 1, and RN[n] = 0 for n < 0 and n ³ N. The transform relationship given by Eqs. (14.17a) and (14.17b) is also valid when s[n] and S[k] are periodic sequences, each of period N. In this case, n and k are permitted to range over the complete set of real integers, and S[k] is referred to as the discrete Fourier series (DFS). The DFS is developed by some authors as a distinct transform pair in its own right [Oppenheim and Schafer, 1975]. Whether or not the DFT and the DFS are considered identical or distinct is not very important in this discussion. The important point to be emphasized here is that the DFT treats s[n] as though it were a single period of a periodic sequence, and all signal processing done with the DFT will inherit the consequences of this assumed periodicity. Properties of the DFT Most of the properties listed in Table 14.5 for the DFT are similar to those of the z-transform and the DTFT, although there are some important differences. For example, Property 5 (time-shifting property), holds for circular shifts of the finite-length sequence s[n], which is consistent with the notion that the DFT treats s[n] as one period of a periodic sequence. Also, the multiplication of two DFTs results in the circular convolution FIGURE 14.7 Relationship between the CT and DT spectra. S k s n e k N N S k e n N j kn N n N j kn N n N [ ] = [ ] = º - [ ] = ( ) [ ] = º - - = - = - Â Â 2 0 1 2 0 1 0 1 1 14 17 1 0 1 1 p p , , , ( . , , , ( a) s n 14.17b)
TABLE 14.5 Properties of the Discrete Fourier Transform(DFT) Finite-Length Sequence(Length N) N-Point DFT (Length N) xx小 可小+k 4+bx4 4(-m 7∑xm)-m 14 x={()+x+ ={4+x24 Properties 15-17 apply only when xin] is real =-( ={+4- x同=-4-m e 2000 by CRC Press LLC
© 2000 by CRC Press LLC TABLE 14.5 Properties of the Discrete Fourier Transform (DFT) Finite-Length Sequence (Length N) N-Point DFT (Length N) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Properties 15–17 apply only when x[n] is real. 15. Symmetry properties 16. 17. x n[ ] X k[ ] x n x n 1[ ], 2[ ] X k X k 1[ ], 2[ ] ax n bx n 1[ ] 2 + [ ] aX k bX k 1[ ] 2 + [ ] X n[ ] Nx k N ( ) -[ ] x n m N ( ) - [ ] W X k N km [ ] W x n N - n [ ] , X k N ( ) - [ ] , x m x n m m N 1 N 0 1 2 = - Â ( ) ( ) - [ ] X k X k 1[ ] 2[ ] x n x n 1[ ] 2[ ] 1 1 0 1 2 N X X k N N , , , = - Â ( ) ( ) - [ ] x*[ ] n X k N * -[ ] ( ) x n N * -[ ] ( ) X*[ ] k 5e{ } x n[ ] X k X k X k ep N N [ ] = [ ] ( ) + * ( ) - { } [ ] 1 2 j(m{ } x[n] X k X k X k op N N [ ] = [ ] ( ) - * ( ) - { } [ ] 1 2 x n x n x n ep N [ ] = [ ] + * ( ) - { } [ ] 1 2 5e{ } X k[ ] x n x n x n op N [ ] = [ ] - * ( ) - { } [ ] 1 2 j(m{ } X[k] X k X k X k X k X k X k X k X k X k X k N N N N N [ ] = * ( ) -[ ] { } [ ] = - { } [( ) ] { } [ ] = - { } [( )] [ ] = -[ ] ( ) { } [ ] = - ( ) - { } [ ] Ï Ì Ô Ô Ô Ô Ô Ó Ô Ô Ô Ô Ô 5 5 ( ( e e m m , , x n x n x n ep N [ ] = { } [ ] + -[( ) ] 1 2 5e{ } X k[ ] x n x n x n op N [ ] = { } [ ] - -[( ) ] 1 2 j(m{ } X[k]