A. Basics of Discrete Fourier Trasform A.1. Defiitio of Discrete Fourier Trasform (8.5) A.2. Properties of Discrete Fourier Trasform (8.6) A.3. Spectral Aalysis of Cotiuous-Time Sigals Usig Discrete Fourier Trasform (1.1, 1.2) A.4. Covolutio Usig Discrete Fourier Trasform (8.6, 8.7) A.5. Samplig of Discrete-Time Fourier Trasform (8.4, 8.5)
A.1. Defiitio of Discrete Fourier Trasform Let x() be a fiite-legth sequece over 1. The discrete Fourier trasform of x() is defied as X(k) x() 1 1 1 k x()exp X(k)exp 2 j 2 j k, k, (A.2) is called the iverse discrete Fourier trasform. 1. A.1.1. Derivatio of Iverse Discrete Fourier Trasform Let us derive (A.2) from (A.1). (A.1) is rewritte as k 1. ote that X(k) is a fiite-legth sequece over k1. x() ca be recostructed from X(k), i.e., (A.1) (A.2)
1 k X(k) X(k) exp 1 m 2 j x(m) exp k 1 k 2 j 1 m km, k 2 x(m) exp j 1. k( m). (A.3) Multiplyig both the sides of (A.3) by exp(j2k/), 1, ad the summig both the sides with respect to k, we obtai (A.4) Chagig the order of the two summatios o the right side of (A.4), we obtai 1 k X(k) exp j 2 k 1 m x(m) 1 k exp j 2 k( m). (A.5) Sice 1 2, m exp j k( m) k, m, (A.6)
we obtai 1 k X(k) exp 2 j k x(). (A.7) (A.2), the iverse discrete Fourier trasform, is derived by dividig both the sides of (A.7) by. A.1.2. Relatio of Discrete Fourier Trasform to Discrete-Time Fourier Series Let us assume that X(k) is the discrete Fourier trasform of x(), x() is x() exteded with period, ad X(k) is the discrete-time Fourier series coefficiet of x(). The, X(k) is equal to X(k) over period k1 multiplied by, i.e., X(k)=X(k), k1. Figure A.1 illustrates this relatio. (A.8)
x() x() X(k) A A X(k) Figure A.1. Relatio to Discrete-Time Fourier Series. A.1.3. Relatio of Discrete Fourier Trasform to Discrete-Time Fourier Trasform We assume that X(k) ad X() are the discrete Fourier trasform ad the discrete-time Fourier trasform of x(), respectively. The, X(k) equals the samples of X() over period [, 2) at iterval 2/, i.e., k k
X(k) 2 X k, k 1. (A.9) Figure A.2 illustrates this relatio. x() X(k) k X() 2 Figure A.2. Relatio to Discrete-Time Fourier Trasform.
If is too small, X(k) may ot reflect X() well. Example. Assume x()=1, M1. (A.1) (1) Fid X(k), the -poit discrete Fourier trasform of x(), where M. (2) Fid X(), the discrete-time Fourier trasform of x(). If =M, does X(k) reflect X() well? Example. The samplig iterval of x() is 2 1 4 secod. Fid the legth rage of the discrete Fourier trasform such that the samplig iterval of X(k) is 2 radias/secod at the most. A.2. Properties of Discrete Fourier Trasform A.2.1. Liearity If x 1 ()X 1 (k) ad x 2 ()X 2 (k), the
a 1 x 1 ()+a 2 x 2 ()a 1 X 1 (k)+a 2 X 2 (k), (A.11) where a 1 ad a 2 are two arbitrary costats. (A.11) holds whe x 1 () ad x 2 () have the same legth iitially or after zero paddig. A.2.2. Circular Shift Assume that x() is a fiite-legth sequece over 1. The circular shift of x() by is defied as x( )=x( ), 1, where x() is x() exteded with period. (A.12) The circular shift ca be carried out i the followig steps (figure A.3). (1) Exted x() with period to obtai x(). (2) Shift x() by to obtai x( ). (3) Limit x( ) to period 1 to obtai x( ).
x() x() 1 2 3 m 1 2 3 m x(2) x(2) 1 2 3 4 5 m 1 2 3 m x(2) x(2) 1 2 3 m 1 2 3 Figure A.3. Circular Shift. m The circular shift ca also be carried out by shiftig ad wrappig
x() (figure A.3). If x()x(k), the 2 ( ) X(k) exp j k, x where is a arbitrary iteger. If x()x(k), the 2 x() exp j k X(kk ), where k is a arbitrary iteger. A.2.3. Reversal If x()x(k), the x(), 1X(k), k1, (A.13) (A.14) (A.15)
where x() ad X(k) are, respectively, x() ad X(k) exteded with period. A.2.4. Cojugatio If x()x(k), the x * ()X(k) *, k1, where X(k) is X(k) exteded with period. From (A.16), the followig coclusios ca be draw: (1) Im[x()]= X(k)=X(k) *, k1. (2) Re[x()]= X(k)=X(k) *, k1. If x()x(k), the x() *, 1X * (k), where x() is x() exteded with period. (A.16) (A.17)
From (A.17), the followig coclusios ca be draw: (1) Im[X(k)]= x()=x() *, 1. (2) Re[X(k)]= x()=x() *, 1. Example. X(k) is the 512-poit discrete Fourier trasform of x(). X(11)=2(1+j). Fid X(51). (1) x() is real. (2) x() is purely imagiary or. A.2.5. Symmetry If x()x(k), the X()x(k), k1, where x() is x() exteded with period. If x()x(k), the (A.18)
X()/, 1x(k), where X(k) is X(k) exteded with period. A.2.6. Circular Covolutio (Sectio A.4) A.2.7. Parseval s Equatio If x()x(k), the (A.19) 1 x() 2 1 1 k X(k) 2. (A.2) A.3. Spectral Aalysis of Cotiuous-Time Sigals Usig Discrete Fourier Trasform A.3.1. Frame The discrete Fourier trasform ca be used i the spectral aalysis of a cotiuous-time sigal (figure A.4).
x c (t) C/D x() y() DFT Y(k) w() Figure A.4. Spectral Aalysis of a Cotiuous- Time Sigal Usig Discrete Fourier Trasform. Each step is discussed as follows (figure A.5). (1) Samplig. Let X c () be the cotiuous-time Fourier trasform of x c (t), X() be the discrete-time Fourier trasform of x() ad T be the samplig iterval. The, X() equals X c () exteded with period 2/T, divided by T ad the expressed i terms of, i.e., X 1 T X. m c 2m T (A.21)
A X c () A/T X() 2 Y() 2 Y(k) 1 k Figure A.5. Illustratio of X c (), X(), Y() ad Y(k).
(2) Widowig. Assume that W() ad Y() are the discrete-time Fourier trasforms of w() ad y(), respectively. The, Y() equals the periodic covolutio of X() ad W() divided by 2, i.e., Y( ) 1 2 2 X( )W( )d. (A.22) (3) Discrete Fourier Trasform. Assume that Y(k) is the -poit discrete Fourier trasform of y(). The, Y(k) equals the samples of Y() over period [, 2) at iterval 2/, i.e., Y(k) Y 2 A.3.2. Effects of Widowig k, k 1. Let us ivestigate the effects of the above widowig further. (A.23) Let x i ()=A i exp(j i ) be a frequecy compoet of x(). The, the discrete-time Fourier trasform of x i () is
X ( ) i m 2A ( i i 2m). (A.24) Assume that y i () is the compoet of y() obtaied by widowig x i (), i.e., y i ()=x i ()w(). The, y i () has the discrete-time Fourier trasform Y i ()=A i W( i ). That is, Y i () equals W() shifted by i ad multiplied by A i. (A.25) From (A.24) ad (A.25), we see that compared with X i (), Y i () has the followig features (figure A.6). 1. Y i () has a distributio bad. (1) The width of the distributio bad equals the width of the mai lobe of W(). I order for Y i () to have a arrow distributio bad, W() should have a arrow mai lobe. (2) The width of the mai lobe of W() depeds o the shape ad the legth of w(). A loger w() results i a arrower mai lobe.
x i () A i X i () 2A i i i + i w() W() E y i () Y i () A i E i i i + Figure A.6. Differeces betwee X i () ad Y i ().
2. Y i () has leakages. (1) The relative amplitudes of the leakages equal the relative amplitudes of the side lobes of W(). For Y i () to have leakages of small relative amplitudes, W() should have side lobes of small relative amplitudes. (2) The relative amplitudes of the side lobes of W() deped o the shape of w() oly. A.3.3. Widows The rectagular widow is defied as 1, w(), The Bartlett widow is defied as 1. otherwise (A.26) w() 2 /( 1), 2 2 /( 1),, ( 1) / 2 ( 1)/2 1. otherwise (A.27)
The Ha widow is defied as.5.5 cos w(), The Hammig widow is defied as.54.46cos w(), The Blackma widow is defied as.42.5cos w(), 2.8cos 1 2, 1 1. otherwise 2, 1 1. otherwise 4, 1 1. otherwise The above widows are illustrated i figure A.7. (A.28) (A.29) (A.3)
Figure A.7. Shapes of Commoly Used Widows.
Table A.1 gives the features of these widows. It ca be see that whe the legth of the widow is fixed, a arrow distributio bad correspods to leakages of large relative amplitudes, ad therefore there exists a tradeoff betwee the width of the distributio bad ad the relative amplitudes of the leakages. Shape of Widow Width of Distributio Bad Peak Relative Amplitude of Leakages Rectagular 4/ 13 db Bartlett 8/(1) 25 db Ha 8/(1) 31 db Hammig 8/(1) 41 db Blackma 12/(1) 57 db Table A.1. Features of Commoly Used Widows.
The Kaiser widow is defied as w() I, 1 I 2 1 1 ( ) 2, 1. otherwise (A.31) Here I ( ) is the zero-order modified Bessel fuctio of the first kid ad is a shape parameter. A.4. Covolutio Usig Discrete Fourier Trasform A.4.1. Defiitio of Circular Covolutio Suppose that x 1 () ad x 2 () are two fiite-legth sequeces over 1, ad x 1 () ad x 2 () are x 1 () ad x 2 () exteded with period, respectively. The circular covolutio of x 1 () ad x 2 () is defied as the periodic covolutio of x 1 () ad x 2 () over a period
1, i.e., x1() x2() x(m)x 1 2( m), 1, which is also writte as m 1 x1() x 2() x (m)x 1 2( m), 1. m (A.32) (A.33) Several poits should be oted about the circular covolutio: (1) x 1 () ad x 2 () should have the same legth iitially or after zero paddig to carry out the circular covolutio. (2) The circular covolutio possesses the commutative property, the associative property ad the distributive property. The circular covolutio ca be carried out i the followig steps. (1) Exted x 2 (m) with period to obtai x 2 (m).
(2) Reflect x 2 (m) about the origi to obtai x 2 (m). (3) Shift x 2 (m) by to obtai x 2 (m). This is equivalet to the shiftig ad the wrappig of x 2 (m), m1. (4) Calculate the circular covolutio at. Example. Fid the circular covolutio of the two sequeces i the followig table. 1 2 otherwise x 1 () 1 x 2 () 3 2 1 A.4.2. Circular Covolutio Usig Discrete Fourier Trasform If x 1 ()X 1 (k) ad x 2 ()X 2 (k), the
x1() x2() X 1(k)X2(k). (A.34) If x 1 ()X 1 (k) ad x 2 ()X 2 (k), the 1 x1()x 2() X1(k) X2(k). (A.35) (A.34) shows that the circular covolutio ca be carried out usig the discrete Fourier trasform (figure A.8). x 1 () DFT IDFT x1() x 2() x 2 () DFT Figure A.8. Circular Covolutio Usig Discrete Fourier Trasform.
A.4.3. Covolutio Usig Discrete Fourier Trasform I this sectio, we will study the relatio betwee covolutio ad circular covolutio ad cosider how to carry out covolutio usig the discrete Fourier trasform. Let x 1 () ad x 2 () be two fiite-legth sequeces over 1 1 ad 2 1, respectively. If f() ad g() are the covolutio ad the -poit circular covolutio of x 1 () ad x 2 (), respectively, the we have g() f ( i i), 1. (A.36) That is, g() equals f() exteded with period ad limited to period 1. Sice f() is a fiite-legth sequece over 1 + 2 2, we ca draw the followig coclusios. If 1 + 2 1, g() equals f(). Otherwise, aliasig occurs. The relatio betwee f() ad g() is illustrated i figure A.9.
f() f() 1 + 2 2 m 1 + 2 2 m 1 + 2 2 m 1 + 2 2 m g() g() 1 + 2 2 1 + 2 1 m m 1 + 2 2 < 1 + 2 1 Figure A.9. Relatio betwee Covolutio ad Circular Covolutio.
(A.36) is proved as follows. Assume that x 2 () is x 2 () exteded with period. The, g() 1 m x (m)x 1 2( m), 1. (A.37) Sice x () 2 i x 2 ( i), (A.38) we obtai x ( m) 2 i x 2 ( i m). Substitutig (A.39) ito (A.37), we obtai (A.39) g() 1 m i x1(m)x2( i m), 1. (A.4)
Chagig the order of the summatios, we obtai g() 1 im Sice x 1 ()= outside 1, x1(m)x 2( i m), 1. (A.41) f () 1 m x 1 (m)x 2 ( m). (A.42) Thus, f ( i) 1 m x 1 (m)x 2 ( i m). (A.43) Accordig to (A.41) ad (A.43), we ca obtai the relatio betwee f() ad g(), i.e., g() f ( i i), 1. (A.44)
As we kow, the circular covolutio ca be carried out usig the discrete Fourier trasform. The, the covolutio ca also be carried out usig the discrete Fourier trasform because it equals the circular covolutio with 1 + 2 1 (figure A.1). This will be sigificat because the discrete Fourier trasform ca be implemeted by a class of fast algorithms. x 1 () 1 x 2 () 2 Zero Paddig Zero Paddig DFT DFT IDFT x 1 ()x 2 () 1 + 2 1 Figure A.1. Covolutio Usig Discrete Fourier Trasform.
A.4.4. Covolutio of a Log Sequece with a short Sequece Usig Discrete Fourier Trasform Assume that x 1 () is a log sequece ad x 2 () is a short sequece over 2 1. The, the two followig methods are usually used to carry out the covolutio of x 1 () ad x 2 (). 1. Overlap-Add Method (figure A.11): (1) x 1 () is segmeted, ad each segmet has legth 1. (2) Each segmet is padded with to poits (= 1 + 2 1). x 2 () is also padded with to poits. Each segmet is covolved with x 2 () usig the discrete Fourier trasform. (3) These covolutios are added. 2. Overlap-Save Method (figure A.12). (1) x 1 () is segmeted, ad each segmet has legth 1. (2) Each segmet is padded with subsequet samples to poits (= 1 + 2 1). x 2 () is padded with to poits. Each segmet is circularly covolved with x 2 () usig the discrete Fourier trasform. (3) The oaliasig segmets of these
1 1 1 x 1 () 1 2 1 x 11 () x 11 () x 2 () 1 2 1 x 12 () 1 2 1 x 12 () x 2 () 1 2 1 x 13 () 1 2 1 x 13 () x 2 () 1 2 1 x 1 () x 2 () Figure A.11. Overlap-Add Method.
1 1 1 x 1 () 1 x 12 () x 13 () 2 1 1 x 11 () 2 1 1 2 1 x 11 () x 2 () 2 1 x 12 () x 2 () x 13 () x 2 () x 1 () x 2 () 1 2 1 1 2 1 1 Figure A.12. Overlap-Save Method.
circular covolutios are combied. A.5. Samplig of Discrete-Time Fourier Trasform If X() is the discrete-time Fourier trasform of x(), X(k) is the samples of X() over period [, 2) at iterval 2/L, ad x() is the iverse discrete Fourier trasform of X(k), the x() i That is, x() equals x() exteded with period L ad the limited to period L1. If x() is withi L1, the x() equals x(). Otherwise aliasig happes. Figure A.13 shows the relatio betwee x() ad x(). (A.45) is proved as follows. x() 1 L L 1 k x( il), X (k)exp 2 j L L 1. k, L 1 (A.45)
x() m 1 X() 2 x() X(k) 1 L m 2/L 2 x() X(k) L 1 m 2/L 2 Figure A.13. Samplig of Discrete-Time Fourier Trasform.
L 1 1 L k 2 X L k exp j 2 L k, L 1. (A.46) Sice X( ) m x(m) exp( jm), (A.47) we obtai 2 X k L m x(m) exp 2 j L Substitutig (A.48) ito (A.46), we obtai km. (A.48) x () 1 L L1 m k m x(m) 2 x(m) exp j L 1 L L1 k 2 exp j L k( k( m), m), L 1 L 1. (A.49)
Sice L 1 1 2 1, m il exp j k( m), L k L, otherwise (A.5) we obtai x() i x( il), L 1. (A.51)