3 : 1
( ). (.. ), : 2
(1, 2 ),,, etc... 1, III ( ) ( ). : 3
,., III. : 4
,Weierstrass : Rudin, Principles of Mathematical Analysis, 3/e, McGraw-Hil, 1976.. Weierstrass (Stone-Weierstrass, ),,. : 5
2π f 0. f 0 (0) = f 0 (2π). f(e jθ ) = f 0 (θ), f (0 θ < 2π). e jθ = z (f(e jθ ) = f(z)),, Stone-Weierstrass : 6
f p(z) = a 0 z N +a 1 z N=1 + +a N 1 z+a N p ( ),. z = e jθ, p(e jθ ) = a 0 e jnθ + a 1 e j(n 1)θ + +a N 1 e jθ +a N.. : 7
e jθ = cosθ +jsinθ, p(e jθ ) = N A m cos(mθ)+ m=0 N B m sin(mθ) m=1. {A m } 0 m N {B m } 1 m N : ( ). : 8
, (Weierstrass ) Stone-Weierstrass, ( ). A m B m,. : 9
t, t, f(t) N m=0 A mcos(mt) + N m=1 B msin(mt)., 2π 0 f(t)dt = c 0 2πA 0 = 2πA 0, A 0 = (1/2π) 2π 0 f(t)dt; 2π 0 f(t)cosmt = 2π 0 A m cos 2 (mt)dt = πa m, A m = (1/π) 2π 0 f(t) cos(mt)dt 2π 0 f(t)sinmt = 2π 0 A m sin 2 (mt)dt = πb m, B m = (1/π) 2π 0 f(t) sin(mt)dt. : 10
, 2π N f(t) A m cos(mt)+ B m sin(mt) A 0 = 1 2π A m = 1 π B m = 1 π m=0 2π 0 2π 0 2π 0 f(t)dt m=1 f(t)cos(mt)dt (1 m < ) f(t)sin(mt)dt (1 m < ) : 11
T, 2π T f(t) A 0 = 1 T A m = 2 T B m = 2 T m=0 T ( 2πm A m cos T t 0 T 0 T 0 ) N + m=1 ( ) 2πm B m sin T t f(t)dt ( ) 2πm f(t)cos T t dt (1 m < ) ( ) 2πm f(t)sin T t dt (1 m < ) : 12
e j 2πm T t f(t) m Z C m = 1 T C m e j 2πm T t T 0 f(t)e 2πm T t dt, m Z, ( ). Euler, m Z. p. 27 3.1 Ω 0, Ω 0 = 2π/T. : 13
f(t)? :. : Stone-Weierstrass, f,.,. III. : 14
,. F(ν) = f(t)? = f(t)e j2πνt dt F(w)e j2πνt dν : 15
f g(ν) = f(t)e j2πνt dt g,f,f[f] (g = F[f]). g h(t) = g(ν)e j2πνt dν h, f, F 1 [g] (h = F[f]). : 16
f Fourier, (F 1 [F[f]] = f)., f F[f].., Ω = 2πν : 17
g(ω) (= g(ω/(2π))) = h(t) = 1 2π g(ω)e jωt dω f(t)e jωt dt. :
g g h(t) = 1 g(ω/(2π))e jωt dω, Ω = 2π 2πν, h(t) =. g(ν)e j2πνt dν, 2π, ( ),. : 19
(Fourier Series, FS) (Fourier Transofrm, FT) (Discrete-Time Fourier Transform, DTFT) (Discrete-Time Fourier Series, DTFS) (Discrete Fourier Transform, DFT) :
( ). 4. 5 ( ),.,. : 21
. x N ( ). Ω = 2π/N. : 22
x : N 1 X[m] = x[k]e jωmk (0 m < N). k=0 : 23
x (X[0],...,X[n 1]) ( ) x[n] = 1 N 1 X[m]e jωmn (0 n < N). N m=0. : 24
( p.29). X[m] = N 1 k=0 x[k]e jωmk (0 m < N) x[n] = 1 N 1 X[m]e jωmn (0 n < N) N m=0 : 25
x (X[0],...,X[n 1]), (X[0],...,X[n 1]) x,., (x[0],...,x[n 1]) (X[0],...,X[N 1]), x ( x )., 5 ( 3.1, 5 ),. : 26
X[m] 0 m < N,, m Z, X[m] = N 1 k=0 x[k]e jωmk X[m]. X. Ω = 2π/N, p Z, X[m + pn] = N 1 k=0 x[k]e jω(m+pn)k = N 1 k=0 x[k]e jωmk e j 2π N }{{ pkn } = N 1 k=0 x[k]e jωmk = X[m], X N., X (X[0],...,X[N 1]). =1 : 27
x F = (x[0],...,x[n 1]), X F = (X[0],...,X[N 1]),, (...,x F,x F,x F,...) (...,X F,X F,X F,...), x F X F,. x X (Analysis), x (Synthesis). : 28
N 1 x[n] = 1 N m=0 X[m]ejΩmn ( ), N = 1 ( ), N > 1,, N = 1. (N = 1 ). N > 1,. : 29
N 1 0 n < N n, z[n] = 1 N 1 X[m]e jωmn (0 n < N) N m=0. n z[n] = x[n]. X[m], z[n] = 1 N 1 N 1 x[k]e jωmk e jωmn = 1 N 1 N 1 N N. m=0 k=0 m=0 k=0 x[k]e jωm(n k) : 30
, k = n k n, ( z[n] = 1 N 1 ) N 1 x[n]+ x[k]e jωm(n k) N m=0 m=0 k n., 1 m, 2, ( z[n] = 1 Nx[n]+ ) N 1 x[k]e jωm(n k) N k n m=0 ( ) = x[n]+ 1 N 1 x[k] e jωm(n k) N. k n m=0 : 31
, 0 n,k < N, n k < N, Ω = 2π/N, e jω(n k) = e j2πn k N 1., 2, Ω = 2π/N, N 1 N 1 e j 2π N m(n k) = m=0 m=0 (e j 2π N (n k) ) m = 1 (e j 2π N (n k)) N 1 e j 2π(n k) N ( ), (e j 2π N (n k)) N = e j2π(n k) = 1,, z[n] = x[n]. : 32
N = 1 N = 1, x x[0]., X[0], X[m] = N 1 k=0 x[k]e jωmk m = 0 N = 1, X[0] = x[0]. N > 1 z[n], z[n] = 1 N 1 N m=0 X[m]ejΩmn, 0 n < N, N = 1 X[0] = x[0], z[0] = x[0]. : 33
, x, X (p. 30 3.2 )., ( ). : 34
: x : X[m] = 1 N 1 x[k]e jωmk (0 m < N). N k=0 x X : N 1 x[n] = X[m]e jωmn (0 n < N). m=0 1/N,. : 35
( 5 ),.,, 5, x X, X = F DFT [x], x = F 1 DFT [X].,,, F 1 DFT [F DFT[x]] = x. : 36
,,, F DFT [F 1 DFT [X]] = X?, F DFT F 1 DFT. e jωnk e jωnk, 1/N F DFT F 1 DFT, F DFT [F 1 DFT [X]] = X., F DFT F 1 DFT., F DFT F 1 DFT. : 37
F DFT [ ] F 1 DFT [ ],, F DFT F 1 DFT = I, F 1 DFT F DFT = I (I, ). X = F DFT [x] x = F 1 DFT [X], x = F 1 DFT [X] X = F DFT[X]. : 38
pp. 29 31, x, x 1, x 2 N. x, x 1, x 2 X, X 1, X 2., X, X 1, X 2, N. : 39
( 3.1 1 ) a 1 x 1 +a 2 x 2 N 1 k=0 (a 1x 1 [k]+a 2 x 2 [k])e jωmk, F DFT [a 1 x 1 +a 2 x 2 ] = a 1 X 1 +a 2 X 2. F 1 DFT [ ], a 1 x 1 +a 2 x 2 = F 1 DFT [a 1X 1 +a 2 X 2 ]. : 40
, 3.2 2,. x N, Ω = 2π/N. c,k Z, X[c,k] = c+n 1 m=c x[m]e jωmk. X[c,k] = N 1 m=0 x[m]e jωmk = X[k]., p Z, c = pn +c 1, 0 c 1 < N., m 1 = m pn, m = m 1 +pn, X[c,k] = c1+n 1 x[m m1=c1 1 + pn]e jω(m1+pn)k, x[m 1 + pn] = x[m 1 ], e jω(m1+pn)k = e jωm1k e jωpnk = e jωm1k (Ω = 2π/N ), X[c,k] = c1+n 1 x[m m1=c1 1 ]e jωm1k. c 1 = 0, c 1 0., X[c,k] = N 1 x[m m1=c1 1]e jωm1k + c1+n 1 m1=n x[m 1]e jωm1k, m 2 = m 1 N, 2 c1 1 m2=0 x[m 2 + N]e jω(m2+n)k = c1 1 m2=0 x[m 2]e jωm2k ( x Ω = 2π/N ). m 2 m 1 1, X[c,k] = N 1 x[m m1=c1 1]e jωm1k + c1 1 m1=0 x[m 1]e jωm1k = N 1 m1=0 x[m 1]e jωm1k, m 1 m,. : 41
, c Z X[k] = c+n 1 m=c x[m]e jωmk,, N x,. : 42
m ( 3.2 2 ) τ m m, Z = F DFT [τ m x], N 1 N 1 Z[n] = (τ m x[k])e jωnk = (x[k m])e jωnk k=0 k=0 N 1 = (x[k m])e jωn(k m) e jωnm = e jωnm X[n] k=0 : 43
m ( 3.2 3 ) τ m m, z = F 1 DFT [τ mx], N 1 N 1 z[n] = (τ m X[k])e jωnk = (X[k m])e jωnk k=0 k=0 N 1 = (X[k m])e jωn(k m) e jωnm = e jωnm x[n] k=0 : 44
( 3.2 4,5 ) x 2 x 2 ( ) X 1 X 2 ( ). N 1 (x 1 x 2 )[n] = x 1 [m]x 2 [n m], m=0 N 1 (X 1 X 2 )[n] = X 1 [m]x 2 [n m] m=0 : 45
,,. : F DFT [x 1 x 2 ] = X 1 X 2,FDFT[ 1 1 X ] N 1 X 2 = x1 x 2 3.2 4, 5,. : 46
N 1 (F DFT [x 1 x 2 ])[n] = (x 1 x 2 )[k]e jωnk k=0 N 1 N 1 = x 1 [m]x 2 [k m]e jωnk k=0 m=0 N 1 N 1 = x 1 [m]e jωnm x 2 [k m]e jωn(k m) k=0 m=0 N 1 N 1 = x 1 [m]e jωnm x 2 [k m]e jωn(k m) m=0 k=0 = X 1 [n]x 2 [n] : 47
( [ ]) 1 F 1 DFT N X 1 X 2 [n] = 1 N 1 ( 1 N N X 1 X 2 )[k]e jωnk = 1 N 2 = 1 N 2 N 1 N 1 k=0 X 1 [m]x 2 [k m]e jωnk k=0 m=0 N 1 N 1 X 1 [m]e jωnm X 2 [k m]e jωn(k m) k=0 m=0 N 1 = 1 X 1 [m]e jωnm 1 N 1 X 2 [k m]e jωn(k m) N N m=0 = x 1 [n]x 2 [n] k=0 : 48
, ( 3.2 6 8 ), N. x[n],, x x, x[n]. x n x[n]. x, n x[n] = x[n],,. 6 7. : 49
N 1 X[ n] = X[ n] = x[m]e jωm( n) = = N 1 m=0 m=0 x[n]e jωmn = (F DFT [ x])[n] N 1 x[m]e jωmn m=0 x[ n] = x[ n] = 1 N 1 X[m]e N jωm( n = 1 N 1 X[m]e jωmn N = 1 N N 1 m=0 m=0 X[m]e jωmn = ( F 1 DFT [ X] ) [n] m=0 : 50
x N 1 X[ n] = X[ n] = m=0 x[m]e jωm( n) N 1 = x[m]e jωmn = X[n]. m=0 3.2 8. : 51
(Discretetime Fourier Transform, DTFT) ( ),. : 52
( T) f(t) m Z C m = 1 T C m e j 2πm T t T 0 f(t)e 2πm T t dt, m Z T = 2π ( )., m n. : 53
( 2π) f(t) C n e jnt n Z C m = 1 2π f(t)e nt dt, n Z 2π 0, e jnt e jnt, (e jnt ) n Z,. : 54
( 2π), f(t) n Z C n = 1 2π C n e jnt 2π 0 f(t)e nt dt, n Z : t f C n ω X x[n]. : 55
X(ω) x[n]e jnω n Z x[n] = 1 2π X(ω)e nω dω, n Z 2π 0 X( f) x( C n )., x X., x. : 56
x x l 1 (x l 1 x[n] <. n= x, [0,2π) ( ; X(e jω ) ). X(ω) = n Z x[n]e jnω : 57
t < 0 t 2π, f 2π. x[n] (n Z) X.. x[n] = 1 2π X(ω)e nω dω. 2π 0 : 58
,. X(ω) = x[n]e jnω n Z x[n] = 1 2π X(ω)e nω dω, n Z 2π 0,. : 59
Logz, Y(z) = (X Log)(z), Y(e jω ) = X(Log(e jω )) = X(ω), X(ω) Y(e jω ), Y(e jω ) = n Z x[n]e jnω, x[n] = 1 2π Y(e jω )e nω dω (n Z). 2π 0,.,. : 60
, x l 1, x, l 1 2π., l 2 (l 1 l 2 ),.. : 61
( ),, MATLAB,, 2000.,, 1993.,, 2010. A. V. Oppenheim and R. W. Schafter, Discrete-Time Signal Processing, International Ed., Pearson, 2010. M. Mandal and A. Asif, Continuous and Discrete Time Signals and Systems, Cambridge University Press, 2007. K. B. Howell, Principles of Fourier Analysis, 2/e, CRC Press, 2017.,.. : 62