EDA365欢迎您登录!
您需要 登录 才可以下载或查看,没有帐号?注册
x
) K8 F5 M5 Q: R+ j* T' D8 \/ @案例分析 Inverse Transform of Vector Padded Inverse Transform of Matrix Conjugate Symmetric Vector
% y; E& q) ?: Q, f b$ G# w+ l& ~5 k- y
案例分析
V x( v; L# r% ~5 u& M- T ]4 @# P! dInverse Transform of Vector3 n5 U1 U7 t3 u8 ^) q, E6 s
3 T4 ]/ j& B" g# l, Z
- % The Fourier transform and its inverse convert between data sampled in time and space and data sampled in frequency.
- %
- % Create a vector and compute its Fourier transform.
- X = [1 2 3 4 5];
- Y = fft(X)
- % Y = 1×5 complex
- %
- % 15.0000 + 0.0000i -2.5000 + 3.4410i -2.5000 + 0.8123i -2.5000 - 0.8123i -2.5000 - 3.4410i ⋯
- %
- % Compute the inverse transform of Y, which is the same as the original vector X.
- ifft(Y)
- % ans = 1×5
- %
- % 1 2 3 4 51 |! a# `4 _4 A: d
+ I" F1 f. b; Y {
* O6 \2 A% K: [6 }, V& F% \) C& y
; a' R0 a# c8 k) a+ q结果如下:8 d1 Q. K! u0 X, _$ C/ B
9 `& o" m$ `- z$ _" Q8 _3 |
ifft_vector
* m* d% k6 x8 m3 K5 W; y, M3 X" C
) D; _' z. A; I% H* ?! LY =
$ h B1 Q- l( ~2 p) d, l0 a
, h- `4 L$ B0 X4 Z7 c, X 1 至 4 列
+ i8 t0 d: C: I7 j# \6 I1 d+ Z
) ]( t# U! U* B. ~7 ? 15.0000 + 0.0000i -2.5000 + 3.4410i -2.5000 + 0.8123i -2.5000 - 0.8123i
6 S: N8 W6 |: A" O. b) T) I
! t8 l6 K& ~9 d7 L( |2 p p 5 列
. P, a+ E; X. ^2 K5 V/ l' |) K4 r
. Q5 U3 R8 ?7 S4 T& c -2.5000 - 3.4410i
8 a/ Z. U2 @: k8 K! E2 F: i' c6 x/ R' `9 F
ans =
' v$ b2 P8 V7 l/ _+ |4 t) j1 q" ?) B' i; M8 p; p) c: B
1 2 3 4 59 _ Y; |7 w9 @4 W1 I& d
5 g& u/ a8 B, H8 }) \& @. v
) P3 y+ u) w+ w5 S) H* @, O, n( z! U
Padded Inverse Transform of Matrix
4 i$ h5 {8 N2 N/ a: |. |. n- clc
- clear
- close all
- % The ifft function allows you to control the size of the transform.
- %
- % Create a random 3-by-5 matrix and compute the 8-point inverse Fourier transform of each row.
- % Each row of the result has length 8.
- Y = rand(3,5)
- n = 8;
- X = ifft(Y,n,2)
- size(X)
1 B2 H m5 k( g/ t/ p, g+ U
1 ] ]- |3 M5 R8 j# \ M
$ z5 u8 C) M& M
* |$ z/ b+ n7 q: \/ b8 R1 V$ p( [9 u1 k
结果如下:+ e# E) y3 s5 e* \9 w
3 F( X6 K. f; P' T5 q: xY =
- x7 U/ k$ {4 P/ w! h# W$ x
9 s) d0 d8 w! c5 a& S2 m/ _9 m! Y 0.8147 0.9134 0.2785 0.9649 0.9572
1 e4 f3 e( q5 ^* e1 I' C: a 0.9058 0.6324 0.5469 0.1576 0.4854; ]8 N6 w" I/ O
0.1270 0.0975 0.9575 0.9706 0.8003
( L I) Z0 v: O7 K U! G5 }. t/ A
X =/ J# n# |% }5 ]
j, n# o# f% |5 _9 Q- s 1 至 4 列# c+ i- U2 \% j" _% r
8 ~2 _3 s& G/ Y2 k: _
0.4911 + 0.0000i -0.0224 + 0.2008i 0.1867 - 0.0064i -0.0133 + 0.1312i
' w7 C) j+ I8 k8 x _2 U7 \ 0.3410 + 0.0000i 0.0945 + 0.1382i 0.1055 + 0.0593i 0.0106 + 0.0015i
4 X6 A1 R3 d3 V% W. ~9 c 0.3691 + 0.0000i -0.1613 + 0.2141i -0.0038 - 0.1091i -0.0070 - 0.0253i
& _$ o( r: G* F/ }
1 G4 t& L/ ^$ Y: \. E 5 至 8 列/ i2 E% ^. M k" _% C9 V
) I" t* D# `' ^# k$ J$ B4 P, e" G
0.0215 + 0.0000i -0.0133 - 0.1312i 0.1867 + 0.0064i -0.0224 - 0.2008i
" I) N2 g- t% J/ {( m8 P) A% \$ d 0.1435 + 0.0000i 0.0106 - 0.0015i 0.1055 - 0.0593i 0.0945 - 0.1382i- N! L7 | `. g7 [# Z) K! P9 q
0.1021 + 0.0000i -0.0070 + 0.0253i -0.0038 + 0.1091i -0.1613 - 0.2141i
' [7 O1 w p; R5 T) d- [# i% r5 q
- i/ L! a2 K6 U' ^6 j5 C% B8 ]9 gans =
3 g' O* c' c! [6 E) S3 \2 F3 _2 o0 o8 _ O& z: M
3 8% O s' L: R1 S2 K9 ?0 m
4 @9 q5 W% l+ }5 [+ T& N7 I- J+ S) y/ ]; t
上面的程序是计算矩阵每一行的8点ifft,故结果是每一行的ifft有8个元素,而计算矩阵 Y 每一行的 ifft,关键语句为:
( l/ c2 h2 a9 {' p4 t" u8 u# e2 Q) V- [. X ]* c% Z0 c
X = ifft(Y,n,2),里面的2,如果去掉2,则是对矩阵Y的每一列计算ifft,测试如下:
" v( \) C4 p8 [7 G7 d5 d& E$ m5 ~: h- ?
- clc
- clear
- close all
- % The ifft function allows you to control the size of the transform.
- %
- % Create a random 3-by-5 matrix and compute the 8-point inverse Fourier transform of each row.
- % Each row of the result has length 8.
- Y = rand(3,5)
- n = 8;
- X = ifft(Y,n)
- size(X)
) g# H; m% `; Q : ^* a1 t. l9 v/ v S! ^/ n4 d
b0 ^ ]* L: {8 R* _
Y =
4 W1 Y" u! X# [% g6 c; W: w. W2 P9 C% U' c, x7 D0 L
0.1419 0.7922 0.0357 0.6787 0.3922
B3 g3 s* N( S+ N# u4 x 0.4218 0.9595 0.8491 0.7577 0.6555$ @4 c8 j* K& v, q7 L. R7 o O
0.9157 0.6557 0.9340 0.7431 0.1712
( f+ y; q( n2 B' x: H6 @) p9 H7 X& n$ ?7 \/ W3 [
X =) J0 W# h! N2 g- W
' C- C' o- H* b1 |0 o7 X! r- E
1 至 4 列
6 s6 C& j# B4 A9 z# k: ^2 h/ p# l
0.1849 + 0.0000i 0.3009 + 0.0000i 0.2274 + 0.0000i 0.2725 + 0.0000i$ r1 j. I! P8 e$ Z$ s7 P
0.0550 + 0.1517i 0.1838 + 0.1668i 0.0795 + 0.1918i 0.1518 + 0.1599i! _8 U7 P" K' B4 P
-0.0967 + 0.0527i 0.0171 + 0.1199i -0.1123 + 0.1061i -0.0080 + 0.0947i) C2 d1 L7 e! i) C; R) O
-0.0195 - 0.0772i 0.0142 + 0.0028i -0.0706 - 0.0417i 0.0179 - 0.0259i7 j! [7 N: s- l* Y1 K3 F' w* `5 e
0.0795 + 0.0000i 0.0611 + 0.0000i 0.0151 + 0.0000i 0.0830 + 0.0000i, P% v0 b' v0 B/ _9 S3 w0 g5 W
-0.0195 + 0.0772i 0.0142 - 0.0028i -0.0706 + 0.0417i 0.0179 + 0.0259i5 P- M( m) d- r2 Y
-0.0967 - 0.0527i 0.0171 - 0.1199i -0.1123 - 0.1061i -0.0080 - 0.0947i
7 _9 I, C5 }& I! M5 Y# k; B4 K. E 0.0550 - 0.1517i 0.1838 - 0.1668i 0.0795 - 0.1918i 0.1518 - 0.1599i
; K2 o: u) w p/ ]* k' b7 d- w0 _1 O% M
5 列
* `+ D" V I, `. b1 \6 c; C7 L6 v0 a) h
0.1524 + 0.0000i* r- S. ^' O# T3 N
0.1070 + 0.0793i& K4 `; S# f6 k# v& b
0.0276 + 0.0819i$ j) Y) e* w# d4 H2 a
-0.0089 + 0.0365i
. h; T- r6 i3 F/ J/ z -0.0115 + 0.0000i1 @" _" Z' d: E% n9 J
-0.0089 - 0.0365i! l, r' ^$ n* K( b
0.0276 - 0.0819i$ r2 P) N1 b: p. r( q9 B6 D; p
0.1070 - 0.0793i
4 H9 |( I4 N# |8 ~; E1 q9 D- q6 f- G
ans =* `/ D: O n* Y7 s
6 Y; l% j3 ^7 C8 \# D3 ^' @
8 59 r& @% ?& b2 @2 k) B
4 Q+ ?5 m/ ?# D* D0 @
- q! L, V# T) y$ W G: R1 F: R! C
Conjugate Symmetric Vector0 s4 n! l% o3 D' u& `2 w5 m
. y* n) f% c, e2 ZFor nearly conjugate symmetric vectors, you can compute the inverse Fourier transform faster by specifying the 'symmetric' option, 7 i8 A+ y1 K% I
which also ensures that the output is real. Nearly conjugate symmetric data can arise when computations introduce round-off error.
4 ^( p* Q- r* ?. `- a4 C7 I. i |& O: A: z9 P4 }
Create a vector Y that is nearly conjugate symmetric and compute its inverse Fourier transform. 8 h# [" g, {( Q- s8 j3 Z
Then, compute the inverse transform specifying the 'symmetric' option, which eliminates the nearly 0 imaginary parts.+ l5 s* c) N6 S9 Y2 A, k& L
9 T+ u- }" j- Z- z, x7 N对于近似共轭对称矢量,您可以通过指定“symmetric”选项来更快地计算逆傅里叶变换,这也确保了输出是真实的。 当计算引入舍入误差时,可能出现几乎共轭的对称数据。! o- G: h. |9 ?2 W& L6 W% u+ ?
/ d! k* |, X! j创建几乎共轭对称的向量Y并计算其逆傅里叶变换。然后,计算指定'对称'选项的逆变换,它消除了近0虚部。
4 }& B% t7 u f( s3 A9 G l' I" T' p7 x
- clc
- clear
- close all
- % For nearly conjugate symmetric vectors, you can compute the inverse Fourier transform faster by specifying the 'symmetric' option,
- % which also ensures that the output is real. Nearly conjugate symmetric data can arise when computations introduce round-off error.
- %
- % Create a vector Y that is nearly conjugate symmetric and compute its inverse Fourier transform.
- % Then, compute the inverse transform specifying the 'symmetric' option, which eliminates the nearly 0 imaginary parts.
- Y = [1 2:4+eps(4) 4:-1:2]
- % Y = 1×7
- %
- % 1.0000 2.0000 3.0000 4.0000 4.0000 3.0000 2.0000 ⋯
- X = ifft(Y)
- % X = 1×7 complex
- %
- % 2.7143 + 0.0000i -0.7213 + 0.0000i -0.0440 - 0.0000i -0.0919 + 0.0000i -0.0919 - 0.0000i -0.0440 + 0.0000i -0.7213 - 0.0000i ⋯
- Xsym = ifft(Y,'symmetric')
- % Xsym = 1×7
- %
- % 2.7143 -0.7213 -0.0440 -0.0919 -0.0919 -0.0440 -0.7213 ⋯
% w, x1 ^! ^$ Y2 l2 b0 ~ 7 o8 l+ f2 ~& @% [1 ?
) w7 F7 ^0 W# R
. o" c8 X* V/ B' ]! I; }+ c! ]2 Q8 H, N, w- A
结果如下:
! O- `2 J, l: q5 F2 [( {& L
5 s/ C8 Q# W+ s+ t% Q( e/ [0 _9 G+ vY =
% C( }' g- m8 o) u
& ]6 @3 x9 A$ h" U) G* K 1.0000 2.0000 3.0000 4.0000 4.0000 3.0000 2.0000
d1 G7 R4 Y A$ ~3 P9 L5 p5 y4 y4 k8 K7 O) M) b/ E* y3 G0 N+ ~
X =
; B. j4 x7 H6 O" V- d+ `! C. Q
1 至 4 列
) J3 t+ `6 _ L, a/ R. |3 z9 Z8 ^; u. _/ ~8 b i0 g0 N
2.7143 + 0.0000i -0.7213 + 0.0000i -0.0440 - 0.0000i -0.0919 + 0.0000i
, ^( c: D; ]: l6 g: o V5 s: c! I$ O2 ^; F2 b }% x6 Y6 R
5 至 7 列3 i0 G! p! {# m7 S% l1 d
! G9 z! l! M; i# K2 g -0.0919 - 0.0000i -0.0440 + 0.0000i -0.7213 - 0.0000i. c+ A% [! e3 G* W& s
8 i. u1 S( ~* o+ G
Xsym =) ^- z# q2 t. ]+ s8 A% ^! K q
+ k; Z( B" h0 E' \ h
2.7143 -0.7213 -0.0440 -0.0919 -0.0919 -0.0440 -0.7213
4 L6 P! O/ ?3 b. T: B2 T
; B+ ^% W' O0 a. }- g) V6 {0 O: P; N5 u$ I
& q# N5 k* F) D) i, u1 @. ~2 ~
) N5 D6 i- s) I& x2 x$ [' p' X/ L8 a7 Q; ^9 a( t
* _- A/ Y5 w% u% y$ n0 u. d# @& I
0 @% L+ E" O1 _ E K: O
0 J4 u" F) A& v, E! \+ e* q$ Z& U: p# e& o) q Y3 y6 [5 Y
|