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案例分析 Inverse Transform of Vector Padded Inverse Transform of Matrix Conjugate Symmetric Vector 0 _6 p f6 A9 X! ]( t7 @( d# i
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案例分析
6 a, f4 _- f1 t L$ hInverse Transform of Vector. s+ P. a {9 A" z4 u
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- % 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 5
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结果如下： r: z1 T1 ^$ T$ ~
- n; h& |' m7 N s0 M/ ]3 k8 r ifft_vector$ e9 f* G# u" {7 K5 p$ o
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Y =( }. I$ h, z# {! m0 x, y( Q6 L; U& m
" v5 X' u8 e' b# E* w& v0 W: ~ 1 至 4 列
( w* ~; ?- T! i" }6 Z# e( A# ]' ?, ^' h" r" o$ }: H! m
15.0000 + 0.0000i -2.5000 + 3.4410i -2.5000 + 0.8123i -2.5000 - 0.8123i
) \; L$ l+ M ^+ N
; p$ B5 m; _) l5 x) m1 u 5 列
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y- \3 o4 s4 ^% r -2.5000 - 3.4410i$ r3 d E" Z# S! ~* v2 o3 j
& v' [5 @1 f6 m" gans =
# x2 ?' L/ u: a5 @: g2 {& y# S! d( i3 y% ?9 [1 i
1 2 3 4 5
& R* g! L1 |# E+ U( {# X
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Padded Inverse Transform of Matrix' M# r. r/ K3 L* a1 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,2)
- size(X)
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8 a+ q) I& e& k2 \: P6 x( O
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1 v8 G2 \& E, G3 J" g结果如下：, }* M9 o! I0 c! j1 l2 g1 A
- x2 P* X4 v# \8 u v& Q5 bY =) c7 R$ t+ Z8 R2 f: X, [' I
\1 N# ?" B4 a$ c: w
0.8147 0.9134 0.2785 0.9649 0.95722 W5 c8 z: H4 f3 @
0.9058 0.6324 0.5469 0.1576 0.4854
: |3 Z/ g2 k' x1 J 0.1270 0.0975 0.9575 0.9706 0.8003
9 s4 }4 i' M# k- D+ L2 O6 ^0 C8 \$ s) B: [7 X
X =0 A6 n: O- x" \. U; k5 H2 }4 O
2 Z) K7 x s- k 1 至 4 列
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0.4911 + 0.0000i -0.0224 + 0.2008i 0.1867 - 0.0064i -0.0133 + 0.1312i
9 b4 x" O2 a( |! Y; s 0.3410 + 0.0000i 0.0945 + 0.1382i 0.1055 + 0.0593i 0.0106 + 0.0015i) B) P0 z' p% A
0.3691 + 0.0000i -0.1613 + 0.2141i -0.0038 - 0.1091i -0.0070 - 0.0253i% r. B0 i/ s; }0 I( {+ n
, A9 G& O" ]1 |; S+ ^- | 5 至 8 列
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0.0215 + 0.0000i -0.0133 - 0.1312i 0.1867 + 0.0064i -0.0224 - 0.2008i
3 a6 l! N7 W# y6 g! g3 F 0.1435 + 0.0000i 0.0106 - 0.0015i 0.1055 - 0.0593i 0.0945 - 0.1382i
8 m! h. c O7 v1 B v" Q. N, ?0 m 0.1021 + 0.0000i -0.0070 + 0.0253i -0.0038 + 0.1091i -0.1613 - 0.2141i2 Z) L9 U5 ?( m1 _' S% O
( ` E* b) n/ j) U" `% D9 F3 V
ans =6 ^% m! _, `! W% o; m
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3 8
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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：, k3 y* q: ^1 d! G* p
) F1 x; Z8 P+ v$ p N. A* OX = 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：* ^7 \0 o" [3 K9 l# r
$ V9 {# r* G6 q: r. P/ A# {- 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)
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) z" X: }4 z( i; H; l2 |
' e3 ~1 z6 Z* C1 m2 PY =3 T: V% P/ E( M7 N+ K' L7 r; Y4 X
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0.1419 0.7922 0.0357 0.6787 0.3922% N: g/ f) F+ J; @& C2 \- H
0.4218 0.9595 0.8491 0.7577 0.6555
' k! P1 p0 z6 x$ h 0.9157 0.6557 0.9340 0.7431 0.17126 x* J! ~" Z- _
: ?. i) n+ v0 q4 D N" Y( S% Q) L7 dX =4 ?$ q% z/ ~7 n( v
$ b' ^ W, u3 N; v& [9 p, X, Y
1 至 4 列
* _& H4 m. o% Y- V* v5 _ G% e( }8 ]2 {' G. V
0.1849 + 0.0000i 0.3009 + 0.0000i 0.2274 + 0.0000i 0.2725 + 0.0000i
# H( N3 ? N4 _1 E- y- G 0.0550 + 0.1517i 0.1838 + 0.1668i 0.0795 + 0.1918i 0.1518 + 0.1599i
5 |% \/ N1 U) ?+ i2 h6 @$ a -0.0967 + 0.0527i 0.0171 + 0.1199i -0.1123 + 0.1061i -0.0080 + 0.0947i0 ` `% S& Z! C; x
-0.0195 - 0.0772i 0.0142 + 0.0028i -0.0706 - 0.0417i 0.0179 - 0.0259i
4 n- k p9 n, k9 D; c 0.0795 + 0.0000i 0.0611 + 0.0000i 0.0151 + 0.0000i 0.0830 + 0.0000i9 k+ R: n. \& a1 P5 d& ]
-0.0195 + 0.0772i 0.0142 - 0.0028i -0.0706 + 0.0417i 0.0179 + 0.0259i
: ^( n' a+ M1 \ -0.0967 - 0.0527i 0.0171 - 0.1199i -0.1123 - 0.1061i -0.0080 - 0.0947i! S5 T$ V, z# b
0.0550 - 0.1517i 0.1838 - 0.1668i 0.0795 - 0.1918i 0.1518 - 0.1599i
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# N- {4 K4 }* j: F4 Y8 [/ R _) H. @/ @ 5 列
6 k m5 z2 P7 I7 b0 z; R4 I! e# ]2 \. x. r7 s. N1 M3 [
0.1524 + 0.0000i
/ [) v, @4 M4 ?5 ~8 S$ N 0.1070 + 0.0793i7 T4 G7 I9 z3 C
0.0276 + 0.0819i! T- ?1 F" X" J8 E# J; s
-0.0089 + 0.0365i+ n& J) a* O( f/ A
-0.0115 + 0.0000i
6 s: f( R. i# H3 [/ d -0.0089 - 0.0365i5 |' v6 y/ i6 F- A6 \
0.0276 - 0.0819i' A" v* B6 U; B. h
0.1070 - 0.0793i. x9 S0 l U' Q/ t$ g: M" ?- [
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ans =
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8 5
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; d8 r9 R4 h3 J- e# g5 WConjugate Symmetric Vector2 }* y- F y; g
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For nearly conjugate symmetric vectors, you can compute the inverse Fourier transform faster by specifying the 'symmetric' option,
7 O0 N# E W4 A1 jwhich also ensures that the output is real. Nearly conjugate symmetric data can arise when computations introduce round-off error.7 f: l* a7 } F! p2 v
, G: G$ M0 P% |, G# P# X$ X8 a1 zCreate a vector Y that is nearly conjugate symmetric and compute its inverse Fourier transform.
& l2 N* j" O# A$ u. h& {7 b& ~Then, compute the inverse transform specifying the 'symmetric' option, which eliminates the nearly 0 imaginary parts.
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: L+ [3 N- m: B, [0 E g: g$ v& }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。2 f4 u& Q! m# q2 u) ^: e. W
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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。
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- 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 ⋯. {; w8 F* d& Y; G' c) p* d6 N- A
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5 _- Z3 \& u f( ?7 W* n0 l9 r: j5 b4 b, i( ^
) U$ Q5 A! @- k% G8 d' Q- m
结果如下：, p+ o3 _# `4 o) m8 w' d0 ]
6 {* Q! W4 C$ ]9 FY =
& j, q; x% B0 j, H9 A
8 [% |8 s( ]% U; M/ s( U 1.0000 2.0000 3.0000 4.0000 4.0000 3.0000 2.0000
4 P. s; g$ O' E8 N" h! h# Y2 o
- `( ?- b1 a+ t cX =
- E( ]/ R' U9 j' ~) T% U; m" R q+ }" V$ ^6 a
1 至 4 列
, M+ s3 N5 Z9 @' t$ q& m. V& Y+ ?
+ B1 o9 D! Y7 M( `) x X 2.7143 + 0.0000i -0.7213 + 0.0000i -0.0440 - 0.0000i -0.0919 + 0.0000i! a( `' o" @) t# Z J- O" r
/ a1 q0 Y8 D/ f5 S. @ J7 L( v
5 至 7 列4 H) o2 N, j4 M9 i
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-0.0919 - 0.0000i -0.0440 + 0.0000i -0.7213 - 0.0000i6 S- x, M0 O/ P3 k2 r1 v, x1 F
2 j/ A+ \" n4 b0 ~% g% C* gXsym =
1 n) v/ l2 q* V+ e: P, n& I# s
0 W) B5 ^/ I& b 2.7143 -0.7213 -0.0440 -0.0919 -0.0919 -0.0440 -0.7213
4 \0 _7 [9 }: {. ]6 K
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发表于 2019-12-11 09:00
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