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! ~6 i( N" _4 a4 [- O案例分析 Inverse Transform of Vector Padded Inverse Transform of Matrix Conjugate Symmetric Vector 3 ?, Q# M* i1 o. `9 @1 t. _& Y2 U5 C9 J0 v
1 q2 T2 o( Z) ]0 S! ?3 S9 o" `案例分析; i; h. x0 S( l" V/ V
Inverse Transform of Vector
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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：
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ifft_vector
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0 Z# j: s( e$ W, X! nY =
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1 至 4 列 U& s; F0 q( c! Q: A
% Y% e* D0 P( u9 d. X 15.0000 + 0.0000i -2.5000 + 3.4410i -2.5000 + 0.8123i -2.5000 - 0.8123i
0 ^& J+ c+ v8 k6 q6 c
$ T. l! D2 x* o& T( p 5 列
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-2.5000 - 3.4410i
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% v# W" j6 L( S' U+ u# C: `" aans =
: U* ^# t d& n8 ]; K# e5 `% H5 t( N: M9 s) ]* {# D9 o+ q/ u
1 2 3 4 55 m: }5 r; Q4 C& u, E( F* T' m
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3 O; U, Q9 I/ T& L/ a$ M9 \9 y4 J1 RPadded Inverse Transform of Matrix! x e ]' v' @7 ] @
- 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)8 Y9 b1 l0 F- h
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结果如下：% v+ T+ [* g& S6 Y, J1 ^) P) i
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Y =$ e, a! i" u0 X, ^* q
# q0 J0 C! L! M 0.8147 0.9134 0.2785 0.9649 0.9572
/ C# r1 Y' k5 T 0.9058 0.6324 0.5469 0.1576 0.4854
1 ^- F2 E; l4 N 0.1270 0.0975 0.9575 0.9706 0.80031 B7 M/ M2 V8 k/ ?
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X =
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0 B' T7 N7 \# ] 1 至 4 列
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( @- ~, `3 I7 t$ L 0.4911 + 0.0000i -0.0224 + 0.2008i 0.1867 - 0.0064i -0.0133 + 0.1312i
5 n! {) R, x2 r. W4 p 0.3410 + 0.0000i 0.0945 + 0.1382i 0.1055 + 0.0593i 0.0106 + 0.0015i; x6 l$ o' ~; N( E% U
0.3691 + 0.0000i -0.1613 + 0.2141i -0.0038 - 0.1091i -0.0070 - 0.0253i! T; y5 ?7 J/ S! ~; {4 m8 Y8 z2 u
6 {, s4 {- S" I3 G8 N 5 至 8 列
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0.0215 + 0.0000i -0.0133 - 0.1312i 0.1867 + 0.0064i -0.0224 - 0.2008i) H% e2 g# u3 t
0.1435 + 0.0000i 0.0106 - 0.0015i 0.1055 - 0.0593i 0.0945 - 0.1382i
7 L( k5 o( K$ A. |/ l/ \) D6 ? 0.1021 + 0.0000i -0.0070 + 0.0253i -0.0038 + 0.1091i -0.1613 - 0.2141i& O) U5 A2 Y) U& r5 c! m
w$ X% R+ l6 V9 y8 v+ N, Z) U. [5 Uans =
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: Q2 o7 Q M* I8 [8 @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：4 g! h* o6 D9 J, }: r8 y
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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：- ~: K$ N+ n9 D
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- 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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Y =
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0.1419 0.7922 0.0357 0.6787 0.3922
, ?$ C1 [6 |0 W# j \- Q' U' F 0.4218 0.9595 0.8491 0.7577 0.6555
; s* M6 [; f& \" I5 u8 ~ 0.9157 0.6557 0.9340 0.7431 0.1712
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X =0 \. A s6 p$ K l7 G5 X( R
, n/ {5 `0 q0 p$ T$ S0 A 1 至 4 列
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; v* ~6 G0 q' s/ p( g 0.1849 + 0.0000i 0.3009 + 0.0000i 0.2274 + 0.0000i 0.2725 + 0.0000i
% ^% m9 K! n: Y. U* x; _ 0.0550 + 0.1517i 0.1838 + 0.1668i 0.0795 + 0.1918i 0.1518 + 0.1599i
3 N7 |5 d8 J" y -0.0967 + 0.0527i 0.0171 + 0.1199i -0.1123 + 0.1061i -0.0080 + 0.0947i1 T. q& f. P1 Z# E
-0.0195 - 0.0772i 0.0142 + 0.0028i -0.0706 - 0.0417i 0.0179 - 0.0259i
1 n1 g& u2 P$ k+ Z 0.0795 + 0.0000i 0.0611 + 0.0000i 0.0151 + 0.0000i 0.0830 + 0.0000i
6 @9 C0 F8 J" r8 b# X0 p -0.0195 + 0.0772i 0.0142 - 0.0028i -0.0706 + 0.0417i 0.0179 + 0.0259i
" D- I0 N g; g -0.0967 - 0.0527i 0.0171 - 0.1199i -0.1123 - 0.1061i -0.0080 - 0.0947i
- t* a, v4 {1 m% w 0.0550 - 0.1517i 0.1838 - 0.1668i 0.0795 - 0.1918i 0.1518 - 0.1599i k; Q- r) J, ~: S: Y2 J7 j) ]4 x
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0.1524 + 0.0000i
. A9 R7 ^1 C5 `& ~ 0.1070 + 0.0793i3 R0 o8 p1 ~% T9 ~, \ N
0.0276 + 0.0819i
6 n3 g B( W& R& i -0.0089 + 0.0365i( l' f7 s& H) k( G
-0.0115 + 0.0000i
: L$ d' n5 V% j3 r+ h, `7 z -0.0089 - 0.0365i
. e' I4 r1 V! n$ X# b3 t' M 0.0276 - 0.0819i; E3 R* j% C( S2 I/ I3 n! r- O* E* ?' @
0.1070 - 0.0793i+ L, W: T3 D! {% W& S
: P. K# M+ I( ^; d% mans =
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/ |. V; s& p# }% lConjugate Symmetric Vector
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For nearly conjugate symmetric vectors, you can compute the inverse Fourier transform faster by specifying the 'symmetric' option, 1 w! P/ L9 d+ V2 l2 |( ^' [* B! b
which also ensures that the output is real. Nearly conjugate symmetric data can arise when computations introduce round-off error.8 v' d9 B% \* }/ {% M4 N# |
4 R( j4 T# }* K9 { t. G j' e* m# XCreate a vector Y that is nearly conjugate symmetric and compute its inverse Fourier transform. ' g4 v; J2 X1 O) _4 L
Then, compute the inverse transform specifying the 'symmetric' option, which eliminates the nearly 0 imaginary parts.
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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。1 F2 m" ^+ j* ~& D6 |
) w$ w& m, Y9 [+ Q) ` Z8 p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。- K" k- f1 J9 o
W( @& ?; x& C* L1 v# ]0 s. z# K
- 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 ⋯
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结果如下：% j- O6 Y+ D6 ?+ ^ I9 p
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Y =1 d8 I$ }5 A* h( |3 `+ ]* r$ D3 ]/ n2 [
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1.0000 2.0000 3.0000 4.0000 4.0000 3.0000 2.0000
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. A* R+ X! @( f, ~X =6 N+ K% t1 A O2 |& e v4 N
, x: z& _# r' R& w* Y# g$ m 1 至 4 列9 a/ K. m5 C9 z; b, a0 O
, t Y0 K, j2 Y
2.7143 + 0.0000i -0.7213 + 0.0000i -0.0440 - 0.0000i -0.0919 + 0.0000i0 H- n# K1 T* q! z$ v4 `
0 i3 c9 W/ {0 {- \ 5 至 7 列
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-0.0919 - 0.0000i -0.0440 + 0.0000i -0.7213 - 0.0000i8 E( c) p# s- Z0 Q- c( s: n% n
7 Q$ I( o3 }: NXsym =( U E3 H; Q, U# z, M8 G; _+ q4 Q
* y5 B4 B. n% Q; V2 b+ a 2.7143 -0.7213 -0.0440 -0.0919 -0.0919 -0.0440 -0.7213. [8 D( {) U$ _% Z8 v, d
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发表于 2019-12-11 09:00
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