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6 S" I, d7 b: j# o) A g8 W案例分析 Inverse Transform of Vector Padded Inverse Transform of Matrix Conjugate Symmetric Vector
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$ u8 x% P# c; N, d) f案例分析
1 h4 o& ], f) y0 G, P c6 [Inverse Transform of Vector8 {: D, p0 W# f: z7 m0 |
0 d8 o+ f$ I9 v- % 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, H) M, _2 k6 R2 v- W/ I; J
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结果如下：7 |7 m; E+ \' @5 D: _; t+ |7 f2 g% C
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ifft_vector% b$ ~, P P/ w* m7 }$ C9 b. Q/ G
8 `% A4 P8 |$ \! {% _Y =3 g% G: f/ T& ^' S
/ `# }. H$ n& i 1 至 4 列
v7 @; n0 X2 Z9 E8 X, B" v% U) F: Z7 N1 Y
15.0000 + 0.0000i -2.5000 + 3.4410i -2.5000 + 0.8123i -2.5000 - 0.8123i; c6 E4 c+ A3 d
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5 列- N0 T! i# O5 D/ a# `# U- R
L5 }' S! \: W8 Z2 d
-2.5000 - 3.4410i5 X& p }. b, Q( o, ^! u3 l
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ans =
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- I$ @* |1 E8 _- g 1 2 3 4 5
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0 Z7 Z# k; B: ^3 wPadded Inverse Transform of Matrix
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- 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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结果如下：
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Y =
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0.8147 0.9134 0.2785 0.9649 0.9572; G' |3 p3 p. r! y: ~& d W* z8 U- h
0.9058 0.6324 0.5469 0.1576 0.4854) q0 m8 d3 V2 O8 w
0.1270 0.0975 0.9575 0.9706 0.8003
" D4 }, U6 a1 q8 P7 Z; m+ z" l
7 ^- b" @- q/ P) s4 N/ m0 f& cX =
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1 至 4 列
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0.4911 + 0.0000i -0.0224 + 0.2008i 0.1867 - 0.0064i -0.0133 + 0.1312i7 {% E3 F: A* w% M$ j: K& |
0.3410 + 0.0000i 0.0945 + 0.1382i 0.1055 + 0.0593i 0.0106 + 0.0015i
) U" }( Z9 Y* z1 B' t 0.3691 + 0.0000i -0.1613 + 0.2141i -0.0038 - 0.1091i -0.0070 - 0.0253i0 h+ S3 l1 W+ m! w/ e' R
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5 至 8 列
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0.0215 + 0.0000i -0.0133 - 0.1312i 0.1867 + 0.0064i -0.0224 - 0.2008i+ f8 e. v) ^+ A- k. e: h) k
0.1435 + 0.0000i 0.0106 - 0.0015i 0.1055 - 0.0593i 0.0945 - 0.1382i9 h1 t! z0 s8 K' _
0.1021 + 0.0000i -0.0070 + 0.0253i -0.0038 + 0.1091i -0.1613 - 0.2141i. U3 F2 @! O6 H7 H
/ Z) u- J, a* f$ vans =
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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：5 T. q5 n# v6 V: U. s2 [: R- _) _
$ L& J/ I% b& D4 ]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：
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- 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) S* Z4 c) a- E& m" N
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# E7 E! K, X2 h. iY =& z" L4 q: I) n& n& _/ Y
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0.1419 0.7922 0.0357 0.6787 0.3922
, L/ i3 _/ {* ]7 p5 S 0.4218 0.9595 0.8491 0.7577 0.6555
: L# ]" j1 A ]$ o 0.9157 0.6557 0.9340 0.7431 0.1712& w# S5 Q7 X& k+ U) p
V$ N8 Q6 f5 n( ]& y
X =
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1 至 4 列 V9 L; G1 k* E# Q+ S: d
9 A" C; T7 g [* } 0.1849 + 0.0000i 0.3009 + 0.0000i 0.2274 + 0.0000i 0.2725 + 0.0000i
7 s3 F5 b% B _* F$ Z 0.0550 + 0.1517i 0.1838 + 0.1668i 0.0795 + 0.1918i 0.1518 + 0.1599i
2 D; ^$ z' V; g, V( S e -0.0967 + 0.0527i 0.0171 + 0.1199i -0.1123 + 0.1061i -0.0080 + 0.0947i- ~- o8 Z. R% X. c# ]+ x% r
-0.0195 - 0.0772i 0.0142 + 0.0028i -0.0706 - 0.0417i 0.0179 - 0.0259i
% `3 M# k, h( M1 L0 t, h 0.0795 + 0.0000i 0.0611 + 0.0000i 0.0151 + 0.0000i 0.0830 + 0.0000i
: x' Z' |2 ^1 J -0.0195 + 0.0772i 0.0142 - 0.0028i -0.0706 + 0.0417i 0.0179 + 0.0259i% u5 \2 E4 I4 U' N8 Y
-0.0967 - 0.0527i 0.0171 - 0.1199i -0.1123 - 0.1061i -0.0080 - 0.0947i
( G4 W; Y) ]7 Q) j* R 0.0550 - 0.1517i 0.1838 - 0.1668i 0.0795 - 0.1918i 0.1518 - 0.1599i2 P0 |) t. R; g$ \$ D5 _, A! Q
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5 列
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% M3 T9 V. O2 a& ] 0.1524 + 0.0000i
+ E8 r, [, p& @# r! t6 c+ }; l 0.1070 + 0.0793i
+ }, ^' l# \' R5 l$ H+ h 0.0276 + 0.0819i2 P; Z+ r+ R. t% t2 s
-0.0089 + 0.0365i
/ _, V) u1 @. y! Q3 a. y7 r, Y -0.0115 + 0.0000i
; t. ]: I9 v8 o' v( R$ b -0.0089 - 0.0365i
+ ]1 d8 }6 F. q( g+ N# A) ?/ Z 0.0276 - 0.0819i6 |; `; V* ]* K" d7 |0 R
0.1070 - 0.0793i+ F1 b: f' m$ ]& V% P
( [% i& [$ H) _+ [ans =
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Conjugate Symmetric Vector
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For nearly conjugate symmetric vectors, you can compute the inverse Fourier transform faster by specifying the 'symmetric' option,
0 k& e7 E F3 ^6 `0 ]' jwhich also ensures that the output is real. Nearly conjugate symmetric data can arise when computations introduce round-off error.
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& Q% m& A" j# s6 m+ PCreate a vector Y that is nearly conjugate symmetric and compute its inverse Fourier transform.
! r% T% x' W/ Q6 H! _/ TThen, compute the inverse transform specifying the 'symmetric' option, which eliminates the nearly 0 imaginary parts.
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0 r# s) B* H' Z6 s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。 p3 ~# K9 n- N
$ K( `7 ]( O d- G" F2 Y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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P: ]. W/ X C- 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 ⋯% {% f b+ x2 F' `7 H1 S$ {2 N
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结果如下：
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5 L7 }# ?4 v4 ?0 HY =
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* Q- b9 t9 J6 j, o! ?% u1 w$ g 1.0000 2.0000 3.0000 4.0000 4.0000 3.0000 2.0000
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X =! n& H6 A& T$ M/ l) m" N
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1 至 4 列
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2.7143 + 0.0000i -0.7213 + 0.0000i -0.0440 - 0.0000i -0.0919 + 0.0000i
" `/ @( m4 s7 M
- A0 |7 n6 [8 e( k! {3 ?- Z% T 5 至 7 列
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-0.0919 - 0.0000i -0.0440 + 0.0000i -0.7213 - 0.0000i. ~: I9 W$ z4 l% v! i
% V. f8 w% T6 N# C0 p" f$ MXsym =
# |# f1 H' W: g$ ^. l
) C# e, ~/ J1 i 2.7143 -0.7213 -0.0440 -0.0919 -0.0919 -0.0440 -0.7213* w9 U' M4 j3 j8 q( ^, K/ x' o3 A* V
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发表于 2019-12-11 09:00
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