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6 N3 A6 _! ~& q7 l3 q- x/ H案例分析 Inverse Transform of Vector Padded Inverse Transform of Matrix Conjugate Symmetric Vector " t; R. q4 n5 |7 F3 t1 r
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案例分析
0 F6 J$ @. t0 \+ V P) [8 xInverse Transform of Vector$ I" A' n- r' \% }* W
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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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1 k K% R! @% G7 E6 ]9 A结果如下：
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ifft_vector
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Y =
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$ h0 l6 `. f& M- e2 [3 S 1 至 4 列
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15.0000 + 0.0000i -2.5000 + 3.4410i -2.5000 + 0.8123i -2.5000 - 0.8123i8 O H5 \" _" b6 ~
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5 列
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-2.5000 - 3.4410i
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: P& T) u3 q( s: J0 b% Vans =- D9 k+ |, ^1 m5 U. w
X% M; C, z3 Z
1 2 3 4 5
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0 p5 w+ P5 ^+ x3 n7 G- P4 _. V6 b4 U$ m6 ^2 F, C" Z( H( f
Padded Inverse Transform of Matrix
- Q8 H, c% v* |+ j# Z- 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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5 d7 w" ^5 D( r' G& i2 q+ c! ?结果如下：
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Y =
+ i9 ]1 R$ i3 U3 }& L1 B
^: {+ T7 T2 W$ _! v8 h 0.8147 0.9134 0.2785 0.9649 0.9572
# y2 f d* _; S0 ^' B 0.9058 0.6324 0.5469 0.1576 0.4854
8 g+ F4 E, ?* B O+ ^ E; }9 m 0.1270 0.0975 0.9575 0.9706 0.8003" W$ P" M9 c" N5 `" P6 x3 h
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X =
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1 至 4 列- _* W" Y/ R ~" q* ]: x
* }& u! f7 e \# w7 m 0.4911 + 0.0000i -0.0224 + 0.2008i 0.1867 - 0.0064i -0.0133 + 0.1312i( C( Q* t3 ?0 x/ {
0.3410 + 0.0000i 0.0945 + 0.1382i 0.1055 + 0.0593i 0.0106 + 0.0015i1 H( Z/ ? ]1 G9 A7 ^2 Q
0.3691 + 0.0000i -0.1613 + 0.2141i -0.0038 - 0.1091i -0.0070 - 0.0253i4 c, ]- q: U/ X6 r( F) Q0 Q. ?: T2 m
% y$ q4 o) F% I A- U- U 5 至 8 列, X: {0 s2 C2 H; d. I t& f
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0.0215 + 0.0000i -0.0133 - 0.1312i 0.1867 + 0.0064i -0.0224 - 0.2008i& R. g8 e( {7 F: Z
0.1435 + 0.0000i 0.0106 - 0.0015i 0.1055 - 0.0593i 0.0945 - 0.1382i+ T: ? G; m" Y8 H
0.1021 + 0.0000i -0.0070 + 0.0253i -0.0038 + 0.1091i -0.1613 - 0.2141i
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ans =/ e+ J% T6 g- T
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; f: F- g6 X0 H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：8 i7 j! Z( n% n; [" `6 |# V
" c" V; c, b2 S! q9 |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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5 m+ \2 @8 g0 \- 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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0.1419 0.7922 0.0357 0.6787 0.3922
9 L! M2 T) k! i7 R5 G& d 0.4218 0.9595 0.8491 0.7577 0.6555
5 N# I5 _ p: s8 i+ e 0.9157 0.6557 0.9340 0.7431 0.17129 u$ ]6 o+ D$ d8 w% E
6 w3 ] e/ Q( \) yX =
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1 至 4 列$ t- j9 t7 J& Q" \: g! Z
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0.1849 + 0.0000i 0.3009 + 0.0000i 0.2274 + 0.0000i 0.2725 + 0.0000i" G: p5 B9 L W# l* y* ]4 w
0.0550 + 0.1517i 0.1838 + 0.1668i 0.0795 + 0.1918i 0.1518 + 0.1599i6 g6 g6 j G i3 `1 j/ I* r7 z
-0.0967 + 0.0527i 0.0171 + 0.1199i -0.1123 + 0.1061i -0.0080 + 0.0947i
' y) R- b% X* k% m8 c, U5 |4 ]" n -0.0195 - 0.0772i 0.0142 + 0.0028i -0.0706 - 0.0417i 0.0179 - 0.0259i) \5 o4 D$ L9 M4 O( r3 q: A
0.0795 + 0.0000i 0.0611 + 0.0000i 0.0151 + 0.0000i 0.0830 + 0.0000i7 \( j% R: X* d r/ w; Q! r8 I
-0.0195 + 0.0772i 0.0142 - 0.0028i -0.0706 + 0.0417i 0.0179 + 0.0259i
2 ?+ G3 q" W. w -0.0967 - 0.0527i 0.0171 - 0.1199i -0.1123 - 0.1061i -0.0080 - 0.0947i/ z: C% v+ j6 E2 b; q& d" R7 \$ Z1 Q
0.0550 - 0.1517i 0.1838 - 0.1668i 0.0795 - 0.1918i 0.1518 - 0.1599i2 C+ M) O7 z) v( w
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0.1524 + 0.0000i- d8 L8 Z/ t" ^. G- m! B% Q" u4 L
0.1070 + 0.0793i
) D% d. C: @4 z 0.0276 + 0.0819i3 k$ p5 }* f7 i+ r1 n
-0.0089 + 0.0365i
/ b3 o( q1 I, f -0.0115 + 0.0000i
- l, ]. j/ W8 `( S" r' X8 z -0.0089 - 0.0365i% N; A( p5 b* m; p, }' U$ I
0.0276 - 0.0819i
" _$ ^6 j+ N3 \6 ?- y 0.1070 - 0.0793i+ C4 \! K7 }/ p5 r& W% H& m
( Z, t( t$ a( w9 y- jans =
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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 T! x1 b9 W& c% y5 ^/ kwhich also ensures that the output is real. Nearly conjugate symmetric data can arise when computations introduce round-off error.; X- n6 t3 n3 r. M7 R1 e9 [ c
- [" w5 j: d( U/ C- H' f: q$ V Y3 zCreate a vector Y that is nearly conjugate symmetric and compute its inverse Fourier transform.
4 j; E1 f& _4 uThen, compute the inverse transform specifying the 'symmetric' option, which eliminates the nearly 0 imaginary parts.7 V$ D! j, m/ f! L$ E
5 s' c( N8 z( R& T' j! E6 p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。
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% L" d7 I8 `3 c( x N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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1 s; F c E4 M; `1 P! i; y- 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 ⋯4 E0 q# |5 |& Y: Z/ U1 z
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" i, Q1 N# I" c6 ]结果如下：
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Y =7 Y; G$ i' V( n1 O4 O% ?7 L
& f" x% p7 ~) D4 c% j 1.0000 2.0000 3.0000 4.0000 4.0000 3.0000 2.0000
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X =
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& K, R7 q& C. s0 s1 b) e+ o 1 至 4 列& G/ ]+ _ L! l0 }0 J& |
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2.7143 + 0.0000i -0.7213 + 0.0000i -0.0440 - 0.0000i -0.0919 + 0.0000i
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5 至 7 列: E7 X. A. d* c9 i7 y2 }
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-0.0919 - 0.0000i -0.0440 + 0.0000i -0.7213 - 0.0000i
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9 {! {: b, m1 D8 @6 ?Xsym =1 E Y! Q* u) j5 }7 \: }* N
) h$ R; r! J* M: F 2.7143 -0.7213 -0.0440 -0.0919 -0.0919 -0.0440 -0.72138 R( _2 w! k0 v5 r' t: X; p
5 u5 y8 F8 Z1 u& C+ M* x
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发表于 2019-12-11 09:00
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