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4 f* M) {) _: z% ~- }5 R案例分析 Inverse Transform of Vector Padded Inverse Transform of Matrix Conjugate Symmetric Vector ~; Y e$ \( Y
$ v( ~$ Q2 E( E: ~! o1 n4 `案例分析' l5 b) p" v# X( K) ?* u0 y% E
Inverse Transform of Vector4 P5 q& ^+ V. D h
& W; r1 z. F {- % 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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5 g q' T2 S" A- W1 d6 Q' |结果如下：# t8 x( U6 y. j" q+ e4 c% E
7 r- M9 ^& ?9 C0 p/ q6 x1 V- G ifft_vector
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: N, K% x! G, [/ C2 ]Y =! N1 n& n. K# V& n0 q( {- ?
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1 至 4 列
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15.0000 + 0.0000i -2.5000 + 3.4410i -2.5000 + 0.8123i -2.5000 - 0.8123i% n" l0 N- _1 D$ F8 _
0 a3 w0 E9 o& `7 C! E
5 列
7 B5 y$ t* z) t2 Z% i( _' \, V
8 e1 ?9 \$ d2 j2 X# n7 C. F- q -2.5000 - 3.4410i6 J7 K! Q5 e" G6 |" Q
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ans =) }/ k1 T$ u* P, o
' a( x( P1 b9 Z& z) @
1 2 3 4 5$ ~+ B3 G0 d5 S+ j c
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6 M2 y( A/ q" e0 Y0 t
Padded 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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& _% v) A- ?6 u% l) M( s9 A结果如下：+ D5 A8 a/ j* N
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Y =. c1 d& k( n& [8 W$ D
. y& @; \: y' m0 q! b9 H1 \6 V- A 0.8147 0.9134 0.2785 0.9649 0.9572
, J* Q U' W; z: z 0.9058 0.6324 0.5469 0.1576 0.4854
8 l% I) Q/ L- R/ K2 y 0.1270 0.0975 0.9575 0.9706 0.8003
- M4 {# k, o* {* R4 A0 x2 m
) o, E D3 b G9 \X =
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1 至 4 列& } i" b$ x: d/ U
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0.4911 + 0.0000i -0.0224 + 0.2008i 0.1867 - 0.0064i -0.0133 + 0.1312i
6 b& t. M( L8 Z) H/ f) i( e7 w 0.3410 + 0.0000i 0.0945 + 0.1382i 0.1055 + 0.0593i 0.0106 + 0.0015i2 v+ c [+ s/ }9 p% B# l0 G
0.3691 + 0.0000i -0.1613 + 0.2141i -0.0038 - 0.1091i -0.0070 - 0.0253i' Z) j! b! h |& Q/ M5 f/ q
: m+ m5 x) ?" V
5 至 8 列
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" B8 H/ h. H; L, M* u" R# t$ W 0.0215 + 0.0000i -0.0133 - 0.1312i 0.1867 + 0.0064i -0.0224 - 0.2008i% J F' a" B. K. j N) e9 Q! a
0.1435 + 0.0000i 0.0106 - 0.0015i 0.1055 - 0.0593i 0.0945 - 0.1382i a0 @6 z/ w% U! C
0.1021 + 0.0000i -0.0070 + 0.0253i -0.0038 + 0.1091i -0.1613 - 0.2141i$ W( V! G& Q+ X/ b# H0 i. U0 _
+ ~- V5 v4 J! A7 M* c
ans =
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3 8
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* b$ A1 X3 f: c# _2 ]' {6 m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( I0 L; Z9 u
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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：" H0 A M6 p7 a
* l3 x4 G- P; D' M- 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)
. X: q( q& `) a2 N& _% G0 S
1 P2 `+ J: Z8 C* {; v8 D u
4 K; L/ F( ~3 R$ i% x: aY =
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0.1419 0.7922 0.0357 0.6787 0.3922
$ x& t% I6 R6 @$ E0 U" z 0.4218 0.9595 0.8491 0.7577 0.6555
# N' V) b. \0 a2 C, \ 0.9157 0.6557 0.9340 0.7431 0.1712
1 R! U# Z; s [2 ^
6 M! D& K% L- L* Q8 ~X =
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1 至 4 列
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: D8 f- X1 T0 e+ }+ w+ K% \! h 0.1849 + 0.0000i 0.3009 + 0.0000i 0.2274 + 0.0000i 0.2725 + 0.0000i
; U8 S/ T' X8 \6 q 0.0550 + 0.1517i 0.1838 + 0.1668i 0.0795 + 0.1918i 0.1518 + 0.1599i
. ~8 _& g" p5 @" F( O -0.0967 + 0.0527i 0.0171 + 0.1199i -0.1123 + 0.1061i -0.0080 + 0.0947i5 O( N2 S. X) \2 r% ?% W# M" d
-0.0195 - 0.0772i 0.0142 + 0.0028i -0.0706 - 0.0417i 0.0179 - 0.0259i
2 Z0 B. ^8 a- F% V1 P 0.0795 + 0.0000i 0.0611 + 0.0000i 0.0151 + 0.0000i 0.0830 + 0.0000i6 O1 B& x' A% C2 j8 g. }
-0.0195 + 0.0772i 0.0142 - 0.0028i -0.0706 + 0.0417i 0.0179 + 0.0259i
: }9 ?( }4 G; N5 ?7 Y -0.0967 - 0.0527i 0.0171 - 0.1199i -0.1123 - 0.1061i -0.0080 - 0.0947i
/ a+ e6 H I0 k1 e 0.0550 - 0.1517i 0.1838 - 0.1668i 0.0795 - 0.1918i 0.1518 - 0.1599i9 {# C! k$ M# k% u% e4 K
1 I6 K; Y+ `; { 5 列9 d- l/ i9 l* K5 y) x; p; ~
% j9 _8 B! d( I# R- A7 M6 `9 L 0.1524 + 0.0000i+ y; H/ v, a" Y: G. n+ G
0.1070 + 0.0793i
4 y1 N! w: J# y4 u9 Y; @ 0.0276 + 0.0819i
2 O2 a. J, w$ s" p- A( b& C -0.0089 + 0.0365i: X0 d+ v+ Z8 T6 y+ q
-0.0115 + 0.0000i9 }' r) Q! ?$ J# s- n
-0.0089 - 0.0365i$ g# M( _( A% @! W6 z) X! e+ i
0.0276 - 0.0819i
6 Q" X% W& n8 I7 [+ Z* ~; s 0.1070 - 0.0793i
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; F* M2 X# n- O o: Sans =
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0 h* ]$ n. B8 J" g6 ?- K4 a. t# LConjugate Symmetric Vector! L: \1 U- o- E5 @
, i0 Q6 Q" J! a# \, _For nearly conjugate symmetric vectors, you can compute the inverse Fourier transform faster by specifying the 'symmetric' option,
( x8 x( X$ H/ M# `- xwhich also ensures that the output is real. Nearly conjugate symmetric data can arise when computations introduce round-off error.+ Z* ^9 s1 Z! v- c
y$ W- E7 V6 J3 e! PCreate a vector Y that is nearly conjugate symmetric and compute its inverse Fourier transform. 3 W+ u/ }: n- A% _0 u3 a- z
Then, compute the inverse transform specifying the 'symmetric' option, which eliminates the nearly 0 imaginary parts.! B; W- s4 |& N2 `- N
b& o( I( d& i+ D+ g' V7 a0 }0 X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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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。
j& Y% n( F6 ^" n$ t: J* O
/ U3 {$ ?$ Q+ i- 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 ⋯9 G5 i/ J% n p) Q! M. V
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) W6 O3 g3 h" Q/ b+ _
7 x6 B- S; ~0 b2 d
, ?4 p. E" y+ _- B. Q7 N. o结果如下：
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' A5 Y. @9 c, t4 @4 j 1.0000 2.0000 3.0000 4.0000 4.0000 3.0000 2.0000
9 \+ u6 E0 i9 x4 E: H: f1 b8 r
: D3 \+ Y. N4 v' ~3 HX =% a% L8 r* n" R$ u& N
* d$ ?, g# C& c, }" n 1 至 4 列 K( {9 e3 w' o
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2.7143 + 0.0000i -0.7213 + 0.0000i -0.0440 - 0.0000i -0.0919 + 0.0000i
4 J l3 k3 m( N' z/ |3 @& Y6 p* l* `! y7 H+ H
5 至 7 列
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0 i. d9 }$ a% V' X -0.0919 - 0.0000i -0.0440 + 0.0000i -0.7213 - 0.0000i
9 S X! s g) c3 B
" k5 @+ }. K1 e2 K: L% b. @6 C8 r0 `Xsym =& D5 l9 e( J: H6 g, H0 _! ^
; z4 x* s, i( N" i4 n: \1 J 2.7143 -0.7213 -0.0440 -0.0919 -0.0919 -0.0440 -0.72135 f5 S+ v/ B: Y! y! b
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发表于 2019-12-11 09:00
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