ω是如何计算的呢？
为了计算离散傅里叶变换，如当N=8时，我们令：
ω = e^(-2πi/N )= e^(-2πi/8) = e^(-πi/4) = cos(-π/4) + i sin(-π/4) = √2 / 2 - i √2 / 2 ≈ 0.7-0.7i
而ω^0~ω^8都以此为基础，以此类推，如ω^0=e^0=1,
ω^1=e^(-πi/4) = cos(-π/4) + i sin(-π/4) = √2 / 2 - i √2 / 2 ≈ 0.7-0.7i,
ω^2=e^(-πi/2)=0+i*(-1)=-i,
……。

[此贴子已经被作者于2020-10-9 23:22编辑过]

总计进行了两次快速傅里叶变换（被乘数和乘数各一次），得到：
{ A7, A6, A5, A4, A3, A2, A1, A0 } = { 12.9+10.9i, 2+7i, 3.1-1.1i, 7, 3.1+1.1i, 2-7i, 12.9-10.9i, 21 }
{ B7, B6, B5, B4, B3, B2, B1, B0 } = { 4.1+6.1i, -2+3i, -0.1-1.9i, 3, -0.1+1.9i, -2-3i, 4.1-6.1i, 9 }

一次对应项相乘，得到向量 {Ck}，即 { C7, C6, C5, C4, C3, C2, C1, C0 }
= { -13.6+123.4i, -25-8i, -2.4-5.8i, 21, -2.4+5.8i, -25+8i, -13.6-123.4i, 189 }

一次逆变换计算出了向量 {ai} 和向量 {bj} 的卷积向量 {ck}，如下所示：
{ c7, c6, c5, c4, c3, c2, c1, c0 } = { 0, 0, 0, 0, 24, 46, 65, 38, 16 }

最后错位相加得到结果。

例如如下向量：
{ A7, A6, A5, A4, A3, A2, A1, A0 } = { 12.9+10.9i, 2+7i, 3.1-1.1i, 7, 3.1+1.1i, 2-7i, 12.9-10.9i, 21 }
显然以7为对称中心，对称项都是共轭复数，只有7和21是不对称的，如何快速得到？
先得到7，还是先得到 3.1+1.1i？然后由对称性和其它算法（而不再进行乘法）推演出来其他项？

改造后的蝶形运算程序和结果：
Private Sub Command1_Click()
Dim xr() As Double, a As String
a = Trim(Text1)
ReDim xr(0 To Len(a) - 1)
For i1 = 0 To Len(a) - 1
xr(i1) = Mid(a, i1 + 1, 1)
  Next
Dim l As Long, le As Long, le1 As Long, n As Long, r As Long, p As Long, q As Long, m As Byte
Dim wr As Double, w1 As Double, wlr As Double, wl1 As Double, tr As Double, t1 As Double
Dim pi As Double, t As Double
Dim xi()
n = Len(a) '求数组大小，其值必须是2的幂
m = 0
l = 2
pi = 3.14159265358979
Do
 l = l + l
 m = m + 1
 Loop Until l > n
 n = l / 2
ReDim xi(n - 1)

l = 1
Do
  le = 2 ^ l
  le1 = le / 2
  wr = 1
  wi = 0
  w1r = Cos(t)
  w1i = -Sin(t)
  r = 0
Do
  p = r
  Do
   q = p + le1
   
   tr = xr(q) * wr - xi(q) * wi
   ti = xr(q) * wi + xi(q) * wr
   
   xr(q) = xr(p) - tr
   xi(q) = xi(p) - ti
   xr(p) = xr(p) + tr
   xi(p) = xi(p) + ti
   
   p = p + le
Loop Until p > n - 1

wr = wr * w1r - wi * w1i
wi = wr * w1i - wi * w1r
r = r + 1
Loop Until r > le1 - 1
l = l + 1
Loop Until l > m

For i = 0 To n - 1 '仅输出模
   xr(i) = Sqr(xr(i) ^ 2 + xi(i) ^ 2)
   Text2 = Text2 & "  " & xr(i)
   Next
End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
End Sub
运算结果：
Text1=00000678,Text2=21  7  9  5  21  7  9  5,
Text1=00000432,Text2=  9  3  1  5  9  3  1  5.

程序出现了错误，怪不得了呀，修改了一下，如下为代码和计算结果：
Private Sub Command1_Click()
Dim xr() As Double, a As String
a = Trim(Text1)
ReDim xr(0 To Len(a) - 1)
For i1 = 0 To Len(a) - 1
xr(i1) = Mid(a, i1 + 1, 1)
  Next
Dim l As Long, le As Long, le1 As Long, n As Long, r As Long, p As Long, q As Long, m As Byte
Dim wr As Double, w1 As Double, wlr As Double, wl1 As Double, tr As Double, t1 As Double
Dim pi As Double, t As Double
Dim xi()
n = Len(a) '求数组大小，其值必须是2的幂
m = 0
l = 2
pi = 3.14159265358979
Do
 l = l + l
 m = m + 1
 Loop Until l > n
 n = l / 2
ReDim xi(n - 1)

l = 1
Do
  le = 2 ^ l
  le1 = le / 2
  wr = 1
  wi = 0
  t = pi / le1
  w1r = Cos(t)
  w1i = -Sin(t)
  r = 0
Do
  p = r
  Do
   q = p + le1
   
   tr = xr(q) * wr - xi(q) * wi
   ti = xr(q) * wi + xi(q) * wr
   
   xr(q) = xr(p) - tr
   xi(q) = xi(p) - ti
   xr(p) = xr(p) + tr
   xi(p) = xi(p) + ti
   p = p + le
Loop Until p > n - 1

wr = wr * w1r - wi * w1i
wi = wr * w1i - wi * w1r
r = r + 1
Loop Until r > le1 - 1
l = l + 1
Loop Until l > m

For i = 0 To n - 1 '仅输出模
   xr(i) = Sqr(xr(i) ^ 2 + xi(i) ^ 2)
   Text2 = Text2 & "  " & xr(i)
   Next
End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
End Sub
计算结果：
Text1=00000678,Text2= 21  5.19615242270664  2.60761871483253  2.80919177949488  21  5.19615242270664  2.60761871483253  2.80919177949488.

改造后输出复数的蝶形运算程序代码及结果：
Private Sub Command1_Click()
Dim xr() As Double, a As String
a = Trim(Text1)
ReDim xr(0 To Len(a) - 1)
For i1 = 0 To Len(a) - 1
xr(i1) = Mid(a, i1 + 1, 1)
  Next
Dim l As Long, le As Long, le1 As Long, n As Long, r As Long, p As Long, q As Long, m As Byte
Dim wr As Double, w1 As Double, wlr As Double, wl1 As Double, tr As Double, t1 As Double
Dim pi As Double, t As Double
Dim xi()
n = Len(a) '求数组大小，其值必须是2的幂
m = 0
l = 2
pi = 3.14159265358979
Do
 l = l + l
 m = m + 1
 Loop Until l > n
 n = l / 2
ReDim xi(n - 1)

l = 1
Do
  le = 2 ^ l
  le1 = le / 2
  wr = 1
  wi = 0
  t = pi / le1
  w1r = Cos(t)
  w1i = -Sin(t)
  r = 0
Do
  p = r
  Do
   q = p + le1
   
   tr = xr(q) * wr - xi(q) * wi
   ti = xr(q) * wi + xi(q) * wr
   
   xr(q) = xr(p) - tr
   xi(q) = xi(p) - ti
   xr(p) = xr(p) + tr
   xi(p) = xi(p) + ti
   p = p + le
Loop Until p > n - 1

wr = wr * w1r - wi * w1i
wi = wr * w1i - wi * w1r
r = r + 1
Loop Until r > le1 - 1
l = l + 1
Loop Until l > m

For i = 0 To n - 1 '仅输出模
   
   Text2 = Text2 & "  " & xr(i) & "+" & xi(i) & "i"
   Next
End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
End Sub
计算结果：
Text1=00000678,Text2= 21+0i  -4.24264068711929+3i  -1.31801948466055+-2.25i  -1.68198051533947+2.25i  -21+0i  4.24264068711929+-3i  1.31801948466055+2.25i  1.68198051533947+-2.25i

跟前面的例子比较如下：
 21+0i  -4.24264068711929+3i  -1.31801948466055+-2.25i  -1.68198051533947+2.25i  -21+0i  4.24264068711929+-3i  1.31801948466055+2.25i  1.68198051533947+-2.25i

 12.9+10.9i, 2+7i, 3.1-1.1i, 7, 3.1+1.1i, 2-7i, 12.9-10.9i, 21

不一样，咋回事呢？希望老师指点！谢谢！

Text1=00000432,Text2=
9+0i  -2.82842712474619+2i  -0.146446609406727+-0.25i  -1.12132034355964+1.5i  -9+0i  2.82842712474619+-2i  0.146446609406727+0.25i  1.12132034355964+-1.5i
和前面的比较：
4.1+6.1i, -2+3i, -0.1-1.9i, 3, -0.1+1.9i, -2-3i, 4.1-6.1i, 9

不一样，显然有错误，咋回事呢？

改造后的代码及运行结果：
 Private Sub Command1_Click()
Dim xr() As Double, a As String
a = Trim(Text1)
ReDim xr(0 To Len(a) - 1)
For i1 = 0 To Len(a) - 1
xr(i1) = Mid(a, i1 + 1, 1)
  Next
Dim l As Long, le As Long, le1 As Long, n As Long, r As Long, p As Long, q As Long, m As Byte
Dim wr As Double, w1 As Double, wlr As Double, wl1 As Double, tr As Double, t1 As Double
Dim pi As Double, t As Double
Dim xi()
n = Len(a) '求数组大小，其值必须是2的幂
m = 0
l = 2
pi = 3.14159265358979
Do
 l = l + l
 m = m + 1
 Loop Until l > n
 n = l / 2
ReDim xi(n - 1)

l = 1
Do
  le = 2 ^ l
  le1 = le / 2
  wr = 1
  wi = 0
  t = pi / le1
  w1r = -Cos(t)
  w1i = Sin(t)
  r = 0
Do
  p = r
  Do
   q = p + le1
   
   tr = xr(q) * wr - xi(q) * wi
   ti = xr(q) * wi + xi(q) * wr
   
   xr(q) = xr(p) - tr
   xi(q) = xi(p) - ti
   xr(p) = xr(p) + tr
   xi(p) = xi(p) + ti
   p = p + le
Loop Until p > n - 1

wr = wr * w1r - wi * w1i
wi = wr * w1i + wi * w1r
r = r + 1
Loop Until r > le1 - 1
l = l + 1
Loop Until l > m

For i = 0 To n - 1 '仅输出模
   xr(i) = Sqr(xr(i) ^ 2 + xi(i) ^ 2)
   Text2 = Text2 & "  " & xr(i) & "+" & xi(i) & "i"
   Next
End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
End Sub

 输入：Text1 =00070608，结果：
     21+0i  5.19615242270664+3i  8.91503909477451+-1.9142135623731i  12.2115949757928+9.49264068711931i  
7+0i  5.19615242270661+-2.99999999999997i  5.62840405338267+1.9142135623731i  12.2115949757928+-9.49264068711934i
与正确值比较：12.9+10.9i, 2+7i, 3.1-1.1i, 7, 3.1+1.1i, 2-7i, 12.9-10.9i, 21也是不对。
输入：Text1 =00030402，结果：
     9+0i  3.46410161513776+2i  2.30993212835938+1.9142135623731i  8.1410633171952+6.3284271247462i  
3+0i  3.46410161513775+-1.99999999999999i  5.08144347718889+-1.9142135623731i  8.14106331719521+-6.32842712474621i
与正确值比较：4.1+6.1i, -2+3i, -0.1-1.9i, 3, -0.1+1.9i, -2-3i, 4.1-6.1i, 9也是不对。

Private Sub Command1_Click()
Dim xr() As Double, a As String
a = Trim(Text1)
b = Trim(Text3)
ReDim xr(0 To Len(a) - 1): ReDim yr(0 To Len(b) - 1): ReDim zr(0 To Len(b) - 1)
For i1 = 0 To Len(a) - 1
xr(i1) = Mid(a, i1 + 1, 1)
yr(i1) = Mid(b, i1 + 1, 1)

  Next
Dim l As Long, le As Long, le1 As Long, n As Long, r As Long, p As Long, q As Long, m As Byte
Dim wr As Double, w1 As Double, wlr As Double, wl1 As Double, tr As Double, t1 As Double
Dim pi As Double, t As Double, tr1 As Double
Dim xi(): Dim yi(): Dim zi()
n = Len(a) '求数组大小，其值必须是2的幂
m = 0
l = 2
pi = 3.14159265358979
Do
 l = l + l
 m = m + 1
 Loop Until l > n
 n = l / 2
ReDim xi(n - 1): ReDim yi(n - 1): ReDim zi(n - 1)

l = 1
Do
  le = 2 ^ l
  le1 = le / 2
  wr = 1
  wi = 0
  t = pi / le1
  w1r = Cos(t)
  w1i = -Sin(t)
  r = 0
Do
  p = r
  Do
   q = p + le1
   
   tr = xr(q) * wr - xi(q) * wi
   ti = xr(q) * wi + xi(q) * wr
   
   tr1 = yr(q) * wr - yi(q) * wi
   ti1 = yr(q) * wi + yi(q) * wr
   
   xr(q) = xr(p) - tr
   xi(q) = xi(p) - ti
   xr(p) = xr(p) + tr
   xi(p) = xi(p) + ti
   
    yr(q) = yr(p) - tr1
   yi(q) = yi(p) - ti1
   yr(p) = yr(p) + tr1
   yi(p) = yi(p) + ti1
   
   p = p + le
Loop Until p > n - 1

wr = wr * w1r - wi * w1i
wi = wr * w1i + wi * w1r
r = r + 1
Loop Until r > le1 - 1
l = l + 1
Loop Until l > m

For i = 0 To n - 1 '仅输出模
   zr(i) = xr(i) * yr(i) - xi(i) * yi(i): zi(i) = xr(i) * yi(i) + xi(i) * yr(i)
   Text2 = Text2 & "  " & zr(i) & "+" & zi(i) & "i"
   Next
End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub


 输入：Text1 =80607000，Text2=20403000,结果：
 189+0i  48.1200576850889+-33.8492424049177i  -9.10907287525383+0.845301243345486i  8.07394476689+-4.83093657874164i
  21+0i  -5.62005768508875+4.15075759508251i  -6.76592712474619+-6.46840725563929i  24.7688582392567+7.92894805108087i
这个与正确值不同，不知道逆变换能不能出来正确结果？