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Fast Fourier transform: Difference between revisions

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Fast Fourier transform (view source)

Revision as of 13:22, 19 May 2021

24,560 bytes added ,  2 years ago
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→‎{{header|Phix}}: added syntax colouring the hard way
m (→‎{{header|Phix}}: added syntax colouring the hard way)
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=={{header|Phix}}==
<!--<lang Phix>(phixonline)-->
<span style="color: #000080;font-style:italic;">--
-- demo\rosetta\FastFourierTransform.exw
-- demo\rosetta\FastFourierTransform.exw
-- =====================================
-- =====================================
--
--
-- Originally written by Robert Craig and posted to EuForum Dec 13, 2001
-- Originally written by Robert Craig and posted to EuForum Dec 13, 2001
--
--</span>
 
constant REAL = 1, IMAG = 2
<span style="color: #008080;">constant</span> <span style="color: #000000;">REAL</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">IMAG</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span>
 
type complex(sequence x)
<span style="color: #008080;">type</span> <span style="color: #004080;">complex</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">)</span>
return length(x)=2 and atom(x[REAL]) and atom(x[IMAG])
<span style="color: #008080;">return</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">2</span> <span style="color: #008080;">and</span> <span style="color: #004080;">atom</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">[</span><span style="color: #000000;">REAL</span><span style="color: #0000FF;">])</span> <span style="color: #008080;">and</span> <span style="color: #004080;">atom</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">[</span><span style="color: #000000;">IMAG</span><span style="color: #0000FF;">])</span>
end type
<span style="color: #008080;">end</span> <span style="color: #008080;">type</span>
 
function p2round(integer x)
<span style="color: #008080;">function</span> <span style="color: #000000;">p2round</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">)</span>
-- rounds x up to a power of two
<span style="color: #000080;font-style:italic;">-- rounds x up to a power of two</span>
integer p = 1
<span style="color: #004080;">integer</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
while p<x do
<span style="color: #008080;">while</span> <span style="color: #000000;">p</span><span style="color: #0000FF;"><</span><span style="color: #000000;">x</span> <span style="color: #008080;">do</span>
p += p
<span style="color: #000000;">p</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">p</span>
end while
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
return p
<span style="color: #008080;">return</span> <span style="color: #000000;">p</span>
end function
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
 
function log2(atom x)
<span style="color: #008080;">function</span> <span style="color: #000000;">log_2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">)</span>
-- return log2 of x, or -1 if x is not a power of 2
<span style="color: #000080;font-style:italic;">-- return log2 of x, or -1 if x is not a power of 2</span>
if x>0 then
<span style="color: #008080;">if</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">></span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
integer p = -1
<span style="color: #004080;">integer</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span>
while floor(x)=x do
<span style="color: #008080;">while</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">x</span> <span style="color: #008080;">do</span>
x /= 2
<span style="color: #000000;">x</span> <span style="color: #0000FF;">/=</span> <span style="color: #000000;">2</span>
p += 1
<span style="color: #000000;">p</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
end while
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
if x=0.5 then
<span style="color: #008080;">if</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0.5</span> <span style="color: #008080;">then</span>
return p
<span style="color: #008080;">return</span> <span style="color: #000000;">p</span>
end if
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
end if
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
return -1
<span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span>
end function
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
 
function bitrev(sequence a)
<span style="color: #008080;">function</span> <span style="color: #000000;">bitrev</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
-- bitrev an array of complex numbers
<span style="color: #000080;font-style:italic;">-- bitrev an array of complex numbers</span>
integer j=1, n = length(a)
<span style="color: #004080;">integer</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
for i=1 to n-1 do
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">deep_copy</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
if i<j then
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
{a[i],a[j]} = {a[j],a[i]}
<span style="color: #008080;">if</span> <span style="color: #000000;">i</span><span style="color: #0000FF;"><</span><span style="color: #000000;">j</span> <span style="color: #008080;">then</span>
end if
<span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">],</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]}</span>
integer k = n/2
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
while k<j do
<span style="color: #004080;">integer</span> <span style="color: #000000;">k</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span>
j -= k
<span style="color: #008080;">while</span> <span style="color: #000000;">k</span><span style="color: #0000FF;"><</span><span style="color: #000000;">j</span> <span style="color: #008080;">do</span>
k /= 2
<span style="color: #000000;">j</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">k</span>
end while
<span style="color: #000000;">k</span> <span style="color: #0000FF;">/=</span> <span style="color: #000000;">2</span>
j = j+k
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
end for
<span style="color: #000000;">j</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">+</span><span style="color: #000000;">k</span>
return a
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
end function
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
 
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
function cmult(complex arg1, complex arg2)
-- complex multiply
<span style="color: #008080;">function</span> <span style="color: #000000;">cmult</span><span style="color: #0000FF;">(</span><span style="color: #004080;">complex</span> <span style="color: #000000;">arg1</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">complex</span> <span style="color: #000000;">arg2</span><span style="color: #0000FF;">)</span>
return {arg1[REAL]*arg2[REAL]-arg1[IMAG]*arg2[IMAG],
<span style="color: #000080;font-style:italic;">-- complex multiply </span>
arg1[REAL]*arg2[IMAG]+arg1[IMAG]*arg2[REAL]}
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">arg1</span><span style="color: #0000FF;">[</span><span style="color: #000000;">REAL</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">arg2</span><span style="color: #0000FF;">[</span><span style="color: #000000;">REAL</span><span style="color: #0000FF;">]-</span><span style="color: #000000;">arg1</span><span style="color: #0000FF;">[</span><span style="color: #000000;">IMAG</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">arg2</span><span style="color: #0000FF;">[</span><span style="color: #000000;">IMAG</span><span style="color: #0000FF;">],</span>
end function
<span style="color: #000000;">arg1</span><span style="color: #0000FF;">[</span><span style="color: #000000;">REAL</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">arg2</span><span style="color: #0000FF;">[</span><span style="color: #000000;">IMAG</span><span style="color: #0000FF;">]+</span><span style="color: #000000;">arg1</span><span style="color: #0000FF;">[</span><span style="color: #000000;">IMAG</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">arg2</span><span style="color: #0000FF;">[</span><span style="color: #000000;">REAL</span><span style="color: #0000FF;">]}</span>
 
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
function ip_fft(sequence a)
-- perform an in-place fft on an array of complex numbers
<span style="color: #008080;">function</span> <span style="color: #000000;">ip_fft</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
-- that has already been bit reversed
<span style="color: #000080;font-style:italic;">-- perform an in-place fft on an array of complex numbers
integer n = length(a)
-- that has already been bit reversed</span>
integer ip, le, le1
<span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
complex u, w, t
<span style="color: #004080;">integer</span> <span style="color: #000000;">ip</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">le</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">le1</span>
 
<span style="color: #004080;">complex</span> <span style="color: #000000;">u</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">w</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span>
for l=1 to log2(n) do
le = power(2, l)
<span style="color: #008080;">for</span> <span style="color: #000000;">l</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">log_2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
le1 = le/2
<span style="color: #000000;">le</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">l</span><span style="color: #0000FF;">)</span>
u = {1, 0}
<span style="color: #000000;">le1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">le</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span>
w = {cos(PI/le1), sin(PI/le1)}
<span style="color: #000000;">u</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">}</span>
for j=1 to le1 do
<span style="color: #000000;">w</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">cos</span><span style="color: #0000FF;">(</span><span style="color: #004600;">PI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">le1</span><span style="color: #0000FF;">),</span> <span style="color: #7060A8;">sin</span><span style="color: #0000FF;">(</span><span style="color: #004600;">PI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">le1</span><span style="color: #0000FF;">)}</span>
for i=j to n by le do
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">le1</span> <span style="color: #008080;">do</span>
ip = i+le1
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">j</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">by</span> <span style="color: #000000;">le</span> <span style="color: #008080;">do</span>
t = cmult(a[ip], u)
<span style="color: #000000;">ip</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">le1</span>
a[ip] = sq_sub(a[i],t)
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">cmult</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ip</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">u</span><span style="color: #0000FF;">)</span>
a[i] = sq_add(a[i],t)
<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ip</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
end for
<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
u = cmult(u, w)
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
end for
<span style="color: #000000;">u</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">cmult</span><span style="color: #0000FF;">(</span><span style="color: #000000;">u</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">w</span><span style="color: #0000FF;">)</span>
end for
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
return a
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
end function
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
 
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
function fft(sequence a)
integer n = length(a)
<span style="color: #008080;">function</span> <span style="color: #000000;">fft</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
if log2(n)=-1 then
<span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
puts(1, "input vector length is not a power of two, padded with 0's\n\n")
<span style="color: #008080;">if</span> <span style="color: #000000;">log_2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)=-</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
n = p2round(n)
<span style="color: #7060A8;">puts</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"input vector length is not a power of two, padded with 0's\n\n"</span><span style="color: #0000FF;">)</span>
-- pad with 0's
<span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p2round</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
for j=length(a)+1 to n do
<span style="color: #000080;font-style:italic;">-- pad with 0's </span>
a = append(a,{0, 0})
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
end for
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">})</span>
end if
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
a = ip_fft(bitrev(a))
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
-- reverse output from fft to switch +ve and -ve frequencies
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">ip_fft</span><span style="color: #0000FF;">(</span><span style="color: #000000;">bitrev</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">))</span>
for i=2 to n/2 do
<span style="color: #000080;font-style:italic;">-- reverse output from fft to switch +ve and -ve frequencies</span>
integer j = n+2-i
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span> <span style="color: #008080;">do</span>
{a[i],a[j]} = {a[j],a[i]}
<span style="color: #004080;">integer</span> <span style="color: #000000;">j</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">+</span><span style="color: #000000;">2</span><span style="color: #0000FF;">-</span><span style="color: #000000;">i</span>
end for
<span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">],</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]}</span>
return a
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
end function
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
 
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
function ifft(sequence a)
integer n = length(a)
<span style="color: #008080;">function</span> <span style="color: #000000;">ifft</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
if log2(n)=-1 then ?9/0 end if -- (or as above?)
<span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
a = ip_fft(bitrev(a))
<span style="color: #008080;">if</span> <span style="color: #000000;">log_2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)=-</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">-- (or as above?)</span>
-- modifies results to get inverse fft
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">ip_fft</span><span style="color: #0000FF;">(</span><span style="color: #000000;">bitrev</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">))</span>
for i=1 to n do
<span style="color: #000080;font-style:italic;">-- modifies results to get inverse fft</span>
a[i] = sq_div(a[i],n)
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
end for
<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_div</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
return a
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
end function
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
 
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
constant a = {{1, 0},
{1, 0},
<span style="color: #008080;">constant</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span>
{1, 0},
<span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span>
{1, 0},
<span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span>
{0, 0},
<span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span>
{0, 0},
<span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span>
{0, 0},
<span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span>
{0, 0}}
<span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span>
 
<span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">}}</span>
printf(1, "Results of %d-point fft:\n\n", length(a))
ppOpt({pp_Nest,1,pp_IntFmt,"%10.6f",pp_FltFmt,"%10.6f"})
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"Results of %d-point fft:\n\n"</span><span style="color: #0000FF;">,</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">))</span>
pp(fft(a))
<span style="color: #7060A8;">ppOpt</span><span style="color: #0000FF;">({</span><span style="color: #004600;">pp_Nest</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #004600;">pp_IntFmt</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%10.6f"</span><span style="color: #0000FF;">,</span><span style="color: #004600;">pp_FltFmt</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%10.6f"</span><span style="color: #0000FF;">})</span>
printf(1, "\nResults of %d-point inverse fft (rounded to 6 d.p.):\n\n", length(a))
<span style="color: #7060A8;">pp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fft</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">))</span>
pp(ifft(fft(a)))</lang>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"\nResults of %d-point inverse fft (rounded to 6 d.p.):\n\n"</span><span style="color: #0000FF;">,</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">))</span>
<span style="color: #7060A8;">pp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ifft</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fft</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)))</span>
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{{out}}
<pre>
Petelomax
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edits

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