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Next: <a href="maxima_86.html#Functions-and-Variables-for-FFTPACK5" accesskey="n" rel="next">Functions and Variables for FFTPACK5</a>, Previous: <a href="maxima_84.html#Introduction-to-fast-Fourier-transform" accesskey="p" rel="previous">Introduction to fast Fourier transform</a>, Up: <a href="maxima_83.html#Numerical" accesskey="u" rel="up">Numerical</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-and-Variables-for-fft"></a>
<h3 class="section">21.2 Functions and Variables for fft</h3>
<a name="polartorect"></a><a name="Item_003a-Numerical_002fdeffn_002fpolartorect"></a><dl>
<dt><a name="index-polartorect"></a>Function: <strong>polartorect</strong> <em>(<var>r</var>, <var>t</var>)</em></dt>
<dd>
<p>Translates complex values of the form <code>r %e^(%i t)</code> to the form
<code>a + b %i</code>, where <var>r</var> is the magnitude and <var>t</var> is the phase.
<var>r</var> and <var>t</var> are 1-dimensional arrays of the same size.
The array size need not be a power of 2.
</p>
<p>The original values of the input arrays are
replaced by the real and imaginary parts, <code>a</code> and <code>b</code>, on return.
The outputs are calculated as
</p>
<div class="example">
<pre class="example">a = r cos(t)
b = r sin(t)
</pre></div>
<p><code><a href="#polartorect">polartorect</a></code> is the inverse function of <code><a href="#recttopolar">recttopolar</a></code>.
</p>
<p><code>load("fft")</code> loads this function. See also <code><a href="#fft">fft</a></code>.
</p>
</dd></dl>
<a name="recttopolar"></a><a name="Item_003a-Numerical_002fdeffn_002frecttopolar"></a><dl>
<dt><a name="index-recttopolar"></a>Function: <strong>recttopolar</strong> <em>(<var>a</var>, <var>b</var>)</em></dt>
<dd>
<p>Translates complex values of the form <code>a + b %i</code> to the form
<code>r %e^(%i t)</code>, where <var>a</var> is the real part and <var>b</var> is the imaginary
part. <var>a</var> and <var>b</var> are 1-dimensional arrays of the same size.
The array size need not be a power of 2.
</p>
<p>The original values of the input arrays are
replaced by the magnitude and angle, <code>r</code> and <code>t</code>, on return.
The outputs are calculated as
</p>
<div class="example">
<pre class="example">r = sqrt(a^2 + b^2)
t = atan2(b, a)
</pre></div>
<p>The computed angle is in the range <code>-%pi</code> to <code>%pi</code>.
</p>
<p><code>recttopolar</code> is the inverse function of <code><a href="#polartorect">polartorect</a></code>.
</p>
<p><code>load("fft")</code> loads this function. See also <code><a href="#fft">fft</a></code>.
</p>
</dd></dl>
<a name="inverse_005ffft"></a><a name="Item_003a-Numerical_002fdeffn_002finverse_005ffft"></a><dl>
<dt><a name="index-inverse_005ffft"></a>Function: <strong>inverse_fft</strong> <em>(<var>y</var>)</em></dt>
<dd>
<p>Computes the inverse complex fast Fourier transform.
<var>y</var> is a list or array (named or unnamed) which contains the data to
transform. The number of elements must be a power of 2.
The elements must be literal numbers (integers, rationals, floats, or bigfloats)
or symbolic constants,
or expressions <code>a + b*%i</code> where <code>a</code> and <code>b</code> are literal numbers
or symbolic constants.
</p>
<p><code>inverse_fft</code> returns a new object of the same type as <var>y</var>,
which is not modified.
Results are always computed as floats
or expressions <code>a + b*%i</code> where <code>a</code> and <code>b</code> are floats.
If bigfloat precision is needed the function <code><a href="#bf_005finverse_005ffft">bf_inverse_fft</a></code> can
be used instead as a drop-in replacement of <code>inverse_fft</code> that is
slower, but supports bfloats.
</p>
<p>The inverse discrete Fourier transform is defined as follows.
Let <code>x</code> be the output of the inverse transform.
Then for <code>j</code> from 0 through <code>n - 1</code>,
</p>
<div class="example">
<pre class="example">x[j] = sum(y[k] exp(-2 %i %pi j k / n), k, 0, n - 1)
</pre></div>
<p>As there are various sign and normalization conventions possible,
this definition of the transform may differ from that used by other mathematical software.
</p>
<p><code>load("fft")</code> loads this function.
</p>
<p>See also <code><a href="#fft">fft</a></code> (forward transform), <code><a href="#recttopolar">recttopolar</a></code>, and
<code><a href="#polartorect">polartorect</a></code>.
</p>
<p>Examples:
</p>
<p>Real data.
</p>
<div class="example">
<pre class="example">(%i1) load ("fft") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
(%i4) L1 : inverse_fft (L);
(%o4) [0.0, 14.49 %i - .8284, 0.0, 2.485 %i + 4.828, 0.0,
4.828 - 2.485 %i, 0.0, - 14.49 %i - .8284]
(%i5) L2 : fft (L1);
(%o5) [1.0, 2.0 - 2.168L-19 %i, 3.0 - 7.525L-20 %i,
4.0 - 4.256L-19 %i, - 1.0, 2.168L-19 %i - 2.0,
7.525L-20 %i - 3.0, 4.256L-19 %i - 4.0]
(%i6) lmax (abs (L2 - L));
(%o6) 3.545L-16
</pre></div>
<p>Complex data.
</p>
<div class="example">
<pre class="example">(%i1) load ("fft") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
(%i4) L1 : inverse_fft (L);
(%o4) [4.0, 2.711L-19 %i + 4.0, 2.0 %i - 2.0,
- 2.828 %i - 2.828, 0.0, 5.421L-20 %i + 4.0, - 2.0 %i - 2.0,
2.828 %i + 2.828]
(%i5) L2 : fft (L1);
(%o5) [4.066E-20 %i + 1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i,
1.55L-19 %i - 1.0, - 4.066E-20 %i - 1.0, 1.0 - 1.0 %i,
1.0 %i + 1.0, 1.0 - 7.368L-20 %i]
(%i6) lmax (abs (L2 - L));
(%o6) 6.841L-17
</pre></div>
</dd></dl>
<a name="fft"></a><a name="Item_003a-Numerical_002fdeffn_002ffft"></a><dl>
<dt><a name="index-fft"></a>Function: <strong>fft</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>Computes the complex fast Fourier transform.
<var>x</var> is a list or array (named or unnamed) which contains the data to
transform. The number of elements must be a power of 2.
The elements must be literal numbers (integers, rationals, floats, or bigfloats)
or symbolic constants,
or expressions <code>a + b*%i</code> where <code>a</code> and <code>b</code> are literal numbers
or symbolic constants.
</p>
<p><code>fft</code> returns a new object of the same type as <var>x</var>,
which is not modified.
Results are always computed as floats
or expressions <code>a + b*%i</code> where <code>a</code> and <code>b</code> are floats.
If bigfloat precision is needed the function <code><a href="#bf_005ffft">bf_fft</a></code> can be used
instead as a drop-in replacement of <code>fft</code> that is slower, but
supports bfloats. In addition if it is known that the input consists
of only real values (no imaginary parts), <code><a href="#real_005ffft">real_fft</a></code> can be used
which is potentially faster.
</p>
<p>The discrete Fourier transform is defined as follows.
Let <code>y</code> be the output of the transform.
Then for <code>k</code> from 0 through <code>n - 1</code>,
</p>
<div class="example">
<pre class="example">y[k] = (1/n) sum(x[j] exp(+2 %i %pi j k / n), j, 0, n - 1)
</pre></div>
<p>As there are various sign and normalization conventions possible,
this definition of the transform may differ from that used by other mathematical software.
</p>
<p>When the data <var>x</var> are real,
real coefficients <code>a</code> and <code>b</code> can be computed such that
</p>
<div class="example">
<pre class="example">x[j] = sum(a[k]*cos(2*%pi*j*k/n)+b[k]*sin(2*%pi*j*k/n), k, 0, n/2)
</pre></div>
<p>with
</p>
<div class="example">
<pre class="example">a[0] = realpart (y[0])
b[0] = 0
</pre></div>
<p>and, for k from 1 through n/2 - 1,
</p>
<div class="example">
<pre class="example">a[k] = realpart (y[k] + y[n - k])
b[k] = imagpart (y[n - k] - y[k])
</pre></div>
<p>and
</p>
<div class="example">
<pre class="example">a[n/2] = realpart (y[n/2])
b[n/2] = 0
</pre></div>
<p><code>load("fft")</code> loads this function.
</p>
<p>See also <code><a href="#inverse_005ffft">inverse_fft</a></code> (inverse transform),
<code><a href="#recttopolar">recttopolar</a></code>, and <code><a href="#polartorect">polartorect</a></code>.. See <code><a href="#real_005ffft">real_fft</a></code>
for FFTs of a real-valued input, and <code><a href="#bf_005ffft">bf_fft</a></code> and
<code><a href="#bf_005freal_005ffft">bf_real_fft</a></code> for operations on bigfloat values. Finally, for
transforms of any size (but limited to float values), see
<code><a href="maxima_86.html#fftpack5_005ffft">fftpack5_fft</a></code> and <code><a href="maxima_86.html#fftpack5_005freal_005ffft">fftpack5_real_fft</a></code>.
</p>
<p>Examples:
</p>
<p>Real data.
</p>
<div class="example">
<pre class="example">(%i1) load ("fft") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
(%i4) L1 : fft (L);
(%o4) [0.0, 1.811 %i - .1036, 0.0, 0.3107 %i + .6036, 0.0,
0.6036 - 0.3107 %i, 0.0, (- 1.811 %i) - 0.1036]
(%i5) L2 : inverse_fft (L1);
(%o5) [1.0, 2.168L-19 %i + 2.0, 7.525L-20 %i + 3.0,
4.256L-19 %i + 4.0, - 1.0, - 2.168L-19 %i - 2.0,
- 7.525L-20 %i - 3.0, - 4.256L-19 %i - 4.0]
(%i6) lmax (abs (L2 - L));
(%o6) 3.545L-16
</pre></div>
<p>Complex data.
</p>
<div class="example">
<pre class="example">(%i1) load ("fft") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
(%i4) L1 : fft (L);
(%o4) [0.5, 0.5, 0.25 %i - 0.25, (- 0.3536 %i) - 0.3536, 0.0, 0.5,
(- 0.25 %i) - 0.25, 0.3536 %i + 0.3536]
(%i5) L2 : inverse_fft (L1);
(%o5) [1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i, - 1.0, - 1.0, 1.0 - 1.0 %i,
1.0 %i + 1.0, 1.0]
(%i6) lmax (abs (L2 - L));
(%o6) 0.0
</pre></div>
<p>Computation of sine and cosine coefficients.
</p>
<div class="example">
<pre class="example">(%i1) load ("fft") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, 5, 6, 7, 8] $
(%i4) n : length (L) $
(%i5) x : make_array (any, n) $
(%i6) fillarray (x, L) $
(%i7) y : fft (x) $
(%i8) a : make_array (any, n/2 + 1) $
(%i9) b : make_array (any, n/2 + 1) $
(%i10) a[0] : realpart (y[0]) $
(%i11) b[0] : 0 $
(%i12) for k : 1 thru n/2 - 1 do
(a[k] : realpart (y[k] + y[n - k]),
b[k] : imagpart (y[n - k] - y[k]));
(%o12) done
(%i13) a[n/2] : y[n/2] $
(%i14) b[n/2] : 0 $
(%i15) listarray (a);
(%o15) [4.5, - 1.0, - 1.0, - 1.0, - 0.5]
(%i16) listarray (b);
(%o16) [0, - 2.414, - 1.0, - .4142, 0]
(%i17) f(j) := sum (a[k]*cos(2*%pi*j*k/n) + b[k]*sin(2*%pi*j*k/n),
k, 0, n/2) $
(%i18) makelist (float (f (j)), j, 0, n - 1);
(%o18) [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]
</pre></div>
</dd></dl>
<a name="real_005ffft"></a><a name="Item_003a-Numerical_002fdeffn_002freal_005ffft"></a><dl>
<dt><a name="index-real_005ffft"></a>Function: <strong>real_fft</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>Computes the fast Fourier transform of a real-valued sequence
<var>x</var>. This is equivalent to performing <code>fft(x)</code>, except that
only the first <code>N/2+1</code> results are returned, where <code>N</code> is
the length of <var>x</var>. <code>N</code> must be power of two.
</p>
<p>No check is made that <var>x</var> contains only real values.
</p>
<p>The symmetry properties of the Fourier transform of real sequences to
reduce he complexity. In particular the first and last output values
of <code>real_fft</code> are purely real. For larger sequences, <code>real_fft</code>
may be computed more quickly than <code>fft</code>.
</p>
<p>Since the output length is short, the normal <code><a href="#inverse_005ffft">inverse_fft</a></code> cannot
be directly used. Use <code><a href="#inverse_005freal_005ffft">inverse_real_fft</a></code> to compute the inverse.
</p>
</dd></dl>
<a name="inverse_005freal_005ffft"></a><a name="Item_003a-Numerical_002fdeffn_002finverse_005freal_005ffft"></a><dl>
<dt><a name="index-inverse_005freal_005ffft"></a>Function: <strong>inverse_real_fft</strong> <em>(<var>y</var>)</em></dt>
<dd><p>Computes the inverse Fourier transform of <var>y</var>, which must have a
length of <code>N/2+1</code> where <code>N</code> is a power of two. That is, the
input <var>x</var> is expected to be the output of <code>real_fft</code>.
</p>
<p>No check is made to ensure that the input has the correct format.
(The first and last elements must be purely real.)
</p>
</dd></dl>
<a name="bf_005finverse_005ffft"></a><a name="Item_003a-Numerical_002fdeffn_002fbf_005finverse_005ffft"></a><dl>
<dt><a name="index-bf_005finverse_005ffft"></a>Function: <strong>bf_inverse_fft</strong> <em>(<var>y</var>)</em></dt>
<dd>
<p>Computes the inverse complex fast Fourier transform. This is the
bigfloat version of <code><a href="#inverse_005ffft">inverse_fft</a></code> that converts the input to
bigfloats and returns a bigfloat result.
</p>
</dd></dl>
<a name="bf_005ffft"></a><a name="Item_003a-Numerical_002fdeffn_002fbf_005ffft"></a><dl>
<dt><a name="index-bf_005ffft"></a>Function: <strong>bf_fft</strong> <em>(<var>y</var>)</em></dt>
<dd>
<p>Computes the forward complex fast Fourier transform. This is the
bigfloat version of <code><a href="#fft">fft</a></code> that converts the input to
bigfloats and returns a bigfloat result.
</p>
</dd></dl>
<a name="bf_005freal_005ffft"></a><a name="Item_003a-Numerical_002fdeffn_002fbf_005freal_005ffft"></a><dl>
<dt><a name="index-bf_005freal_005ffft"></a>Function: <strong>bf_real_fft</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>Computes the forward fast Fourier transform of a real-valued input
returning a bigfloat result. This is the bigfloat version of
<code>real_fft</code>.
</p>
</dd></dl>
<a name="bf_005finverse_005freal_005ffft"></a><a name="Item_003a-Numerical_002fdeffn_002fbf_005finverse_005freal_005ffft"></a><dl>
<dt><a name="index-bf_005finverse_005freal_005ffft"></a>Function: <strong>bf_inverse_real_fft</strong> <em>(<var>y</var>)</em></dt>
<dd><p>Computes the inverse fast Fourier transform with a real-valued
bigfloat output. This is the bigfloat version of <code>inverse_real_fft</code>.
</p>
</dd></dl>
<a name="Item_003a-Numerical_002fnode_002fFunctions-and-Variables-for-FFTPACK5"></a><hr>
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<p>
Next: <a href="maxima_86.html#Functions-and-Variables-for-FFTPACK5" accesskey="n" rel="next">Functions and Variables for FFTPACK5</a>, Previous: <a href="maxima_84.html#Introduction-to-fast-Fourier-transform" accesskey="p" rel="previous">Introduction to fast Fourier transform</a>, Up: <a href="maxima_83.html#Numerical" accesskey="u" rel="up">Numerical</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
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