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<html><head><title>Time/Frequency Transformation</title>
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<a name = "118"><h2>Time/Frequency Transformation</h2></a>
Nyquist provides functions for FFT and inverse FFT operations on streams of audio data.
Because sounds can be of any length, but an FFT operates on a fixed amount of data, FFT
processing is typically done in short blocks or windows that move through the audio. Thus,
a stream of samples is converted in to a sequence of FFT frames representing short-term
spectra.
<p>
Nyquist does not have a special data type corresponding to a sequence of FFT frames.
This would be nice, but it would require creating a large set of operations suitable for
processing frame sequences. Another approach, and perhaps the most "pure" would
be to convert a single sound into a multichannel sound, with one channel per bin of the
FFT.
<p>
Instead, Nyquist violates its "pure" functional model and resorts to objects
for FFT processing. A sequence of frames is represented by an XLISP object. Whenever you
send the selector <code>:next</code> to the object, you get back either NIL, indicating the
end of the sequence, or you get an array of FFT coefficients.
<p>
The Nyquist function <code>snd-fft</code> (mnemonic, isn't it?) returns one of the frame sequence
generating objects. You can pass any frame sequence generating object to another function,
<code>snd-ifft</code>, and turn the sequence back into audio.
<p>
With <code>snd-fft</code> and <code>snd-ifft</code>, you can create all sorts of interesting processes. The main
idea is to create intermediate objects that both accept and generate sequences of frames.
These objects can operate on the frames to implement the desired spectral-domain
processes. Examples of this can be found in the file
<code>demos/fft_tutorial.htm</code><a name="index898"></a><a name="index899"></a><a name="index900"></a>,
which is part of the standard Nyquist release. The documentation for <code>snd-fft</code> and
<code>snd-ifft</code> follows.
<p>
<dl>
<dt>
<code>snd-fft(<a name="index901"></a><a name="index902"></a><i>sound</i>, <i>length</i>, <i>skip</i>, <i>window</i>)</code> [SAL]<br>
<code>(snd-fft <i>sound</i> <i>length</i> <i>skip</i> <i>window</i>)</code> [LISP]<dd>This
function performs an FFT on the first samples in <i>sound</i> and returns a Lisp array of <code>FLONUM</code>s.
The function modifies the <i>sound</i>, violating the normal rule that sounds are immutable in Nyquist, so
it is advised that you copy the sound using <code>snd-copy</code> if there are any other references to
<i>sound</i>. The length of the FFT is specified by <i>length</i>, a <code>FIXNUM</code> (integer) which must
be a power of 2. After
each FFT, the sound is advanced by <i>skip</i> samples, also of type <code>FIXNUM</code>. Overlapping FFTs,
where <i>skip</i> is less than <i>length</i>, are allowed.
If <i>window</i> is not <code>NIL</code>, it must be a sound.
The first <i>length</i> samples of <i>window</i> are multiplied by <i>length</i> samples of <i>sound</i> before
performing the FFT. When there are no more samples in <i>sound</i> to transform,
this function returns <code>NIL</code>. The coefficients in the returned array, in order, are the DC coefficient,
the first real, the first imaginary, the second real, the second imaginary, etc.
The last array element corresponds to the real coefficient at the Nyquist frequency.<br><br>
<dt><code>snd-ifft(<a name="index903"></a><a name="index904"></a><a name="index905"></a><i>time</i>, <i>srate</i>, <i>iterator</i>, <i>skip</i>, <i>window</i>)</code> [SAL]<br>
<code>(snd-ifft <i>time</i> <i>srate</i> <i>iterator</i> <i>skip</i> <i>window</i>)</code> [LISP]<dd>This function performs an IFFT on a sequence of spectral frames obtained from <i>iterator</i>
and returns a sound. The start time of the sound is given by <i>time</i>. Typically, this would be computed
by calling <code>(local-to-global 0)</code>. The sample rate is given by <i>srate</i>. Typically, this would
be <code>*sound-srate*</code>, but it might also depend upon the sample rate of the sound from which the
spectral frames were derived. To obtain each frame, the function sends the message <code>:next</code> to the
<i>iterator</i> object, using XLISP's primitives for objects and message passing. The object should return
an array in the same format as obtained from <code>snd-fft</code>, and the object should return <code>NIL</code>
when the end of the sound is reached. After each frame is inverse transformed into the time domain, it is
added to the resulting sound. Each successive frame is added with a sample offset specified by <i>skip</i>
relative to the previous frame. This must be an integer greater than zero and less than the frame (FFT) size.
If <i>window</i> is
not <code>NIL</code>, it must be a sound. This window signal is multiplied by the inverse transformed frame
before the frame is added to the output sound. The length of each frame should be the same power of 2.
The length
is implied by the first array returned by <i>iterator</i>, so it does not appear as a parameter. This length
is also the number of samples used from <i>window</i>. Extra samples are ignored, and window is padded
with zeros if necessary, so be sure <i>window</i> is the right length. The resulting sound is computed on
demand as with other Nyquist sounds, so <code>:next</code> messages are sent to <i>iterator</i> only when new
frames are needed. One should be careful not to reuse or modify <i>iterator</i> once it is passed to
<code>snd-ifft</code>.
</dl>
<p>
<hr>
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