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.
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.
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
:next to the
object, you get back either NIL, indicating the end of the sequence,
or you get an array of FFT coefficients.
The Nyquist function
snd-fft (mnemonic, isn't it?) returns one
of the frame sequence generating objects. You can pass any frame
sequence generating object to another function,
turn the sequence back into audio.
snd-ifft, 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
nyquist/lib/fft/fft_tutorial.htm, which is part of the
standard Nyquist release. The documentation for
snd-fft(sound, length, skip, window)[SAL]
(snd-fft sound length skip window)[LISP]
FLONUMs. The function modifies the sound, violating the normal rule that sounds are immutable in Nyquist, so it is advised that you copy the sound using
snd-copyif there are any other references to sound. The length of the FFT is specified by length, a
FIXNUM(integer) which must be a power of 2. After each FFT, the sound is advanced by skip samples, also of type
FIXNUM. Overlapping FFTs, where skip is less than length, are allowed. If window is not
NIL, it must be a sound. The first length samples of window are multiplied by length samples of sound before performing the FFT. When there are no more samples in sound to transform, this function returns
NIL. 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.
snd-ifft(time, srate, iterator, skip, window)[SAL]
(snd-ifft time srate iterator skip window)[LISP]
(local-to-global 0). The sample rate is given by srate. Typically, this would be
*sound-srate*, 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
:nextto the iterator object, using XLISP's primitives for objects and message passing. The object should return an array in the same format as obtained from
snd-fft, and the object should return
NILwhen 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 skip relative to the previous frame. This must be an integer greater than zero and less than the frame (FFT) size. If window is not
NIL, 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 iterator, so it does not appear as a parameter. This length is also the number of samples used from window. Extra samples are ignored, and window is padded with zeros if necessary, so be sure window is the right length. The resulting sound is computed on demand as with other Nyquist sounds, so
:nextmessages are sent to iterator only when new frames are needed. One should be careful not to reuse or modify iterator once it is passed to
There are a number of functions defined to make spectral processing
easier in XLISP and SAL. The general approach, as described above, is
to create an iterator object that returns spectral frames. To avoid
using the XLISP object system directly, a more functional interface is
defined, especially for SAL users.
sa-init function creates an iterator, and
sa-next retrieves spectral frames. Various functions are also
provided to transform these into amplitude (magnitude) spectra, plot
them and perform other operations.
Some examples that use these spectral processing functions can be found
in the Nyquist extension “fftsal” (use the
NyquistIDE's Window : Nyquist Extensions menu item to download it; it
will then be in your
nyquist/lib/fftsal directory. You can find
descriptions of the examples in
sa-init(resolution: hz, fft-dur: dur, skip-period: skip, window: window-type, input: input)[SAL]
(sa-init :resolution hz :fft-dur dur :skip-period skip :window window-type :input input)[LISP]
resolutionkeyword parameter gives the width of each spectral bin in Hz. It may be
nilor not specified, in which case the resolution is computed from
fft-dur. The actual resolution may be finer than the specified resolution because fft sizes are rounded to a power of 2. The
fft-duris the width of the FFT window in seconds. The actual FFT size will be rounded up to the nearest power of two in samples. If
fft-durwill be calculated from
resolution. If both
nilor not specified, the default value is 1024 samples, corresponding to a duration of 1024 / signal-sample-rate. If both
fft-durare specified, the
resolutionparameter will be ignored. Note that
resolutionare reciprocals. The
skip-periodspecifies the time interval in seconds between successive spectra (FFT windows). Overlapping FFTs are possible. The default value overlaps windows by 50%. Non-overlapped and widely spaced windows that ignore samples by skipping over them entirely are also acceptable. The
windowspecifies the type of window. The default is raised cosine (Hann or "Hanning") window. Options include
nilmean a rectangular window. The
inputcan be a string (which specifies a sound file to read) or a Nyquist SOUND to be analyzed. The return value is an XLISP object that can be called to obtain parameters as well as a sequence of spectral frames. Normally, you will set a variable to this result and pass the variable to
sa-next, described below.
sa-init(see above). The return value is
nil, but information is printed.
sa-init(see above). The return value is an array of FLONUMs representing the discrete complex spectrum.
sa-frame. The ith bin is stored at index i. The size of the array is the FFT size / 2 + 1.
sa-normalize(frame [, max])[SAL]
(sa-normalize frame [max])[LISP]
sa-magnitude. If max (a FLONUM) is provided, the spectrum will be normalized to have a maximum value of max, which defaults to 1.
(sa-plot sa-obj frame)[LISP]
sa-magnitude. The sa-obj parameter should be the same value used to obtain the frame.
sa-print(file, sa-obj, frame, cutoff: cutoff, threshold: threshold)[SAL]
(sa-print sa-obj file frame :cutoff cutoff :threshold threshold)[LISP]
sa-normalize). The file is either a file opened for writing or
Tto print to the console. The caller is responsible for closing the file (eventually). The sa-obj parameter should be the same value used to obtain the frame. If cutoff, a FLONUM, is provided, only the spectrum below cutoff (Hz) will be printed. If threshold, a FLONUM, is provided, the output may elide bins with values below the threshold.
i * bin-width.