fft in Constant 2009


ocessing, the folding that we might understand as a ‘centre of envelopment' in action.
The Fast Fourier Transform: transformations between time
and space
I have been arguing that the complications of the mathematics
and the convoluted nature of the code or hardware used in DSP,
stems from an intensive movement or constitutive difference that is
interiorised. We can trace this interiorisation in the DSP used in
wireless networks. I do not have time to show how this happens
in detail, but hopefully one example of DSP that occurs but in the
video codecs and wireless networks will illustrate how this happens
in practice.
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Late in the encoding process, and much earlier in the decoding
process in contemporary wireless networks, a fairly generic computational algorithm comes into action: the Fast Fourier Transform
(ffT). In some ways, it is not surprising to find the ffT in wireless networks or in digital video. Dating from the mid-1960s, ffTs
have long been used to analyse electrical signals in many scientific
and engineering settings. It provides the component frequencies of
a time-varying signal or waveform. Hence, in ‘spectral analysis', the
ffT can show the spectrum of frequencies present in a signal.
The notion of the Fourier transform is mathematical and has been
known since the early 19th century: it is an operation that takes
an arbitrary waveform and turns it into a set of periodic waves (sinusoids) of different frequencies and amplitudes. Some of these sinusoids
make more important contributions to overall shape of the waveform
than others. Added together again, these sine or cosine waves should
exactly re-constitute the original signal. Crucially, a Fourier transform can turn something that varies over time (a signal) into a set of
simple components (sine or cosine waves) that do not vary over time.
Put more technically, it switches between ‘time' and ‘frequency' domains. Something that changes in time, a signal, becomes a set of
distinct components that can be handled separately. 4
In a way, this analysis of a complex signal into simple static component signals means that DSP does use the set-based approaches I
described earlier. Once a complex signal, such as an image, has been
analysed into a set of static components, we can imagine code that

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Humanities and social science work on the Fast Fourier Transform is hard to find, even
though the ffT is the common mathematical basis of contemporary digital image,
video and sound compression, and hence of many digital multimedia (in JPEG, MPEG
files, in DVDs). In the early 1990s, Friedrich Kittler wrote an article that discussed
it {Kittler, 1993 #753}. His key point was largely to show that there is no realtime
in digital signal processing. The ffT works by defining a sliding window of time for
a signal. It treats a complicated signal as a set of blocks that it lifts out of the time
domain and transforms into the frequency domain. The ffT effectively plots an event
in time as a graph in space. The experience of realtime is epiphenomenal. In terms of
the ffT, a signal is always partly in the future or the past. Although Kittler was not
referring to the use of ffT in wireless networks, the same point applies – there is no
realtime communication. However, while this point about the impossibility of realtime
calculation was important to make during the 1990s, it seems well-established now.

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would select the most important or relevant components. This is precisely what happens in video and sound codecs such as MPEG and
MP3.
The ffT treats sounds and images as complicated superimpositions of waveforms. The envelope of a signal becomes something that
contains many simple signals. It is interesting that wireless networks
tend to use this process in reverse. It deliberately takes a well-separated and discrete set of signals – a digital datastream – and turns it
into a single complex signal. In contrast to the normal uses of ffT in
separating important from insignificant parts of a signal, in wireless
networks, and in many other communications setting, ffT is used to
put signals together in such a way as to contain them in a single envelope. The ffT is found in many wireless computation algorithms
because it allows many different digital signals to be put together on
a single wave and then extracted from it again.
Why would this superimposition of many signals onto a single complex waveform be desirable? Would it not increase the possibilities of
confusion or interference between signals? In some ways the ffT is
used to slow everything down rather than speed it up. Rather than
simply spatialising a duration, the ffT as used in wireless networks
defines a different way of inhabiting the crowded, noise space of electromagnetic radiation. Wireless transmitters are better at inhabiting
crowded signal spectrum when they don't try to separate themselves
off from each other, but actually take the presence of other transmitters into account. How does the ffT allow many transmitters to
inhabit the same spectrum, and even use the same frequencies?
The name of this technique is OFDM (Orthogonal Frequency Division Multiplexing). OFDM spreads a single data stream coming
from a single device across a large number of sub-carriers signals (52
in IEEE 802.11a/g). It splits the data stream into dozens of separate signals of slightly different frequency that together evenly use
the whole available radio spectrum. This is done in such a way that
many different transmitters can be transmitting at the same time,
on the same frequency, without interfering with each other. The advantage of spreading a single high speed data stream across many
signals (wideband) is that each individual signal can carry data at a
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much slower rate. Because the data is split into 52 different signals,
each signal can be much slower (1/50). That means each bit of data
can be spaced apart more in time. This has great advances in urban
environments where there are many obstacles to signals, and signals
can reflect and echo often. In this context, the slower the data is
transmitted, the better.
At the transmitter, a reverse ffT (IffT) is used to re-combine
the 50 signals onto 1 signal. That is, it takes the 50 or so different
sub-carriers produced by OFDM, each of which has a single slightly
different, but carefully chosen frequency, and combines them into one
complex signal that has a wide spectrum. That is, it fills the available
spectrum quite evenly because it contains many different frequency
components. The waveform that results from the IffT looks like
'white noise': it has no remarkable or outstanding tendency whatsoever, except to a receiver synchronised to exactly the right carrier
frequency. At the receiver, this complex signal is transformed, using ffT, back into a set of 50 separate data streams, that are then
reconstituted into a single high speed stream.
Even if we cannot come to grips with the techniques of transformation using in DSP in any great detail, I hope that one point stands
out. The transformation involves ‘c'hanges in kind. Data does not
simply move through space. It changes in kind in order to move
through space, a space whose geography is understood as too full of
potential relations.
Conclusion
A couple of points in conclusion:
a. The spectrum of different wireless-audiovisual devices competing
to do more or less the same thing, are all a reproduction of the
same. Extensive movement associated with wireless networks and
digital video occur in various forms. firstly in the constant enveloping of spaces by wireless signals, and secondly in the dense

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population of wireless spectrum by competi

 

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