*fft* in Constant
2009

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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 wav

pproaches 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 num

paced 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 t

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