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Unformatted text preview: a hundred different
frequencies or a hundred different colors. With current technology, each color can carry
data through the optical ﬁber at a rate of 100 gigabits per second, yielding a net transfer
rate of 10 terabits per second.
In the case of ﬁber, it is tempting to suggest that one should just keep adding channels,
to make the net data transfer rate approach inﬁnity. Unfortunately, for ﬁber, whose carrier
frequencies are on the order of 1014 hertz, there are technological problems that limit the
number of carriers. For wireless transmission systems, whose carrier frequencies are on
the order of 109 hertz (i.e., one to a few gigahertz), current technology can easily achieve
a more fundamental limit: for wireless systems, there is a fundamental tradeoff between
the distance between carrier frequencies, and the maximum data rate per carrier. 3 SECTION 14.2. DISCRETE FREQUENCY To understand these limitations, we will need a new tool, the Fourier series. The Fourier
series is a way to express any periodic sequence as a weighted sum of cosines and sines or,
equivalently, as a weighted sum of complex exponentials (taking advantage of the familiar
identities relating sines and cosines to complex exponentials).
Before describing the Fourier series, we discuss the notion of discrete frequencies and
how they arise from a continuous waveform. 14.2 Discrete Frequency If we have some periodic function in continuous time, t, say, x(t) = cos(2π f t), we say that
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it has a frequency, f , and a period, T = f . The notion of frequency for such a periodic
continuous waveform is easy to interpret in part because it is welldeﬁned for all values of
t. But how do we go about deﬁning frequency for a discretized function?
To discretize this continuous function into a set ot discrete voltage samples, we sample
it at some other sampling frequency, fs , resulting in a discret...
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This document was uploaded on 02/26/2014 for the course CS 6.02 at MIT.
 Fall '13
 HariBalakrishnan

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