# 0.3 Modelling corruption  (Page 3/11)

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## Multipath interference

In some situations, an electromagnetic wave can propagate directly from one place to another. For instance, when aradio signal from a spacecraft is transmitted back to Earth, the vacuum of space guarantees that the wave will arrivemore or less intact (though greatly attenuated by distance). Often, however, the wave reflects, refracts, or diffracts,and the signal arriving is quite different from the one that was sent.

These distortions can be thought of as a combination of scaled and delayed reflections of the transmitted signal, which occurwhen there are different propagation paths from the transmitter to the receiver. Between two transmission towers, for instance, the paths may include one alongthe line-of-sight, reflections from the atmosphere, reflections from nearby hills, and bounces from a field or lake betweenthe towers. For indoor digital TV reception, there are many (local)time-varying reflectors, including people in the receiving room, nearby vehicles, and the buildings of an urban environment. [link] , for instance, shows multiple reflections that arrive after bouncing off a cloud, after bouncing offa mountain, and others that are scattered by multiple bounces from nearby buildings.

The strength of the reflections depends on the physical properties of the reflecting surface, while the delay of thereflections is primarily determined by the length of the transmission path. Let $s\left(t\right)$ be the transmitted signal. If $N$ delays are represented by ${\Delta }_{1},\phantom{\rule{4pt}{0ex}}{\Delta }_{2},...,\phantom{\rule{4pt}{0ex}}{\Delta }_{N}$ , and the strengths of the reflections are ${h}_{1},\phantom{\rule{4pt}{0ex}}{h}_{2},...,\phantom{\rule{4pt}{0ex}}{h}_{N}$ , then the received signal $r\left(t\right)$ is

$r\left(t\right)={h}_{1}s\left(t-{\Delta }_{1}\right)+{h}_{2}s\left(t-{\Delta }_{2}\right)+...+{h}_{N}s\left(t-{\Delta }_{N}\right).$

As will become clear in "Convolution in Time: It's What Linear Systems Do" , this model of the channel has the form of a linear filter(since the expression on the right hand side is a convolution of the transmitted signal and the ${h}_{i}$ 's). This is shown in part (a) of [link] . Since this channel model is a linear filter,it can also be viewed in the frequency domain, and part (b) shows its frequency response.When this is combined with the BPF and the spectrum of the signal (shown in (c)), the result is the distorted spectrumshown in (d).

What can be done?

If the kinds of distortions introduced by the channel are known (or can somehow be determined), then thebandpass filter at the receiver can be modified in order to undo the effects of the channel. This can beseen most clearly in the frequency domain, as in [link] . Observe that the BPF is shaped (part (d)) to approximately invert the debilitatingeffects of the channel (part (a)) in the frequency band of the signal and to remove all the out-of-band frequencies.The resulting received signal spectrum (part (e)) is again a close copy of the transmitted signal spectrum, in stark contrast tothe received signal spectrum in [link] where no shaping was attempted.

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