mohatta2015.pdf

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One extreme is to be given a limit on complexity (e.g., cost) and then design the
equalizer that optimizes performance for that cost. Another extreme is to be told
a performance requirement (e.g., bit error rate) and then design an equalizer that
minimizes cost while still meeting that requirement.
Alas, life is rarely so simple. As for cost, there is usually some flexibility. As for
performance, there is usually a number of performance requirements. Sometimes
there is a limiting performance requirement, such that if that one is met, the other
ones will be met as well.
So where do these requirements come from? Some come from a standardization
organization which sets performance requirements for standard-compliant devices.
Some come from the industry, such as cellular operators, which want good performing
cell phones in their network. Then there is the need to be competitive with
other companies who make devices with equalizers. Whether their devices perform
much better or much worse than yours can have an impact on which devices get
purchased (or not, if other features are more important).
Back to equalizer selection. One useful tool in the selection process is to compare
the performance of different equalization approaches in scenarios of interest. There
are often channel conditions and services of interest identified by standardization
bodies, the customer, or you. There is also usually an operating point, such as an
acceptable error rate for speech frames. If a much more complicated equalizer gives
a very small gain in performance at the operating point, then it may not be worth
the effort.
8.2.3 Radio aspects
In addition, there are radio aspects we have not considered. To understand these,
we have to think of the received samples as complex numbers, with a real and
imaginary part. Thus, each number is a vector in the complex plane as shown in
Fig. 8.2. The horizontal axis corresponds to the real part, also referred to as the
in-phase (I) component. The vertical axis corresponds to the imaginary part, also
referred to as the quadrature (Q) component. We can also think in terms of polar
coordinates, with amplitude and phase.
One radio aspect is frequency offset. Like commercial radio stations, the receiver
must tune to the proper radio channel. If it is slightly off, a frequency offset is
introduced. If the offset is small, it can be modeled as multiplication by a complex
sinusoid:
r m = [COS(2K f 0 mT) + j sm(2n fomT)][cs k + ds m _i] + n m , (8.10)
where f 0 is the frequency offset in cycles per second (or Hertz) and T is the symbol
period. We can think of this as introducing a constant rotation in the complex
plane, with the phase changing linearly with time. Automatic frequency control or
AFC is used to estimate this offset and de-rotate the received samples to remove
the frequency offset.
Another radio aspect is phase noise. It can be modeled as a time-varying additional
phase term to the sinusoids for frequency offset, giving
r m = [cos(2nf 0 mT+e(mT))+jsm(2nfomT s +e(mT))\[cs m +ds m - 1 ]+n m , (8.11)