[opendtv] Re: COFDM and equalization
- From: Eory Frank-p22212 <Frank.Eory@xxxxxxxxxxxx>
- To: "'opendtv@xxxxxxxxxxxxx'" <opendtv@xxxxxxxxxxxxx>
- Date: Mon, 19 Apr 2004 23:33:41 -0700
Henry Baker wrote:
>I have a very dim understanding of COFDM.
>I understand that it involves splitting up the
>bandwidth into thousands of independent narrow
>channels, and that FFT's can be used to efficiently
>do a lot of the processing.
So far your understanding is far from dim...
>The question is the following: if the channel is dispersive,
>meaning that the delays are frequency-dependent, then a
>COFDM whose channels are narrow enough shouldn't care at all,
>since it is unlikely that a single narrow channel would be
>wide enough to disperse the frequencies within the channel.
Not exactly. Individual carriers suffer some damage -- in magnitude, phase or
both -- due to the dispersive channel. If you examine the spectrum of the
overall channel under frequency-selective fading conditions, you can readily
see which carriers are attenuated and which are boosted by the non-ideal
characteristics of the channel.
>Many equalizers are trying to recover the proper shape of
>a signal, but if the signal is very narrow-band, the signal
>is always a good approximation to a sine wave, no matter
>what happens to it in the channel.
It had better not be a good approximation to a sine wave! Those individual
carriers are modulated by data, so they should be QAM constellations, albeit
very narrowband ones.
>So why would a COFDM ever need equalization?
To minimize the mean square error and therefore the BER of each individual
carrier.
>(I realize that multipath could involve polarization and
>cancellation, but this merely kills the capacity of one
>or more of the individual narrow-band channels, which should
>be taken care of by FEC, not equalization.)
A plain-vanilla COFDM equalizer -- the simple variety implied by the DVB-T spec
-- is simply a complex spectrum analyzer (i.e., FFT) and a complex divider. The
scattered pilots (training signals) are interpolated in frequency and in time,
to build a complex-valued channel estimate for each individual data carrier --
an estimate that is updated very frequently. The equalizer simply divides the
data carriers, D(f), by the channel estimates, C(f): Deq(f) = Din(f)/C(f).
The FEC indeed comes into play in equalization. At frequencies where C(f) has
attenuation, this division process enhances the noise, thus degrading the BER
for those data carriers. At frequencies where C(f) has gain, the opposite is
true. These effects can be reflected in the confidence values (soft decisions)
given to the FEC decoder for each carrier, so that the FEC has the most
accurate knowledge of which carriers' symbol decisions are less reliable and
which are more reliable. Spectral nulls due to multipath don't necessarily wipe
out particular carriers, but they do degrade them.
An optimum equalizer should not simply apply the inverse channel response to
the data -- should not simply flatten the overall frequency response -- but
should instead minimize the mean square error in the symbol decisions. This is
one difference between the plain-vanilla approach and more sophisticated
approaches.
But the plain-vanilla approach works well and has a relatively modest hardware
(silicon area) cost. It is not the "magic" of the equalizer that makes this so,
but the brilliant signal structure -- pilots and guard interval -- that allow a
simple, straightforward equalizer approach to perform acceptably well for most
channels. The system was designed to allow a low-cost simple approach to "just
work."
-- Frank
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