Building Blocks
Equalizers are yet another area where advances in digital technologies
have reaped big dividends for modern communications systems.
By Rob Howald
It wasn't very long ago that people in this country were satisfied with reliable
voice communication using that revolutionary instrument, the telephone. There are
still plenty of places in the world where the system here is enviable by comparison.
Now, I'm still annoyed that the area codes changed on us a few years ago. Taken for
granted in this country, the network and its twisted-pair termination is only described
nowadays in terms of how much multimedia streams can possibly be plowed through this
seemingly overmatched channel. The only time you hear about a simple voice telephone
is if it's a cellular one, or when a highly placed federal government employee is
using it to fill his campaign coffers. Clearly, the appetite for lots of data has
outpaced the ability of existing networks to deliver it. At this point, the telephone
channel, by default, has necessarily been a focal point. It's the only thing that
everybody is connected to. The cable channel, the next most-connected network, is
also
hoping to cash in on this data lust. Unfortunately, this system was not designed
with web surfers in mind, either. The fact is, if this desire for gobs of instantaneous
content had been foreseen, there's a good chance you wouldn't be waiting in frustration
when you point your browser to a web page.
Pipe-busters
To the rescue come technologies that turn the existing trickling pipe into a neighborhood
water main, in the form of asymmetric digital subscriber line (ADSL), hybrid
fiber-coax
(HFC), and fiber-to-the-curb (FTTC). These technologies are all competitors. Thanks
to them, today's consumer dreams about Doom played across the neighborhood (for our
more mature audience, Doom is a PC/video game), or interacting with Oprah over the
PC (for our audience living under a rock, she's a TV talk show host) are approaching
reality, in part, with fascinating advances in the ability of digital receivers to
perform in sometimes horrific channels. Among major technology improvements
that
have aided this effort, perhaps only forward error correction (FEC) and combined
modulation and coding have played a bigger role than equalization. In last month's
piece, the fundamental concepts of channel problems, intersymbol interference (ISI),
and pulse shaping were discussed. In this column, equalization will be outlined.
The $100k engineer
Maybe you keep an eye out for salary surveys in EE Times or IEEE, maybe you don't.
If you are in the latter group, my boss
would like to speak to you. It has become
evident that there is a shortage of skilled, ASIC design engineers. The shortage
exists in both the digithead-only type, and the even more rare breed, the mixed-signal
designer. The result has been substantial climbs in average income in these disciplines.
The growth and development of digital receiver technology is a big reason why. The
major technology improvements described above are all functions that must be performed
in the digital realm, with perhaps some
small exceptions to the modulation itself.
Equalization is, by its nature, superbly suited to discrete-time operation in digital
receivers. Mixed-signal designers are elusive, and the need is driven by the fact
that regardless of how many fancy digital functions are embedded, there is generally
an analog channel to be driven or sampled, and sometimes (yikes!) an RF channel.
Consider the ISI concept presented in the last month's column (I warned you to save
it!). The basic nature of this
problem is that ideal rectangular pulses have infinite
bandwidth. Not many channels do. However, a wide enough channel will appear essentially
unbandlimited if the baud rate is low enough. However, this is spectrally inefficient.
Instead, the pulses are bandlimited, such as in raised cosine filtering, or, in a
more passive sense, by just bandpass filtering the transmit output. In either case,
pulses become rounded, and their energy dispersed into adjacent symbol times. For
Nyquist shaping, there is still
ideally no interference among symbols. But things
happen in crummy channels. Primarily under consideration are distortions of amplitude,
phase response versus frequency, and reflections. Essentially, the frequency response
is not flat, the phase response is not linear, and multipath versions may appear
at the receiver.
The most basic equalizer form would reverse these distortions, thus reinstituting
an ideal frequency response to the detector. Consider that the degraded response
is known,
and the result will be adjacent symbols each multiplied by some different
constant associated with the distortion, and summed with the desired symbol. This
is mathematically expressed as follows:
Complete ISI removal is an approach of a zero-forcing (ZF) linear equalizer. Here,
the desired symbol is ak, with k representing the k-th sample of an infinite symbol
sequence. The aj represents adjacent
symbols in time. Kj represents the multiplicative
constant caused by distortion on the j-th sample. Of course, the sequence is not
infinite, and even if it were, all the terms wouldn't be significant enough to consider.
In the assumption above, the Kj are known. A digitally implemented equalizer, implemented
as a tapped delay line (see Figure 1), also called a transversal filter, would have
multiplying coefficients (-Kj) summed in at the adjacent symbol offset time, j. For
example, if five symbols on
either side of the desired symbol being sampled affected
the output, ideal equalization from an ISI-only perspective would sum the sample
taken to get:
Providing the distortion could be properly characterized and taught to the equalizer,
this equalization would remove the ISI associated with the distortion, with the goal
of recovering detection performance.
In practice, this finite impulse response (FIR) filter is the implementation of choice.
The complex issue of filter length requirements and trade-offs is outside the scope
of this column.
Linear equalization
Every channel can be characterized by an equivalent discrete-time transfer function,
like the response of a filter. The transfer function of the complete channel consists
of the cascade of a transmit pulse shaping
filter, a D/A converter usually with (x/sin
x) correction prior, the channel itself, and receiver filtering. This equivalent
channel response has a mathematical form expressed by a frequency response He(z),
where z is used to imply the discrete time-frequency domain variable (the z-transform).
As described in last month's introduction, there is a frequency response form that
must be maintained in order for there to be zero ISI. The zero ISI condition exists
when the transfer function, replicated in the
frequency domain at every multiple
of the sampling frequency, sums to a flat spectrum. When that is not the case, ISI
will occur.
There are two basic approaches to linear equalization, aimed at different ýoptimizingý
criteria. The most intuitively satisfying is for the equalizer to be the reciprocal
of the equivalent transfer function to produce zero ISI. This is essentially the
idea described in the introductory example above. Calling the equalizer transfer
function Lzf(z), we have:
Thus, the resulting cascade will again have the net flat response, and zero ISI,
because Lzf(z)He(z) = 1. There is a problem with this ZF approach when spectral nulls
exist, since the gain of Lzf(z) in those regions becomes very high. This results
in excessive noise enhancement, since the channel noise will also be passing through
the equalizer.
The ZF criteria aims to eliminate ISI completely.
However, because of the problems
cited, it may be more desirable from an error probability standpoint to allow some
ISI if it means less noise enhancement, and an overall smaller mean-square-error
(MSE) at the equalizer output from the exact transmitted symbol. This approach is
appropriately called the MSE criteria. While the ideal criteria would be minimum
detection error probability, this is not straightforward to determine. A reasonable
design goal is to minimize the combined error of AWGN and ISI at
the slicer input.
Under this criteria, the equalizer transfer function becomes:
where the channel AWGN is white, with two-sided density No/2, and Ps representing
the average transmit power. Thus, the second term in the denominator is basically
an SNR term to modify the straight ZF equalizer to improve overall MSE.
Decision-feedback equalization
As the name implies, decision
feedback equalization (DFE) uses decisions ý on the
symbol stream ý to correct distorted sequences. The logic behind this is clear: If
you know the correct answer to a bunch of symbols, and you know the channel sums
these past symbols into a symbol presently being detected, just subtract what you
know about those symbol values. There are two key points. First, you have to know
the correct answer. DFE analysis' work for low-error-rate assumptions. Because the
DFE is a feedback loop, incorrect decisions
leading to error propagation are a concern.
The second point is what to do about symbols on the other side of the desired pulse,
since the feedback equalizer only corrects for previously detected pulses, or postcursor
ISI. The answer is the old linear equalizer, in either form, where it is referred
to as the forward filter. Thus, the DFE topology looks like Figure 2. It turns out
that the ZF problem of nulls is no longer an issue when a linear equalizer is implemented
in a DFE, because the linear
equalizer in this case is only concerned with eliminating
half of the ISI, the precursor ISI. However, a ZF forward filter is still outperformed
in MSE by a DFE, which has a forward filter designed around the MSE criteria.
When operating as it is supposed to, the DFE provides significant advantages in terms
of noise enhancement for severe ISI. This is because the spectral null issue is no
longer, and
because correct decisions result in noise-free feedback. Another important
point with regard to the DFE is its relationship to optimal reception. One problem
with the equalization idea is that the optimal fundamental receiver structure is
well known for an AWGN channel, and an equalizer removes the ýwhitenessý of the noise.
Fortunately, it turns out that in a DFE approach, the forward filter output is compatible
with the well-known optimal receiver structures. This is because its noise component
is white
and Gaussian when the forward filtering is designed to minimize mean square
error.
The derivation of the DFE transfer function is more complex.1 Using a ZF approach
in the forward filter results in a cascade of the ZF filter mentioned above and a
linear predictor, which has a well-known solution. The feedback filter falls out
of the derivation as part of the linear predictor solution.
Advanced topics
The items mentioned previously are basic equalizer
functionality. Import-ant items
that must also be mentioned are adaptive equalization, and blind equalization. Tossing
in some reality, with some of these troublesome channels, the distorted response,
He(z), isn't known ahead of time. It has to be learned, and even after that, it may
change slowly over time. Adaptive equalizers are employed to generate and maintain
equalizer performance under these dynamic conditions. The learning is done through
a training sequence, a known data pattern, which is also
periodically transmitted
as overhead to adapt the equalizer to the changing channel.
Blind equalization is a type of adaptive equalizer that doesn't use a training sequence,
and thus adapts "blindly." It is a hot research topic currently, because of the obvious
advantages of zero-calibration, and because of the complexity.
More down-to-earth, fractionally spaced equalizers (FSEs) provide a topology that
is robust to symbol timing variation. Transversal filters operated at the symbol
rate do
not satisfy the Nyquist criteria for signal replication. While we are not
trying to replicate signals in these filters, the FSE topology allows the filter
to equalize beyond the Ny-quist frequency of half the symbol rate, hardening the
system to aliasing effects. The well traveled "T/2-spaced" equalizer topology is
one implementation of an FSE. Several significant papers have proved the superiority
of a fractionally spaced approach over a baud-rate sample time approach under the
nonideal conditions
often encountered in practice.2
This piece wouldn't be complete without mentioning an important transmission scheme,
known as orthogonal frequency division multiplexing (OFDM). It is known as discrete
multitone, or DMT, in ADSL lingo. While maybe not apparent from the above discussion,
ugly channels can make equalizers very complex. So complex, in fact, that transmission
schemes such as OFDM have been designed to eliminate the complexity by changing the
signaling format from the basic single
carrier QAM-modulated signal, to multiple,
low data rate carriers, each carrying a fraction of the total throughput over a narrow
bandwidth. If the channels are made narrow enough, then the lousy channel may look
nice, at least, over these narrow segments. Then at the receiver, only individual
gain adjusts are required if the subchannel processing is adequately delayed to compensate
for arrival-time variation. In this fashion, DMT-based ADSL makes the most of the
very nasty (for high-speed digital)
twisted-pair channel. QAM-based ADSL modems (also
called CAP in some literature) use the advanced equalizers. While DMT is more complex
than single carrier, the argument is that you are trading off a simple, off-the-shelf
IFFT/FFT function used in DMT to generate multiple carriers, and simple postprocessing
for an otherwise very complex single carrier receiver.
What next?
Just when it seems every possible excuse for a channel is being tried, a new one
comes along. Equalizers
are key to making this happen. My favorite is meteor-burst
communication. There are people working on using the power lines to pass data. Again,
this leverages something that goes to everyone's house. If only the plumbing system
wasn't grounded.
Robert Howald is a staff engineer in the transmission network systems group at
General Instrument's communications division in Hatboro, PA. He has a BSEE and an
MSEE from Villanova University, and is currently a PhD candidate at Drexel University.
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