In fact, the field has been declared dead again and again. I think that there is a moral in that, because each time that it's declared dead, something new comes along.

My recollection is that information theory was declared dead at MIT in the Sixties specifically because we knew how to get to channel capacity, or at least to "practical" channel capacity. You got to R_0 (which we then called R_comp) by using sequential decoding.

In my first practical work at Codex Corporation I worked on a power-limited additive white Gaussian noise channel, the deep space channel. We eventually implemented sequential decoding on that channel, and we basically thought that that was the end of the story. Of course since then there have been a few developments. Little things like the Viterbi algorithm.

In fact, the Viterbi algorithm, with a very long constraint-length convolutional code, and with sophisticated concatenated coding, was what was used to save the Galileo mission recently. So, even on the power-limited white Gaussian noise channel the field was not dead then, and it has continued to evolve over all these years.

We can now recognize that we really knew how to approach capacity then only on power-limited Gaussian channels. On the bandwidth-limited Gaussian channel, there have been tremendous developments in recent years, and I would assert that only in the past few years have we really learned how to approach the capacity of the bandlimited Gaussian channel.

The essential breakthrough from a coding point of view was of course trellis codes. Trellis codes were completely unanti- cipated until Ungerboeck invented them. They led to a whole field of Euclidean-space coding that we had several sessions on at Trondheim, and that is a very active field of research today.

We still do not have good methods of constructing complicat- ed trellis codes, so that is a very good open area for research.

Then there was a little thing called shaping that was not even recognized as a problem until the past couple of years. What is shaping? Ordinarily with trellis codes you use a QAM constellation with a uniform probability distribution over the points. But a uniform distribution loses one and a half dB compared to a Gaussian distribution over the signal points. So, somehow you have to develop a method for get- ting a non- uniform distribution over the points in your signal constellation. These methods collectively are known as shaping. Shaping really has only been worked on in the last five years. It is a wholly new topic, but there is not too much to it. I do not recommend that graduate students go into it, but it just illustrates that there can be new aspects to what was thought to be an old problem.

All of this is still for ideal Gaussian channels. For a real Gaussian channel like the telephone-line channel, you have to deal with intersymbol interference, with equalization, with the fact that the channel does not have an ideal brickwall response. Methods for doing equalization in combination with powerful coding and shaping have been developed extremely recently and are a very actively developing area in communications.

Precoding is really about how do you realize Shannon's prescription of doing water pouring across the channel. First of all, you have to measure the channel, so you have to develop line probing techniques to measure what is going on across the channel. You really need to know the signal- to- noise ratio as a function of frequency. Then, how are you going to combat this? There is a relatively straightforward way that you would think of, for instance, if you read Gallager's text, which is to slice the channel up into narrow little bands and do multi-carrier modulation, with individual modulation in each of the bands. That is not a bad approach for some applications, but on the telephone- line channel, for instance, it simply involves too much delay for the kinds of computer applications that modems are used for.

So, what is a single-carrier way of doing this? Well, various techniques collectively known as precoding (but different from other kinds of precoding that you know about) have been developed.

Many of you may know that there is has been a telephone-line modem standard under development in the past three years, originally called V.fast but now just ratified three weeks ago under the number V.34. During the course of that development, I believe that there were something like four different versions of precoding, each one better that the last, each one building on the last. Rajiv Laroia had a pa- per at Trondheim on the final version that is in the standard. So there have been tremendous developments in the area of doing combined coding, shaping and equalization.

Theoretically, this raises a lot of interesting questions. I think that one of the lessons of my experience in this field has been the interplay between theory and practice, and how practical advances lead to interesting hard questions for information theory. This precoding algorithm that I have mentioned is based on the idea of decision-feedback equalization, except it puts the decision feedback in the transmitter in a certain way. Now, there are various theorems of one degree of rigor or another which indicate that all you need to get to channel capacity is decision- feedback equalization. You do not need to do maximum- likelihood sequence detection to get to channel capacity. I do not believe that anybody has made this proposition rigorous. All of the arguments basically use the assumption that you make in decision feedback that all decisions are correct. Under that ideal assumption, in fact, you can show that you can sometimes do better than capacity. So, to develop a really solid proof that decision-feedback- equalization performance is sufficient to get to capacity, at least on high-SNR channels, would be an example of an in- teresting theoretical problem that is directly suggested by these re- cent practical advances.

I could say a lot more on the same theme about open ques- tions in Euclidean-space coding, about connections with com- plexity, connections with system theory, and maybe I will as the discussion goes on. But, the overall message that I would like to leave with you is this. About every ten years I hear the statement that information theory is dead. At the same time, practical people are off working away trying to come even close to realizing what information theory says is possible. Great advances have been made, and they in turn have thrown back rich new questions for Shannon theory people to work on. So, I just view the field as following its nose in the way that a discipline does, and I see it as being as healthy now as I have ever seen it.