For Gerhard Kramer, choosing which engineering field to specialise in came naturally: the most challenging one – from a mathematics perspective. Since the pre-internet days, he's been using maths to pursue the limits within the communications field, helping to advance optical fibre networks and to provide more efficient, reliable and secure transmission of data.
“I guess I was always interested in what seemed difficult,” Gerhard chuckles, sitting in his large, bright office at Technische Universität München (TUM) in the heart of the Bavarian capital. Now Professor and Head of the Institute for Communications Engineering at TUM, he started his career studying electrical engineering in his native Canada, then completed a doctoral degree in Switzerland. Attracted to mathematics as well as to the practical relevance of engineering, he was able to find an outlet for both interests in the communications field.
“At the time when I did my Bachelor’s degree, back in 1991, communications was considered the toughest topic from the math perspective,” he says, adding quickly: “Over time, though, you realise how limited you are and you respect it more.”
Realising the limits – and approaching them – has been a common thread throughout Gerhard’s research. At the heart of this pursuit lies a mathematical branch called information theory, which – among other things – deals with the ultimate limits for communicating data efficiently, reliably and securely.
Advancing optical fibre networks
One of Gerhard’s big contributions to the field has been his research on the capacity of optical fibre networks, while working for the US-based Bell Labs of the telecommunications company Alcatel-Lucent during the 2000s.
Fibre-optics technology, which uses pulses of light to transmit data, has progressed immensely over the past 15-20 years. Optical fibres have largely outcompeted copper wires – which use electrical energy – for sending data over long distances, enabling the high speed of today’s internet traffic.
“What’s slowly been happening, though, is that we’ve started to hit the limit of optical fibres – the maximum rate at which you can transmit data over one fibre,” Gerhard explains. “That’s not necessarily a real problem, because they are so thin that you can use many of them at the same time. But it’s not very economical to use for instance a hundred fibres, because you’d need a hundred lasers, a hundred amplifiers and so on. So it makes much more sense to try to increase the capacity of one fibre.”
Along with colleagues at Bell Labs, Gerhard pioneered a very accurate method for estimating the capacity limit of fibre channels. Predicting a capacity well above experimental observations, the team created quite some excitement in the world of fibre-optics. Their work also helped explain the reasons for the fibre capacity limitations, as dictated by physical laws. Based on this new knowledge, they presented a model for a so-called transparent optical network, which bypasses the need for decoding and recoding of bits at intermediate nodes.
“It doesn’t do what’s called optical-electronic-optical conversion – taking the optical signal and converting it to an electronic signal. Instead, you load the information on a wavelength, and the next node will just route that wavelength directly through. It’s simpler, it’s like a switch,” Gerhard explains. “This is the way optical networks look set to be going in the near future.”
Breaking equations into bits
After a year-and-a-half-long professorship – his first – at the University of Southern California, Gerhard was nominated by TUM for an Alexander von Humboldt Professorship, a prestigious German award designed to recruit top-notch international researchers. Already familiar with the language thanks to his German parents, Gerhard accepted the position and moved to Munich in 2010. He now leads a group of 10 PhD students and five postdocs, who work on different topics within the communications field.
A common denominator in their research on various communication channels is the use of mathematical models – within the context of engineering.
“Understanding the channel is quite crucial. You need a good model, and then you need to understand the mathematics behind it and break the equation down into its fundamental pieces. We want to use those fundamental pieces to communicate data efficiently, reliably and with high speed,” Gerhard says.
The equations come from physical laws of the channel material, for instance the silica glass used in optical fibres. “We take the math, developed by others, and use it for a purpose that the mathematicians weren’t thinking about,” he adds.
“What we want to accomplish is to understand how to signal across the channel, so that we can communicate with little loss of energy, and get the same information coming out at the receiver end as what we sent at the transmitter end.”
Dealing with strange behaviours
Although optical fibres have been used for long-haul telecommunications for some time, scientists don’t fully understand the fibre channel. “For example, we don’t know if the capacity increases when we put more energy into it. That always happens for so-called linear channels; the more energy you put in, the faster you can communicate over the channel,” Gerhard says.
Communication channels such as copper-wired or wireless telephone networks operate at low energy and can be described by linear equations, meaning – in the simplest of terms – that if you change the input, you’ll get a proportional change in the output. What’s special about the fibre channel is that an enormous amount of energy is pumped into a very thin piece of glass; in fact, the power per unit area in the glass fibre is greater than on the surface of the sun. This gives rise to strange behaviours in the channel – and to challenges for the engineers.
“Because you’re putting in so much energy, the atoms in the fibre can’t react in a harmonic way and start behaving inharmoniously. That causes non-linear behaviour. So rather than a linear relationship you get a quadratic relationship between the amplitude input and certain phase shifts in the output of the channel. And this non-linearity is very hard to deal with,” Gerhard says.
Surfing strange waves
Even so, members of his team are starting to discover new ways to communicate across fibre channels. Looking very promising in terms of boosting transmission rates, these non-standard methods involve wave forms that differ from those used in traditional channels.
“Think of a radio channel tuned to for instance 90 MHz, which is the frequency of the wave that’s transmitted. The waves have a sinusoidal shape, and we separate the channels by how fast the waves go up and down,” Gerhard explains. “For non-linear channels, using these sinusoids isn’t good enough. There are other types of wave forms that travel better through this medium, but it’s not completely understood which those are. One type is called solitons. Soliton wave forms can travel through the fibre medium in a very efficient way without losing energy.”
The soliton phenomenon was, in fact, first observed in water 180 years ago – in the form of a strange wave hitting the Union Canal in Scotland. “Usually waves in the ocean die off, but this one didn’t – it just kept on going and going without losing energy,” Gerhard muses.
“It’s remarkable that the same style of mathematical equation governs phenomena as diverse as water waves as well as electromagnetic and light waves. That’s also why mathematics is so powerful; the same methods apply to many problems, in this particular case, wave equations.”
Overriding signal noise
While fibre-optics is the channel of choice for sending data at lightning speed and with low loss over thousands of kilometres, on land or through oceans, other channels are better for other types of communication. Short-range wireless communication is one example – and another area in which Gerhard’s researchers are chipping away at various challenges.
One challenge is how to design signals that stand out from the noise in wireless channels, where electromagnetic or air pressure waves spread out in all directions, bouncing off walls to create noisy versions of the signal. Again, it’s all about getting the same information to the receiver as what the transmitter sent – and getting the channel to perform near its ultimate limit.
Designing smart codes
Another variation of this problem is one of the hottest topics in the communications field – code design. “Code design is about designing check sums so that you can build a receiver that uncovers things in an efficient way, in terms of complexity, energy, computation and so on,” Gerhard says.
Check sums are a smarter way of encoding data than, for instance, using feedback mechanisms or repetition, which take time. Gerhard explains with a simple example of how to get a message across: “If I say everything twice, hopefully my message will get across more reliably than if I say it just once. But that’s very inefficient – the repetition takes twice as long. Instead, I could say two things followed by their sum, for example ‘one-two, three’ or ‘two-four, six’. The sum is a check for you. If you heard ‘two-three, six’, you know a mistake was made.”
Although the example is a simple one, that’s exactly what coding is – building many different sums or linear combinations of bits of information. The sums can be designed in a sophisticated way so that large amounts of data can be transmitted without mistakes and at a fast rate.
“The problem with these sums is how to recover the correct answer if you start making mistakes. That’s the hard part – not the encoding but the decoding under errors,” Gerhard says. He also has a student working in this area, using a set of practical codes called turbo codes that have helped revolutionise the coding world over the past 20 years.
Others in his group work on the equally hot area of secrecy communications, covering topics such as information hiding, encryption and authentication.
Finding comfort in maths
Apart from guiding the general direction of research in his group – and teaching Bachelor’s and Master’s courses – Gerhard gets involved with specific research projects as much as he can. “I try to ‘get my hands dirty’ if I have the time. It also depends on the topic – on some topics I can do more than on others,” he says.
And in the fast-lane that is the communications field, research training goes both ways between mentor and mentee. “If things are unclear to me, then of course my research students have to explain it to me,” says Gerhard, who still feels most at home in the information theoretic arena.
“I find math comforting in that it gives you the guidelines; you have logic and rules and you have to deal with those rules. That’s easier, I think, for some people to work with. And engineering – electrical engineering and communications – was always the right thing for me.”
– By Lillian Sando