Ever wonder how OFDM-based systems estimate the channel response? I know I have. In fact it was the topic of my M.Sc. dissertation. The idea is quite simple: we dedicate a fraction of what I will loosely call the degrees of freedom of the system to the task at hand. As a case in point, a portion of the orthogonal subcarriers within an OFDM bandwidth may be used to estimate and/or track the frequency selective channel. So let me define energy and frequency as two degrees of freedom. For example, a widely accepted and also widely employed means of acquiring channel state information (CSI) at the receiver of wireless links is to multiplex known pilot symbols into the transmission data stream; a technique referred to as pilot symbol aided modulation (PSAM). PSAM directly utilizes both energy and frequency, and the channel response is then obtained using either Bayesian or ML techniques. I should note that using pilot symbols has been around for a long time; in fact it dates all the way back to the early days of FM radio. Interestingly it has proven to be so robust that it is even used in modern systems such as WiMAX/802.16. The degrees of freedom in PSAM have been optimized in almost every way an engineer can imagine: to minimize error rate, to maximize capacity, to minimize MSE, etc.
So what else can we do? Well this brings me back to what I called degrees of freedom. I forgot to tell you that PSAM is designed for open-loop systems. The TX allocates energy and frequency to the pilot symbols without any knowledge of the channel within which the pilots propagate through. Intuitively one would expect that the degrees of freedom be uniformly distributed: both in the sense of energy and in the sense of frequency. This is in fact the case and is a well established theoretical fact: equi-power, uniformly frequency-spaced pilots are optimal in the SER, capacity and even MSE sense. Well what if the TX somehow had CSI through, e.g., an ideal feedback link? You expect that the TX now makes a more intelligent decision on distributing the degrees of freedom, right? For example say a certain subcarrier frequency is experiencing a deep fade, the TX may opt to allocate a pilot symbol to this fade instead of a data symbol. Clearly this will be at the expense of channel MSE (uniform pilots are always MSE optimum). So given a certain number of data symbols M, and a certain number of pilot symbols P, where M+P=const, we clearly have a fundamental trade-off we may optimize.
I hope I have convinced you that uniform pilots are no longer (at least intuitively) optimal in any sense. So if you’ve managed to read this far and are curious to find out what the optimal pilots are I encourage you to check out [J1] on my website. There I look at pilot symbols that obtain maximal SNR at the RX. For my MIMO course project, and based on [J1], I extended to pilot allocations that optimize for minimum SER and also for allocations that maximize ergodic capacity. I’ve also got some nice results using vector quantization and Alamouti STBC. I’m putting some finishing touches on these results and will hopefully submit a finalized version to IEEE Transactions on Vehicular Technology later this month. Of course I’ll have a preprint up afterwards, so stay tuned!
