Archive for the 'Miscellaneous' Category

Overview of MIMO Broadcast Channel Capacity Results

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The broadcast channel is a communication channel with a single transmitter that transmits independent information to multiple receivers. In information theory, capacity of a communication channel is defined as the maximum rate of information that can be transmitted from the transmitter to the receiver reliably. By reliable transmission we mean that the information can be decoded at the receiver with arbitrarily small probability of error. Let us assume that the broadcast channel has $N$ receivers, then a rate vector ${\bf R} = [R_1, R_2, \ldots, R_N]$ is said to be achievable on the broadcast channel if information can be transmitted reliably at rate $R_n$ from the transmitter to the receiver $n$,$n=1,2,\ldots, N$, simultaneously. The capacity of broadcast channel is defined as the convex hull of all achievable rate vectors. Finding the capacity of a general broadcast capacity has been a long standing open problem in information theory. In this article we will discuss the multiple input multiple output broadcast channel (MIMO-BC) whose capacity region has been found recently by Weingarten et. al.

The MIMO-BC model is as follows. A transmitter with $M_t$ antennas wants to transmit independent information to $N$receivers, where $n^{th}$ receiver is equipped with $M_n$ antennas $n=1,2,\ldots,N$. Let ${\bf x}$ be the signal transmitted by the transmitter, then the received signal ${\bf y}_n$received by receiver $n$ is given by \[{\bf y}_n = H_n{\bf x} + {\bf v}_n,\] where $H_n$ is a $M_n\times M_t$ channel matrix and ${\bf v}_n$ is the additive white Gaussian noise (AWGN) with covariance matrix ${\bf N}_n, \ n=1,2,\ldots,N$.

For MIMO-BC, several transmission strategies have been proposed in literature, for example, superposition coding and dirty paper coding. In superposition coding, if ${\bf x}_n$ is the signal for receiver $n$ then ${\bf x}= \sum_{n=1}^N {\bf x}_n$ is transmitted by the transmitter. Dirty paper coding (DPC) is a technique developed by Costa for single input single output (SISO) AWGN channels when there is an interference signal present at the receiver together with AWGN. We assume the interference signal is known non causally at the transmitter. In this setting Costa showed that the capacity is $\frac{1}{2}\log(1+SNR)$ which is exactly equal to the capacity of AWGN channel without any interference. Thus, DPC is shown to completely eliminate the effect of interference. Using DPC for the MIMO BC, if the signal for the first receiver is ${\bf x}_1$, then the signal ${\bf x}_2$ for receiver $2$ is generated using ${\bf x}_1$ as the interference signal. Similarly, the signal ${\bf x}_n$ for the receiver $n$ is generated using ${\bf x}_1,\ldots,{\bf x}_{n-1}$ as the interference signals and ${\bf x} = \sum_{n=1}^N{\bf x}_n$ is transmitted from the transmitter. From now on we consider $N=2$ in this article for simplicity. With $N=2$,the rates $R_1$ and $R_2$, simultaneously achievable for receiver $1$and $2$, are given by \[R_1^{DPC} = \frac{1}{2}\log\frac{\det\left({\bf S}_1+ {\bf S}_2 + {\bf N}_1\right)}{\det\left({\bf S}_2+{\bf N}_1\right)}\] and \[R_2^{DPC} = \frac{1}{2}\log\frac{\det\left({\bf S}_2 + {\bf N}_2\right)} {\det\left({\bf N}_2\right)},\] where ${\bf S}_n$ is the covariance matrix of ${\bf x}_n$. Changing the order of coding, i.e. first generating ${\bf x}_2$ and depending on ${\bf x}_2$ generating ${\bf x}_1$, \[R_1^{DPC} = \frac{1}{2}\log\frac{\det\left({\bf S}_1 + {\bf N}_1\right)}{\det\left({\bf N}_1\right)}\] and \[R_2^{DPC} = \frac{1}{2}\log\frac{\det\left({\bf S}_1+ {\bf S}_2 + {\bf N}_2\right)}{\det\left({\bf S}_1+{\bf N}_2\right)}.\] We denote the convex hull of all rate vectors achievable with DPC as $R^{DPC}$.

As shown above, finding the achievable rate region of MIMO-BC is easy using different transmission strategies but proving that they are capacity achieving, i.e. rate more than the rate achieved by a particular transmission strategy is not achievable with reliable transmission constraint, is quite difficult. Weingarten solved this problem in their paper and showed that rates achieved with DPC are optimal for MIMO-BC.

Here we present the argument for symmetric MIMO Broadcast channels from Weingarten et. al.’s paper, where the number of antennas at the transmitter and each receiver are equal. We also only consider two receivers here for simplicity, the general case follows similarly. As before let the received signal at receiver $n$ be $Y_n$, where \[{\bf y}_n = {\bf H}_n{\bf x}_n + {\bf v}_n, \ n=1,2.\] Since we are assuming ${\bf H}_n$ to be a square matrix, without loss we can multiply the received signal by ${\bf H}_n^{-1}$ without changing the capacity. Note that ${\bf H}_n$ is invertible with probability $1$. This is formally defined as aligned MIMO BC (AMIMO-BC) with \[{\bf y}_n = {\bf x}_n + {\bf v}_n, \ n=1,2.\] Then a transformation of AMIMO-BC is considered, called degraded and aligned MIMO BC (DAMIMO-BC), where \[{\bf y}_n = {\bf H}_n{\bf x}_n + {\bf v}^{*}_n, \ n=1,2.\] with $cov ({\bf v}^{*}_1) \le cov({\bf v}^{*}_2)$, $cov(.)$ denotes the covariance matrix. For any two matrices, ${\bf A}, {\bf B}$,by ${\bf A} \le {\bf B}$ we mean ${\bf A} -{\bf B}$ is a negative semidefinite matrix. It is easy to see that DAMIMO-BC is a degraded BC channel, i.e. one receiver receives less noisy signal than the other. For the degraded BC it is known that superposition coding is optimal, however, it is not known whether Gaussian signaling is optimal or not.

{\it Remark:} For a single input single output degraded BC (SISO-DBC), where the transmitter and each receiver has a single antenna, Bergmans showed that Gaussian signaling is optimal, however, the proof of Bergmans does not extend to the DAMIMO-BC case.

The optimality of Gaussian signalling is shown as follows. Let $R^{G}$ be the set of all achievable rate vectors $\{R_1^G, R_2^G\}$ with Gaussian superposition coding for DAMIO-BC and let ${\bf B}_1$ and ${\bf B}_2$ be the input covariance matrices achieving $R_1^G$ and $R_2^G$. To show that any rate vector which does not belong to $R^G$ is not achievable, an enhanced DAMIMO-BC is introduced, which is defined as \[{\bf y}_n = {\bf x}_n + {\hat {\bf v}}_n, \ n=1,2.\] where \[cov({\hat {\bf v}}_1) \le cov({\bf v}^{*}_1), cov({\hat {\bf v}}_1) + cov({\hat {\bf v}}_2) \le cov({\bf v}^{*}_1) + cov({\bf v}^{*}_2).\] Clearly, any rate vector achievable on DAMIMO-BC is also achievable on the enhanced DAMIMO-BC, thus the capacity region of DAMIMO-BC is contained inside the capacity region of enhanced DAMIMO-BC.

Next, it is shown that for any DAMIMO-BC there exits an enhanced DAMIMO-BC such that the following properties hold
1. Proportionality: There exits $\alpha_1 \ge 0$, such that \[\alpha_1\left(B_1 + cov({\hat {\bf v}}_1)\right) = cov({\hat {\bf v}}_{2}).\]
2. Rate Preservation: The rate vector $(R_1, R_2)$ achievable on enhanced DAMIMO-BC with Gaussian signaling is equal to rate vector $(R_1^G,R_2^G)$ obtained with Gaussian signaling for DAMIMO-BC.
Then using the proportionality and rate preservation properties, it is shown that any rate vector which does not belong to $R^G$ is not achievable on the enhanced DAMIMO-BC, using Bergmans proof for SISODBC. An important point to note here is that the Bergmans proof works
because of the proportionality property, which is hard to show for DAMIMO-BC. Since the capacity region of DAMIMO-BC is contained inside the capacity region of enhanced DAMIMO-BC, it follows that any rate vector which does not belong to $R^G$ is not achievable and thus Gaussian signaling is optimal.

Coming back to the original AMIMO-BC, to show that DPC is optimal, the authors show that any rate vector which lies outside the region $R^{DPC}$ also lies outside the capacity region. Toward that end, it is shown that for any vector ${\bf R}$ which lies outside the region $R^{DPC}$ there exists an enhanced DAMIMO-BC whose capacity region contains the capacity region of AMIMO-BC, but does not contain ${\bf R}$. Recall that we found out the capacity region of enhanced DAMIMO-BC and showed that Gaussian signaling is optimal, thus it follows that any rate vector ${\bf R}$ which lies outside the region $R^{DPC}$ also lies outside of the capacity region of AMIMO-BC and we conclude that DPC is the optimal transmission strategy.

Write a good paper or file a good patent? Or both?

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As a grad student and possibly a system engineer in the near future, I have been thinking about the importance between the two: writing a good paper or filing a good patent? If you are capable of doing both, what is your choice?

Writing a patent is quite different from writing a paper. A good paper should have a clear derivation of a good algorithm that has never been proven before. It should be academically challenging, mathematically tractable & reversible and conceptually innovative. A good example is Teletar’s MIMO capacity paper, Zheng & Tse’s D-MG tradeoff paper and the GLP paper by Dr. Love and Dr. Heath.

A good patent, however, is quite different. A good patent should have broad (as broad as possible) claims which are important in implementing and making real systems / devices. It focuses on new concepts, new methods and new apparatus that have never been claimed before. A good example of it is a series of handover patents by Qualcomm. Qualcomm’s MAHO (mobile assisted handover) patent and so called “501” (soft handover) patents has the following first claims:
[MAHO] 1. A method of mobile assisted handoff in a cellular communication system comprising a plurality of mobile stations and a land system and a plurality of cells, comprising the steps of:
assigning a mobile station a list of cells to measure, wherein said list is divided into two sections, a steady section and an alternating section;
measuring quality level of each assigned cell;
reporting said quality levels to said land system;
transferring a cell from the alternating section of the list to the steady section of the list when said cell has a quality level higher than one of the cells in the steady section, wherein the cell with the lowest quality level in the steady section is transferred to said alternating section; and
changing said cells listed in said alternating section every predetermined period of time.

[501] 1. In a cellular telephone system in which a mobile system user and another system user communicate user information signals therebetween via at least one of a plurality of geographically separated cell-sites each defining a respective geographic service area, a system for directing communications between said mobile system user and said another system user via said at least one of said plurality of cell-sites as said mobile system user changes cell-site service areas, comprising:
means for, while said mobile system user is in a service area of one cell-site and communicating user information signals with said another system user via said one cell-site, determining a transition of said mobile system user from said one cell-site service area to a service area of another cell-site, and for providing a handoff request identifying said another cell-site;
means responsive to said handoff request for coupling a communication of said user information signals between said mobile system user and said another system user via said another cell-site while said mobile system user and said another system user continue in communication of said user information signals via said one cell-site such that said mobile system user and said another system user concurrently communicate said user information signals through said one cell-site and said another cell-site; and
means responsive to said coupling of communication of said user information signals between said mobile system user and said another system user via said another cell-site for terminating said communication of said user information signals between said mobile system user and said another system user via said one cell-site with said communication of said user information signals continuing between said mobile system user and said another system user via said another cell-site.

As you see, MAHO and ‘501’ patents contain almost every possible method for the soft handover (SO) and mobile assisted handover (MAHO). It means that if you build a system with handover algorithms at cell edge, you have to design a system without SO and MAHO, to avoid those patents. You can’t either receive two pilot sequences from two base stations simultaneously, or compare the signal strength at the mobile side between the two. Then, what can you do? Will you pay millions of dollars to the legal owner of it, or try not to use both SO and MAHO?

This is the main reason why the WiMAX system uses only hard handover (HO) and why the cell edge performance of that system doesn’t quite good. Another bad new is that even SO and MAHO was flied in 1992 and 1994 respectively (the patent right is valid for 15 years) there are tons of patents which covers another aspect of SO and MAHO. It will be the same in CDMA, LDPC, and probably MIMO.

As explained above, a well-written patent influence on the whole system design. In addition, it presents a huge amount of money to the legal owner (usually a company) and to the inventor (as some form of incentives). For example, Samsung pays about several hundred million dollars to Qualcomm as the CDMA patent right PER YEAR. And when they make a cell phone, they must use Qualcomm’s MSM chip and pay 5.25% - 5.75% of whole retail price as a royalty. Samsung makes at least 100 million cellphones per year.. imagine the amount of money that the CDMA patent makes. Yes, it is quite huge, and a little part of it (which is given to the inventor) is also huge.

The reason that I am writing this article is not to debase the importance of writing a good paper. I think it is one of the most important things to do as a grad student. (Especially the best paper award is a life-long honor. Isn’t it, Kaibin, Ramya, Chan-Byoung?) But it could be better, if you think about filing a patent for your brilliant idea BEFORE you submit it as a paper. (Usually if a thing is already presented and published before, it can’t be the claim of a patent.) You could be a millionaire, or if not so, there is nothing to lose.

I think a good patent may not guarantees a good paper. But a good paper may guarantees a good patent, because the patent can scale down the novel idea in the paper into a practical method and apparatus. One good example is the Alamouti Code. S. Alamouti and V. Tarokh have filed a patent (US 6185258) in Sep. 16, 1997, just before they submit the famous Alamoti code paper.
[Alamoti] 1. An arrangement comprising:
a coder responsive to incoming symbols, forming a set of channel symbols that incorporate redundancy, where the coder employs replications and, at least for some of the channel symbols, forms a negative of an incoming symbol, forms a complex conjugate of an incoming symbol, or forms a negative complex conjugate of an incoming symbol; and
an output stage that applies said channel symbols to at least one transmitter antenna to form at least two distinct channels over a transmission medium.


As you see in the first claim above, technically any diversity based STBC may infringe their right. It is really terrifying. And it is more terrifying if I can find a patent that has a claim like this: “a device that uses multiple transmit antennas and multiple receive antennas for communication.” It will be all MIMO companies’ nightmare.

The Gold Prize

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As you may know, I was out of town last week and got back to Austin last Saturday. I’m rather inclined to doubt it but I was awarded the Gold Prize in the 2007 humantech paper contest sponsored by Samsung.

The contest receives about 900 submissions every year in fields like including signal processing, analog circuit design, communications and networking, computer science, computer engineering, mechanical engineering, material science and process, physical devices, and physical science. Only six students are awarded the Gold prize annually, and thirty four graduate students share the Silver, Bronze and honor prizes.  There was only one Gold prize winner in the communications & networking area in the last five years.

I’ve never been awarded this kind of big prize in my life. Anyway, I will treat you guys soon.

V2V Communications in Japan

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Recently, there has been increased discussion on Vehicle-to-Vehicle (V2V) communications. I find this area to be interesting and challenging to the wireless engineer because it opens a completely new field for radio environment, demands, application, etc. (Yes, systems like trunked radio has been around for few decades but I assume we’re talking more than a simple FM system.) Since vehicles are central to our lives, especially in the US, this is probably the best place for applications to emerge. Being a gearhead certainly adds to my interest.

I cannot say that I fully follow the V2V activities, but the following news caught my attention which seems to indicate that there is a strong movement towards V2V communication and ITS in Japan.

On Feb. 4, 2008, Denso announced the beginning of Vehicle-to-Vehicle communication tests on public roads in Japan (see press release here). It uses 669 - 679 MHz UHF and 5.8GHz band DSRC. The V2V communication is apparently part of the Japanese Ministry of Internal Affairs and Communications’ initiative “Ubiquitous ITS”, which includes digital video broadcast (or VICS, vehicle information and communication system), 60-76GHz mm-wave sensor application, cellular service, short range applications (expressway tollgates, DSRC, wireless service transaction, etc).

The reason for using 669-679MHz above is only to avoid interference with the TV broadcast in the 700MHz band, of which 715-725MHz is expected to open for ITS use in 2011 when the analog TV broadcast goes off air in the 700MHz band.

Some interesting areas of ITS are identified here at the ITS Japan website.Traditional wireless systems (cellular, WiFi, etc) has always had a  clear demand: cellular service to provide bidirectional voice services, and WiFi to provide fast download and so-so upload. For ITS, at least to me, it is not clear what kind of data traffics are needed, that is to say, what type of wireless technologies should be used to support the service. Look at it another way, this area is wide open with possible applications, technologies to support it, and perhaps new business models.

So I thought it would be interesting to throw out a question: What can we do in ITS?

Does MIMO enable certain applications? Or will new applications drive towards new technology?

Some links for further reading:

Automated Theorem Prover

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Ever wondered what does it take to construct a proof for a theorem, in my mind a lot of hard work, mathematical ability and sometimes stroke of luck. However things are generally not that simple, at times figuring out a proof can drive you nuts, make you crazy, especially if the statement of theorem looks obvious. To relieve us from the misery of figuring out a proof people are developing automated theorem provers and in future we might just have to type in some equations into a software and boom it gives a proof out.

For a preliminary solver, check out this link.

Part I : Introduction to antenna design in cellular systems

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I have received a lot of questions on MIMO antenna design and how it applies to cellular systems. I hope this (long overdue) blog post helps motivate the need for antenna design in cellular systems and answer some common questions regarding the challenges faced by antenna designers. The details of antenna design, especially for base stations, will be discussed in a future blog post.
So here goes ….

‘Antenna design for cellular systems’ is a very broad and generic term. I would like to point out here that antenna design requirements and capabilities for base stations and mobile terminals are very different. Let us start with the requirements of a base station antenna. First of all, it needs to be directional, i.e. the antenna should have a high gain in the direction facing cellular sector being illuminated, as shown in Fig. 1(a). Secondly, the major beam of the antenna should not be parallel to the ground. It should be tilted down slightly to ensure that the users on the ground get coverage (see Fig. 1(b)). The third requirement deals with the space occupied by a base station antenna. It is widely assumed that we can afford to have more antenna elements, larger inter-element spacing between the elements, and broadly speaking, larger antenna elements as compared to the receiver (mobile terminal). However, this is not completely true. As cellular service providers and base station antenna manufacturers will tell you, it is very expensive to mount antennas on towers, rooftops, etc. The cost can range from anything between $500 - $3000 a month. Sometimes, the charge is on the basis of the sq. footage or antenna units mounted. Hence, the all-so-common assumption of having any number of antenna elements on the base station side, separated by large inter-element spacings is actually quite impractical. This is where antenna design comes into picture. Our aim is to try and design base station antennas that satisfy all the above mentioned requirements, while ensuring that performance is not affected. When we say performance, we mean the average data rates obtained in the cell and the outage probabilities that a base station antenna design will yield.

Fig 1: Directional base station antennas.

Now we come to the part where MIMO fits into all this. Current (single antenna) cellular technologies like CDMA, GSM, EDGE etc. cannot support the growing number of cellular service users, coupled with the large demand for high data rate applications. Cellular service providers are now looking to MIMO as a solution to meet this demand for high data rates. The inclusion of MIMO in standards like WiMAX and 3GPP-LTE has increased pressure on the cellular service providers to make the shift (and make it soon). The only problem that the providers have with making this shift is that it might require that the transceivers at the base stations be upgraded to support MIMO. It is not clear if the base station antennas should be changed or if the existing setup will suffice for MIMO as well. We in WSIL are currently investigating if the existing base station antenna designs will work well for MIMO. This will make the transition for cellular service providers a whole lot easier.

Coming now to the mobile terminals. These devices, as we well know, are placed and held at random orientations. This implies that the antenna used should be such that it can receive (and transmit) equally well in all directions. This is the first requirement for antennas for a mobile terminal. Note that omnidirectional antennas generally do not perform as well as directional antennas, making this a challenge for the antenna designer. The second requirement for the mobile terminal antenna is that it should occupy as little space as possible. This is where low profile microstrip (patch) antennas, planar inverted-F antennas (PIFAs), etc. come into picture. Note that these are embedded antennas, whereas the earlier models of cell phones used whip, retractables, etc. The third requirement is that the cell phone antenna should be able to support a large number of functions apart from just supporting phone calls, like FM or bluetooth, for example. This requires that the antenna used should be a multi-band antenna supporting all the frequencies needed or that we use different antennas for each frequency. It is obvious that the single antenna is a much better option, both in terms of cost and space occupied. Designing multi-band antennas is a challenge for the designer if the number of bands to be supported is greater than two. The larger the number of frequencies required, the greater the difficulty in design.

I will conclude with a brief statement on why cellular antenna design is important. An antenna designer spends a lot of time and effort trying to optimize antenna parameters like cross-pol pattern discrimination, the port-to-port isolation, etc. These parameters need to be optimized by (in some cases) minutely tweaking the physical structure of the antenna. This process can at times take days to accomplish. By using effective antenna design techniques, we can optimize these parameters using simulations and can give a conservative estimate beyond which the performance will not be affected. This in turn, saves time and effort (and money!).

That’s it for now. I will post another blog on some details of antenna design and, in particular, the work we do in WSIL on this topic.

Food for the Eyes

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Slightly off topic… I came across this news today. A cool application of MEMS sure to fire up some imagination or vision!?

The news article here

We’ve all seen something similar in some SciFi movie. Imagine the things you can do. Connect wirelessly to display information about your location within your vision, a cheat sheet display for your next big talk, etc…

Unfortunately, it appears that there is no clean way to feed power and information signal to it other having a wire dangling from your eyes (ouch!). Maybe a RFID-like passive display could be of immediate use.

To give a MIMO flavor, maybe some form of MIMO visual communication is possible…

What’s Your Erdos Number?

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Some of you may be familiar with the Erdos number in mathematics. This number is sort of a “degrees of separation”, in publication terms, from one Paul Erdos. I did a little surfing and found out that Prof. Heath’s Erdos number is about to become 6, if a recently submitted article is accepted.

Claude Shannon has an Erdos number of 3. Also, Bob Gallager published a couple papers with Shannon, one of them being:

Claude E. Shannon, Robert G. Gallager, Elwyn R. Berlekamp: Lower Bounds to Error Probability for Coding on Discrete Memoryless Channels. I Information and Control 10(1): 65-103 (1967)

One of Gallager’s students was Randall Berry:

Randall A. Berry, Robert G. Gallager: Communication over fading channels with delay constraints. IEEE Transactions on Information Theory 48(5): 1135-1149 (2002)

Finally, a recently submitted article to IEEE Communications magazine is co-authored by Prof. Heath and Prof. Berry, collaboraters on the DARPA IT-MANET project:

J. G. Andrews, N. Jindal, M. Haenggi, R. Berry, S. Jafar, D. Guo, S. Shakkottai, R. W. Heath, Jr., M. Neely, S. Weber, A. Yener, P. Stone, “Rethinking Information Theory for Mobile Ad Hoc Networks,'’ submitted to IEEE Communications Magazine Dec. 2007.

If this is accepted, it will give anyone connected to Prof. Heath an Erdos number of at most 7. It’s amazing how connected we all are. Albert Einstein has an Erdos number of 2, so (at most) 9 papers separate us from Albert Einstein. Isn’t that crazy?

If you’re not a member of WSIL, what is your Erdos number?

IEEE 802.16j Draft 2 in balloting

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The second draft of IEEE 802.16j, the Multihop Relay amendment to IEEE 802.16e/Mobile WiMAX, is currently being voted on by IEEE 802.16 members. The first draft only garnered a 67% approval while needing 75%. Over 1050 comments have been resolved since Draft 1, and so far the turnout has looked positive—6 people have changed their vote from Disapprove to Approve so far; 2 changed from Abstain to Approve; 1 that did not vote before now Approves.

Votes from the first ballot carry over if the person does not vote again, which means if no one else votes between now and January 14th–when the ballot closes–Draft 2 will have a 69.85% approval, meaning it will fail again. However, by the number of changes in votes so far, I’m betting this draft will be approved by a narrow margin.

Optimizing Pilot Locations Using Feedback in OFDM Systems

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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!