Renaldi Winoto EE229B Project Peak-to-Average Power Control in OFDM Systems I. Introduction High Peak-to-Average Power Ratio (PAPR) has been recognized as one of the major practical problem involving OFDM modulation. High PAPR results from the nature of the modulation itself where multiple subcarriers/sinusoids are added together to form the signal to be transmitted. When N sinusoids add, the peak magnitude would have a value of N, where the (rms) average might be quite low due to destructive interference between the sinusoids. High PAPR signals are usually undesirable for it usually strains the analog circuitry. High PAPR signals would require a large range of dynamic linearity from the analog circuits, which usually results in expensive devices, and higher power consumption/lower efficiency (for example, power amplifier has to operate with larger backoff to maintain linearity). It’s quite straightforward to observe that in an OFDM systems, some input sequences would result in higher PAPR than others. For example, an input sequence that requires all subcarriers to transmit their maximum amplitudes would certainly result in a high output PAPR. Thus by limiting the possible input sequences to a smallest subset, it should be possible to obtain output signals with a guaranteed low output PAPR. At the same time, by limiting input sequences to a smaller subset we also provide the system with more resiliency in the presence of noise – this idea is central in the area of error-control codes. Thus the question is can we set up a framework such that we can jointly optimize the selection of codewords based on both their PAPR and minimum distance performance. In this project we will conduct a literature survey of the various techniques proposed to establish PAPR control in OFDM modulation. The emphasis will certainly be on those techniques which uses codes designed to have both low PAPR along with good error-correction capabilities. We will also briefly mention other methods that have been proposed in the literature. These methods also indirectly take advantage of the redundancy afforded by error-control coding performed on the input data sequences – in that the PAPR reduction system is allowed to alter the transmitted signal a little bit (introduce bit error) with the hope of optimizing for PAPR. In most of the main results presented here, the goal is to convey the main fundamental ideas and intuitive understanding of the concept introduced using simple examples, rather than dwell into the details of the more involved results. This is done primarily to minimize the amount of terminology and notation that have to be introduced in this short report but also to give a complete overview of the techniques known today for PAPR reduction.II. OFDM Modulation and Notation It is useful to briefly review the OFDM modulation as a way to set up notation for the succeeding sections. For an n-subcarrier OFDM signal with carrier frequencies fo + j.fs, with 0 ≤ j < n, we have n carrier sinusoids of the formtjffisoe)(2 +π. Suppose at a particular symbol period, a length-n information vector ].....[110 −=nmmmm is to be transmitted. Then the resulting OFDM modulated signal can be written as tjffinjjmsoemtS)(210)()(+−=∑=π Each of the message symbol mj can be a general complex number. However in all the papers reviewed here a PSK modulation is assumed. This is done to ease the mathematical formulation of the problem and also the derivation of the results, as in PSK modulation all the subcarriers would have average envelope power of n. The instantaneous power of the signal S(m)(t) can be written as : 2)()())(()( tStPmmℜ= Whereas the envelope power is defined as : 2)()()()( tStPmmenv= We can easily see that )()()()(tPtPmenvm≤ The Peak-to-Average Power Ratio (PAPR) can be defined as : )(sup1)()(10tPnmPAPRmtfs<≤= Whereas the Peak-to-Mean Envelope Power Ratio (PMEPR) can be defined as : )(sup1)()(10tPnmPMEPRmenvtfs<≤= In the literature the term PMEPR and PAPR is sometimes used interchangeably. It is more convenient to work with PMEPR than with PAPR, thus in most of the theoretical literature PMEPR is used. And since PMEPR upper-bounded PAPR, the use of PMEPR can be well justified. If error-control coding is to be used, the length n vector c from a code C is directly modulated - one coordinate of the vector c, to one subcarrier in the OFDM modulationIII. Practical Methods of PAPR Reduction in the literature Several approaches to reduce PAPR have been suggested in the literature. The simplest, and somewhat crude approach involves clipping the signal. High signal peaks, although possible, has a relatively low probability of occurring. As such, clipping these few peaks would not significantly impair the performance (in terms of BER) of the system. However clipping is a highly nonlinear operation and it has an adverse impact towards the spectral purity of this signal (spectral regrowth problem). For that reason this approach is usually not suitable for wireless application where spectral emission is tightly controlled. The paper by Li and Cimini [1] is just one example of such approach. In this paper the spectral regrowth problem is somewhat alleviated by a post-filtering operation to remove the out-of-band harmonics caused by the clipping operation. However by filtering the clipped signal, there is a possibility of peaks regrowing in the time domain. This fact really underlies the fundamental problem of signal clipping approach in PAPR reduction – it is still a relatively zero-sum game between achieving good spectral purity and low PAPR. The method of tone injection reserves a tone (in the OFDM modulation) to be injected to the system to cancel out high peaks. The difficult part in this method is how to select the optimal input sequences to be modulated on this reserved tone to maximally reduce the PAPR. Tellado and Cioffi[10] proposed this method and uses LP optimization to get the optimal input sequence. While both approaches mentioned above are viable options for systems, it does still incur some costs. Clipping would inevitably introduce error, effectively reducing the resiliency of the system in the presence of noise. In the tone injection method, a dedicated tone has to be reserved. In both these approaches, ultimately PAPR reduction would reduce the capacity of the system. One PAPR reduction scheme that does not constitute a constant reduction of