Introduction

Peak to Average Power Ratio (also known as Peak to Mean Envelope Power Ratio or PMEPR) is a metric of wireless signals which define the variations in the transmit power for that signal. Large variations in output power are a cause of concern; they can cause PA saturation, non-linearity and other effects. To combat this, the power amplifier has to operate in the linear range; this in turn causes significant inefficiency of operation and requires more expensive PAs. Large PAPR is especially of a concern in multi-carrier modulation techniques such as OFDM . For example, Wimax preambles in 2048 subcarrier ofdma configuration are between 4 to 5dB. Note that these sequences have been chosen for good PAPR characteristics amongst others.

Basics

What is PAPR

Consider a signal which is comprised of N sub-carriers at a spacing f_c. Each tone contains a complex symbol s_j. The composite signal is the array of complex symbols S = {s₀,s₁...s_N-1}, which are generated by modulating the input block C = {c₀,c₁...c_M-1}. This signal is transmitted over a time symbol of duration T_s. The complex signal envelope (prior to modulating on the carrier is given by . The peak to average power ratio is then given by . Here P_avg is given by the well known formula .

The diagram below shows the signal power envelope for a signal consisting of 64 separate tones. Note the large swings in output power.

PAPR occurs because, in a multi-carrier environment, the different sub-carriers are out of phase with each other. Thus, at each instant they are offset with respect to each other at different phase values. However, there may come a point when all of them achieve the maximum value simultaneously; this will cause the output envelope to suddenly shoot up. This causes a 'peak' in the output envelope.

Estimation of PAPR

To estimate PAPR, one can take all the possible code words, compute the PAPR for a given code word using the formula above and then take the expectation of this. i.e. , where Σ is the constellation of available output symbol words corresponding to all possible input code words C. However, this will require an exhaustive search of all possible code words; for block codes with large N or for trellis codes, this could be computationally impossible.

A simple approach was used by [Tellambura] where it is assumed that the complex constellation is a BPSK modulation. Tellambura shows that the PAPR can then be written as where β₀ is 1 and β_k is given by where k > 0. This is subsequently maximized by standard numerical techniques.

The polynomial approach

Tarokh and Jafarkhani propose a technique for the analysis of a signal comprising of N symbols from a constellation C. Their analysis follows from the fact that that the expression for PAPR given above can be re-written as , where f_o is the output carrier frequency and f_s is the frequency offset for the sth sub-carrier. Noting that f_o >> f_s, we set ψ = f_o / f_s and get . In other words, we can write the PAPR as the max of the polynomial cp₀ + cp₁.z + cp₂z² ... + cp_N-1z^N-1 where z = exp(2 π j t) and cp_k = c_kexp( 2 π ψ t). In other words, to compute the PAPR, we must find that vector Z = { 1, z, z², ...} for a given C = {c₀, c₁ , c₂, ...} that maximizes the inner product CZ^T. Tarokh and Jafarkhani suggest various methods to compute this, including a trellis coding technique. See [Tarokh] for more details

Techniques for PAPR reduction

Many techniques have been suggested for PAPR reduction, with different levels of success and complexity. The interested reader is refered to the survey by Han and Lee and other papers for further reading. We will summarize a few well-known and analyzed techniques in the subsequent sections.

Clipping

Clipping simply takes the output signal and clips the peak amplitude to the desired level. However, clipping creates distortion, which in turn creates noise. The noise can be both in-band and out-of-band. The out-of-band noise can be filtered, but this filtering operation (after the clipping) can cause regrowth of the signal so that the output of the filter has a worse peak than the original signal. There are iterative clip and filter approaches, but a large number of iterations are required for good results.

A more original idea proposed by Chen and Haimovich is to put the filtering and noise cancellation in the receiver. The logic is that, since clipping is a deterministic process, the receiver can determine the right noise cancellation technique to use.

Block coding techniques

Block coding simply suggests using codes in a given coding technique which have a low PAPR. For example, consider we have an R(3,4) coding technique and input code words of 6 bits in length. There are 256 possible output code words out of which, say, 150 code words have low PAPR. We then start handling smaller input words, which can be mapped into one of those 6 bit inputs which generate the low PAPR code words.

The obvious side-effect of this technique is to reduce coding efficiency. Even more problematic is that a general search of a large code-word space to compute the PAPR of each output code word is very expensive computationally; for large block codes it is simply out of the question. It has been found that Golay complementary codes have very attractive PAPR (almost 2) and it is possible to generate golay complementary code pairs from ordinary Reed-Muller block codes. However, this technique is severely limited in that it can only handle a small number of sub-carriers and is thus not frequently used.

Techniques based on phase shifting

Tarokh and Jafarkhani propose a method where, instead of transmitting S, they transmit SΦ^T where Φ is a vector of optimally chosen phase shift angles { φ₀,φ₁,...φ_M-1}. There is no direct method for computing the optimal phase shift vector Φ, however, there are iterative algorithms to compute them for different modulation techniques. Tarokh and Jafarkhani have reported results for 48 sub-carriers, which in practise is a fairly good block size for OFDM type systems. The authors demonstrate a PAPR reduction of 4.22dB for 8PSK modulation.

Phase shifting can also be done in blocks, instead of individual sub-carriers. This technique also goes by the name partial transmit sequences. In this technique, S is broken into a set of blocks, {S₀, S₁,...S_c-1} of size k = S/C, and then the entire block is weighted using an optimal weight vector B={b_₀, ₁..._c-1}. Block partitioning can be interleaved, adjacent, or psuedo-random.
The obvious bottleneck is again the search time taken to find an optimal weight vector. To reduce the possible number of combinations, the phase shift values are taken from
P = {e^j2πl/c, l = 0,1,...c-1}. This also means that the side-information to be transmitted about the phase shifts used is limited to c log(l). For l=0, there is no shift. One technique, called iterative flipping, works as follows

Start with an initial vector {1,1,1,1,1,1,1}
Fix the first value b₀ to 1, and then search for optimal setting for the second value b₁, which yields the lowest PAPR.
After finding the optimal value of b₁, fix it and go on to b₁

For c=2, the second term is obtained by flipping the sign of the first term, hence the term iterative flipping. This method is a heuristic search and obviously does not give the global optimum. Han and Lee have suggested an improvement where instead of changing just one index at one time, they search for the optimum out of all possible vectors with upto r changes from the current one i.e. a Hamming Distance of r.

An alternative to this method, known as Selective Mapping is to construct K different phase shift vectors Φ_m = { φ⁰_m,φ¹_m,...φ^M-1_m}. For each input block of N symbols, we generate K candidate blocks by multiplying the input block with the k possible sets of phase shifts Φ_k. The output block SΦ_k^*^T with the least PAPR is transmitted. Here k^* is the index of the optimal phase shift vector for this particular input block.

In all the above techniques, it is necessary to transmit side information to the receiver about the transformation affected on the original block. This side information is an overhead, since it consumes bandwidth and energy to transmit it.

Tone injection and reservation

In these techniques, a specific time-domain signal is added to the original signal which improves the PAPR of the combined output. To accomodate this special signal, some provision must be made so that it can be easily separated and removed by the receiver. In the tone injection method, a subset of the sub-carriers in the OFDMA signal is reserved. In the tone injection technique, the basic signal constellation itself is mapped to a larger constellation i.e. each point in the original constellation can map to a subset of points in the larger constellation. The choice of which exact point to choose in the larger constellation is exercised so as to minimize PAPR.

Tone injection/reservation is a surprisingly effective technique, simply because it can be shown that computation of the optimal signal c for any input signal x is a problem in convex_optimization and thus can be computed near-optimally in finite time. It is the currently supported technique for wimax.

References


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[Tellambura]Tellambura, C. "Computation of the continuous-time PAR of an OFDM signal with BPSK subcarriers",
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[ToneResv] Paterson, K.G.; Tarokh, V., "On the existence and construction of good codes with low peak-to-average power ratios", 
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