HSCTechnicalWiki

where f_o is the output carrier frequency and f_s is the frequency offset for the sth sub-carrier. In other words, we can write the PAPR as the max of the polynomial {cp}_0 + {cp}_1z + {cp}_2 z^2 \ldots + {cp}_{N-1} z^{N-1} where z = exp(2 \pi j t) and {cp}_k = c_k exp( 2 \pi \psi t) . To compute PAPR we must find that vector Z = \{ 1, z, z^2, \ldots\} for a given C = {c_0, c_1 , c_2, \ldots} 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

\begin{eqnarray} PAPR(C) = \sum_{k=0}^{N-1} \beta_k \cos(2 \pi k t)
\beta_0 = 1
\beta_k = \frac{2}{N} \sum_{m=0}^{N-1-k} c_m c_{m+k} \end{eqnarray}

This is subsequently maximized by standard numerical techniques.

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. In other words, we can write the PAPR as the max of the polynomial {cp}_0 + {cp}_1z + {cp}_2 z^2 \ldots + {cp}_{N-1} z^{N-1} where z = exp(2 \pi j t) and {cp}_k = c_k exp( 2 \pi \psi t) . In other words, to compute the PAPR, we must find that vector Z = \{ 1, z, z^2, \ldots\} for a given C = {c_0, c_1 , c_2, \ldots} 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

g(S=s_0,s_1,\ldots,) = \max_{0 \le t \le T_s} \frac{ \Re \left[\sum_{k=0}^{N-1} s_i exp\{-i 2\pi k f_s t\} \right]^2}{P_{avg}} \\

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 \beta_0 is 1 and \beta_k is given by \frac{2}{N} \sum_{m=0}^{N-1-k} c_m c_{m+k} where k > 0. This is subsequently maximized by standard numerical techniques.

g(S=s_0,s_1,\ldots,) = \max_{0 \le t \le T_s} \frac{ \Re \left[\sum_{k=0}^{N-1} s_i exp\{-i 2\pi k f_s t\} \right]^2}{P_avg} \\

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 \Sigma 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.

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 = s_0,s_1 \ldots s_{N-1} , which are generated by modulating the input block C = \{c_0,c_1 \ldots 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 \Re \sum_{k=0}^{N-1} s_i exp(-i2\pi k f_s t) .

The peak to average power ratio is then given by \max_{0 \le t \le T_s} \Re \sum_{k=0}^{N-1} s_i exp(-i2\pi k f_s t)/P_{avg} .

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 = s_0,s_1 \ldots s_{N-1} , which are generated by modulating the input block C = \{c_0,c_1 \ldots 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 \Re \sum_{k=0}^{N-1} s_i exp(-i2\pi k f_s t) . The peak to average power ratio is then given by \max_{0 \le t \le T_s} \Re \sum_{k=0}^{N-1} s_i exp(-i2\pi k f_s t)/P_{avg} .

P = e^{j2\pi l/c}, l = 0,1,\ldots,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

An alternative to this method, known as Selective Mapping is to construct K different phase shift vectors \Psi_m = \{ {\psi^0}_m,{\psi^1}_m, \ldots \psi^{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 \phi_k. The output block S {\phi_k}^{*T} with the least PAPR is transmitted. Here k^{*} is the index of the optimal phase shift vector for this particular input block.

An alternative to this method, known as Selective Mapping is to construct K different phase shift vectors \Psi_m = \{ {\psi^0}_m_,{\psi^1}_m, \ldots \psi^{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 \phi_k. The output block S {\phi_k}^{*T} with the least PAPR is transmitted. Here k^{*} is the index of the optimal phase shift vector for this particular input block.

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 = s_0,s_1 \ldots s_{N-1} , which are generated by modulating the input block C = \{c_0,c_1 \ldots 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 \mathcal{R} \sum_{k=0}^{N-1} s_i exp(-i2\pi k f_s t) . The peak to average power ratio is then given by \max_{0 \le t \le T_s} \mathcal{R} \sum_{k=0}^{N-1} s_i exp(-i2\pi k f_s t)/P_{avg} .

• Fix the first value b_0 to 1, and then search for optimal setting for the second value b_1, which yields the lowest PAPR.
• After finding the optimal value of b_1, fix it and go on to b_2

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 \Psi_m = \{ {\psi^0}_m_,{\psi^1}_m, \ldots \psi^{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 \phi_k. The output block S {\phi_k}^{*T} with the least PAPR is transmitted. Here k^* is the index of the optimal phase shift vector for this particular input block.

Tarokh and Jafarkhani propose a method where, instead of transmitting S, they transmit S \Phi^T where \Phi is a vector of optimally chosen phase shift angles \{\phi_0,\phi_1,\ldots \phi_{M-1}\} . There is no direct method for computing the optimal phase shift vector \Phi, 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, \{S_0, S_1,...S_{c-1}\} of size k = S/C, and then the entire block is weighted using an optimal weight vector B= \{b_0, b_1, \ldots, b_{c-1}\}. Block partitioning can be interleaved, adjacent, or psuedo-random.

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 = s_0,s_1 \ldots s_{N-1} , which are generated by modulating the input block C = \{c_0,c_1 \ldots 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 \mathcal{R} \sum_{k=0}^{N-1} s_i exp(-i2\pi k f_s t) . The peak to average power ratio is then given by \max{0 \le t \le T_s} \mathcal{R} \sum_{k=0}^{N-1} s_i exp(-i2\pi k f_s t)/P_{avg} .

Here P_{avg} is given by the well known formula P_{avg} = \Vert S \Vert = \sum {s_k}^2

PAPR occurs when, 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.

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. Attach:papr4.png Δ, 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.

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 Înter% Attach:papr6.png Δ, 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 \psi = f_o / f_s and get Attach:papr7.png Δ. In other words, we can write the PAPR as the max of the polynomial {cp}_0 + {cp}_1z + {cp}_2 z^2 \ldots + {cp}_{N-1} z^{N-1} where z = exp(2 \pi j t) and {cp}_k = c_k exp( 2 \pi \psi t) . In other words, to compute the PAPR, we must find that vector Z = \{ 1, z, z^2, \ldots\} for a given C = {c_0, c_1 , c_2, \ldots} 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

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 = s_0,s_1 \ldots s_{N-1} , which are generated by modulating the input block C = \{c_0,c_1 \ldots 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 \mathcal{R} \sum_{k=0}^{N-1} s_i exp(-i2\pi k f_s t) . The peak to average power ratio is then given by \frac{\mathcal{R} \sum_{k=0}^{N-1} s_i exp(-i2\pi k f_s t)}{P_{avg}} . Here P_{avg} is given by the well known formula P_{avg} = \Vert S \Vert = \sum {s_k}^2

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 = s_0,s_1 \ldots s_{N-1} , which are generated by modulating the input block C = \{c_0,c_1 \ldots 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 Înter% Attach:papr1.png Δ. The peak to average power ratio is then given by \mathcal{R} \sum_{k=0}^{N-1} s_i exp(-i2\pi k f_s t) . Here P_{avg} is given by the well known formula P_{avg} = \Vert S \Vert = \sum {s_k}^2

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 = s_0,s_1 \ldots s_{N-1} , which are generated by modulating the input block C = \{c_0,c_1 \ldots 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 Înter% Attach:papr1.png Δ. The peak to average power ratio is then given by \mathcal{R} \sum_{k=0}^{N-1} s_i exp(-i2\pi k f_s t) . Here P_{avg} is given by the well known formula Attach:papr3.png Δ.

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 = s_0,s_1 \ldots s_{N-1} , which are generated by modulating the input block C = {c_0,c_1 \ldots 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 Înter% Attach:papr1.png Δ. The peak to average power ratio is then given by Înter% Attach:papr2.png Δ. Here P_{avg} is given by the well known formula Attach:papr3.png Δ.