The proposed PAPR reduction techniques are based on current ACE methods expanded to FBMC applications. Equation 17 allows for scaling of the clipped portion {\u0109}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right] by a single scaling value across the entire frame, i.e. all symbols in the frame will be scaled by the same value. This strategy does not lead to the best performance since the frame is made up of M overlapping symbols which may be scaled independently.
The main goal is to reduce the number of ACE iterations required to obtain a low PAPR, by finding an extended vectorial scaling factor. This scaling vector allows for independent scaling of each overlapping symbol thereby adding an additional degree of freedom at the cost of additional signal processing.
We can rewrite Equation 17 using a different scaling factor μ_{
m
} for each corresponding symbol of the frame as
\begin{array}{ll}{\mathit{\u015d}}_{l}& ={\mathit{s}}_{l}+{\mathit{\u0108}}_{l}^{T}\mathit{\mu},\phantom{\rule{2em}{0ex}}\end{array}
(18)
\begin{array}{ll}{\mathit{s}}_{l}& ={\left[s\left[\phantom{\rule{0.3em}{0ex}}0\right],\cdots \phantom{\rule{0.3em}{0ex}},s\phantom{\rule{0.3em}{0ex}}\left[(M1)N/2+L1\right]\right]}^{T},\phantom{\rule{2em}{0ex}}\end{array}
(19)
{\mathit{\u0108}}_{l}=\left[\begin{array}{cccccccc}{\u0109}_{l,0}\left[\phantom{\rule{0.3em}{0ex}}0\right]& {\u0109}_{l,0}\left[\phantom{\rule{0.3em}{0ex}}1\right]& \cdots & {\u0109}_{l,0}[\phantom{\rule{0.3em}{0ex}}N/2]& \cdots & {\u0109}_{l,0}[\phantom{\rule{0.3em}{0ex}}L1]& 0& \cdots \\ 0& 0& \cdots & {\u0109}_{l,1}\left[\phantom{\rule{0.3em}{0ex}}0\right]& \cdots & {\u0109}_{l,1}[\phantom{\rule{0.3em}{0ex}}LN/21]& \cdots & \cdots \\ \vdots & \ddots & \vdots \\ 0& \cdots & {\u0109}_{l,M1}\left[\phantom{\rule{0.3em}{0ex}}0\right]& \cdots & {\u0109}_{l,M1}[\phantom{\rule{0.3em}{0ex}}L1]\end{array}\right]
(20)
{\u0109}_{l,m}\left[\phantom{\rule{0.3em}{0ex}}n\right]=\sum _{k=0}^{N1}{\theta}_{k+m}{C}_{l,m}\left[\phantom{\rule{0.3em}{0ex}}k\right]p\phantom{\rule{0.3em}{0ex}}\left[nm\frac{N}{2}\right]{e}^{\phantom{\rule{0.3em}{0ex}}j2\pi k\frac{n}{N}}
(21)
and
\mathit{\mu}={\left[\begin{array}{ccc}{\mu}_{0}& {\mu}_{1}& \cdots {\mu}_{M1}\end{array}\right]}^{T}.
(22)
From Equation 18, it should be clear that the new baseband FBMC signal, after PAPR reduction, is a function of the individual overlapping clipped symbols, and the rows of matrix {\mathit{\u0108}}_{l} scaled by their relevant scaling factor μ_{0}⋯μ_{M  1} and added to the original FBMC signal s_{
l
}.
To obtain the discrete FBMC clipped symbol components of {\mathit{\u0108}}_{l}, the individual FBMC symbols must be modulated in a noncontiguous manner. This is required to take into account the effect that each clipped signal has on the original signal looking for the optimal μ_{
m
}for 0 ≤ m ≤ M  1. This process will reduce much of the efficiency associated with the polyphase implementation.
Figure 6 illustrates the effective overlap of the clipped portions superimposed over the original baseband signal. Each sample of {\widehat{\mathit{s}}}_{l} can then be seen as a function of the original baseband signal s_{
l
}as well as 2K samples of the overlapping clipped symbols {\u0109}_{l,m}\left[\phantom{\rule{0.3em}{0ex}}n\right]. Convergence time can therefore be decreased by optimizing the μ_{
m
} to increase the effect that the clipped signals have on reducing the peaks in the original baseband signal. However, the μ_{
m
} must be scaled carefully in order to avoid peak regrowth.
4.1 LPbased optimization
The authors propose two methods, based on an LP optimization formulation, for finding the optimal scaling factors μ_{
m
}. In order to provide the largest amount of PAPR reduction, the optimization aims to maximize the μ_{
m
} scaling factors. The scaling factor is applied to the negative of the clipped portion of the signal, and therefore, this is equivalent to minimizing the peaks of the signal. Constraints, however, have to be placed on the μ_{
m
} values in order to prevent peak regrowth. These constraints provide tight bounds on the LP formulation which guarantee convergence to a lower or equal PAPR per iteration.
The second term of the right hand side of Equation 18 can be written as
\begin{array}{c}{\mathit{\u0108}}_{l}^{T}\mathit{\mu}=\sum _{m=\u230a\frac{n}{N/2}\u230b(2K1)}^{\u230a\frac{n}{N/2}\u230b}{\mu}_{m}{\u0109}_{l,m}\left[n{}_{N/2}+\left(\u230a\frac{n}{N/2}\u230bm\right)\frac{N}{2}\right],\end{array}
(23)
with
n=\phantom{\rule{2.77626pt}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,\cdots \phantom{\rule{0.3em}{0ex}},(M1)N/2+L1],\phantom{\rule{1.6em}{0ex}}m=\phantom{\rule{2.77626pt}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,1,2,\cdots \phantom{\rule{0.3em}{0ex}},M1].
(24)
It should be noted that, in order to keep the symbol causal and limit the frame length, the summation limits in Equation 23 are limited by the constraints for m in Equation 24. As an example, for a critically sampled FBMC system with N = 16 subcarriers and a frame length of M = 25,
\begin{array}{ll}{\left\{{\mathit{\u0108}}_{l}^{T}\mathit{\mu}\right\}}_{150}& ={\mu}_{11}{\u0109}_{l,11}\left[\phantom{\rule{0.3em}{0ex}}62\right]+{\mu}_{12}{\u0109}_{l,12}\left[\phantom{\rule{0.3em}{0ex}}54\right]+{\mu}_{13}{\u0109}_{l,13}\left[\phantom{\rule{0.3em}{0ex}}46\right]\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{\mu}_{14}{\u0109}_{l,14}\left[\phantom{\rule{0.3em}{0ex}}38\right]+{\mu}_{15}{\u0109}_{l,15}\left[\phantom{\rule{0.3em}{0ex}}30\right]+{\mu}_{16}{\u0109}_{l,16}\left[\phantom{\rule{0.3em}{0ex}}22\right]\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{\mu}_{17}{\u0109}_{l,17}\left[\phantom{\rule{0.3em}{0ex}}14\right]+{\mu}_{18}{\u0109}_{l,18}\left[\phantom{\rule{0.3em}{0ex}}6\right]\phantom{\rule{2em}{0ex}}\end{array}
(25)
with {·}_{
n
} representative of the n th component of the vector.
4.1.1 LP formulation 1
In order to prevent peak regrowth, the constraints for each sampling point are written in terms of the μ_{
m
} scaling factors. The constraints for sample n of frame l can be written, with n_{peak} the position of the peak as follows:
\begin{array}{cc}\left\left{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]+{\left\{{\mathit{\u0108}}_{l}^{T}\mathit{\mu}\right\}}_{n}\right\right& \le \left\left{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}{n}_{\text{peak}}\right]+{\left\{{\mathit{\u0108}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak}}}\right\right,\\ n& =\phantom{\rule{2.77626pt}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,1,2,\cdots \phantom{\rule{0.3em}{0ex}},(M1)N/2+L1],\phantom{\rule{1em}{0ex}}\end{array}
(26)
with
\begin{array}{ll}{\left\{{\mathit{\u0108}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak}}}& =\sum _{m=\u230a\frac{{n}_{\text{peak}}}{N/2}\u230b(2K1)}^{\u230a\frac{{n}_{\text{peak}}}{N/2}\u230b}{\mu}_{m}{\u0109}_{l,m}\left[{n}_{\text{peak}}{}_{N/2}\right.\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\left(\right)close="]">\left(\u230a\frac{{n}_{\mathit{\text{peak}}}}{N/2}\u230bm\right)\frac{N}{2}& .\phantom{\rule{2em}{0ex}}\end{array}\n
(27)
Equation 26 implies that at each sample n, the magnitude of the baseband transmit signal s_{
l
}[ n] combined with the scaled 2K clipped symbols overlapping at n\left({\left\{{\mathit{\u0108}}_{l}^{T}\mathit{\mu}\right\}}_{n}\right) cannot exceed the value of the peak of the frame combined with the scaled clipped symbols overlapping at the peaks position (the right hand side of Equation 26). As a consequence, the sample cannot be increased to a magnitude larger than that of the peak after it has been reduced. These constraints limit possible peak regrowth and place a tight upper bound on the problem formulation.
In order to reduce the complexity of an LPbased method, it is desirable to have purely real samples. The output of the FBMC modulator is however a complex baseband signal s[ n]. For this reason, utilizing a similar approach to that of [15], the magnitude of the sample s[ n] and the projections of the clipped symbols onto the original signal {\stackrel{\u0304}{c}}_{m}\left[\phantom{\rule{0.3em}{0ex}}n\right] are used in order to obtain real values for the optimization problem. These projections provide good approximations of the magnitude of the clipped symbols in the same direction as the original signal and can be calculated as [15]
{\stackrel{\u0304}{c}}_{l,m}\left[\phantom{\rule{0.3em}{0ex}}n\right]=\frac{\Re \phantom{\rule{0.3em}{0ex}}\left\{{\u0109}_{l,m}\left[\phantom{\rule{0.3em}{0ex}}n\right]\xb7{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\phantom{\rule{0.3em}{0ex}}\right\}}{\left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\left\right}.
(28)
Equation 28 can be used to estimate Equation 26 by replacing {\left\{{\mathit{\u0108}}_{l}^{T}\mathit{\mu}\right\}}_{n} with {\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{n}. The new equation is therefore purely real in nature as it is a representative of magnitude only and can be written as
\begin{array}{l}\left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\left\right+{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{n}\le \left\right{s}_{l}\left[{n}_{\text{peak}}\right]\left\right+{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak}}},\phantom{\rule{3em}{0ex}}\\ \text{for}\phantom{\rule{2.77626pt}{0ex}}n=0,\cdots \phantom{\rule{0.3em}{0ex}},(M1)N/2+L1,\phantom{\rule{2em}{0ex}}\end{array}
(29)
with {\stackrel{\u0304}{C}}_{l} a matrix of the same form as Equation 20 with \u0109 replaced by \stackrel{\u0304}{c}. After moving the constants to the right hand side and the variables to the left, Equation 29 can be written as
{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{n}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak}}}\le \left\right{s}_{l}\left[{n}_{\text{peak}}\right]\left\right\left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\left\right.
(30)
As mentioned, Equation 29 provides a tight upper bound for the LP problem. To guarantee convergence, a lower bound is also required in the optimization formulation. A good lower bound can be formulated by ensuring that the negative of the absolute value of the scaled samples at each sampling point is greater than the negative of the absolute value of the scaled peak. This can be formulated by adding an extra set of (M  1)N/2 + L constraints,
\begin{array}{l}\left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\left\right+{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{n}\ge \left\right{s}_{l}\left[{n}_{\text{peak}}\right]\left\right+{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak}}}\phantom{\rule{3em}{0ex}}\\ \text{for}\phantom{\rule{2.77626pt}{0ex}}n=0,\cdots \phantom{\rule{0.3em}{0ex}},(M1)N/2+L1,\phantom{\rule{2em}{0ex}}\end{array}
(31)
As in Equation 30, Equation 31 can be written with all constants on the right and variables on the left resulting in the final optimization formulation being written as
\text{max}\phantom{\rule{0.3em}{0ex}}\text{z}={\mu}_{0}+{\mu}_{1}+\cdots +{\mu}_{M1}
(32)
s.t.
\begin{array}{l}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{n}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak}}}\le \left\right{s}_{l}\left[{n}_{\text{peak}}\right]\left\right\left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\left\right,\phantom{\rule{3em}{0ex}}\\ \text{for}\phantom{\rule{2.77626pt}{0ex}}n=0,\cdots \phantom{\rule{0.3em}{0ex}},(M1)N/2+L1,\phantom{\rule{2em}{0ex}}\end{array}
(33)
\begin{array}{l}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{n}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak}}}\ge \left\right{s}_{l}\left[{n}_{\text{peak}}\right]\left\right+{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\left\right\phantom{\rule{3em}{0ex}}\\ \text{for}\phantom{\rule{2.77626pt}{0ex}}n=0,\cdots \phantom{\rule{0.3em}{0ex}},(M1)N/2+L1,\phantom{\rule{2em}{0ex}}\end{array}
(34)
{\mu}_{m}\ge 0,\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{2.77626pt}{0ex}}m=0,\cdots \phantom{\rule{0.3em}{0ex}},M1.
(35)
This algorithm is an approximation due to the samples being complex in nature and the projections being an approximation of the ‘growth’ of the clipped signals in the direction of the original baseband signal.
4.1.2 LP formulation 2
The aim of LP2 is to minimize the PAPR of sections of the frame instead of over the entire frame as in LP1. This approach therefore handles smaller sections and can achieve superior PAPR performance with a lower average transmit power increase. This is done by changing the optimization formulation. Instead of attempting to reduce a single peak in the frame, the goal is to reduce M peaks in overlapping sections of length L. The objective function is also scaled based on the magnitude of the peaks. This allows for higher priority on the larger peaks. In order to do this, we define a new peak position vector n_{
peak
} which can be defined as
{n}_{\text{peak}}={\left[\begin{array}{cccc}{n}_{\text{peak 0}}& {n}_{\text{peak 1}}& \cdots & {n}_{\text{peak}\phantom{\rule{1em}{0ex}}M1}\end{array}\right]}^{T},
(36)
with n_{peak 0} the position of the peak on the interval 0 ≤ n ≤ L  1, n_{peak 1} the position of the peak on the interval N/2 ≤ n ≤ L + N /2  1 and so on. These peaks set the limits for the intervals of length N/2.
The new optimization formulation can be set up as follows:
\text{max}\phantom{\rule{0.3em}{0ex}}\text{z}={\mu}_{0}+{\mu}_{1}+\cdots +{\mu}_{M1}
(37)
subject to the following constraints
\begin{array}{l}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{n}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak 0}}}\le \left\right{s}_{l}\left[{n}_{\text{peak 0}}\right]\left\right\left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\left\right,\phantom{\rule{3em}{0ex}}\\ \text{for}\phantom{\rule{2.77626pt}{0ex}}n=0,1,\cdots \phantom{\rule{0.3em}{0ex}},L1,\phantom{\rule{2em}{0ex}}\end{array}
(38)
\begin{array}{l}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{n}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak 0}}}\ge \left\right{s}_{l}\left[{n}_{\text{peak 0}}\right]\left\right+\left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\left\right,\phantom{\rule{3em}{0ex}}\\ \text{for}\phantom{\rule{2.77626pt}{0ex}}n=0,1,\cdots \phantom{\rule{0.3em}{0ex}},L1,\phantom{\rule{2em}{0ex}}\end{array}
(39)
\begin{array}{l}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{n}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak}\phantom{\rule{1em}{0ex}}1}}\le \left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}{n}_{\text{peak 1}}\right]\left\right\left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\left\right,\phantom{\rule{3em}{0ex}}\\ \text{for}\phantom{\rule{2.77626pt}{0ex}}\frac{N}{2}\le n\le L+\frac{N}{2}1,\phantom{\rule{2em}{0ex}}\end{array}
(40)
\begin{array}{l}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{n}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak}\phantom{\rule{1em}{0ex}}1}}\ge \left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}{n}_{\text{peak 1}}\right]\left\right+\left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\left\right,\phantom{\rule{3em}{0ex}}\\ \text{for}\phantom{\rule{2.77626pt}{0ex}}\frac{N}{2}\le n\le L+\frac{N}{2}1,\phantom{\rule{2em}{0ex}}\end{array}
(41)
\begin{array}{l}\begin{array}{c}\vdots \\ {\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{n}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak}\phantom{\rule{1em}{0ex}}M1}}\le \left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}{n}_{\text{peak}\phantom{\rule{1em}{0ex}}M1}\right]\left\right\left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\left\right,\\ \phantom{\rule{2.77626pt}{0ex}}\text{for}\phantom{\rule{2.77626pt}{0ex}}\frac{N}{2}(M1)\le n\le \frac{N}{2}(M1)+L1,\end{array}\end{array}
(42)
\begin{array}{ll}{\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{n}& {\left\{{\stackrel{\u0304}{C}}_{l}^{T}\mathit{\mu}\right\}}_{{n}_{\text{peak}\phantom{\rule{1em}{0ex}}M1}}\ge \left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}{n}_{\text{peak}\phantom{\rule{1em}{0ex}}M1}\right]\left\right+\left\right{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\left\right,\\ \phantom{\rule{1.3em}{0ex}}\text{for}\phantom{\rule{2.77626pt}{0ex}}\frac{N}{2}(M1)\le n\le \frac{N}{2}(M1)+L1,\end{array}
(43)
{\mu}_{m}\ge 0,\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{2.77626pt}{0ex}}m=0,\cdots \phantom{\rule{0.3em}{0ex}},M1.
(44)
The objective function of Equation 37 can also be scaled by the relevant peak magnitudes to which the symbols contribute. This adds priority to larger peaks. This can be done by noting the projections at the peak positions of the clipped symbols. The larger peaks will result in larger projections, and therefore, these projections can be used as a scaling factors in the objective function of Equation 37 in order to speed up convergence time to a lower PAPR.
4.2 SGP extension
Two additional methods are proposed based on a more direct extension of the SGP ACE method to FBMC. These methods do not require the use of a linear programming methodology, which can have high computational complexity depending on the problem. Once again, both methods focus on finding an optimal scaling factor μ or scaling factor vector μ.
4.2.1 SGP single scaling
This method follows closely to that of the SGP proposed in [15] for OFDM systems. The focus is on finding a single scaling factor μ by which we can scale the entire clipped frame {\u0109}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right] given by Equation 17. This approach can be seen as suboptimal as the component clipped signals can be scaled independently for greater effect.
The proposed single scaling method does not differentiate between the individual symbols in the FBMC frame.
The implementation of the single scaling SGP method is relatively straight forward and can be seen as a generalization of the OFDM method in [15] to an FBMC implementation. Once the clipped portion of the signal across the entire frame is obtained, it can be superimposed over the original signal to obtain the projections. This is illustrated in Figure 7.
The projections are obtained in a manner similar to Equation 28 but with a single vector spanning across the entire frame. In order to obtain the best suited scaling factor μ for the entire clipped frame {\u0109}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right], we observe only the projections in the steady state part of the frame, as illustrated by the solid line of the arc in Figure 7. The samples in the transient state are often small in magnitude as they do not contain 2K overlapping symbols and as can be seen in Figure 7 and are scaled by a small magnitude of the corresponding filter response. The signal behaviour in the transient is therefore erratic and not accurate for use in calculating the μ value. The steady state section begins approximately a full filter length into the frame and ceases a filter length from the end of the frame.
The single scaling SGP method for FBMC requires some alterations of the method in [15] and can be summarized as follows:

1.
Find the peak of the frame, P _{
l
}, inside the steady state region (the peak should almost always exist in the steady state region of the transmit data block) as well as the peak position n _{peak}.

2.
Calculate the projections {\stackrel{\u0304}{c}}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right] of the clipped frame {\u0109}_{l}\left[n\right] only along the viable interval by performing the dot product
{\stackrel{\u0304}{c}}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]=\frac{\Re \phantom{\rule{0.3em}{0ex}}\left\{{\u0109}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\xb7{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\phantom{\rule{0.3em}{0ex}}\right\}}{\left{s}_{l}\right[\phantom{\rule{0.3em}{0ex}}n\left]\right}
(45)
for
LN/2\le n\le (M1)N/2.
(46)

3.
Consider only the values of {\stackrel{\u0304}{c}}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right] that are positive and therefore result in magnitude growth.

4.
Compute the scaling factor for each of these projections
\mu \left[\phantom{\rule{0.3em}{0ex}}n\right]=\frac{{P}_{l}\left{s}_{l}\right[\phantom{\rule{0.3em}{0ex}}n\left]\right}{{\stackrel{\u0304}{c}}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]{\stackrel{\u0304}{c}}_{l}\left[\phantom{\rule{0.3em}{0ex}}{n}_{\text{peak}}\right]},\phantom{\rule{1em}{0ex}}n\ne {n}_{\text{peak}.}
(47)

5.
Use the minimum value of μ [ n] (so that no peak regrowth occurs) as the scaling factor for the whole frame, namely
\mu =\text{min}(\mu [n\left]\phantom{\rule{0.3em}{0ex}}\right).
(48)

6.
Scale the clipped signal {\u0109}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right] by μ and add it to the original signal to obtain the new transmit signal
{\u015d}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]={s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]+\phantom{\rule{0.3em}{0ex}}\mu {\u0109}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right],\phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}n=\phantom{\rule{2.77626pt}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,1,\cdots \phantom{\rule{0.3em}{0ex}},(M1)N/2+L1].
(49)

7.
If μ is negative, stop the iterative ACE algorithm as any further PAPR reduction attempts will result in peak regrowth.
4.2.2 Overlapping SGP
As previously mentioned, the single scaling SGP method presented above scales the clipped signal, which spans the entire frame, by a single scaling factor. This method can be expanded to consider the impact of the individual symbols on the original baseband signal and obtain an optimal scaling vector μ by which we can scale each individual symbol separately allowing for higher degrees of freedom.
Once again, we obtain the discrete clipped FBMC symbols in isolation, namely {\u0109}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]. As in Figure 8, each clipped symbol only has an effect on a portion of length L on the original transmit signal.
However, when observing the impulse response of the prototype filter as shown in [5], only the middle half of the impulse response actually has significant values. The rest of the impulse response values are almost negligible. This is also illustrated in Figure 8 where the solid line of {\u0109}_{l,4}\left[\phantom{\rule{0.3em}{0ex}}n\right] and {\u0109}_{l,5}\left[\phantom{\rule{0.3em}{0ex}}n\right] represent the significant portion of the clipped signals. Therefore, to calculate a more accurate estimate for μ, we only consider the middle section of length N of the clipped symbols. The peak value of the original transmit signal s_{
l
}[ n] is then calculated between these intervals. This is used to calculate a ratio between the new positive projections and the peak on this interval. In order to calculate the projections {\stackrel{\u0304}{c}}_{l,m}\left[\phantom{\rule{0.3em}{0ex}}n\right] of {\u0109}_{l,m}\left[\phantom{\rule{0.3em}{0ex}}n\right] onto the phase angle of the original transmit signal, the dot product can be used. This calculates only the component of {\u0109}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right] projected on the same phase angle as the original signal. In order to mitigate peak regrowth, the goal is to reduce the magnitude of the peak as much as possible, whilst simultaneously limiting new projections that occur on other samples that may result in peak regrowth. Once again, only positive projections are considered as they indicate peak regrowth.
As can be seen by Figure 8, the significant portion of the clipped signals only overlap on an interval of length N whereas the actual clipped signals overlap over the interval L= K N. The scaling of the clipped portions still has an effect on a length L of the original transmit signal.
The overlapping SGP technique can be summarized as follows:

1.
Obtain the clipped symbols {\u0109}_{l,0}\left[\phantom{\rule{0.3em}{0ex}}n\right],\cdots \phantom{\rule{0.3em}{0ex}},{\u0109}_{l,M1}\left[\phantom{\rule{0.3em}{0ex}}n\right].

2.
Begin by setting m = 0.

3.
Obtain the projections of {\u0109}_{l,m}\left[\phantom{\rule{0.3em}{0ex}}n\right] on the meaningful intervals of length N as
{\stackrel{\u0304}{c}}_{l,m}\left[\phantom{\rule{0.3em}{0ex}}k\right]=\frac{\Re \phantom{\rule{0.3em}{0ex}}\left\{{\u0109}_{l,m}\left[\phantom{\rule{0.3em}{0ex}}k\right]\xb7{s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]\phantom{\rule{0.3em}{0ex}}\right\}}{\left{s}_{l}\right[\phantom{\rule{0.3em}{0ex}}n\left]\right}
(50)
with
n=\mathit{\text{mN}}/2+k,\phantom{\rule{1em}{0ex}}3N/2\le k\le 5N/21.
(51)

4.
Find the peak P _{
m
} on the meaningful interval and its corresponding positions n _{peakm} as
{P}_{m}=\text{max}\left{s}_{l}\right[\phantom{\rule{0.3em}{0ex}}n\left]\right
(52)
with
\begin{array}{c}\mathit{\text{mN}}/2+3N/2\phantom{\rule{0.3em}{0ex}}\le n\le \mathit{\text{mN}}/2+5N/2,\\ \phantom{\rule{5.5em}{0ex}}m\phantom{\rule{0.3em}{0ex}}=[\phantom{\rule{0.3em}{0ex}}0,1,\cdots \phantom{\rule{0.3em}{0ex}},M1].\end{array}
(53)

5.
Compute the scaling factors for all of the projections by only considering the positive projections, on the meaningful interval calculated in Equation 50, as
{\mu}_{m}\left[\phantom{\rule{0.3em}{0ex}}n\right]=\frac{{P}_{m}\left{s}_{l}\right[\phantom{\rule{0.3em}{0ex}}n\left]\right}{{\stackrel{\u0304}{c}}_{l,m}\left[\phantom{\rule{0.3em}{0ex}}n\right]{\stackrel{\u0304}{c}}_{l,m}\left[\phantom{\rule{0.3em}{0ex}}{n}_{\text{peak}m}\right].}
(54)

6.
Use the minimum value of μ _{
m
}[ n] as the scaling factor for symbol m
{\mu}_{m}=\text{min}\left({\mu}_{m}\right[\phantom{\rule{0.3em}{0ex}}n\left]\phantom{\rule{0.3em}{0ex}}\right).
(55)

7.
Scale the clipped symbol {\u0109}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right] with its relevant scaling factor μ _{
m
} and add it to the original signal s _{
l
}[ n] to obtain the new transmit signal {\u015d}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right] as
{\u015d}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]={s}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right]+\phantom{\rule{0.3em}{0ex}}{\mu}_{m}{\u0109}_{q}\left[\phantom{\rule{0.3em}{0ex}}n\right].
(56)

8.
Return to step 3, set m = m + 1 and update s _{
l
}[ n] with {\u015d}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right] obtained in step 7.

9.
Once a scaling factor has been obtained for each clipped symbol, a full overlapping SGP iteration has been completed.

10.
If the PAPR meets requirements, transmits the new updated {\u015d}_{l}\left[\phantom{\rule{0.3em}{0ex}}n\right], otherwise, repeats the clipping process.
It should be clear from the methodology described above for the overlapping SGP method, scaling values are obtained for each symbol starting at symbol 0 and the transmit signal is updated on a symbol by symbol basis. The updating therefore starts at symbol 0 and propagates through the entire frame until symbol M1 has been updated. This can be considered as a forward progression through the frame. Each consecutive symbol therefore depends on the previous symbol to calculate its respective projections and scaling factor. If a scaling value smaller than zero is obtained for a specific symbol, the scaling factor for that symbol is set to zero as it indicates that scaling will result in peak regrowth. However, the process does not terminate. There may be other symbols that can reduce the PAPR, and so, the scaling factors for other symbols are continued.
The fact that the overlapping SGP proposal depends only on previous symbols implies that a framebased method of execution is not required. In fact, the overlapping SGP method can be implemented with a delay of only one filter length, namely L. It can also be seen at step 6 that the minimum value of μ_{
m
}[ n] is chosen whereas the proposed optimization methods aim to maximize the μ_{
m
} values. The justification for the difference is that the minimization criterion is built into the optimization problem in the form of constraints placed at each sample point, namely Equations 33 to 34 and Equations 38 to 43.