Information-bearing symbols, resulting from modulation performed by any type of constellation, are decomposed into several blocks. Each block of symbols is inserted in parallel into a modulation system as shown in Figure 1. At the output of the modulator, the GWMC signal can be expressed as 19 :

In order to apply the generalized central limit theorem (G-CLT) in Section 3, our system must satisfy the following two assumptions:

As a consequence,

E C m , n R C p , q R = σ c 2 2 δ m , p δ n , q ,

E C m , n I C p , q I = σ c 2 2 δ m , p δ n , q ,

E C m , n R = E C m , n I = 0 ,

E C m , n R C p , q I = E C m , n R E C p , q I = 0 .

The symbols from constellation diagrams of the usual digital modulation schemes (quadrature amplitude modulation (QAM), quadrature phase shift keying (QPSK), etc.) satisfy these conditions.

As discussed in Appendix 2B, Assumption 2 is sufficient to ensure Lyapunov’s conditions, which ensures the validity of a variant of the CLT that considers independent random variables which are not necessarily identically distributed 20 . The reader can refer to Appendices Appendix 1 and Appendix 2A for more details.

For a finite observation duration of NT, we can define the PAPR for the GWMC signal expressed in Equation 1 as follows:

PAPR c N = max t ∈ [ 0 , NT ] | X ( t ) | 2 P c , mean .

The index c corresponds to the continuous-time context and the exponent N is the number of GWMC’s symbols considered in our observation.

For the discrete-time context, the PAPR can be defined as follows:

The subscript d corresponds to the discrete-time context, and P is the number of samples in period T.

In fact, the discrete mean power is defined as

P d , mean = lim K → + ∞ 1 2 K + 1 ∑ k = - K K E | X ( k ) | 2 .

Siclet has derived in his thesis 21 the mean power of a discrete BFDM/QAM (biorthogonal frequency division multiplexing) signal that is expressed as X [ k ] = ∑ m = 0 M - 1 ∑ n = - ∞ + ∞ f m [ k - nP ] , such that f _m [ k] is an analysis filter. His derivation does not use the exponential property of f _m [ k]. Then, in our case, we can follow the same method to get Equation 11.

To simplify the derivation of an approximation of the PAPR distribution, we will consider the discrete-time definition of the PAPR.

The CDF or its complementary function is usually used in the literature as a performance criterion of the PAPR. The CDF is the probability that a real-valued random variable (the PAPR here) featuring a given probability distribution will be found at a value less than or equal to γ, which can be expressed in our case as

Pr PAPR d N ≤ γ = Pr max k ∈ [ [ 0 , NP - 1 ] ] | X ( k ) | 2 P d , mean ≤ γ .

Note that PAPR c , sup = sup I PAPR c ( I ) ≤ PAPR c , bound . For proof and a full demonstration, please refer to Appendix 4.

Based on Lemma 2, there is a finite PAPR _c,sup such that, for any γ > PAPR _c,sup and any observation duration I, Pr(PAPR _c (I) ≤ γ) = 1, the CCDF is then equal to zero beyond the PAPR _c,sup (see Figure 2). We can also mention that the bound goes to infinity when the number of carriers M goes to infinity, the CCDF is then equal to 1 for any sufficiently small γ (see Figure 2).

In what follows, we give an approximation of the CDF of the PAPR for γ sufficiently small compared with PAPR _c,sup. We leave to further work a mathematical quantification of how small γ should be. For finite N, the experiments show a good fit with our prediction.

By considering an observation duration limited to N GWMC symbols of P samples each, and by approximating the samples X (0), X (1), X (2),…, X (N P - 1) as being independent (which is the approximation made in 8 in the absence of oversampling), the CDF of the PAPR for the GWMC signal can be expressed as follows:

Pr PAPR d N ≤ γ = Pr max k ∈ [ [ 0 , NP - 1 ] ] | X ( k ) | 2 ≤ γ P d , mean = Pr ∀ k ∈ [ [ 0 , NP - 1 ] ] | X ( k ) | 2 ≤ γ P d , mean ≈ ∏ k ∈ [ [ 0 , NP - 1 ] ] Pr | X ( k ) | 2 ≤ γ P d , mean .

We should now look for the distribution law of |X (k)|². We first find the distribution of the real part of X (k) and, after that, do the same for the imaginary part.

Let us put

X m R ( k ) = ∑ n ∈ Z C m , n R g m , n R ( k ) - C m , n I g m , n I ( k ) X m I ( k ) = ∑ n ∈ Z C m , n R g m , n I ( k ) + C m , n I g m , n R ( k ) , such that X R ( k ) = ∑ m = 0 M - 1 X m R ( k ) X I ( k ) = ∑ m = 0 M - 1 X m I ( k ) .

We have the random variables X 0 R ( k ) , X 1 R ( k ) , X 2 R ( k ) , … , X M - 1 R ( k ) that are independent with zero mean and satisfy Lyapunov’s condition by assuming Assumption 2. Thus, for large M, we can apply the G-CLT (see Appendix 2) and get

∑ m = 0 M - 1 X m R ( k ) ∼ N 0 , σ c 2 2 ∑ n ∈ Z ∑ m = 0 M - 1 | g m , n ( k ) | 2 ︸ σ X 2 ( k ) 2

X R ( k ) ∼ N 0 , σ X 2 ( k ) 2 .

Following the same steps, we get

X I ( k ) ∼ N 0 , σ X 2 ( k ) 2 .

Then X (k) follows a complex Gaussian process with zero mean and variance σ X 2 ( k ) = σ c 2 ∑ n ∈ Z ∑ m = 0 M - 1 | g m ( k - nP ) | 2 . Hence,

| X ( k ) | 2 ∼ χ 2 with two degrees of freedom . Denoting x ( k ) = P d , mean σ X 2 ( k ) ,

we obtain from Equations 16 and 20 the following result:

Note that the approximate distribution of the PAPR depends not only on the family of modulation functions (g _m ) _{m ∈ [ [0,M - 1] ]} but also on the number of carriers M used. In addition, it depends on the parameter N, the number of GWMC symbols observed. Taking into account the simplifying assumption of our derivations, we can easily study the PAPR performance of any multi-carrier modulation (MCM) system. The approximate expression of the CDF of the PAPR will be compared to the empirical CDF featured in Section 4.1. In Section 3.3, we deduce the behaviour of the PAPR distribution for an infinite observation period.

By considering an infinite number of GWMC symbols observed, we can define the PAPR as being

Pr PAPR d ∞ ≤ γ = Pr max k ∈ N | X ( k ) | 2 P d , mean ≤ γ .

To get the PAPR distribution in this case, we can let N go to infinity in Equation 21; then, we obtain the following formula:

In fact, we have

Pr ( PAPR d ∞ ≤ γ ) = ∏ k ∈ N 1 - e - x ( k ) γ , with x ( k ) = ∑ m = 0 M - 1 ∥ g m ∥ 2 P ∑ n ∈ Z ∑ m = 0 M - 1 | g m ( k - nP ) | 2 .

And we have ∑ m = 0 M - 1 ∥ g m ∥ 2 = ∑ m = 0 M - 1 ∑ n ∈ Z ∑ k = 0 P - 1 | g m ( k - nP ) | 2 , then A 2 MP < ∑ m = 0 M - 1 ∥ g m ∥ 2 < B 2 MP ,

and 1 B 2 MP < 1 M ∑ n ∈ Z ∑ m = 0 M - 1 | g m , n ( k ) | 2 < 1 A 2 MP .

From Equations 25 and 26, we get

A 2 B 2 < x ( k ) < B 2 A 2 ,

hence

1 - e - x ( k ) γ < 1 - e - B 2 A 2 γ < 1 .

Thus, for an infinite observation duration, and for γ sufficiently small compared with PAPR _c,sup, we have

Pr ( PAPR d ∞ ≤ γ ) = 0 .

The CCDF is then equal to 1 for any γ (see Figure 3).

Equation 29 can be interpreted by the fact that we cannot control the PAPR for an infinite number of GWMC symbols, because we will inevitably have some GWMC symbols with large peaks. Thus, for M ≥ 8 and γ sufficiently small compared with PAPR _c,sup and an infinite observation duration, there is no family of modulation functions (g _m ) _{m ∈ [ [0,M - 1] ]} that can avoid the large peaks. Based on this fact, we expect to optimize the PAPR within a finite number of GWMC symbols. As the number of GWMC symbols decreases, PAPR optimization gets better.

We know that the PAPR is a random variable, that is why the derivation of its distribution law may be complex. Especially if you have to repeat the derivation each time the family of modulation functions changes. Equation 21 is a closed form approximation, so it allows us to directly find an approximate CDF of the PAPR without complex operations. To illustrate this, we consider the following examples.

The CCDF of the PAPR depends on the number of carriers M, the observation duration N T and the parameter γ to which the PAPR is compared. For an infinite number of carriers, the PAPR is always large and the CCDF is then equal to 1 for any γ. If the number of carriers M is finite, the CCDF of the PAPR is equal to 0 for γ greater than PAPR _c,sup. And for γ less than PAPR _c,sup, the CCDF depends on the observation duration: when we consider an infinite observation duration, the CCDF is equal to 1, but if we limit the observation duration to a finite number of symbols, the CCDF is defined by Equation 21. Figure 6 presents all the possible cases.

To reduce the occurrence probability of the peaks, several techniques can be used 24 25 26 . These techniques take place before and after the modulation. But as we can see in Equation 21, the distribution of the PAPR depends on the family of modulation functions used in the system. Thus, we can act also on the modulation system to reduce the PAPR. In fact, to confirm this idea, we can compare the PAPR performance of the OFDM/OQAM 13 to that of the wavelet OFDM 27 . The first variant uses the same modulation basis as the conventional OFDM (Fourier basis) and does not show any improvement on the PAPR’s performance 22 , unlike the second that uses the wavelet basis and shows an improvement of up to 2 dB 28 .

Let us model the PAPR’s optimization problem. It consists in finding the optimal functions g _m that maximize the CDF of the PAPR in Equation 21. By trying to do that, we notice that we should do the optimization at some particular points of the functions g _m (the sampling points), while these points are not chosen in a particular way. To adjust the optimization problem, we approximate the denominator of x (k). Let us put

S p = ln Pr PAPR d N ≤ γ = ∑ k = 0 NP - 1 ln 1 - e - ∑ m = 0 M - 1 ∥ g m ∥ 2 P ∑ n ∈ Z ∑ m = 0 M - 1 | g m , n ( k T P ) | 2 γ ,

and we put

h ( x ) = ln 1 - e - ∑ m = 0 M - 1 ∥ g m ∥ 2 P ∑ n ∈ Z ∑ m = 0 M - 1 | g m , n ( x ) | 2 γ ,

We have h: [ 0,N T] → R is piecewise continuous over [ 0,N T], and we have

NT NP S p = NT NP ∑ k = 0 NP - 1 h k NT NP is a Riemann sum, then

T P S p ≈ ∫ 0 NT h ( x ) dx ⇒ 1 P S p ≈ 1 T ∫ 0 NT h ( x ) dx ⇒ S p = P T ∫ 0 NT h ( x ) dx + o ( P ) ⇒ Pr PAPR d N ≤ γ = e P T ∫ 0 NT h ( x ) dx + o ( P ) .

Maximizing Pr PAPR d N ≤ γ is equivalent to maximizing ∫ 0 NT h ( x ) dx , and it is equivalent to maximizing ∫ 0 T h ( x ) dx since h is a periodic function of T. Then, the quantity that we should maximize over the functions (g _m ) _{m ∈ [ [0,M - 1] ]} is

∫ 0 T ln 1 - e - ∑ m = 0 M - 1 ∥ g m ∥ 2 P ∑ n ∈ Z ∑ m = 0 M - 1 | g m , n ( t ) | 2 γ dt .

To express the optimization problem that can lead to a new modulation system featuring an optimally reduced PAPR, we have to express the constraints on the functions (g _m ) _{m ∈ [ [0,M - 1] ]}. Let us define the following constraints:

• C ₀ = {(g _m ) _{m ∈ [ [0,M - 1] ]}/for M large

∑ m = 0 M - 1 E | ∑ n ∈ Z C m , n R g m , n R ( t ) - C m , n I g m , n I ( t ) | 3 3 σ c 2 2 ∑ m = 0 M - 1 ∑ n ∈ Z | g m ( t - nT ) | 2 < ε M ≪ 1 , ∑ m = 0 M - 1 E | ∑ n ∈ Z C m , n R g m , n I ( t ) + C m , n I g m , n R ( t ) | 3 3 σ c 2 2 ∑ m = 0 M - 1 ∑ n ∈ Z | g m ( t - nT ) | 2 < ε M ′ ≪ 1 }

• C ₀ is the condition that we need to apply CLT. We considered Assumption 2 because the expression of C ₀ is not intuitive.

•  C 1 = ( g m ) m ∈ [ [ 0 , M - 1 ] ] / for large M max m , t ∑ n ∈ Z | g m ( t - nT ) | = B 1 < + ∞ and min m , t ∑ n ∈ Z | g m ( t - nT ) | 2 = A 2 > 0 . C ₁ is the set of functions that satisfy Assumption 2. C ₁ is stronger than C ₀.

•  C 2 = ( g m ) m ∈ [ [ 0 , M - 1 ] ] / ∀ m ∈ [ [ 0 , M - 1 ] ] ∑ n ∈ Z | g m , n ( t ) | 2 = ∑ n ∈ Z | g 0 , n ( t ) | 2 . C ₂ is stronger than C ₀ and C ₁ but easy to manipulate.

These conditions define the choice of the functions (g _m ) _{m ∈ [ [0,M - 1] ]}. Figure 7 shows the inclusion relationship between the different conditions.

Thus, the optimization problem can be expressed as follows:

Optimization problem (i)

maximize g m m ∈ [ [ 0 , M - 1 ] ] ∫ 0 T ln 1 - e - ∑ m = 0 M - 1 ∥ g m ∥ 2 P ∑ n ∈ Z ∑ m = 0 M - 1 | g m , n ( t ) | 2 γ dt , subject to g m m ∈ [ [ 0 , M - 1 ] ] ∈ C i .

The present study proposes to act on the modulation system to reduce the PAPR. Looking for the optimal family of modulation functions is a new approach to deal with the PAPR problem. Its advantage is that we can use, at the same time, one of the previous PAPR reduction techniques to get an even smaller PAPR.

In probability theory, the CLT states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases to infinity. Under stronger assumptions, the Berry-Esseen theorem, or Berry-Esseen inequality, specifies the rate at which this convergence takes place by giving a bound on the maximal error of approximation between the normal distribution and the true distribution of the scaled sample mean. In the case of independent samples, the convergence rate is 1 M . In general, we have a good approximation for the Gaussian curve for M > 8.

In our case, we perform simulations by considering the number of carriers to be equal to M = 64 > 8. Under this assumption, the following Lyapunov’s condition:

lim M → + ∞ ∑ m = 0 M - 1 E | ∑ n ∈ Z C m , n R g m , n R ( t ) - C m , n I g m , n I ( t ) | 3 3 σ c 2 2 ∑ m = 0 M - 1 ∑ n ∈ Z | g m ( t - nT ) | 2 = 0 and lim M → + ∞ ∑ m = 0 M - 1 E | ∑ n ∈ Z C m , n R g m , n I ( t ) + C m , n I g m , n R ( t ) | 3 3 σ c 2 2 ∑ m = 0 M - 1 ∑ n ∈ Z | g m ( t - nT ) | 2 = 0

becomes

∑ m = 0 M - 1 E | ∑ n ∈ Z C m , n R g m , n R ( t ) - C m , n I g m , n I ( t ) | 3 3 σ c 2 2 ∑ m = 0 M - 1 ∑ n ∈ Z | g m ( t - nT ) | 2 < ε M ≪ 1 and ∑ m = 0 M - 1 E | ∑ n ∈ Z C m , n R g m , n I ( t ) + C m , n I g m , n R ( t ) | 3 3 σ c 2 2 ∑ m = 0 M - 1 ∑ n ∈ Z | g m ( t - nT ) | 2 < ε M ′ ≪ 1 .