3.1 General definition of the continuous-time PAPR
For a finite observation duration of NT, we can define the PAPR for the GWMC signal expressed in Equation 1 as follows:
(8)
The index c corresponds to the continuous-time context and the exponent N is the number of GWMC’s symbols considered in our observation.
Lemma 1. (Mean power of GWMC signal)
(9)
The mean power P
c,mean
is defined over an infinite integration time, because our scenario assumes an infinite transmission time. The details of its derivation, in order to obtain Equation 9, are explained in Appendix 3.
For the discrete-time context, the PAPR can be defined as follows:
Definition 2.(Discrete-time PAPR of GWMC signal for a finite observation duration)
(10)
(11)
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
(12)
Siclet has derived in his thesis [21] the mean power of a discrete BFDM/QAM (biorthogonal frequency division multiplexing) signal that is expressed as 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.
3.2 Approximation of the CDF 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
(13)
Lemma 2 (Upper bound of the PAPR) For a finite number of carriers M, and a family of functions (g
m
)m ∈ [ [0,M-1] ]that satisfiesand, we have for any observation duration I, and any input symbols,
(14)
(15)
Note that . For proof and a full demonstration, please refer to Appendix 4.
Based on Lemma 2, there is a finite PAPRc,sup such that, for any γ > PAPRc,sup and any observation duration I, Pr(PAPR
c
(I) ≤ γ) = 1, the CCDF is then equal to zero beyond the PAPRc,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 PAPRc,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:
(16)
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
We have the random variables , , , 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
(17)
(18)
Following the same steps, we get
(19)
Then X (k) follows a complex Gaussian process with zero mean and variance . Hence,
(20)
we obtain from Equations 16 and 20 the following result:
Theorem 1. (PAPR distribution of GWMC signal for a finite observation duration)
For large M, with the considered simplifying assumptions, we have
(21)
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.
3.3 PAPR distribution for an infinite observation period
By considering an infinite number of GWMC symbols observed, we can define the PAPR as being
(22)
To get the PAPR distribution in this case, we can let N go to infinity in Equation 21; then, we obtain the following formula:
Corollary 1. (PAPR distribution of GWMC signal for an infinite observation duration) For large M, with the considered simplifying assumptions, we have
In fact, we have
(24)
(25)
(26)
From Equations 25 and 26, we get
(27)
hence
(28)
Thus, for an infinite observation duration, and for γ sufficiently small compared with PAPRc,sup, we have
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 PAPRc,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.