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# EVM derivation of multicarrier signals to determine the operating point of the power amplifier considering clipping and predistortion

- Ali Cheaito
^{1}Email authorView ORCID ID profile, - Jean-François Hélard
^{1}, - Matthieu Crussière
^{1}and - Yves Louët
^{2}

**2016**:281

https://doi.org/10.1186/s13638-016-0771-5

© The Author(s) 2016

**Received:**5 May 2016**Accepted:**10 November 2016**Published:**7 December 2016

The Erratum to this article has been published in EURASIP Journal on Wireless Communications and Networking 2017 2017:78

## Abstract

The high peak-to-average power ratio (PAPR) remains a major drawback of multicarrier modulations. Hence, the nonlinear characteristics of power amplifiers (PA) results in strong distortion and low power efficiency when multicarrier modulations are used. In this paper, the impact of the nonlinearities on the amplified multicarrier signal is analyzed, considering both PAPR limitation using clipping and PA linearization. Therefore, we derive the expression of the error vector magnitude (EVM) of the amplified signal with and without the use of polynomial predistortion and/or clipping techniques. We provide analytical EVM expressions that depend on the PA and predistortion characteristics, as well as the PAPR and the average power of both input and clipped signals. These expressions are general formulas which allow to measure in-band distortion at the PA output. The simulation results show that the proposed expressions present perfect accuracy. Moreover, the trade-off between the PA linearity and efficiency is investigated considering the performance of the clipping and predistortion techniques. An analytical expression which gives the optimal input back-off (IBO) maximizing the PA efficiency with respect to any EVM constraint is provided. Finally, the predistortion complexity is discussed aiming at reducing it with respect to an EVM constraint.

## Keywords

- EVM
- PAPR
- OFDM
- Clipping
- Predistortion

## 1 Introduction

Multicarrier modulations are a key technology extensively deployed in wireless communication systems such as WiFi, WiMAX, DVB, and LTE. Furthermore, multicarrier modulations may be a worthy candidate for the next generation (5G) cellular systems. Orthogonal frequency division multiplexing (OFDM) wavelet packets multicarrier modulation (WP-MCM) and filter bank multicarrier (FBMC) are the current state-of-the art multicarrier techniques. All these techniques suffer from high peak-to-average power ratio (PAPR) of the transmitted signal which prevents to feed the power amplifier (PA) at its optimal point, hereby lowering its power efficiency. Considering the PA characteristics and the PA power efficiency, it is well-known that the closer the average power of the transmitted signal to the nonlinear zone of the PA characteristics, the higher the PA efficiency. So, a trade-off between the linearity and the efficiency of the PA should be carefully considered. In the literature, one can find two main strategies to deal with this problem: PAPR reduction of the transmitted signal and linearization of the PA. The PAPR reduction approach aims at reducing the dynamic range of the transmitted signal. The most popular PAPR reduction techniques are clipping [1], coding [2], selected mapping (SLM) [3], tone reservation (TR) [4], and active constellation extension (ACE) [5]. The second approach is the linearization which tries to compensate for the nonlinearity of the PA. Among the large variety of linearization techniques, the most popular ones are digital predistortion (DPD) [6], linear amplification with nonlinear component (LINC) [7], and feedback [8]. In our study, we consider the clipping technique as a PAPR reduction technique and the predistortion as a linearization technique. Although PAPR reduction and linearization techniques work in a complementary way [9, 10], it turns out that they are often designed separately and implemented independently in conventional systems [11–16]. Nevertheless, recent studies have investigated some joint design of these techniques in order to enhance their interoperability [17–20]. In this paper, we propose to go a step beyond this strategy by introducing a new adaptive approach which controls the clipping and predistortion functions in a flexible way. Our aim is to maximize the PA efficiency and minimize the predistortion complexity with respect to the linearity constraint, as represented by the error vector magnitude (EVM) of the amplified signal, and according to some predefined parameters. These parameters are the PA characteristics, the transmit power, the PAPR of the input signal, and the target EVM. The EVM is a common figure of merits used to evaluate the quality of communication systems. Indeed, most of wireless communication standards such as the IEEE802.11a standard [21], the IEEE802.16e WiMAX standard [22], and the LTE standard [23] have already specified their requirements in terms of the EVM.

Our objective is to derive a flexible transmitter model able to update its parameters according to incoming requirements and outside environment. Hence, this work is a very important step in the analytical study of the global optimization approach of the transmitter efficiency and linearity.

In this context, we are involved in this paper in the analytical derivation of the EVM of multicarrier signals. In [24, 25], we already derived EVM expressions of clipped predistorted multicarrier signals amplified by Rapp PA model. In this paper, we study the amplitude probability distribution of the signal after applying the clipping technique. After that, we derive the EVM expression of nonlinear amplified multicarrier signals using a memoryless polynomial PA model when clipping is activated or not. Then, we focus on the impact of the predistortion technique on the EVM expression. Once again, we derive the EVM of multicarrier signals when the memoryless polynomial predistortion is activated. Therefore, our first contribution consists in providing EVM expressions as a function of the transmitted power, the clipping threshold, and the PA and predistortion characteristics, as well as the PAPR of the input and clipped signals. As far as theoretical EVM derivations are concerned, some upper-bounds can be found in [26, 27], and other contributions, as [28–30], give closed-form EVM expressions but modeling the RF front end as a simple clipping. To the best of our knowledge, despite numerous studies in the literature about PAPR, no analytical expression of the EVM, taking into account all the above mentioned parameters, exists in the literature. Moreover, we investigate the theoretical analysis of the PA linearity-efficiency and the predistortion complexity. We first examine the PA linearity-efficiency trade-off. Thereby, we provide an analytical expression which gives the optimal input back-off (IBO) and clipping threshold which must be taken to maximize the PA efficiency for any EVM constraint. Secondly, we discuss the trade-off between the PA linearity and the predistortion complexity aiming at reducing the predistortion complexity with respect to an EVM constraint. Therefore, we seek the optimal configuration of clipping and predistortion which maximizes the PA efficiency taking into account the predistortion complexity and satisfying the EVM constraint.

The remainder of the paper is organized as follows. In Section 2, we introduce our system model. Then, the theoretical derivations of the EVM with the simulation results in both cases without and with predistortion are presented in Section 3 and 4, respectively. Afterwards, the theoretical analysis of the trade-off between the PA linearity-efficiency and the complexity of the predistortion is presented in Section 5. Finally, a conclusion and perspectives are drawn in Section 6.

## 2 System model

*x*

_{1}(

*t*), generated by the system, becomes

*x*

_{2}(

*t*) after clipping, and

*x*

_{3}(

*t*) after the predistortion operation. The output of the PA is

*z*(

*t*).

### 2.1 Clipping

One of the major drawbacks of multicarrier waveforms is the high PAPR of the signals obtained in the time domain.

###
**Definition 1**

*T*, and is given by

where *E*{.} is the expectation function.

*A*

_{max}, without affecting the phase

*ϕ*(

*x*). Thus, the clipped signal,

*x*

_{2}(

*t*), is represented as

This technique results in both in-band and out-of-band distortions because of its nonlinear operation. Additionally, it changes the amplitude probability distribution function of the signal which is discussed in the next subsection.

### 2.2 Signal amplitude distribution after clipping

where \(P_{x_{1}}\) is the average power of the input signal *x*
_{1}(*t*) and *r* the magnitude of the input voltage, i.e., *r*=|*x*
_{1}(*t*)|. However, after clipping, the PDF of the signal amplitude changes and is stated in the following lemma.

###
**Lemma 1**

where *δ*(*r*)is the Dirac delta function and \(PAPR_{[x_{1}]}\) is the deterministic maximum PAPR value of *x*
_{1}(*t*). Please note that for a given number of subcarriers and a type of modulation, there is a maximum value that PAPR can reach. For example, given that *N* is the number of subcarriers and M-QAM is the modulation type, the PAPR upper bound is given by \(\frac {3N (\sqrt {M-1})}{(\sqrt {M+1})}\).

###
*Proof*

*x*

_{1}(

*t*)|<

*A*

_{max}. In this case, \( f_{x_{1}}\left (r\right) = f_{x_{2}}\left (r\right)\). However, if |

*x*

_{1}(

*t*)|>

*A*

_{max}, then |

*x*

_{2}(

*t*)| will be equal to

*A*

_{max}. Therefore, we obtain

*P*{

*r*>

*A*

_{max}} is the probability that

*r*, the amplitude of

*x*

_{1}(

*t*), is larger than the clipping threshold

*A*

_{max}. Thus, we can write

where *R*
_{max} is the maximum amplitude of *x*
_{1}(*t*). Hence, calculating the integral in Eq. (6) we get the expression in Eq. (4). Note that the signal amplitude in practice does not tend to infinity and is limited to a maximum value *R*
_{max} which depends on the system parameters as the number of subcarriers. □

*x*

_{1}(

*t*), and the clipped signal

*x*

_{2}(

*t*), are presented in Fig. 2.

Undoubtedly, the average power of the clipped signal, \(P_{x_{2}}\), is different from \(P_{x_{1}}\) as we will see in the following lemma.

###
**Lemma 2**

*λ*between the average power after and before clipping is given by

###
*Proof*

*x*

_{2}(

*t*) into Eq. (8), we show that

□

Note that the PAPR of the clipped signal, \(\text {PAPR}_{[x_{2}]}\), is \(\frac {A^{2}_{\max }}{P_{x_{2}}}\). Thus, the ratio \(\frac {A^{2}_{\max }}{P_{x_{1}}}\), which will be used in the following analytical derivations, is equal to \(\lambda \ \text {PAPR}_{[x_{2}]}\).

### 2.3 Nonlinear behavioral model of the PA

*H*

_{ PA }(

*r*) by the following odd order polynomial

^{1}

where *L*
_{
p
} denotes the nonlinear model order, *b*
_{2l+1} are the nonlinear PA characteristics coefficients, and *r* is the amplitude of the input voltage.

### 2.4 Predistortion model

*H*

_{ PD }(

*r*) which is exactly the inverse of the PA transfer function. Thus, the concatenation of these two functions will ideally be equivalent to a linear function. Since the PA model is a polynomial function, the inverse predistortion function can also be expressed as a polynomial. Therefore, the predistortion function is expressed as follows

with *K*
_{
p
} the nonlinear model order and *a*
_{2k+1} the nonlinear polynomial coefficients.

### 2.5 EVM definition

*X*

_{ k }and

*Z*

_{ k }the

*k*th complex symbols of the signal before clipping and after amplification,

*x*

_{1}(

*t*) and

*z*(

*t*), respectively. In this figure, a unitary amplification gain is assumed for the clarity of the representation. Note that thanks to the Parseval’s theorem which state that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its Fourier transform, we can emphasize that the calculation of the EVM can equivalently been led in time or frequency domains. In the sequel, EVM derivations will be based on the following definition of the EVM in the time domain.

###
**Definition 2**

*z*(

*t*) is expressed as follows

with *ε*(*r*) the stationary random variable modeling the signal error and \({{f}_{x_{1}} \left (r\right)}\) the PDF of the amplitude of signal *x*
_{1} as defined in Eq. (3). Then, Eq. (13) represents the second order moment of the magnitude error |*ε*(*r*)| and *E*{|*x*
_{1}(*t*)|^{2}} is the average signal power.

## 3 EVM expressions without predistortion

In this section, we present the results of the EVM derivations when predistortion is not activated, in both cases with and without clipping, as a function of the PA characteristics, the average power and the PAPR of both input and clipped signals.

### 3.1 Analytical EVM derivations

#### 3.1.1 Without clipping

Firstly, we start by deriving the EVM expression when clipping and predistortion are deactivated. Using Eqs. (3), (10), and (12), we derive a closed-form expression of the EVM. Our main result is presented in the following theorem.

###
**Theorem 1**

*γ*represents the incomplete gamma function defined as in (43) (see Appendix 1), with

Note that the sum in (14) is taken over all combinations of nonnegative integer exponents *p*
_{1} through \(p_{2L_{p} -1}\) such that the sum of all *p*
_{2l+1} is equal to *2*.

###
*Proof*

See Appendix 1. □

This theorem provides an analytical EVM expression in the form of a series expansion involving gamma functions and depending on different parameters. We can see that the in-band distortion of the amplified signal depends on the PA characteristics, as well as the PAPR of the signal *x*
_{1}(*t*) and the average power \(P_{x_{1}}\). Note that this expression will be useful in the following analytical derivations of the EVM with clipping.

#### 3.1.2 With clipping

We now lead the EVM calculation considering that the clipping is activated. In this case, using Lemma 1, Lemma 2, Eq. (10), and Eq. (12), we derive a closed-form expression of the EVM which is presented in the following theorem.

###
**Theorem 2**

###
*Proof*

See Appendix 2. □

Eventually, as in the case without clipping, Theorem 2 gives an EVM expression in the form of a series expansion composed of Gamma functions and depending on several parameters. These parameters are the PA order *L*
_{
p
}, the PA coefficients *b*
_{2l+1}, the average power and the PAPR of both input and clipped signals, as well as, the clipping threshold, and *λ* the ratio between \(P_{x_{2}}\) and \(P_{x_{1}}\).

### 3.2 Simulation results and analysis

^{5}randomly generated OFDM symbols with 1024 subcarriers associated to 16-QAM modulation symbols. The polynomial coefficients

*b*

_{2l+1}of the PA function have been derived by identification of two AM/AM characteristic. The first one is for a Rapp’s SSPA model with knee factor

*p*=2 and the second one is a real PA for digital video broadcasting-terrestrial (DVB-T). It is designed for use in the 174 to 230 MHz VHF broadcast band, where it can deliver 50 W [33]. Figure 4 represents the gain and power efficiency of a real DVB-T PA as a function of the output power at 202 MHz. We use a high-order polynomial function (

*L*

_{ p }=6) to achieve a satisfactory fitting accuracy. Figures 5 and 6 depict the theoretical and simulated EVM as a function of the input power back-off (IBO) when clipping is activated or not using a Rapp-modeled PA and DVB-T PA, respectively. We consider clipping thresholds \(\frac {{A_{\max }}^{2}}{P_{r}}\) of 5, 6, 7, and 8 dB. From the curves in Figs. 5 and 6, it can be verified that the more the clipping threshold decreases, the more the EVM increases. In addition, one can notice that using the Rapp-modeled PA, the EVM tends to be zero for high IBO when clipping is deactivated. However, such a behavior is not observed in the case of the DVB-T PA. This is due to the nonlinear characteristics of the DVB-T PA even within the expected linear region, which results in additional distortion. Finally, from the comparison between the theoretical and simulated curves, we conclude that our analytical results in Theorem 1 and Theorem 2 perfectly match the simulated EVM for both PAs. This proves the accuracy of our proposed analytical EVM derivations with and without clipping.

## 4 EVM expressions with predistortion

*H*

_{ EQ },

### 4.1 Analytical EVM derivations

#### 4.1.1 Without clipping

In this case, using Eqs. (3), (12), and (20), we calculate the analytical EVM expression and the solution is given in Theorem 3.

###
**Theorem 3**

###
*Proof*

See Appendix 3. □

As in the case without predistortion, this theorem provides an EVM expression in the form of a series expansion based on gamma functions. Besides, it now depends on the order and coefficients of both the predistortion and the amplifier, as well as the PAPR of the input signal *x*
_{1}(*t*) and its average power \(P_{x_{1}}\).

#### 4.1.2 With clipping

In this subsection, we derive the EVM expression when both clipping and predistortion techniques are activated using Eqs. (12) and (20), Lemma 1, and Lemma 2.

###
**Theorem 4**

###
*Proof*

See Appendix 4. □

Eventually, as in the case without clipping, Theorem 4 gives an EVM expression in the form of a series expansion depending on the same parameters. In addition, it now depends on the PAPR of the clipped signal *x*
_{2}(*t*), the clipping threshold, and *λ* the ratio between \(P_{x_{2}}\) and \(P_{x_{1}}\).

### 4.2 Simulation results and analysis

*K*

_{ p }=5 and clipping thresholds \(\frac {{A_{\max }}^{2}}{P_{r}}\) of 5, 6, 7, and 8 dB. Note that the indirect learning is used to extract the coefficients

*a*

_{2k+1}[34]. First of all, we can see from the curves in Figs. 7 and 8 that the EVM of the predistorted and amplified signal in the case of DVB-T PA is similar to the EVM in the case of a modeled PA. This is due to the high linearization order which provides a linear response of the transmitter chain in both cases. In addition, we remark that the lower the clipping threshold, the higher the EVM. We note also that the EVM increases rapidly when the IBO is lower than the clipping threshold. This is explained by the fact that the peaks of the signal are amplified in the saturation zone of the PA. Once again, we can observe that the theoretical curves perfectly match the simulated ones validating the consistency and accuracy of the proposed EVM expressions.

## 5 Theoretical analysis of the trade-off between PA linearity-efficiency and predistortion complexity

In the previous section, we have analytically proven that the linearity of the PA measured by the EVM metric depends on the performance of clipping and predistortion. In this section, we show that the performance of clipping and predistortion also impacts the PA’s efficiency. Thus, we theoretically analyze the trade-off between the PA linearity and efficiency considering the clipping and predistortion techniques. Secondly, as far as the complexity and the power consumption of the digital signal processing in the baseband are concerned, we expand our study to minimize the complexity of the predistortion technique with respect to an EVM constraint. Therefore, we seek to provide a joint configuration of the predistortion and clipping techniques which maximizes the PA power efficiency taking into account the complexity of the predistortion technique and considering an EVM constraint.

### 5.1 Definition of PA efficiency

*η*

_{ DC }is defined as the ratio between the output power

*P*

_{out}and the DC power

*P*

_{DC}.

*β*equals 0.50, 0.66, and 0.78 for classes A, AB, and B PAs, respectively. In Fig. 10, the power efficiency in Eq. (30) is plotted versus the OBO for classes A, B, and AB PAs. Since, the relationship between OBO and IBO is proportional, we can notice that the more the IBO decreases, the more the efficiency increases. However, from Figs. 6 and 8, as expected, we note that the more the IBO decreases, the more the EVM increases. This means that the linearity is degraded while the power efficiency is improved. Therefore, a trade-off between the PA efficiency and linearity must be considered. This aspect will be discussed in the next subsection.

### 5.2 Trade-off between PA linearity-efficiency and predistortion complexity considering clipping and predistortion

Wireless communication standards impose significant constraints on the transmitter’s linearity, and at the same time require high efficiency. The EVM is a common figure of merits used to evaluate the linearity of communication systems and has become a mandatory part of some communication standards, e.g., [23]. In the rest of this subsection, we seek to maximize the power efficiency of the PA while respecting an EVM limitation and considering a given predistortion complexity.

In the first part, we investigate the PA linearity-efficiency trade-off. For this reason, we propose to control the clipping technique and adapt the transmit power in order to maximize the PA efficiency with respect to an EVM constraint. Therefore, we provide an analytical expression which gives the optimal input power back-off (IBO) and clipping threshold which must be taken to maximize the PA efficiency for any EVM constraint. In the next part, we investigate the complexity of the predistortion technique given that the complexity of the clipping technique is negligible compared to the complexity of the predistortion. So we propose to control the clipping and predistortion techniques in order to find the optimal configuration which maximizes the PA efficiency taking into account the predistortion complexity and satisfying the EVM constraint.

#### 5.2.1 Control the clipping technique

From Eq. (30), we state that maximizing the efficiency is equivalent to minimizing the IBO. Consequently, we seek the minimum IBO and the corresponding clipping threshold with respect to a specific EVM constraint. For this reason, we plot the EVM as a function of the IBO for different clipping thresholds using a DVB-T PA [33].

###
**Corollary 1**

###
*Proof*

According to the results of Theorem 4 represented in Fig. 11 and [26], we can state that the maximum possible power efficiency, avoiding PA saturation, is achieved when peak power of the amplified signal coincides with the saturation power. So by replacing the \(\text {PAPR}_{[x_{2}]}\) by the IBO in Eq. (25), we get Eq. (31). □

A simplification of Eq. (31) is given by the following corollary.

###
**Corollary 2**

###
*Proof*

If the linearization technique is sufficiently accurate, the distortion of the PA could be negligible compared to the clipping distortion. Therefore, the expression of the EVM can be approximated by neglecting the integral *I*
_{9} in Eq. (58) and assuming that \(\frac {{H_{EQ}(A_{\max })}^{2}}{{P_{x_{1}}}} = \lambda \text {PAPR}_{[ x_{2} ]}\). Moreover, according to Corollary 1, \(\text {PAPR}_{[ x_{2} ]} = \text {IBO}\), so after doing some maths we obtain Eq. (32). □

From Fig. 11, we show that Corollary 2 provides a perfect approximation of Corollary 1. We remark that the approximated EVM given by Eq. (32) is slightly less than the exact one given by Eq. (31). This is explained by the considered simplification of the PA distortion. Indeed, the importance of Corollary 2 is that it does not depend on the linearization technique. So, for any linearization technique, using the clipping as a PAPR reduction, Corollary 2 is valid. However, the less accurate the used linearization technique is, the less correct the approximation is.

#### 5.2.2 Control the clipping and predistortion techniques

where *X*
_{
N×1} represents the predistortion output vector and \(\mathbf {R}_{N\times K_{p}}\) is the matrix which contains all the linear and product terms of the amplitude signal *r, r*
^{3},…,*r*
^{2k+1} for all the input data samples. Note that *N* represents the number of used input/output data samples and *K*
_{
p
} is the number of involved predistortion coefficients. \(\mathbf {A}_{K_{p}\times 1} \) represents the unknown coefficients vector.

*x*

_{3}(

*n*), is used as the expected output, while the output of the PA,

*z*(

*n*), is used as the input of the model. Using the least square algorithm, the coefficients vector can be estimated from

where **Z** is the PA output matrix in a similar form to the matrix **R**, and (.)^{
H
} represents the Hermitian transpose. Now, we have to compute the computational complexity of the matrix operations in Eq. (34). Indeed, there are one matrix inversion and three matrix-matrix multiplications. In fact, the inversion of a *K*
_{
p
} × *K*
_{
p
} matrix is approximately equivalent to *K*
_{
p
}
^{3} number of multiplication operations. However, multiplying a *K*
_{
p
}×*N* matrix by a *N*×*K*
_{
p
} matrix requires (*N*−1)×*K*
_{
p
}×*K*
_{
p
} additions and *K*
_{
p
}×*N*×*K*
_{
p
} multiplications.

This leads to the conclusion that the computational complexity of model extraction actually depends on the nonlinear model order of the predistortion *K*
_{
p
} and the number of the training samples *N*. Thereby, the more we decrease the number of predistortion coefficients and the training samples, the more we decrease the computational complexity. Consequently, thanks to our EVM expressions we can minimize the computational complexity of the predistortion technique by controlling *K*
_{
p
}. Thus, we have to choose the minimum number of coefficients with respect to our EVM constraint.

*%*, we notice from Fig. 13 that different solutions can be adopted according to the transmission constraints and limits. So, in terms of PA efficiency and without taking into account the predistortion complexity, the optimal IBO, with respect to the EVM constraint, is 7 dB which implies a \(\text {PAPR}_{[ x_{2} ]} = 7\) dB. However, if we do not have sufficient computational capacity for the predistortion technique, we can decrease the predistortion order and choose for example

*K*

_{ p }=2 instead of choosing

*K*

_{ p }=5. In this case, the optimal IBO, that satisfies the computational complexity and respects the EVM constraint, becomes 7.6 dB. Although the PA efficiency decreases from 32 to 30

*%*, we significantly mitigate the predistortion complexity. Therefore, the number of multiplication operations decreases from 55

*N*+125 to 10

*N*+8 and the number of addition operations decreases from 50

*N*−30 to 8

*N*−6. So thanks to our proposed theorems, we can decrease, in this scenario, the number of multiplication operations five times and the number of addition operations six times.

## 6 Conclusions

In this paper, the EVM expression of a multicarrier signal distorted by a polynomial PA model is derived with or without the use of a clipping technique. Then, the impact of a polynomial predistortion on the EVM expressions is investigated. The provided analytical expressions of the EVM are based on series expansions, and depend on the PA characteristics and the predistortion characteristics, as well as the PAPR and the average power of both input and clipped signals. Simulation results compared to our proposed model confirm the accuracy of our derived analytical expressions. Therefore, it is no more necessary to realize a PA prototype to know if it respects the EVM standard. This analytical expression could be favorably integrated in simulation models to validate the PAs behavior. Moreover, the trade-off between the PA linearity and efficiency is discussed. We showed that our theoretical EVM expressions are very useful for optimizing transmitter efficiency and linearity. Indeed, an analytical expression which gives the optimal IBO maximizing the PA efficiency with respect to any EVM constraint is provided. Then, the complexity of the predistortion technique is investigated aiming at reducing it.

It is worthwhile to note that our proposed theoretical analyses could be very useful for optimizing future transmitter efficiency and linearity and the computational complexity of the predistortion technique. These analytical results are very interesting and could be applied in the field of broadcasting for the deployment of DVB-T2 transmitters as well as in LTE cellular networks. Future works should investigate out-of-band distortions, e.g., adjacent channel power ratio (ACPR), and take into account the memory effects of the PA.

## 7 Endnote

^{1} Note that the odd order comes from the bandpass assumption as explained in [32] p. 161.

## 8 Appendix 1: Proof of the EVM derivation when clipping and predistortion are deactivated

*m*

_{2}, can be expressed as follows

*m*and any nonnegative integer

*n*, the multinomial formula is given by

*p*

_{1}through

*p*

_{ m }such that the sum of all

*p*

_{ i }is

*n*. That is, for each term in the expansion, the exponents of the

*x*

_{ i }must add up to

*n*. Hence, we can obtain an expansion of

*m*

_{2}as

Then, using Eq. (12), the EVM expression can be written as Eq. (14).

## 9 Appendix 2: Proof of the EVM derivation when clipping is activated

*I*

_{1}is similar to derivation of

*m*

_{2}in the case without clipping which is previously presented. Thus, using Lemma 2,

*I*

_{1}can be computed as

*I*

_{2}is an integral to be calculated. After expanding the squared term in

*I*

_{2}and letting \(u = \frac {r^{2}}{P_{x_{1}}}\) we get

*I*

_{3}and

*I*

_{5}can be written as

*I*

_{4}can be calculated as

Finally, compiling *I*
_{1} and *I*
_{2}, the EVM expression can be written as Eq. (18).

## 10 Appendix 3: Proof of the EVM derivation when predistortion is activated

*I*

_{6}is easily obtained by applying integration by parts

*I*

_{7}, and after integration we have

*I*

_{8}, and after integration we get

Finally, after compiling these integrals, the EVM expression can be expressed as Eq. (21).

## 11 Appendix 4: Proof of the EVM derivation when clipping and predistortion are activated

However, the derivation of *I*
_{9} is similar to the derivation of Eq. (50). Moreover, the derivation of *I*
_{10} is previously presented, which is exactly the same calculation of *I*
_{2}. Hence, the EVM expression can be expressed as Eq. (25).

## Notes

## Declarations

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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