Effects of the adjacent channels on IP3 and IP5 of RF amplifier

Abstract

This paper discusses the impact of the adjacent channels on the 1-dB compression point, the IP3 and the IP5 of Radio Frequency (RF) amplifier by n-tone test. By combining the theoretical derivation and software simulation, the model analysis for the third/five order polynomial nonlinear amplifier has been achieved. Moreover, the control variable method is adopted to draw the curves for the input/output signals. The research shows that the 1-dB compression point, the IP3 and the IP5 drop as n increases, and they all have symmetry for a given n. The fifth-order polynomial nonlinear amplifier model is proposed, the research shows that the adjacent channels have a great impact on the 1-dB compression point, the IP3 and the IP5 of the desired channel. This effect must be taken into account in actual RF amplifier designs and wireless communication architectures.

Keywords

RF amplifier the nonlinear effect adjacent channels intermodulation

1. Introduction

The frequency spectrum is a valuable resource, for optimum spectrum utilization, the available spectrum is usually subdivided into several channels, resulting in channels very close or slightly overlapped [1,2]. For example, in Long Term Evolution (LTE) system, the subcarrier spacing is just 15 kHz, and the subcarrier bandwidth is also 15 kHz [3,4]. It’s obvious that intermodulation interference becomes more severe when these channels are operating at the same time [5]. In another hand, for a RF power amplifier, linearity and power efficiency tend to be mutually exclusive, so that any increase of the linearity by amplifying in linear region is usually achieved at the expense of the efficiency and conversely [6]. So for high power efficiency and the signal-to-noise ratio (SNR), the input power back-off is relatively low, giving rise to worse linear performance of the amplifier [7,8].

Generally, linearity and power efficiency are two mutually exclusive criteria in the amplifier design. That is to say that linearity is improved at the expense of power efficiency. On the contrary, the improvement of efficiency will also damage the linearity to some extent. The non-linearity of power amplifier is inherent, its existence has a great influence on adjacent (or separated) channels. It will also produce an irreducable interference to the signal in the band. Thus, the signal-to-noise ratio is greatly reduced, and the quality of transmission information is affected. In addition, the rapid progress of communication technology makes the current spectrum resources increasingly scarce and precious. In order to improve its utilization efficiency, the frequency interval between channels is getting smaller and smaller, which poses a more severe challenge to the linearity of amplifiers [9]. At the same time, to pursue enough high output power for the wireless communication, the amplifier always works in the nonlinear region, or even works near the saturation region. Thus, the amplifier shows relatively large nonlinear effect. In theory, high spectral efficiency can be achieved by efficient modulation methods such as M-element orthogonal amplitude modulation and orthogonal phase-shift keying. However, because of its uncertain signal envelope, it is easy to be affected by the nonlinear characteristics of the amplifier. As a result, when the signal is input to the amplifier, it will lead to serious intermodulation distortion phenomenon, and the output spectrum of the amplifier will spread out greatly, which will have a great impact on the adjacent signals. Therefore, in order to improve the efficiency of spectrum utilization, the interval between the signal frequencies of each channel becomes very small, or even overlap. Furthermore, to obtain a high enough output power, it is necessary to increase the amplitude of the input signal to some extent, so that a large power can be obtained to overcome the transmission loss of the signal in the air medium. All these will pose severe challenges to the nonlinear effect of power amplifier.

Moreover, IP3 and IP5 are the important parameter of linearity. Intermodulation distortion will cause crosstalk of adjacent channels, which will reduce the spectrum utilization and worsen the bit error rate. The greater the power of adjacent channels, the greater the values of IP3 and IP5, the better the linearity of power amplifier [10]. In addition, 1-dB compression point is another parameter for the nonlinear of power amplifier. Generally, the greater the input power at the 1-dB compression point, the wider the linear range of the power amplifier can work, that is to say, the better the linearity of the power amplifier. On the contrary, the smaller the linear interval in which the power amplifier can operate, the less linear the power amplifier will be.

At present, the nonlinear characteristics of power amplifiers are often described by the third-order polynomial nonlinear function model. However, the model is no longer accurate enough under large input excitation, because the effects of the neglected higher-order distortion terms are already apparent. Therefore, the nonlinear function model of the fifth order polynomial is proposed, and the nonlinear effect of the power amplifier under the excitation of multiple input signals is deduced based on this model.

In both cases above, it is essential to investigate the nonlinear effect of the RF amplifier. Traditionally, the nonlinearity of a RF amplifier is described by the 1-dB compression point and the third-order intercept point (IP3), which are defined by using the single-tone and the two-tone test, respectively [11,12]. However these are inadequate in the condition of multi-channel and low input power back-off, because the magnificent of higher order intermodulation products is comparable to that of the desired channel. So the fifth-order intercept point (IP5) is also researched.

In this paper, the fifth-order polynomial nonlinear amplifier model is proposed, and we will discuss the phenomenon of gain compression and intermodulation interference by the 1-dB point, the IP3 and the IP5. Unlike the traditional analysis, the measure method is not limited to single-tone test and two-tone test but extended to n-tone test. The research shows that the adjacent channels have a great impact on the 1-dB compression point, the IP3 and the IP5 of the desired channel. This effect must be taken into account in actual RF amplifier designs and wireless communication architectures.

2. Analysis

When multiple channel signals simultaneously excite a power amplifier, due to the inherent nonlinear characteristics of the power amplifier, the frequency combination of some channel signals may happen to fall on a certain channel signal frequency, which will greatly affect the quality of the channel signal. The spectra of multiple channels is shown in Fig. 1, where n is total number of channels, m is an arbitrary channel, ω₁ is the angular frequency of the first channel and Δ is the channel spacing, and Δ is normally far less than ω₁. It can be shown that the output spectrum has a significant change compared with the input spectrum for the reason that the nonlinear effect of the RF amplifier. Figure 1 illustrates this phenomenon vividly. There are n channels on the left side of the figure, and the angular frequency of the first channel is ω₁, m represents any channel among n channels, denoted as useful channel, and the difference of the angular frequency of adjacent channels is Δ, which Δ is usually far less than ω₁.

In order to characterize the nonlinearity, the input/output characteristic of the RF amplifier can be approximated by a fifth-order polynomial as the following formula. $\begin{eqnarray}y(t)=a_{0}+a_{1}x(t)+a_{2}x^{2}(t)+a_{3}x^{3}(t)+a_{4}x^{4}(t)+a_{5}x^{5}(t)\end{eqnarray}$ (1) where y (t) and x (t) are the amplifier output and input voltages, respectively. And a ₀ is the direct current component of the output, a _i (i = 2, 3, 4, 5) are nonlinear coefficients, considering saturation characteristics of an amplifier, the sign of a ₃ and a ₅ are given minus [13].

Here the input signal is composed of n channels as: $\begin{eqnarray}\displaystyle \hspace{-18.0pt}x(t)=A_{1}\cos {\omega}_{1}t+A_{2}\cos {\omega}_{2}t+A_{3}\cos {\omega}_{3}t+\cdots +A_{m}\cos {\omega}_{m}t+\cdots +A_{n-1}\cos {\omega}_{n-1}t+A_{n}\cos {\omega}_{n}t & & \displaystyle\end{eqnarray}$ (2) where A _k and ω_k for k = 1, 2, …, m, …, n − 1, n, represent the magnitude and the angular frequency of the kth channel signal respectively, and the difference between the angular frequencies of the adjacent channel is Δ. To simplify the analysis, it is assumed that all channel signals have the same magnitude, that is to say A _k = A for any k, and the following discussion is based on this assumption.

Fig. 1.

The N channels through a RF nonlinear amplifier.

Taking one channel as the research object, denoted as the channel m, here m ∈ [1,2,3…n −1, n], and its angular frequency is ω_m. When the desired channel m and the rest of channels all together excited the amplifier, the output spectrum will have many frequency components, including frequency component ω_m. Obviously, the magnitude of frequency component ω_m is affected by remaining n −1 channels because of the nonlinearity of the amplifier. In order to calculate the impact of the remaining channels, substituting (2) into (1), leads to $\begin{eqnarray}\displaystyle y(t) & = & \displaystyle a_{1}A\cos {\omega}_{m}t-|a_{3}|\frac{3A^{3}}{4}\cos {\omega}_{m}t-|a_{5}|\frac{5A^{5}}{8}\cos {\omega}_{m}t\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -|a_{3}|\frac{3A^{3}}{8}(n^{2}-n-2m^{2}+2mn+2m-2)\cos {\omega}_{m}t\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -|a_{5}|\frac{5A^{5}}{8}\left(\frac{11}{24}n^{4}-\frac{1}{4}n^{3}+\frac{1}{24}n^{2}-\frac{1}{4}n+\frac{1}{4}m^{4}-\frac{1}{2}m^{3}-\frac{1}{4}m^{2}+\frac{1}{2}m+\frac{1}{2}mn^{3}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -\left.\frac{1}{4}m^{2}n^{2}-\frac{1}{2}nm^{3}+\frac{1}{4}mn^{2}+\frac{3}{4}nm^{2}+\frac{1}{4}mn-1\right)\cos {\omega}_{m}t+\mathit{others}.\end{eqnarray}$ (3)

There are six items on the right-hand side of Eq. (3), the first item represents the linearly amplified desired channel, the second and third terms are the products of nonlinear effect on the desired channel itself, the fourth and fifth items are the products of intermodulation interference among channels, and the last item represents other frequency components. Here the fourth item is defined as the IM3 whose mechanism is that the remaining channel signals transformed to the frequency of the desired channel signal through third-order nonlinear distortion item [12]. Similarly, the fifth item is defined as the IM5 whose mechanism is that the remaining channel signals transformed to the frequency of the desired channel signal through fifth-order nonlinear distortion item. Particularly, the IM3 and the IM5 defined here are different from the conventional definition. Clearly, the effects defined in the conventional definitions are not located in the desired channel. The new definition given here are defined the intermodulation effects of the remaining channels on the desired channel signal.

From the equation (3), it is found that the total output of the frequency component ω_m, the IM3 and the IM5 are all the functions of three variables which are m, n and A. We can research the relationship between the total output, the IM3 and the IM5 with two variables by fixed the third variable, respectively. Obviously here are three cases.

First case, for a given magnitude A, the total output, the IM3 and the IM5 are all in a relationship with the total number of channels n and the channel m, as shown in Fig. 2, 3, 4, respectively. In Fig. 2, it is found that with the increasing of n, there is a sharp decline in the output. Besides, for one of the given n (n > 2), the curve of the output is concave and symmetrical, it means that the channels whose position is symmetrical have the same output, and the channel whose position is in central is the most significantly influenced by the rest of channels. In Fig. 3, it shows that the IM3 rises sharply when n increases. Besides, the curve of the IM3 is convex and symmetrical for one of the given n (n > 2). In Fig. 4, it is clearly that the IM5 and the IM3 have almost the same property, the only difference is that the IM5 is large in ascension more range than the IM3 as n increases, and it can be inferred that the IM5 is greater than the IM3 when n is large enough.

Second case, for a given total number of channels n, the total output, the IM3 and the IM5 are all in a relationship with the magnificent A and the channel m, as shown in Fig. 5, 6, 7, respectively. In Fig. 5, the curve of the output is convex for any channel m, and the output of the channels whose position is symmetrical are completely the same. In addition, the closer the channel m to the channel [0.5n] + 1 is, where “[ ]” denotes floor function, the smaller the peak of the curve is, and the earlier the block effect appears. In Fig. 6, it shows that the IM3 rises sharply as A increases, and the IM3 of the channels whose position is symmetrical are the same. Besides, for a given A, the closer the channel m to the channel [0.5n] + 1 is, the larger the IM3 is. In Fig. 7, the curve of the IM5 is similar to that of the IM3 except that the IM5 is far more than IM3 when A is large enough.

Fig. 2.

The total output of channels for a given magnitude A, A = 1, when n is 1,2…10.

Fig. 3.

The IM3 of channels for a given magnitude A, A = 1, when n is 1,2…10.

Third case, for a special channel m, [0.5n] + 1, the total output, the IM3 and the IM5 are all in a relationship with the total number of channels n and the magnificent A, as shown in Fig. 8, 9, 10, respectively. In Fig. 8, the curve of the output is convex for any n, and it sharply shrinks toward the origin as n increases, it means that the phenomenon of the gain compression become significant when n is large enough. In Fig. 9, the IM3 grows rapidly with the increasing of n and A. In Fig. 10, the curve of the IM5 is similar to that of the IM3 except that the IM5 is far more than IM3 when n is large enough.

Fig. 4.

The IM5 of channels for a given magnitude A, A = 1, when n is 1,2…10.

Fig. 5.

The total output versus the magnitude A for a given n, n = 10, when m is 1,2…10.

Fig. 6.

The IM3 versus the magnitude A for a given n, n = 10, when m is 1,2…10.

Fig. 7.

The IM5 versus the magnitude A for a given n, n = 10, when m is 1,2…10.

Fig. 8.

The total output versus the magnitude A for a given m, m = [0.5n] + 1, when n is 1,2…10.

Fig. 9.

The IM3 versus the magnitude A for a given m, m = [0.5n] + 1, when n is 1,2…10.

Fig. 10.

The IM5 versus the magnitude A for a given m, m = [0.5n] + 1, when n is 1,2…10.

The discussion above describes some properties of the total output, the IM3 and the IM5. Now we can obtain the 1-dB compression point, the IP3 and the IP5 by using the total output, the IM3 and the IM5, respectively.

The 1-dB compression point can be easily obtained as following. $\begin{eqnarray}20\lg (a_{1}A_{1-\text{dB}})-20\lg (y(t)-\mathit{others})=1\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}A_{1-\text{dB}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-21.0pt}=\left\{\frac{-3a_{3}(n^{2}-n-2m^{2}+2mn+2m)-\left[9a_{3}^{2}(n^{2}-n-2m^{2}+2mn+2m)^{2}-160^{\ast }(1-10^{-\frac{1}{20}})a_{1}a_{5}\left(\frac{11}{24}n^{4}-\frac{1}{4}n^{3}+\frac{1}{24}n^{2}-\frac{1}{4}n+\frac{1}{4}m^{4}-\frac{1}{2}m^{3}-\frac{1}{4}m^{2}+\frac{1}{2}m+\frac{1}{2}mn^{3}-\frac{1}{4}m^{2}n^{2}-\frac{1}{2}nm^{3}+\frac{1}{4}mn^{2}+\frac{3}{4}nm^{2}+\frac{1}{4}mn\right)\right]^{\frac{1}{2}}}{10a_{5}\left(\frac{11}{24}n^{4}-\frac{1}{4}n^{3}+\frac{1}{24}n^{2}-\frac{1}{4}n+\frac{1}{4}m^{4}-\frac{1}{2}m^{3}-\frac{1}{4}m^{2}+\frac{1}{2}m+\frac{1}{2}mn^{3}-\frac{1}{4}m^{2}n^{2}-\frac{1}{2}nm^{3}+\frac{1}{4}mn^{2}+\frac{3}{4}nm^{2}+\frac{1}{4}mn\right)}\right\}^{\frac{1}{2}}.\end{eqnarray}$ (5)

The IP3 can be derived as following. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle a_{1}A_{\mathit{IP}3}=\frac{3}{8}|a_{3}|(n^{2}-n-2m^{2}+2mn+2m-2)A_{\mathit{IP}3}^{3}\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{\mathit{IP}3}=\left\{\frac{8a_{1}}{3|a_{3}|(n^{2}-n-2m^{2}+2mn+2m-2)}\right\}^{\frac{1}{2}}.\end{eqnarray}$ (7)

The IP5 can be derived as following. $\begin{eqnarray}\displaystyle a_{1}A_{\mathit{IP}5} & = & \displaystyle \frac{5}{8}|a_{5}|\left(\frac{11}{24}n^{4}-\frac{1}{4}n^{3}+\frac{1}{24}n^{2}-\frac{1}{4}n+\frac{1}{4}m^{4}-\frac{1}{2}m^{3}-\frac{1}{4}m^{2}+\frac{1}{2}m+\frac{1}{2}mn^{3}-\frac{1}{4}m^{2}n^{2}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -\left.\frac{1}{2}nm^{3}+\frac{1}{4}mn^{2}+\frac{3}{4}nm^{2}+\frac{1}{4}mn-1\right)A_{\mathit{IP}5}^{5}\end{eqnarray}$ (8) $\begin{eqnarray}\hspace{-18.0pt}A_{\mathit{IP }5}=\left(\frac{8a_{1}}{5|a_{5}|\left(\frac{11}{24}n^{4}-\frac{1}{4}n^{3}+\frac{1}{24}n^{2}-\frac{1}{4}n+\frac{1}{4}m^{4}-\frac{1}{2}m^{3}-\frac{1}{4}m^{2}+\frac{1}{2}m+\frac{1}{2}mn^{3}-\frac{1}{4}m^{2}n^{2}-\frac{1}{2}nm^{3}+\frac{1}{4}mn^{2}+\frac{3}{4}nm^{2}+\frac{1}{4}mn-1\right)}\right)^{\frac{1}{4}}.\end{eqnarray}$ (9)

The relationship among the total number of channels n, the channel m and the 1-dB compression point is shown in Fig. 11. Clearly, when n is small, the 1-dB compression point drops rapidly, but when n is large enough, the 1-dB compression point declines slowly. Further observation can be found that the surface of the 1-dB compression point is symmetrical about the plane 1 + n − 2m = 0, besides, the curve of the1-dB compression point is concave for any n.

The IP3 and the IP5 are shown in Fig. 12 and 13, respectively. Obviously, the IP3 and the IP5 are all meaningless when n is equal to one, because there is no adjacent channels. Similar to the 1-dB compression point, they also drops rapidly when n is small, and they declines slowly when n is large enough, besides, the surfaces of the IP3 and the IP5 are all symmetrical about the plane 1 + n − 2m = 0, and the curves of the IP3 and IP5 are all concave for any n. It should be point out that the IP3 is bigger than the IP5 for a given n and m.

Fig. 11.

The relationship among the total number of channels n, the channel m and the 1-dB compression point.

Fig. 12.

The relationship among the total number of channels n, the channel m and the IP3.

Fig. 13.

The relationship among the total number of channels n, the channel m and the IP5.

3. Conclusions

In this paper, the fifth-order polynomial nonlinear amplifier model is proposed, and the impact of the adjacent channels on the 1-dB compression point, the IP3 and the IP5 of RF amplifier are investigated by n-tone test. The results show that the 1-dB compression point, the IP3 and the IP5 are deceased with the increasing of n, but their descending trend is slow. It means that the sensitivity of the nonlinearity to n become smaller and smaller as n increases. Besides, the channels whose position is symmetrical have the same the 1-dB compression point, the IP3 and the IP5, and the channel whose position is in central has the smallest the 1-dB compression point, the IP3 and the IP5.

It is also shown that both the total number of channels N and the input signal amplitude A will deteriorate the linearity of useful channel signals, and the channel signals with symmetrical positions have the same properties, and the channel signals in the most central location are most affected by the intermodulation effect. When the total number of channels n is equal to 1, the third-order intercept IP3 is meaningless, because there are only useful channel signals and no other channel signals at this time, so there will be no intermodulation between channel signals. Similar to the characteristics of 1-dB gain compression point, when the total number of channels n is small, the third-order intercept point IP3 will decrease rapidly with the increase of the total number of channels n. However, when the total number of channels n is large enough, the third-order intercept point IP3 will decrease slowly with the increase of the total number of channels n. Meanwhile, for any total number of channels n, the output curve of IP3 is concave and symmetric. That is to say, the output surface of the third-order intercept IP3 is also symmetric about the plane of 1 + n − 2m = 0. Meanwhile, when the total number of channels n is equal to 1, IP5 is meaningless, because there are only useful channels at this time, and there are no other channels, so there will be no intermodulation interference between channels. Similar to the characteristics of 1-dB gain compression point, when the total number of channels n is small, IP5 will decline rapidly. While when the total number of channels n is large enough, IP5 will decline slowly. And the output surface of IP5 is symmetric about the plane of 1 + n − 2m = 0. At the same time, the output curve of IP5 is concave for the total number of any channel n. In summary, the properties of IP5 and IP3 are similar, but the surface of IP5 is generally below the IP3 curve.

In summary, the nonlinearity get worse when the channel moves from both sides to the central. These are very useful to RF amplifier designs and wireless communication architectures.

Footnotes

Acknowledgement

The authors gratefully acknowledge financial support from the National Natural Science Foundation (62161046), the West Light Youth Talent Program of the Chinese Academy of Sciences (1_14) and the Natural Science Foundation of Qinghai Province, (2021-ZJ-910).

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