Analysis of MIMO-OFDM system impaired by nonlinear dual-band power amplifiers

Abstract

This paper presents a theoretical analysis of multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing in the presence of nonlinear dual-band power amplifiers. A two-dimensional memory polynomial model is used to describe the dual-band power amplifier. It is analytically shown that the effect of nonlinearity impairment in the transmitter can be modeled by a complex coefficient and a nonlinear additive noise. An analytical formulation is provided for symbol error rate (SER) of this system considering a frequency selective MIMO channel. The experimental results have been deployed to extract a realistic model for a dual-band amplifier. Also, the model of the linearized power amplifier has been extracted after applying digital pre-distortion (DPD). The proposed model is validated by simulation for 2 × 2 and 2 × 4 MIMO systems. The results show that the analytical model maps well onto the simulation. Also, it is shown that the DPD is able to compensate for the nonlinearity effect of the dual-band power amplifier as the SER is improved.

Appendices

Appendix 1: Derivation for complex coefficients, $\mu_{b q,k}^{i}$

In order to calculate $\mu_{b q,k}^{i}$, from (11) we have:

$$E\left[ {\beta_{bq,k}^{i} a_{bk}^{i*} } \right] = \mu_{bq,k}^{i} P_{ib}$$

(22)

In (22), $\beta_{b q,k}^{i}$ and $a_{b k}^{i}$ can be replaced by (10) and its IDFT representation, respectively. Thus,

$$E\left[ {\beta_{q,k}^{i} a_{k}^{i *} } \right] = E\left[ {\mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{p1 = 0}^{p} \mathop \sum \limits_{n = 0}^{N - 1} C_{bp, p1,q }^{i} x_{b\left[ n \right]}^{i} \left| {x_{b\left[ n \right] }^{i} } \right|^{p - p1} \left| {x_{bb\left[ n \right]}^{i} } \right|^{p1} e^{ - j2\pi kn/N} \times \sum\limits_{v = 0}^{N - 1} {\left( {x_{b\left[ \upsilon \right]}^{i} } \right)^{ * } e^{j2\pi k\upsilon /N} } } \right]$$

(23)

Also, it can be rewritten as,

$$E\left[ {\beta_{q,k}^{i} a_{k}^{i *} } \right] = E\left[ {\mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{p1 = 0}^{p} \mathop \sum \limits_{n = 0}^{N - 1} \mathop \sum \limits_{\upsilon = 0}^{N - 1} C_{bp, p1,q }^{i} x_{b\left[ n \right] }^{i} (x_{b \left[ \upsilon \right]}^{i} )^{ *} \left| {x_{b \left[ n \right]}^{i} } \right|^{p - p1} \left| {x_{bb \left[ n \right]}^{i} } \right|^{p1} e_{{}}^{{ - j2\pi k\left( {n - \upsilon } \right)/N}} } \right]$$

(24)

As $x_{b \left[ \upsilon \right]}^{i}$ and $x_{b \left[ n \right]}^{i}$ are Gaussian random variable, they are independent when $n \ne \upsilon$. So the expectation value is zero unless $n = \upsilon$ therefore,

$$E\left[ {\beta_{q,k}^{i} a_{k}^{i*} } \right] = E\left[ {\mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{p1 = 0}^{p} \mathop \sum \limits_{n = 0}^{N - 1} \mathop \sum \limits_{\upsilon = 0}^{N - 1} C_{bp, p1, q }^{i} x_{b \left[ n \right]}^{i} (x_{b \left[ n \right]}^{i} )^{*} \left| {x_{b \left[ n \right]}^{i} } \right|^{p - p1} \left| {x_{bb \left[ n \right]}^{i} } \right|^{p1} } \right]$$

(25)

$$E\left[ {\beta_{q,k}^{i} a_{k}^{i *} } \right] = E\left[ {\mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{p1 = 0}^{p} \mathop \sum \limits_{n = 0}^{N - 1} \mathop \sum \limits_{\upsilon = 0}^{N - 1} C_{bp, p1, q }^{i} \left| {x_{b \left[ n \right]}^{i} } \right|^{p - p1 + 2} \left| {x_{bb \left[ n \right]}^{i} } \right|^{p1} } \right]$$

(26)

Considering the central momentum of Rayleigh- distributed random variable, $E\left[ { \left| x \right|^{p} } \right] = \sigma^{p} 2^{{\frac{p}{2}}} \varGamma \left( {1 + \frac{p}{2}} \right)$ where $\sigma^{2} = P_{i b} /\left( {2N} \right)$, (22) can be written as:

$$\mu_{b q,k}^{i} P_{b i} = N\mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{p1 = 0}^{p} C_{b p, p1, q }^{i} \left[ {\varGamma \left( {2 + \frac{p - p1}{2}} \right)\varGamma \left( {1 + \frac{p}{2}} \right)\left( {{\raise0.7ex\hbox{${P_{b i} }$} \!\mathord{\left/ {\vphantom {{P_{b i} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{{\frac{p - p1}{2} + 1}} \left( {{\raise0.7ex\hbox{${P_{bb i} }$} \!\mathord{\left/ {\vphantom {{P_{bb i} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{{\frac{p1}{2}}} } \right]$$

(27)

Consequently (12) is extracted.

Appendix 2: Derivation for Mean of Nonlinear Noise

In this part we prove that the mean of nonlinear noise, $\eta _{b k}^{i}$, is zero. For this purpose, we have,

$$E\left[ {\eta _{b k}^{i} } \right] = E\left[ {\beta_{b k}^{i} } \right] - \mu_{b k}^{i} E\left[ { a_{b k}^{i} } \right]$$

(28)

As $a_{b k}^{i}$ is a zero mean random variable, $E\left[ {\eta _{b k}^{i} } \right] = E\left[ {\beta_{b k}^{i} } \right]$

Also, $\beta_{b k}^{i} = \mu_{b k}^{i} a_{b k}^{i} + \eta _{b k}^{i}$ where $\beta_{b k}^{i} = \mathop \sum \limits_{q = 0}^{Q - 1} \beta_{b q,k}^{i} e_{{}}^{ - j2\pi kq/N}$, So,

$$\beta_{b k}^{i} = \mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{p1 = 0}^{p} \mathop \sum \limits_{n = 0}^{N - 1} \varphi_{b p, p1 }^{i} e_{{}}^{ - j2\pi kn/N} x_{b \left[ n \right]}^{i} \left| {x_{b \left[ n \right]}^{i} } \right|^{p - p1} \left| {x_{bb \left[ n \right]}^{i} } \right|^{p1}$$

(29)

Where $\varphi_{p,k}^{i} = \mathop \sum \limits_{q = 0}^{Q - 1} C_{b p, p1, q }^{i} e_{{}}^{ - j2\pi kq/N}$

$$E\left[ {\beta_{b k}^{i} } \right] = \mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{p1 = 0}^{p} \mathop \sum \limits_{n = 0}^{N - 1} \varphi_{b p, p1 }^{i} e_{{}}^{ - j2\pi kn/N} E\left[ {x_{b \left[ n \right]}^{i} \left| {x_{b \left[ n \right]}^{i} } \right|^{p - p1} \left| {x_{bb \left[ n \right]}^{i} } \right|^{p1} } \right]$$

(30)

Since $x_{b \left[ n \right]}^{i} \left| {x_{b \left[ n \right]}^{i} } \right|^{p - p1} \left| {x_{bb \left[ n \right]}^{i} } \right|^{p1}$ is circularly symmetric, its expectation value is zero. So, $E\left[ {\beta_{b k}^{i} } \right] = 0$

Appendix 3: Derivation for Variance of Nonlinear Noise

In order to calculate the variance of the nonlinear noise, starting from $\beta_{b k}^{i} = \mu_{b k}^{i} a_{b k}^{i} + \eta _{b k}^{i}$, and considering $\eta _{b k}^{i}$ as a zero-mean random variable, we have,

$$\sigma_{NLbk }^{2i} = E\left[ { \left| {\eta _{bk}^{i} } \right|^{2} } \right] = E\left[ { \left| {\beta_{bk}^{i} } \right|^{2} } \right] - \left| {\mu_{bk}^{i} } \right|^{2} P_{b i}$$

(31)

where $\left| {\beta_{bk}^{i} } \right|^{2} = \left( {\beta_{bk}^{i} } \right)\left( {\beta_{bk}^{i} } \right)^{*}$. By replacing $\beta_{b k}^{i}$ from (29),

$$\begin{aligned} E\left[ { \left| {\beta_{b k}^{i} } \right|^{2} } \right] = \mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{p1 = 0}^{p} \mathop \sum \limits_{{n_{1} = 0}}^{N - 1} \varphi_{b p, p1 }^{i} e^{{ - j2\pi kn_{1} /N}} E\left[ {x_{{b \left[ {n_{1} } \right]}}^{i} \left| {x_{{b \left[ {n_{1} } \right]}}^{i} } \right|^{p - p1} \left| {x_{{bb \left[ {n_{1} } \right]}}^{i} } \right|^{p1} } \right] \hfill \\ \times \mathop \sum \limits_{pp = 0}^{P - 1} \mathop \sum \limits_{pp1 = 0}^{pp} \mathop \sum \limits_{{n_{2} = 0}}^{N - 1} \varphi_{b pp, p p1 }^{i*} e_{{}}^{{ + j2\pi kn_{2} /N}} E\left[ {x_{{b \left[ {n_{2} } \right]}}^{i*} \left| {x_{{b \left[ {n_{2} } \right]}}^{i} } \right|^{pp - pp1} \left| {x_{{bb \left[ {n_{2} } \right]}}^{i} } \right|^{pp1} } \right] \hfill \\ \end{aligned}$$

(32)

(32) can be rearranged as,

$$\begin{aligned} \left| {\beta_{k}^{i} } \right|^{2} = \mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{p1 = 0}^{p} \mathop \sum \limits_{{n_{1} = 0}}^{N - 1} \mathop \sum \limits_{pp = 0}^{P - 1} \mathop \sum \limits_{pp1 = 0}^{pp} \mathop \sum \limits_{{n_{2} = 0}}^{N - 1} \varphi_{bp, p1 }^{i} \varphi_{bpp, pp1 }^{i*} E[ e_{{}}^{{ - j2\pi k(n_{1} - n_{2} )/N}} \hfill \\ \times x_{{b \left[ {n_{1} } \right]}}^{i} \left| {x_{{b \left[ {n_{1} } \right]}}^{i} } \right|^{p - p1} \left| {x_{{bb \left[ {n_{1} } \right]}}^{i} } \right|^{p1} x_{{b \left[ {n_{2} } \right]}}^{i*} \left| {x_{{b \left[ {n_{2} } \right]}}^{i} } \right|^{pp - pp1} \left| {x_{{bb \left[ {n_{2} } \right]}}^{i} } \right|^{pp1} ] \hfill \\ \end{aligned}$$

(33)

The expectation value is zero unless $n_{2} = n_{1} = n$, so,

$$E\left[ { \left| {\beta_{bk}^{i} } \right|^{2} } \right] = N \times E\left[ {\mathop \sum \limits_{n}^{N - 1} \mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{p1 = 0}^{p} \mathop \sum \limits_{pp = 0}^{P - 1} \mathop \sum \limits_{pp1 = 0}^{pp} \varphi_{b p, p1 }^{i} \varphi_{b pp, p p1 }^{i*} \left| {x_{b \left[ n \right]}^{i} } \right|^{{p + pp - \left( {p1 + pp1} \right) + 2}} \left| {x_{bb \left[ n \right]}^{i} } \right|^{p1 + pp1} } \right]$$

(34)

and,

$$\begin{aligned} E\left[ { \left| {\beta_{b k}^{i} } \right|^{2} } \right] = N \times \mathop \sum \limits_{n}^{N - 1} \mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{p1 = 0}^{p} \mathop \sum \limits_{pp = 0}^{P - 1} \mathop \sum \limits_{pp1 = 0}^{pp} \varphi_{b p, p1 }^{i} \varphi_{b pp, p p1 }^{i*} \hfill \\ \times \varGamma \left( {2 + \frac{{p + pp - \left( {p1 + pp1} \right)}}{2}} \right)\varGamma \left( {1 + \frac{p1 + pp1}{2}} \right)\left( {{\raise0.7ex\hbox{${P_{b i} }$} \!\mathord{\left/ {\vphantom {{P_{b i} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{{\frac{{p + pp - \left( {p1 + pp1} \right) + 2}}{2}}} \left( {{\raise0.7ex\hbox{${P_{bb i} }$} \!\mathord{\left/ {\vphantom {{P_{bb i} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{{\frac{p1 + pp1}{2}}} \hfill \\ \end{aligned}$$

(35)

Therefore, the (14) is extracted.