Analysis of nonlinear crosstalk impairment in MIMO-OFDM systems

Abstract

An analytical modeling and performance analysis for a multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) transceiver exhibiting crosstalk and nonlinear distortion in transmitting chain is presented. For this purpose, a two-dimensional memory polynomial is used to describe the dynamic nonlinear MIMO transmitter. It is shown that the impairment of nonlinear crosstalk can be characterized by two complex attenuations and a nonlinear additive noise. Moreover, an analytical formulation for the symbol error rate (SER) of MIMO-OFDM system considering a frequency selective channel is provided. Experimental results have been employed to extract a realistic model for the nonlinear transmitter in the presence of crosstalk. Also, a model of the linearized transmitter has been calculated after experimentally applying digital predistortion (DPD). The proposed model for the system is validated by the simulation results and a good agreement between the analytical and simulation results is observed. Finally, it is shown that nonlinear crosstalk degrades SER of the system, whereas, the DPD can effectively compensate for the nonlinearity and crosstalk infections, which, in turn, improves SER of the MIMO-OFDM system.

Appendix

1.1 Derivation of complex attenuation coefficients $\mu_{q,k}^{i,i}$ and $\mu_{q,k}^{i,j}$

1.1.1 Derivation of $\mu_{q,k}^{i,i}$

As mentioned in Sect. 2, $\mu_{q,k}^{i,i}$ is weakly dependent on data and its variance is negligible. So, we can assume it as an unknown variable rather than a random one. It can be shown with simulation. So by multiplying (17) by $a_{k}^{i*}$ and calculating its expected value of expression we have;

$$E\left[ {\beta_{q,k}^{i} a_{k}^{i *} } \right] = \mu_{q,k}^{i,i} P_{a}^{i}$$

(36)

Substituting $\beta_{q,k}^{i}$ from (16) and $a_{k}^{i*}$ with its IDFT representation form, (36) can be rewritten as:

$$E\left[ {\beta_{q,k}^{i} a_{k}^{i *} } \right] = E\left[ {\mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{\upsilon = 0}^{N - 1} \mathop \sum \limits_{n = 0}^{N - 1} b_{pq }^{i} \gamma_{p \left[ n \right]}^{i} \tilde{x}_{\left[ n \right]}^{i} (x_{\left[ \upsilon \right]}^{i} )^{ *} e_{{}}^{ - j2\pi kn/N} e_{{}}^{j2\pi k\upsilon /N} } \right]$$

(37)

It is known that IDFT components of i.i.d random sequence is uncorrelated. As mentioned in Sect. 2, $x_{\left[ n \right]}^{i}$ is modeled as a Gaussian random variable for the large number of N. Therefore, any function of $x_{{\left[ {n_{i} } \right]}}^{i}$ and $x_{{\left[ {n_{j} } \right]}}^{i}$ ($n_{i} \ne n_{j}$) are independent and $\left| {x_{{\left[ {n_{i} } \right]}}^{i} } \right|$ is a Rayleigh distributed random variable. So, expected value of any terms in (37) with $n \ne \upsilon$ is zero, and just the terms corresponding to $\upsilon = n$ are remained. Also,

$$E\left[ {\gamma_{p \left[ n \right]}^{i} \tilde{x}_{\left[ n \right]}^{i} (x_{\left[ n \right]}^{i} )^{ *} } \right] = E\left[ {\left| {x_{\left[ n \right]}^{i} } \right|^{p + 2} } \right] + E\left[ {\frac{p}{2}\alpha^{ij} \left| {x_{\left[ n \right]}^{i} } \right|^{p} x_{\left[ n \right]}^{j} \left( {x_{\left[ n \right]}^{i} } \right)^{ *} } \right] + E\left[ {\frac{p}{2}\left| {x_{\left[ n \right]}^{i} } \right|^{p} x_{\left[ n \right]}^{i} \left( {\alpha^{ij} x_{\left[ n \right]}^{j} } \right)^{ *} } \right] + E\left[ {\alpha^{ij} \left| {x_{\left[ n \right]}^{i} } \right|^{p} x_{\left[ n \right]}^{j} \left( {x_{\left[ n \right]}^{i} } \right)^{ *} } \right] + E\left[ {\frac{p}{2}\alpha^{ij2} \left| {x_{\left[ n \right]}^{i} } \right|^{p} \left| {x_{\left[ n \right]}^{j} } \right|^{2} } \right] + E\left[ {\frac{p}{2}\left| {\alpha^{ij} } \right|^{2} \left| {x_{\left[ n \right]}^{i} } \right|^{p - 2} \left( {\left( {x_{\left[ n \right]}^{j} } \right)^{ *} x_{\left[ n \right]}^{i} } \right)^{2} } \right]$$

(38)

$x_{\left[ n \right]}^{i}$ and $x_{\left[ n \right]}^{j}$ considered as two circularly symmetric independent complex-valued Gaussian random variables. Also, we know that, $\left| {x_{\left[ n \right]}^{i} } \right|^{p}$ and $\left| {x_{\left[ n \right]}^{j} } \right|^{p}$ are random positive numbers. Therefore (38) is reduced to (39).

$$E\left[ {\gamma_{p \left[ n \right]}^{i} \tilde{x}_{\left[ n \right]}^{i} (x_{\left[ n \right]}^{i} )^{ *} } \right] = E\left[ {\left| {x_{\left[ n \right]}^{i} } \right|^{p + 2} } \right] + E\left[ {\frac{p}{2}\left| {\alpha^{ij} } \right|^{2} \left| {x_{\left[ n \right]}^{i} } \right|^{p} \left| {x_{\left[ n \right]}^{j} } \right|^{2} } \right]$$

(39)

By using the central moment of Rayleigh random variable and performing some mathematical manipulations, we have:

$$\mu_{q,k}^{i,i} P_{i} = \mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{n = 0}^{N - 1} b_{pq }^{i} \left[ {\varGamma \left( {2 + \frac{p}{2}} \right)\left( {{\raise0.7ex\hbox{${P_{a}^{i} }$} \!\mathord{\left/ {\vphantom {{P_{a}^{i} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{{\frac{p}{2} + 1}} + \frac{p}{2}\left| {\alpha^{ij} } \right|^{2} \varGamma \left( {1 + \frac{p}{2}} \right)\left( {{\raise0.7ex\hbox{${P_{a}^{i} }$} \!\mathord{\left/ {\vphantom {{P_{a}^{i} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{{\frac{p}{2}}} \left( {{\raise0.7ex\hbox{${P_{a}^{j} }$} \!\mathord{\left/ {\vphantom {{P_{a}^{j} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{{}} } \right]$$

(40)

Thus, (18) can be obtained.

1.1.2 Derivation of $\mu_{q,k}^{i,j}$

The method of calculation of $\mu_{q,k}^{i,j}$ is similar to $\mu_{q,k}^{i,i}$. Here, the expectation of $\beta_{q,k}^{i} a_{k}^{j *}$ is calculated and regarding to the (17) we have:

$$E\left[ {\beta_{q,k}^{i} a_{k}^{j *} } \right] = \mu_{q,k}^{i,j} P_{a}^{j}$$

(41)

By expanding (41), using (16) and IDFT form of $a_{k}^{j *}$:

$$E\left[ {\beta_{q,k}^{i} a_{k}^{i *} } \right] = \mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{\upsilon = 0}^{N - 1} \mathop \sum \limits_{n = 0}^{N - 1} b_{pq }^{i} e_{{}}^{ - j2\pi kn/N} e_{{}}^{j2\pi k\upsilon /N} .E\left[ {\gamma_{p \left[ n \right]}^{i} \tilde{x}_{\left[ n \right]}^{i} (x_{\left[ \upsilon \right]}^{j} )^{ *} } \right]$$

(42)

With the same reasoning mentioned for $\mu_{q,k}^{i,i}$ in (38), the expected value of any terms in (42) with $n \ne \upsilon$ is zero. So, just the terms corresponding to $\upsilon = n$ are remained. After some mathematic manipulation and removing zero terms in expectation, we have,

$$E\left[ {\gamma_{p \left[ n \right]}^{i} \tilde{x}_{\left[ n \right]}^{i} (x_{\left[ n \right]}^{j} )^{ *} } \right] = \alpha^{ij} \frac{3p}{2}E\left[ {\left| {x_{\left[ n \right]}^{i} } \right|^{p} } \right]E\left[ {\left| {x_{\left[ n \right]}^{j} } \right|^{2} } \right]$$

(43)

So,

$$\begin{aligned} \mu_{q,k}^{i,j} P_{j} & = \mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{n = 0}^{N - 1} b_{pq }^{i} \left[ {\frac{3p}{2}\alpha^{ij} \varGamma \left( {1 + \frac{p}{2}} \right)\left( {{\raise0.7ex\hbox{${P_{a}^{i} }$} \!\mathord{\left/ {\vphantom {{P_{a}^{i} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{{\frac{p}{2}}} \left( {{\raise0.7ex\hbox{${P_{a}^{j} }$} \!\mathord{\left/ {\vphantom {{P_{a}^{j} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)} \right] \\ & = N\mathop \sum \limits_{p = 0}^{P - 1} b_{pq }^{i} \frac{3p}{2}\alpha^{ij} \left[ {\varGamma \left( {1 + \frac{p}{2}} \right)\left( {{\raise0.7ex\hbox{${P_{a}^{i} }$} \!\mathord{\left/ {\vphantom {{P_{a}^{i} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{{\frac{p}{2}}} \left( {{\raise0.7ex\hbox{${P_{a}^{j} }$} \!\mathord{\left/ {\vphantom {{P_{a}^{j} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)} \right] \\ \end{aligned}$$

(44)

Therefore, $\mu_{q,k}^{i,j}$ can be expressed as (19).

1.2 Derivation of mean of nonlinear noise $\eta _{k}^{i}$

In this section, it is proven that the mean of the nonlinear noise component is zero. For this purpose, by using (17), the expected value of $\eta _{k}^{i}$ is calculated as;

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

(45)

where;

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

(46)

And the input symbols are zero mean random variables. By substituting (16) in (46), and defining $\varphi_{p,k}^{i} = \mathop \sum \limits_{q = 0}^{Q - 1} b_{pq }^{i} e_{{}}^{ - j2\pi kq/N}$, we have,

$$\beta_{k}^{i} = \mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{n = 0}^{N - 1} \varphi_{p,k}^{i} e_{{}}^{ - j2\pi kn/N} \left( {x_{\left[ n \right]}^{i} + \alpha^{ij} x_{\left[ n \right]}^{j} } \right)\left[ {\left| {x_{\left[ n \right]}^{i} } \right|^{p} + \frac{p}{2}\left| {x_{\left[ n \right]}^{i} } \right|^{p - 2} \left( {\left( {x_{\left[ n \right]}^{i} } \right)^{ *} \alpha^{ij} x_{\left[ n \right]}^{j} + x_{\left[ n \right]}^{i} \left( {\alpha^{ij} x_{\left[ n \right]}^{j} } \right)^{ *} } \right)^{{}} } \right]$$

(47)

where the expectation is equal to zero if $\ne \upsilon$, it is considered that $\upsilon = n$,

$$E\left[ {\beta_{k}^{i} } \right] = \mathop \sum \limits_{p = 0}^{P - 1} \mathop \sum \limits_{n = 0}^{N - 1} \varphi_{p,k}^{i} e_{{}}^{ - j2\pi kn/N} E\left[ {\left( {x_{\left[ n \right]}^{i} + \alpha^{ij} x_{\left[ n \right]}^{j} } \right)\left[ {\left| {x_{\left[ n \right]}^{i} } \right|^{p} + \frac{p}{2}\left| {x_{\left[ n \right]}^{i} } \right|^{p - 2} \left( {\left( {x_{\left[ n \right]}^{i} } \right)^{*} \alpha^{ij} x_{\left[ n \right]}^{j} + x_{\left[ n \right]}^{i} \left( {\alpha^{ij} x_{\left[ n \right]}^{j} } \right)^{*} } \right)^{{}} } \right]} \right]$$

(48)

Since $x_{\left[ n \right]}^{i}$ and $x_{\left[ n \right]}^{j}$ are considered as two circularly symmetric independent zero mean variables, (48) equals to zero. Hence, the mean of the nonlinear noise, $\eta _{k}^{i}$, is equal to zero.

1.3 Derivation of the variance of nonlinear noise

In this case, as previously mentioned, $\eta _{k}^{i}$ is assumed independent from input data. So, regarding to (17), we have;

$$\sigma_{Nl\,k}^{2} \,^{i} = E\left[ { \left| {\eta _{k}^{i} } \right|^{2} } \right] = E\left[ { \left| {\beta_{k}^{i} } \right|^{2} } \right] - \left| {\mu_{k}^{i,i} } \right|^{2} P_{a}^{i} - \left| {\mu_{k}^{i,j} } \right|^{2} P_{a}^{j}$$

(49)

Starting from (47),

$$\left| {\beta_{k}^{i} } \right|^{2} = \left( {\beta_{k}^{i} } \right)\left( {\beta_{k}^{i} } \right)^{*} = \mathop \sum \limits_{{p_{1} = 0}}^{P - 1} \mathop \sum \limits_{{n_{1} = 0}}^{N - 1} \mathop \sum \limits_{{p_{2} = 0}}^{P - 1} \mathop \sum \limits_{{n_{2} = 0}}^{N - 1} \varphi_{{p_{1} ,k}}^{i} \varphi_{{p_{2} ,k}}^{i} \,^{*} .e_{{}}^{{ - j2\pi k(n_{1} - n_{2} )/N}} A_{{n_{1} ,n_{2} ,p_{1} ,p_{2} }}$$

(50)

where,

$$\begin{aligned} A_{{n_{1} ,n_{2} ,p_{1} ,p_{2} }} & = \left\{ {\left( {x_{{\left[ {n_{1} } \right]}}^{i} + \alpha^{ij} x_{{\left[ {n_{1} } \right]}}^{j} } \right)\left[ {\left| {x_{{\left[ {n_{1} } \right]}}^{i} } \right|^{{p_{1} }} + \frac{{p_{1} }}{2}\left| {x_{{\left[ {n_{1} } \right]}}^{i} } \right|^{{p_{1} - 2}} \left( {\left( {x_{{\left[ {n_{1} } \right]}}^{i} } \right)^{ *} \alpha^{ij} x_{{\left[ {n_{1} } \right]}}^{j} + x_{{\left[ {n_{1} } \right]}}^{i} \left( {x_{{\left[ {n_{1} } \right]}}^{j} } \right)^{ *} } \right)} \right]} \right\} \\ & \quad \times \left\{ {\left( {x_{{\left[ {n_{2} } \right]}}^{i} \,^{ *} + \alpha^{ij *} x_{{\left[ {n_{2} } \right]}}^{j} \,^{ *} } \right)\left[ {\left| {x_{{\left[ {n_{2} } \right]}}^{i} } \right|^{{p_{2} }} + \frac{{p_{2} }}{2}\left| {x_{{\left[ {n_{2} } \right]}}^{i} } \right|^{{p_{2} - 2}} \left( {\left( {\alpha^{ij} x_{{\left[ {n_{2} } \right]}}^{j} } \right)^{ *} x_{{\left[ {n_{2} } \right]}}^{i} + \alpha^{ij} x_{{\left[ {n_{2} } \right]}}^{j} \left( {x_{{\left[ {n_{2} } \right]}}^{i} } \right)^{ *} } \right)} \right]} \right\} \\ \end{aligned}$$

(51)

Also, since the expected value is equal to zero where $n_{1} \ne n_{2}$, it is considered that $n_{1} = n_{2} = n$. By doing some straight mathematical manipulations and removing the zero term, we have:

$$E\left[ {A_{{n,n,p_{1} ,p_{2} }} } \right] = E\left[ {\left| {x_{\left[ n \right]}^{i} } \right|^{{p_{1} + p_{2} + 2}} } \right] + \left( {\frac{1}{2}\left( {p_{1} \alpha^{ij2} + p_{2} \alpha^{ij *2} } \right)\left| {\alpha^{ij} } \right|^{2} \left( {1 + \frac{{p_{1} }}{2} + \frac{{p_{2} }}{2} + \frac{{p_{1} p_{2} }}{2}} \right)} \right)E\left[ {\left| {x_{\left[ n \right]}^{j} } \right|^{2} } \right]E\left[ {\left| {x_{\left[ n \right]}^{i} } \right|^{{p_{1} + p_{2} }} } \right] + \left( {\frac{{p_{1} p_{2} }}{2}\left| {\alpha^{ij} } \right|^{4} } \right)E\left[ {\left| {x_{\left[ n \right]}^{j} } \right|^{4} } \right]E\left[ {\left| {x_{\left[ n \right]}^{i} } \right|^{{p_{1} + p_{2} - 2}} } \right]$$

(52)

Using the central moment of Rayleigh distributed random variable results in:

$$E\left[ {A_{{n,n,p_{1} ,p_{2} }} } \right] = \varGamma \left( {2 + \frac{{p_{1} + p_{2} }}{2}} \right)\left( {{\raise0.7ex\hbox{${P_{a}^{i} }$} \!\mathord{\left/ {\vphantom {{P_{a}^{i} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{{\frac{{p_{1} + p_{2} + 2}}{2}}} + \left( {Real\left( {\alpha^{ij2} } \right)\frac{{\left( {p_{1} + p_{2} } \right)}}{2} + \left| {\alpha^{ij} } \right|^{2} \left( {1 + \frac{{p_{1} }}{2} + \frac{{p_{2} }}{2} + \frac{{p_{1} p_{2} }}{2}} \right)} \right) \varGamma \left( {1 + \frac{{p_{1} + p_{2} }}{2}} \right)\left( {{\raise0.7ex\hbox{${P_{a}^{i} }$} \!\mathord{\left/ {\vphantom {{P_{a}^{i} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{{\frac{{p_{1} + p_{2} }}{2}}} \left( {{\raise0.7ex\hbox{${P_{a}^{j} }$} \!\mathord{\left/ {\vphantom {{P_{a}^{j} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right) + 6\left( {\frac{{p_{1} p_{2} }}{2}\alpha^{ij4} } \right)\varGamma \left( {\frac{{p_{1} + p_{2} }}{2}} \right)\left( {{\raise0.7ex\hbox{${P_{a}^{i} }$} \!\mathord{\left/ {\vphantom {{P_{a}^{i} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{{\frac{{p_{1} + p_{2} - 2}}{2}}} \left( {{\raise0.7ex\hbox{${P_{a}^{j} }$} \!\mathord{\left/ {\vphantom {{P_{a}^{j} } N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)^{2}$$

(53)

Therefore for the large value of N, we have

$$E\left[ { \left| {\beta_{k}^{i} } \right|^{2} } \right] = N\mathop \sum \limits_{{p_{1} = 0}}^{P - 1} \mathop \sum \limits_{{p_{2} = 0}}^{P - 1} \varphi_{{p_{1} ,k}}^{i} \varphi_{{p_{2} ,k}}^{i} \,^{ *} E\left[ {A_{{n,n,p_{1} ,p_{2} }} } \right]$$

(54)

By substituting (54) into (49) the variance of nonlinear noise, $\sigma_{Nl\,k}^{2} \,^{i}$, is calculated.