Performance Analysis of Full-Rate Coordinate Interleaved Orthogonal Designs Over Time-Selective Fading Channels

Abstract

Coordinate interleaved orthogonal designs (CIOD) (Khan and Rajan in IEEE Trans Inform Theory 52(5):2062–2091, 2006) can offer desirable properties, such as full rate, full diversity and single-symbol maximum likelihood decoding for two, three and four transmit antennas under quasi-static fading channels. When fading is time-selective, zero-forcing decoder can applied to achieve good performance while still maintain low decoding complexity. In this paper, theoretical analysis of symbol error rate performance for CIOD codes over time-selective fading channels with a zero-forcing linear receiver is derived. Firstly, a closed-form expression (i.e., not in integral form) is derived for the average symbol pair-wise error probability (SPEP) in time-selective frequency-nonselective independent identically distributed (i.i.d.) Rayleigh fading channels. Then, the SPEP is used to derive a tight upper bound (UB) for the symbol-error rate (SER) of CIOD codes. Simulation results indicate that our theoretical UB often coincides (within 0.05 dB) with the true SER obtained via Monte-Carlo simulation. The UB can thus be used to accurately predict and optimize the performance of CIOD codes over time-selective fading channels.

Appendix

To proof Theorem 1, we firstly introduce Lemma 1 as follow:

Lemma 1

Let $X_1 =\frac{\left| {g_1 } \right| ^{2}}{\lambda _1 }\frac{SNR}{4}\Delta _{I}^2 =\frac{\left| {g_1 } \right| ^{2}}{\lambda _1 }c_{I}$ and $X_2 =\frac{\left| {g_2 } \right| ^{2}}{\lambda _3 }\frac{SNR}{4}\Delta _Q^2 =\frac{\left| {g_2 } \right| ^{2}}{\lambda _3 }c_Q$ then $X_{1}$ and $X_{2}$ are independent random variables whose probability density functions (PDFs) respectively are

$$\begin{aligned} p_{X_1} (x)&= \sum _{k=0}^{2N_{R} -1} {\left( {{\begin{array}{l} {2N_{R} -1} \\ {2N_{R} -1-k} \\ \end{array}}} \right) } \frac{\rho ^{2k}\left( {1-\rho ^{2}} \right) ^{2N-1-k}}{k\hbox {!}c_{I}^{k+1}}x^{k}\exp \left( {\frac{-x}{c_{I}}} \right) \end{aligned}$$

(29)

$$\begin{aligned} p_{X_2 } (y)&= \sum _{k=0}^{2N_{R} -1} {\left( {{\begin{array}{l} {2N_{R} -1} \\ {2N_{R} -1-k} \\ \end{array}}} \right) } \frac{\rho ^{2k}\left( {1-\rho ^{2}} \right) ^{2N-1-k}}{k\hbox {!}c_Q^{k+1}}y^{k}\exp \left( {\frac{-y}{c_Q }} \right) \end{aligned}$$

(30)

Proof

The proof of Eqs. (29) and (30) is the same, so we consider the proof of Eq. (29). Let us introduce the random variables $\xi _{1k}$ and $\xi _{2k}, \,k = 1,..., N_{R}$

$$\begin{aligned} \xi _{1k} =\frac{h_{1k} (1)-\rho h_{1k} (2)}{\sqrt{1-\rho ^{2}}};\,\xi _{2k} =\frac{h_{2k} (2)-\rho h_{2k} (1)}{\sqrt{1-\rho ^{2}}} \end{aligned}$$

(31)

where we assume $\rho ^{2} < 1$. By construction, $\xi _{1k}$ and $\xi _{2k}$ are independent and identically distributed with the same pdf as $h_{1k}$(2) and $h_{2k}$(1); furthermore, $\xi _{1k}$ is independent of $h_{1k}$(2), $\xi _{1k}$ and is independent of $h_{2k}$(1). Plugging (31) into $X_{1}$ leads to

$$\begin{aligned} X_1&= \left| {\rho \sqrt{\lambda _1 }+\sqrt{1-\rho ^{2}}\frac{\sum _{k=1}^{N_{R}} {\left( {h_{1k}^{*} (2)\xi _{1k} +h_{2k} (1)\xi _{2k}^{*}} \right) }}{\sqrt{\lambda _1 }}} \right| ^{2}c_{I} \end{aligned}$$

(32)

To simplify this expression further, observe that the fraction in (32) can be expressed as an inner product

$$\begin{aligned} z_1 +\hbox {j}z_2 =\frac{\sum _{k=1}^{N_{R}} {\left( {h_{1k}^{*} (2)\xi _{1k} +h_{2k} (1)\xi _{2k}^{*}} \right) }}{\sqrt{\lambda _1 }}=\left\langle {\frac{{\mathbf {h}}}{\left\| {\mathbf {h}} \right\| },{\mathbf {e}}} \right\rangle \end{aligned}$$

(33)

where $z_{1}, \,z_{2}$ are real variables, ${\mathbf {h}}=\left[ {h_{11} (2),h_{21}^{*} (1),\ldots ,h_{1N_{R}} (2),h_{2N_{R}}^{*} (1)} \right] ^{\mathrm{T}}$ and ${\mathbf {e}}=\left[ {\xi _{11}^{*} ,\xi _{21} ,\ldots ,\xi _{1N_{R}}^{*} ,\xi _{2N_{R}}} \right] ^{\mathrm{T}}$. Since e is symmetric, the distribution of $z_{1}$ + j$z_{2}$ reduces to the distribution of $h_{1k}$(1), independent of h (and, thus, also independent of $\left\| {\mathbf {h}} \right\| )$. Thus, (33) simplifies to

$$\begin{aligned} X_1&= \left| {\rho \sqrt{\lambda _1 }+\sqrt{1-\rho ^{2}}\left( {z_1 +\hbox {j}z_2 } \right) } \right| ^{2}c_{I}\end{aligned}$$

(34)

$$\begin{aligned} X_1&= \left( {A+Z_1 } \right) ^{2}+Z_2^2 \end{aligned}$$

(35)

where $A=\sqrt{\lambda _1 \rho ^{2}c_{I}}$ and $Z_i =\sqrt{\left( {1-\rho ^{2}} \right) c_{I}}z_i , \,i = 1, 2$. Therefore, given $A, \,X_{1}$ has a noncentral chi-square distribution with two degrees of freedom, and PDF

$$\begin{aligned} p_{X_1 \left| A \right. } \left( {x\left| a \right. } \right) =\frac{1}{\left( {1-\rho ^{2}} \right) c_{I}}\exp \left( {-\frac{x+a^{2}}{\left( {1-\rho ^{2}} \right) c_{I}}} \right) J_{0} \left( {\frac{2\hbox {j}a}{\left( {1-\rho ^{2}} \right) c_{I}}\sqrt{x}} \right) \end{aligned}$$

(36)

where $J_{0}(\cdot )$ is the zeroth-order Bessel function of the first kind, and $\hbox {j}=\sqrt{-1}$.

However, $A=\sqrt{\lambda _1 \rho ^{2}c_{I}}$ is Rayleigh distributed with 2$N_{R}$ degrees of freedom with PDF

$$\begin{aligned} p_A (a)=\frac{2}{\rho ^{4N_{R}}c_{I}^{2N_{R}} \Gamma \left( {2N_{R}} \right) }a^{4N_{R} -1}\exp \left( {-\frac{a^{2}}{\rho ^{2}c_{I}}} \right) \end{aligned}$$

(37)

Integrating the product of (36) and (37) over the variable from 0 to $\infty$ leads to the following PDF for $x$ as

$$\begin{aligned} p_{X_1 } (x)&= \int \limits _0^\infty {p_{X_1 \left| A \right. } \left( {x\left| a \right. } \right) } p_A (a)da\end{aligned}$$

(38)

$$\begin{aligned} p_{X_1 } (x)&= \frac{2}{\left( {1-\rho ^{2}} \right) \rho ^{4N_{R}}c_{I}^{2N_{R} +1} \Gamma \left( {2N_{R}} \right) }\exp \left( {-\frac{x}{\left( {1-\rho ^{2}} \right) c_{I}}} \right) \nonumber \\&\quad \times \int _0^\infty {a^{4N_{R} -1}\exp \left( {-\frac{a^{2}}{\left( {1-\rho ^{2}} \right) \rho ^{2}c_{I}}} \right) J_0 \left( {\frac{2\hbox {j}\sqrt{x}}{\left( {1-\rho ^{2}} \right) c_{I}}a} \right) } da \end{aligned}$$

(39)

From (40), by using [17, equation(6.631.1)] we obtain

$$\begin{aligned} p_{X_1 } (x)=\frac{\left( {1-\rho ^{2}} \right) ^{2N_{R} -1}}{c_{I}}\exp \left( {-\frac{x}{\left( {1-\rho ^{2}} \right) c_{I}}} \right) {}_1 F_1 \left( {2N_{R} ,1;\frac{\rho ^{2}}{\left( {1-\rho ^{2}} \right) c_{I}}x} \right) \end{aligned}$$

(40)

where $_1 F_1 \left( {\alpha ,\beta ;z} \right)$ is Kummer confluent hypergeometric function [17]. By applying equations given in [18, 19] we have

$$\begin{aligned} {}_1 F_1 \left( {2N_{R} ,1;\frac{\rho ^{2}}{\left( {1-\rho ^{2}} \right) c_{I}}x} \right)&= \exp \left( {\frac{\rho ^{2}x}{\left( {1-\rho ^{2}} \right) c_{I}}} \right) \nonumber \\&\times \sum _{k=0}^{2N_{R} -1} {\left( {{\begin{array}{l} {2N_{R} -1} \\ {2N_{R} -1-k} \\ \end{array}}} \right) } \frac{\rho ^{2k}}{k\hbox {!}\left( {1-\rho ^{2}} \right) ^{k}c_{I}^k }x^{k} \end{aligned}$$

(41)

Plugging (41) into (40) we obtain

$$\begin{aligned} p_{X_1 } (x)=\sum _{k=0}^{2N_{R} -1} {\left( {{\begin{array}{l} {2N_{R} -1} \\ {2N_{R} -1-k} \\ \end{array}}} \right) } \frac{\rho ^{2k}\left( {1-\rho ^{2}} \right) ^{2N-1-k}}{k\hbox {!}c_{I}^{k+1}}x^{k}\exp \left( {\frac{-x}{c_{I}}} \right) \end{aligned}$$

(42)

The proof of Eq. (30) is similar. The Lemma 1 is demonstrated completely.

Remark 1

It is very interesting that although the calculation process from (31) to (42) is performed under the assumption $\rho ^{2} <$ 1, the final result (42) is general and can apply to any value of $\rho$ (which includes $\rho ^{2} = 1$). The proof of this observation is not difficult, so it is omitted here for brevity.

Next, we use Lemma 1 to demonstrate Theorem 1. Since $X_{1}, \,X_{2}$ are independent random variables, thus the PDF of $\omega$, which $\omega =X_{1}+X_{2}$, is expressed by

$$\begin{aligned} p_W (\omega )=\int \limits _0^\omega {p_{X_1 } (x)p_{X_2 } \left( {\omega -x} \right) dx} \end{aligned}$$

(43)

Plugging Eqs. (29) and (30) into Eq. (43), we obtain

$$\begin{aligned} p_W (\omega )=\sum _{k_1 =0}^{2N_{R} -1} {\sum _{k_2 =0}^{2N_{R} -1} {a_1 a_2 } \frac{\rho ^{2k_1 +2k_2 }\left( {1-\rho ^{2}} \right) ^{4N_{R} -2-k_1 -k_2 }}{k_1 !k_2 !c_{I}^{k_1 +1} c_Q^{k_2 +1}}} e^{\frac{-\omega }{c_Q }}\int \limits _0^\omega {e^{-\mu x}x^{k_1 }\left( {\omega -x} \right) ^{k_2 }dx}\nonumber \\ \end{aligned}$$

(44)

where $\left( {{\begin{array}{l} n \\ a \\ \end{array}}} \right) {\triangleq } \frac{n!}{a!\left( {n-a} \right) !}, \,a_1 =\left( {{\begin{array}{l} {2N_{R} -1} \\ {2N_{R} -1-k_1 } \\ \end{array}}} \right) , \,a_2 =\left( {{\begin{array}{l} {2N_{R} -1} \\ {2N_{R} -1-k_2 } \\ \end{array}}} \right)$, and $\mu =\frac{1}{c_{I}}$ $-\frac{1}{c_Q }$.

By applying Newton’s binomial

$$\begin{aligned} x^{k_1 }\left( {\omega -x} \right) ^{k_2 }=\sum _{k_3 =0}^{k_2 } {\left( {{\begin{array}{l} {k_2 } \\ {k_3 } \\ \end{array}}} \right) ({-1})^{k_3 }\omega ^{k_2 -k_3 }x^{k_1 +k_3 }} \end{aligned}$$

(45)

we obtain

$$\begin{aligned} p_W (\omega )&= \sum _{k_1 =0}^{2N_{R} -1} {\sum _{k_2 =0}^{2N_{R} -1} {a_1 a_2 } \frac{\rho ^{2k_1 +2k_2 }\left( {1-\rho ^{2}} \right) ^{4N_{R} -2-k_1 -k_2 }}{k_1 !k_2 !c_{I}^{k_1 +1} c_Q^{k_2 +1}}} e^{\frac{-\omega }{c_Q }}\sum _{k_3 =0}^{k_2 } {\left( {{\begin{array}{l} {k_2 } \\ {k_3 } \\ \end{array}}} \right) } ({-1})^{k_3 }\omega ^{k_2 -k_3 }\nonumber \\&\quad \times \int \limits _0^\omega {e^{-\mu x}x^{k_1 +k_3 }dx} \end{aligned}$$

(46)

Then, using Eq. [17, equation (3.351.1)],

$$\begin{aligned} \int \limits _0^\alpha {x^{n}e^{-\mu x}dx} =\frac{n!}{\mu ^{n+1}}-e^{-\mu \alpha }\sum _{k=0}^n {\frac{n!}{k!}\frac{\alpha ^{k}}{\mu ^{n-k+1}}} \end{aligned}$$

(47)

we obtain

$$\begin{aligned} p_W (\omega )&= \sum _{k_1 =0}^{2N_{R} -1} \sum _{k_2 =0}^{2N_{R} -1} {a_1 a_2 } \frac{\rho ^{2\left( {k_1 +k_2 } \right) }\left( {1-\rho ^{2}} \right) ^{4N_{R} -2-k_1 -k_2 }}{k_1 !k_2 !c_{I}^{k_1 +1} c_Q^{k_2 +1}}\nonumber \\&\quad \times \sum _{k_3 =0}^{k_2 } {\left( {{\begin{array}{l} {k_2 } \\ {k_3 } \\ \end{array}}} \right) ({-1})^{k_3 }\frac{\left( {k_1 +k_3 } \right) !}{\mu ^{k_1 +k_3 +1}}\omega ^{k_2 -k_3 }e^{\frac{-\omega }{c_Q }}} \nonumber \\&\quad -\sum _{k_1 =0}^{2N_{R} -1} \sum _{k_2 =0}^{2N_{R} -1} a_1 a_2 \frac{\rho ^{2\left( {k_1 +k_2 } \right) }\left( {1-\rho ^{2}} \right) ^{4N_{R} -2-k_1 -k_2 }}{k_1 !k_2 !c_{I}^{k_1 +1} c_Q^{k_2 +1}}\sum _{k_3 =0}^{k_2 } \left( {{\begin{array}{l} {k_2 } \\ {k_3 } \\ \end{array}}} \right) ({-1})^{k_3 }\nonumber \\&\quad \times \sum _{k_4 =0}^{k_1 +k_3 } {\frac{\left( {k_1 +k_3 } \right) !}{k_4 !\mu ^{k_1 +k_3 -k_4 +1}}} \omega ^{k_2 -k_3 +k_4 }e^{\frac{-\omega }{c_{I}}} \end{aligned}$$

(48)

The Theorem 1 is demonstrated completely.