On Throughput-Reliability Tradeoff of Two-Layer D-BLAST Transmission Scheme

Abstract

A contemporary tradeoff analysis concept, recognized as throughput-reliability tradeoff (TRT) analysis, has pioneered for general block fading MIMO systems. The tribute of this analysis is its ability to reveal the reciprocity between outage probability (OP), signal-to-noise ratio (SNR) and transmission rate (R) unlike diversity-multiplexing tradeoff. The present work considers two-layer D-BLAST transmission scheme for two varieties of receiver. First, TRT analysis of D-BLAST with group detection receiver is investigated where each contented diagonal of D-BLAST is detected at once. Later, TRT analysis of D-BLAST with successive interference cancellation receiver detecting a symbol at once is investigated. Based on the derived expressions, the interplay between SNR, R and OP are explored. We verify theoretically derived expressions through simulation using MATLAB programs.

Appendices

Appendix 1: Proof of Lemma 2

The pdf $p(\mathbf{\lambda })$ is $K_0 e^{- \mathop { \sum } \limits _{k=1}^{M} \lambda _k} {\lambda _1}^{N - 1} \mathop {\prod } \limits _{k=2}^{M} {\lambda _k}^{M -2}$. By the definition of $\mu$, we have $\mathbf{\lambda }= (2^{\mathbf{\mu } R}-1)/\rho$. The derivative $\frac{{d\mathbf{\lambda }}}{{d\mathbf{\mu }}}$ is $\frac{R}{\rho } [\ln (2)] 2^{\mathbf{\mu } R}$. Using the change of variable the pdf $p (\mathbf{\mu })$ through $p( \mathbf{\lambda })$ is equal to $p( \mathbf{\mu })\frac{{d\mathbf{\lambda }}}{{d\mathbf{\mu }}}$. Implementing this step, we will then have $p(\mathbf{\mu }) = {K_0}\frac{{{R^{{M}}}}}{{{\rho ^{{M}}}}}{\left[ {\ln \left( 2 \right) } \right] ^{{M}}} {e^{ - \mathop {\sum }\limits _{k = 1}^{{M}} ( {\frac{{{2^{{\mu _{_k}}R}} - 1}}{\rho }} )}} {2^{\mathop {\sum } \limits _{k = 1}^{{M}} {\mu _k}R}}{\left( {\frac{{{2^{{\mu _1}R}} - 1}}{\rho }} \right) ^{{M} - 1}} \mathop {\prod }\limits _{k = 2}^{{M}} {\left( {\frac{{{2^{{\mu _k}R}} - 1}}{\rho }} \right) ^{{N} - 2}}$. Further, after simplification this equation can be written as given in Lemma 2.

Appendix 2: Proof of Theorem 1

In order to prove Eq. (22), first we have to derive

$$\begin{aligned}&\mathop {\liminf }\limits _{\rho \rightarrow \infty } \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }},\end{aligned}$$

(42)

$$\begin{aligned}&\mathop {\limsup }\limits _{\rho \rightarrow \infty } \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }} \end{aligned}$$

(43)

and equate it for the values of g(k) by the outage probability inequalities of Eq. (7). By looking at Eqs. (7) and (19) the lower and upper bounds formulted are given following two subsections. For the analysis purpose let all the eigenvalues of Eq. (19) are ordered and represented as $\lambda _1 \le \lambda _2 \le \cdots \le \lambda _{M}$. Their probability density function (pdf) of $(\lambda _1,\lambda _2,\ldots ,\lambda _{M})$ is given by Eq. (20). The TRT of D-BLAST with GD is derived through the following subsections lower and upper bound.

1. Lower Bound Following Eq. (7), the lower bound of the outage probability is

$$\begin{aligned} {P_{otg}} \ge {P_{1,otg}} \buildrel \textstyle .\over = \Pr \left\{ {\log \left( {\mathop {\prod }\limits _{k = 1}^{{M}} \left( {1 + \rho {\mathrm{{\lambda }}_k}} \right) } \right) < \frac{{RT}}{2}} \right\} \end{aligned}$$

(44)

Let us introduce a variable related to the channel coefficients and the transmission rate,

$$\begin{aligned} \beta _k=\log (1 + \rho \lambda _k)/R, \end{aligned}$$

where $k=1,2,\ldots ,M$. In addition, since the eigenvalues are considered in order, the values of $\beta _k$ are also ordered, given as

$$\begin{aligned} {\mathcal {B}} = \left\{ {\beta |{\beta _{{M}}} \ge \cdots {\beta _2} \ge {\beta _1} \ge 0,\frac{T}{2} - \mathop {\sum }\limits _{k = 1}^{{M}} {\beta _k} > 0} \right\} . \end{aligned}$$

(45)

When $\beta _k$ is substituted for $\lambda _k$ then pdf $p(\lambda _1,\lambda _2,\ldots ,\lambda _{M})$ is then written in terms of $\beta _k$ using Lemma 2. The resultant equation is multiplied with $2^{c(k) \times R}$ to have,

$$\begin{aligned} \begin{aligned} {P_{otg}}{2^{ - c\left( k \right) R}} \ge {2^{ - c\left( k \right) R}}K{R^{{M}}}{\rho ^{ - \left( {{M}{N} - {M} + 1} \right) }}\mathop {\int }\limits _{\mathcal {B}}^{} {2^{\mathop {\sum } \limits _{k = 1}^{{M}} {\beta _k}R}} {\left( {{2^{{\beta _1}R}} - 1} \right) ^{{N} - 1}}\mathop {\prod }\limits _{k = 2}^{{M}} {\left( {{2^{{\beta _k}R}} - 1} \right) ^{{N} - 2}} \\ {e^{ - \mathop {\sum }\limits _{k = 1}^{{M}} \left( {\frac{{{2^{{\beta _k}R}} - 1}}{\rho }} \right) }}d{\mathrm{{\beta }}_1}d{\mathrm{{\beta }}_2} \ldots \mathrm{{d}}{\mathrm{{\beta }}_{{M}}} \end{aligned} \end{aligned}$$

(46)

where $K=K_0 [\ln (2)]^{M} / \rho ^{M}$ and $K_0$ is an normalizing factor. We now lookup for a subset region ${{\mathcal {B}}_{\varepsilon _1}} \subset {\mathcal {B}}$ such that

$$\begin{aligned} {{\mathcal {B}}_{{\varepsilon }_1}} \,\triangleq\, \left\{ \beta \epsilon {\mathcal {B}} \vert {\varepsilon }_1 > 0, {\mathrm s.t.} \frac{\log \rho }{R} - {\varepsilon }_1 \ge {\beta }_{\max } \right\} \end{aligned}$$

(47)

where $\beta _{\max }=\max \lbrace \beta _1, \beta _2, \ldots , \beta _{M} \rbrace$. The Eq. (46) is valid for regardless of the choice for c(k), so we consider c(k) as an arbitrary function. We will then have

$$\begin{aligned} \begin{aligned} P_{otg} 2^{-c(k)R} \ge K 2^{-c(k)R} \rho ^{-(M N - M + 1)} R^{M} e^{\frac{1}{\rho }} \int \limits _{{\mathcal {B}}_{\varepsilon _1}} (1-2^{-\varepsilon _1 R})^{MN-2M+N-1} (2^{\beta _1 R})^{N} \\ \prod \limits _{k=2}^{M} (2^{\beta _1 R})^{N-1} e^{(- 2^{-\varepsilon _1 R})} d\beta _1 d\beta _2 \ldots d\beta _{M}. \end{aligned} \end{aligned}$$

(48)

Assigning ${\beta _1}{N} + \left( {{N} - 1} \right) \mathop {\sum }\limits _{k = 2}^{{M}} {\beta _k} = f\left( {{\beta _1},{\beta _2}, \ldots ,{\beta _{{M}}}} \right)$ will lead us to have compressed equation

$$\begin{aligned} \begin{aligned} P_{otg} 2^{-c(k)R} \ge K 2^{-c(k)R} \rho ^{-(M N - M + 1)} R^{M} e^{\frac{1}{\rho }} (1-2^{-\varepsilon _1 R})^{MN-2M+N-1} \\ \int \limits _{{\mathcal {B}}_{\varepsilon _1}} (2^{(f(\beta _1,\ldots ,\beta _{M}))R}) e^{(-M 2^{\varepsilon _1 R})} d\beta _1 d\beta _2 \ldots d\beta _{M}. \end{aligned} \end{aligned}$$

(49)

Now, we define $f(\beta ^{\sup })$ as

$$\begin{aligned} f\left( {{\beta ^{\sup }}} \right) \buildrel \Delta \over = \mathop {\sup }\limits _{{{\mathcal {B}}_{\varepsilon _1}}} \left( {{\beta _1}{N} + \left( {{N} - 1} \right) \mathop {\sum }\limits _{k = 2}^{{M}} {\beta _k}} \right) \end{aligned}$$

(50)

Because of continuity of f, for some $\varepsilon _1 > 0$, there exist a neighborhood ${\mathcal {I}}_{\varepsilon _2} \subset {\mathcal {B}}$, in which $f\left( {{\beta _1}, \ldots ,{\beta _{{M}}}} \right) \ge f\left( {{\beta _{{ \varepsilon _1}}}} \right) - { \varepsilon _2}$. So in the intersection region ${\mathcal {I}}_{\varepsilon _2} \cap {\mathcal {B}}_{\varepsilon _1}$ we will have

$$\begin{aligned} \begin{aligned} \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }} \ge \frac{\log Q}{{\log \rho }} - \left( {{M}{N} - {M} + 1} \right) + \left( {f\left( {{\beta ^{\sup }}} \right) - c\left( k \right) } \right) \frac{R}{{\log \rho }} \end{aligned} \end{aligned}$$

(51)

where $Q= Ke^{({\frac{1}{\rho }-2^{-\varepsilon _1 R}})} {(1-2^{-\varepsilon _1 R})}^{MN-2M+N-1} 2^{-\varepsilon _2 R} R^{M} Vol \lbrace {\mathcal {B}}_{\varepsilon _1} \cap {\mathcal {I}}_{\varepsilon _2} \rbrace$ is a constant and $\mathrm{Vol} \lbrace {\mathcal {B}}_{\varepsilon _1} \cap {\mathcal {I}}_{\varepsilon _2} \rbrace = \int \limits _{{\mathcal {B}}_{\varepsilon _1} \cap {\mathcal {I}}_{\varepsilon _2}} d\beta _1 d\beta _2 \ldots d\beta _{M}$. To obtain $f(\beta ^{\sup })$ we have to partition the operating region accordingly by $\frac{\log \rho }{R}$ under the constraint ${\mathcal {B}}_{\varepsilon _1}$. Then the operating region ${\mathcal {R}}_\delta (k)$ for integer k values and $\delta \ge \varepsilon _1$ an arbitrary positive constant is defined as

$$\begin{aligned} {{\mathcal {R}}_\delta }\left( k \right) \buildrel \Delta \over = \left\{ {\begin{array}{ll} {\frac{1}{\delta }> \frac{{\log \rho }}{R}> \frac{T}{{2k}} + \delta }&{}{k = 0}\\ {\frac{T}{{2k}} - \delta> \frac{{\log \rho }}{R}> \frac{T}{{2\left( {k + 1} \right) }} + \delta }&{}{ 0< k < {M}}\\ {\frac{T}{{2k}} - \delta> \frac{{\log \rho }}{R} > \frac{1}{{2}} + \delta }&{}{ k = {M}} \end{array}} \right. \end{aligned}$$

(52)

It is now required to determine the function $f(\beta ^{\sup })$ in different operating regions, whereas in this work we have three operating regions. When the system is operating in ${\mathcal {R}}_\delta (0)$ region, we have

$$\begin{aligned} f\left( {{\beta ^{\sup }}} \right) = \left( {{N} - 1} \right) \left( {\frac{{{M} + 1}}{2}} \right) \end{aligned}$$

(53)

whose supremum happens at $\beta = (0,\ldots ,\frac{T}{2})$ where $T=M + 1$. The second case is when the system is operating in ${\mathcal {R}}_\delta (k)$ region, we then have

$$\begin{aligned} f\left( {{\beta ^{\sup }}} \right) = \left( {{N} - 1} \right) \left( {\frac{{{M} + 1}}{2}} \right) \end{aligned}$$

(54)

where the supremum happens at

$$\begin{aligned} {\beta = \left( {0, \ldots ,\frac{T}{2} - k\left( {\frac{{\log \rho }}{R} + \varepsilon } \right) ,\overbrace{\frac{{\log \rho }}{R} + \varepsilon , \cdots ,\frac{{\log \rho }}{R} + \varepsilon }^{k\;times}} \right) }. \end{aligned}$$

Finally, the operating region is in ${\mathcal {R}}_\delta (k)$ where $k=M$, whose supremum happens at

$$\begin{aligned} {\beta = \left( {\frac{T}{2} - {M}\left( {\frac{{\log \rho }}{R} + \varepsilon } \right) ,\overbrace{\frac{{\log \rho }}{R} + \varepsilon , \cdots ,\frac{{\log \rho }}{R} + \varepsilon }^{({M} - 1)\;times}} \right) } \end{aligned}$$

for which the $f(\beta ^{\sup })$ value is

$$\begin{aligned} f\left( {{\beta ^{\sup }}} \right) = {N}\left( {\frac{{{M} + 1}}{2}} \right) . \end{aligned}$$

(55)

By letting the function $f(\beta ^{\sup })$ in different operating regions with assumption $\varepsilon \le \delta$ guarantees that $\frac{{{M} + 1}}{2} - k\left( {\frac{{\log \rho }}{R} + \varepsilon } \right) > 0$ and as ${\mathcal {R}}(k)$ shown in (23). We will then have

$$\begin{aligned} \mathop {\lim \inf }\limits _{\rho \rightarrow \infty } \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }}\ge & {} - \left( {{M}{N} - {M} + 1} \right) \nonumber \\&+ \left( {\left( {{N} - 1} \right) \left( {\frac{{{M} + 1}}{2}} \right) - c\left( k \right) } \right) \mathop {\lim \inf }\limits _{\rho \rightarrow \infty } \frac{R}{{\log \rho }}\end{aligned}$$

(56)

$$\begin{aligned} \mathop {\lim \inf }\limits _{\rho \rightarrow \infty } \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }}\ge & {} - \left( {{M}{N}} \right) + \left( {{N}\left( {\frac{{{M} + 1}}{2}} \right) - c\left( k \right) } \right) \mathop {\lim \inf }\limits _{\rho \rightarrow \infty } \frac{R}{{\log \rho }}. \end{aligned}$$

(57)

2. Upper Bound We now turn our interest towards upper bound of the outage probability, given as

$$\begin{aligned} {P_{otg}} \le 2{P_{1,otg}} = 2\Pr \left\{ {\log \left( {\mathop {\prod }\limits _{k = 1}^{{M}} \left( {1 + \rho {\mathrm{{\lambda }}_k}} \right) } \right) < \frac{{RT}}{2}} \right\} \end{aligned}$$

(58)

by Eq. (7). We will now consider for change of variable and redefine for $\log \left( {1 + \rho {\lambda _k}} \right) /R = {\beta _k}$, where $k = 1,2, \ldots ,{M}.$ The ${\beta _k}$s are ordered as like the eigenvalues, given by,

$$\begin{aligned} {\mathcal {B}} = \left\{ {\beta |{\beta _{{M}}} \ge \cdots {\beta _2} \ge {\beta _1} \ge 0,\frac{T}{2} - \mathop {\sum }\limits _{i = 1}^{{M}} {\beta _{k > 0}}} \right\} . \end{aligned}$$

(59)

Using Lemma 2, the pdf of ${\lambda _1} \le {\lambda _2} \le \cdots \le {\lambda _{{M}}}$ is written in terms of $\beta$ and substituted for outage probability after multiplying with ${2^{ - c\left( k \right) R}}$ gives

$$\begin{aligned} \begin{aligned} {P_{otg}}{2^{ - c\left( k \right) R}} \le {2^{ - c\left( k \right) R}}2K{R^{{M}}}{\rho ^{ - \left( {{M}{N} - {M} + 1} \right) }} \mathop {\int }\limits _{\mathcal {B}}^{} {2^{\mathop {\sum } \limits _{k = 1}^{{M}} {\beta _k}R}} {\left( {{2^{{\beta _1}R}} - 1} \right) ^{{N} - 1}}\mathop {\prod }\limits _{k = 2}^{{M}} {\left( {{2^{{\beta _k}R}} - 1} \right) ^{{N} - 2}} \\ {e^{ - \mathop {\sum }\limits _{k = 1}^{{M}} \left( {\frac{{{2^{{\beta _{_k}}R}} - 1}}{\rho }} \right) }}d{\beta _1}d{\beta _2} \ldots \mathrm{{d}}{\beta _{{M}}} \end{aligned} \end{aligned}$$

(60)

where ${{K = {K_0}{{\left[ {\ln \left( 2 \right) } \right] }^{{M}}}}/ {{\rho ^{{M}}}}}$. Considering $c\left( k \right)$ as an arbitrary function at this point of time, we write this inequality as ${P_{otg}}{2^{ - c\left( k \right) R}} \le B{}_1 + {B_2}$. The functions $B{}_1$ and $B{}_2$ are the subsets of outage probability that are defined as,

$$\begin{aligned}&{{B_1} \buildrel \Delta \over = {2^{ - c\left( k \right) R}}\int \limits _{{{\mathcal {B}}_1}} {p\left( {{\beta _1},{\beta _2}, \ldots ,{\beta _{{M}}}} \right) d{\beta _1}d{\beta _2}} \ldots d{\beta _{{M}}}}\nonumber \\&{{B_2} \buildrel \Delta \over = {2^{ - c\left( k \right) R}}\int \limits _{{{\mathcal {B}}_2}} {p\left( {{\beta _1},{\beta _2}, \ldots ,{\beta _{{M}}}} \right) d{\beta _1}d{\beta _2} \ldots d{\beta _{{M}}}} } \end{aligned}$$

(61)

where the outage region which is the limit of integration is then defined as

$$\begin{aligned}&{\mathcal {B}}_1 \,\triangleq\, \left\{ \beta \epsilon {\mathcal {B}} \vert \beta _{M} > \frac{\log \rho }{R} + \varepsilon \right\} \nonumber \\&{\mathcal {B}}_2 \,\triangleq\, \left\{ \beta \epsilon {\mathcal {B}} \vert \beta _{M} \le \frac{\log \rho }{R} + \varepsilon \right\} \end{aligned}$$

(62)

With the defined subsets we will have

$$\begin{aligned} \begin{aligned} \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }} \le \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \frac{{\log \left( {1 + {{{B_1}} / {{B_2}}}} \right) }}{{\log \rho }} + \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \frac{{\log \left( {1 + {B_2}} \right) }}{{\log \rho }} \end{aligned} \end{aligned}$$

(63)

Solving for $B_1$ and $B_2$ individually and later substituting in the Eq. (63) yields the result. Now considering $B_1$ for the specified region ${\mathcal {B}}_1$, we get

$$\begin{aligned} \begin{aligned} B_1 \le K 2^{-c(k) R} 2 \rho ^{-(M N - M + 1)} R^{M} e^{\frac{1}{\rho }} \mathop {\int }\limits _{{\mathcal {B}}_1} 2^{f(\beta _1,\beta _2,\ldots , \beta _{M})R} e^{-2^{\varepsilon R}} d\beta _1 d\beta _2 \ldots d\beta _{M} \end{aligned} \end{aligned}$$

(64)

where $f \left( {{\beta _1},{\beta _2}, \ldots ,{\beta _M}} \right)$ redefined as $( N \beta _1 + (N-1) \sum \limits _{k=2}^M \beta _k ).$ Realizing that ${{\mathcal {B}}_1}$ is a subset of ${B_1}$ which belongs to $\left\{ {\beta |1 \ge {\beta _k} \ge 0,\forall k} \right\}$ and hence $Vol\left\{ {{{\mathcal {B}}_1}} \right\} \le 1.$ This leads to the result

$$\begin{aligned} \begin{aligned} B_1 \le K 2 \rho ^{-(M N - M + 1)} R^{M} e^{\frac{1}{\rho }} {2^{\varepsilon R}}^{\frac{f\left( {{{\mathcal {B}}_1}^{\sup }} \right) - c(k)}{\varepsilon }} e^{-2^{\varepsilon R}} Vol \lbrace {\mathcal {B}}_1 \rbrace \end{aligned} \end{aligned}$$

(65)

where $f\left( {{{\mathcal {B}}_1}^{\sup }} \right)$ is defined as

$$\begin{aligned} f\left( {{{\mathcal {B}}_1}^{\sup }} \right) \,\triangleq\, \mathop {\sup }\limits _{{{\mathcal {B}}_1}} f\left( {{\beta _1},{\beta _2}, \ldots ,{\beta _{{M}}}} \right) . \end{aligned}$$

(66)

Now, we have to look at other subset ${B_2}$ of the Eq. (63) for

$$\begin{aligned} {\mathcal {B}}_{\varepsilon _3} \,\triangleq\, \left\{ \beta \epsilon {\mathcal {B}}_2 \vert \frac{log \rho }{R} - \beta _{M}> \varepsilon _3, \beta _1>\varepsilon _3 \right\} \end{aligned}$$

(67)

is given by,

$$\begin{aligned} \begin{aligned} B_2 \ge 2 K 2^{-c(k) R} \rho ^{-(M N - M + 1)} R^{M} e^\frac{1}{\rho } \int \limits _{{\mathcal {B}}_{\varepsilon _3}} (1-2^{-\varepsilon _3 R})^{(M N - 2 M +N - 1)} 2^{f(\beta _1,\beta _2,\ldots ,\beta _{M})R} \\ e^{-M 2^{-\varepsilon _3 R}} d \beta _1,d \beta _2,\ldots ,d \beta _{M}. \end{aligned} \end{aligned}$$

(68)

For the function given in Eq. (67), there exists a supremum value, defined as,

$$\begin{aligned} f\left( {{{\mathcal {B}}_2}^{\sup }} \right) \buildrel \Delta \over = \mathop { \sup }\limits _{\beta \epsilon {{\mathcal {B}}_{{ \varepsilon _{_3}}}}} f\left( {{\beta _1}, \cdots ,{\beta _{{M}}}} \right) . \end{aligned}$$

(69)

Then for any $\varepsilon _4>0$, there exists a neighborhood $I_{\varepsilon _4}$ belongs to $\beta ^{\sup }$ such that,

$$\begin{aligned} f\left( {{\beta _1}, \ldots ,{\beta _{{M}}}} \right) \ge f\left( {{\beta ^{\sup }}} \right) - { \varepsilon _4} \end{aligned}$$

(70)

This condition let us to write

$$\begin{aligned} \begin{aligned} B_2 \ge 2 K \rho ^{-(M N-M+1)} R^{M} e^{\frac{M}{\rho }} e^{-M 2^{-\varepsilon _4R}} (1-2^{-\varepsilon _4 R})^{(M N -2M + N - 1)} 2^{(f(\beta ^{\sup }) - \varepsilon _4 - c(k))R} \\ Vol \lbrace {\mathcal {B}}_{\varepsilon _4} \cap I_{\varepsilon _{4}} \rbrace . \end{aligned} \end{aligned}$$

(71)

From Eqs. (64) and (71) we will have

$$\begin{aligned} \begin{aligned} \frac{B_1}{B_2} &\le {\left( 1-{(2^{\varepsilon R})}^{-\frac{\varepsilon _4}{\varepsilon }}\right) }^{-(M N - 2 M + 1)} e^{\frac{1}{\rho } (1-M)} {(2^{\varepsilon R})}^{\frac{f\left( {{{\mathcal {B}}_1}^{\sup }} \right) - f(\beta ^{\sup })+ \varepsilon _4}{\varepsilon }} \\ &\quad \times e^{-(2^{\varepsilon R + M 2 ^{-\varepsilon _3 R}})} \frac{Vol \lbrace {\mathcal {B}}_1 \rbrace }{Vol \lbrace {\mathcal {B}}_3 \cap I_{\varepsilon _4} \rbrace } \end{aligned} \end{aligned}$$

(72)

This means that

$$\begin{aligned} \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \quad \frac{{\log \left( {1 + {B_1}/{B_2}} \right) }}{{\log \rho }} = 0 \end{aligned}$$

(73)

because the value $2^{\varepsilon R}$ decays exponentially in right-hand side while it grows polynomially with the same variable in Eq. (72). For the second term in Eq. (63), we note that

$$\begin{aligned} \begin{aligned} B_2 \le 2 K R^{M} \rho ^{-(M N - M + 1)} e^{\frac{M}{\rho }} 2^{(f({\beta ^{\sup }}) - c(k))R} {(1-2^{-\varepsilon _3 R})}^{M N- 2M + 1} e^{-M 2^{-\varepsilon _3 R}} \end{aligned} \end{aligned}$$

(74)

is the supremum of $\frac{\log B_2}{\log \rho }$ in the region ${\mathcal {B}}_{\varepsilon _3}$. We will have

$$\begin{aligned} \begin{aligned} \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }} \le \frac{{2K{R^{{M}}}}}{{\log \rho }} - \left( {{M}{N} - {M} + 1} \right) + \left( {f\left( {{\beta ^{\sup }}} \right) - c\left( k \right) } \right) \frac{R}{{\log \rho }} \end{aligned} \end{aligned}$$

(75)

where $f(\beta ^{\sup })$ redefined as

$$\begin{aligned} f(\beta ^{\sup }) \,\triangleq\, \sup \limits _{{\mathcal {B}}_2}^{} f(\beta _1, \beta _2, \ldots , \beta _{M}). \end{aligned}$$

(76)

To obtain value for Eq. (92) we have to let $f(\beta ^{\sup })$ in three operating regions as mentioned earlier, thus we get

$$\begin{aligned} \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }}\le & {} - \left( {{M}{N} - {M} + 1} \right) \nonumber \\&+ \left( {\left( {{N} - 1} \right) \left( {\frac{{{M} + 1}}{2}} \right) - c\left( k \right) } \right) \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \frac{R}{{\log \rho }},\end{aligned}$$

(77)

$$\begin{aligned} \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }}\le & {} - \left( {{M}{N}} \right) + \left( {{N}\left( {\frac{{{M} + 1}}{2}} \right) - c\left( k \right) } \right) \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \frac{R}{{\log \rho }}. \end{aligned}$$

(78)

The Eqs. (56), (57), (77), and (78) are TRT expressions that belong to the operating region given in Eq. (23) for the D-BLAST transmission employing GD at the receiver and completes the proof.

Appendix 3: Proof of Lemma 3

After simplification the proof required can be written as,

$$\begin{aligned} \det \left( \mathbf{I}_N + \frac{\rho }{M} \mathbf{G} \mathbf{G}^H \right) = \prod \limits _{k=1}^M \left[ 1 + \varGamma _k \right] \end{aligned}$$

(79)

Now

$$\begin{aligned} \varOmega _M \,\triangleq\, \left( \mathbf{I}_N + \frac{\rho }{M} \mathbf{G} \mathbf{G}^H \right) = \frac{\rho }{M} \sum \limits _{i=1}^{M} {g}_i {g}_i^H + \mathbf{I}_N \end{aligned}$$

(80)

where $\mathbf{g}_i$ is the ith column of matrix $\mathbf{H}$. For single transmit and N receive antennas, it can be written that

$$\begin{aligned} \det \left( 1 + \frac{\rho }{M} \mathbf{g}_i^{H}{} \mathbf{g}_i \right) = 1 + \varUpsilon _1 \end{aligned}$$

Assuming it is true for $M-1$ transmitting antennas, we can write

$$\begin{aligned} \left( \varOmega _{M-1} \right) = \prod \limits _{k=1}^{M-1} \left[ 1 + \varUpsilon _k \right] \end{aligned}$$

(81)

From Eqs. (80) and (81), we have

$$\begin{aligned} \varOmega _M&= \varOmega _{M-1} + \frac{\rho }{M} \mathbf{g}_M\mathbf{g}_M^H\end{aligned}$$

(82)

$$\begin{aligned} \mathbf{P}&= \mathbf{Q} + \frac{\rho }{M} \mathbf{g}_M \mathbf{g}_M^H \end{aligned}$$

(83)

where $\mathbf{P} \,\triangleq\, \varOmega _M$ and $\mathbf{Q} \,\triangleq\, \varOmega _{M-1}$. By applying the matrix inversion lemma given in [3], we can write

$$\begin{aligned} \left[ \mathbf{Q} + \frac{\rho }{M}{} \mathbf{g}_M\mathbf{g}_M^H \right] ^{-1} \mathbf{g}_M= & {} \frac{\mathbf{Q}^{-1}{} \mathbf{g}_M}{1+\frac{\rho }{M}\mathbf{g}_M^H\mathbf{Q}^{-1}{} \mathbf{g}_M}\end{aligned}$$

(84)

$$\begin{aligned} \mathbf{P}^{-1}{} \mathbf{g}_M= & {} \frac{\mathbf{Q}^{-1}{} \mathbf{g}_M}{1+ \varGamma _M} \end{aligned}$$

(85)

The matrix inverse $\mathbf{P}^{-1}$ is

$$\begin{aligned} \mathbf{P}^{-1} \,\triangleq\, \frac{\mathbf{P}_{adj}}{\det \left( P \right) } \end{aligned}$$

where $\mathbf{P}_{adj}$ is the adjoint of matrix $\mathbf{P}$; thus, we can write Eq. (85) as

$$\begin{aligned} \frac{\mathbf{P}_{adj}}{\det \mathbf{P}}{} \mathbf{g}_M &{} = \frac{\mathbf{Q}_{adj}}{\det \mathbf{Q}} \frac{\mathbf{g}_M}{1+ \varGamma _M} \\ \\ &{} = \mathbf{Q}_{adj} \frac{\mathbf{g}_m}{\prod \limits _{i=1}^m \left( 1 + \varGamma _i \right) } \end{aligned}$$

(86)

where (86) is obtained through exchanging $\det \mathbf{Q}$ using (81). To prove Eq. (79), we need to show that

$$\begin{aligned} \mathbf{P}_{adj}{} \mathbf{g}_M = \mathbf{Q}_{adj}{} \mathbf{g}_M \end{aligned}$$

(87)

Expanding Eq. (83), we will have

$$\begin{aligned}\mathbf{P} &{} = \mathbf{Q} + \frac{\rho }{M}{} \mathbf{g}_M\mathbf{g}_M^H \\ &{} = \left[ \mathbf{q}_1 + \frac{\rho }{M} \mathbf{g}_M H_{M1}^*, \mathbf{q}_2+\frac{\rho }{M}{} \mathbf{g}_M H_{M2}^*, \cdots , \mathbf{q}_N + \frac{\rho }{M} \mathbf{g}_M H_{MN}^* \right]\end{aligned}$$

(88)

where $\mathbf{q}_i , \forall i$ is the ith column of $\mathbf{Q}$ and $H_{ij}$ is the ijth element of matrix $\mathbf{H}$. The Eq. (87) can be proved by showing that jth element of $\mathbf{P}_{adj}{} \mathbf{g}_m$ is equal to the jth element of $\mathbf{Q}_{adj}{} \mathbf{g}_j$ for $j= 1,\ldots ,M$. The jth element of $\mathbf{P}_{adj}{} \mathbf{g}_m$ is given by $det \left( \mathbf{P}_j \right)$, where $\mathbf{P}_j$ is obtained by replacing the jth column of matrix $\mathbf{P}$ by $\mathbf{g}_m$. Similarly, the jth element of $\mathbf{Q}_{adj}{} \mathbf{g}_m$ is given by $det \left( \mathbf{Q}_j \right)$. Accordingly, by replacing the jth column by $\mathbf{g}_m$, we get

$$\begin{aligned} \det \left( \mathbf{P}_j \right) = \det \left[ \mathbf{q}_1 + \frac{\rho }{M} \mathbf{g}_m H_{m1}^{*}, \cdots , \mathbf{g}_m, \cdots , \mathbf{q}_N + \frac{\rho }{M} \mathbf{g}_m H_{mN}^{*} \right] \end{aligned}$$

(89)

By the fact that the determinant of a matrix will not change if a column is multiplied by a constant is added to another column, we can write the above equation as

$$\begin{aligned} \det \left( \mathbf{P}_j \right) &{} = \det \left[ \mathbf{q}_1, \mathbf{q}_2, \ldots , \mathbf{g}_m, \ldots , \mathbf{q}_N \right] \\ &{} = det \left( \mathbf{Q}_j \right) \end{aligned}$$

(90)

Similarly, we can show that $det \left( \mathbf{P}_j \right) = det \left( \mathbf{Q}_j \right)$ for $i= 1,2, \ldots , N, i = j$. This completes the proof.

Appendix 4: Proof of Theorem 2

In order to prove Eq. (37), first we have to derive

$$\begin{aligned}&\mathop {\liminf }\limits _{\rho \rightarrow \infty } \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }},\end{aligned}$$

(91)

$$\begin{aligned}&\mathop {\limsup }\limits _{\rho \rightarrow \infty } \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }} \end{aligned}$$

(92)

and equate it for the values of g(k) by the outage probability inequalities of Eq. (7). By looking at Eqs. (7) and (35) the lower and upper bounds formulted are given following two subsections. For the analysis purpose let all the eigenvalues of Eq. (35) are ordered and represented as $\lambda _1 \le \lambda _2 \le \cdots \le \lambda _{M}$. Their probability density function (pdf) of $(\lambda _1,\lambda _2,\ldots ,\lambda _{M})$ is given by Eq. (36). The TRT of D-BLAST with SIC is derived through the following subsections lower and upper bound.

1. Lower Bound    Following Eq. (7), the lower bound of the outage probability is

$$\begin{aligned} {P_{otg} \ge P_{1,otg} = \Pr \left\{ \frac{1}{T} \log \left( \prod \limits _{k=1}^q \left( 1 + \frac{\rho }{M} \lambda _k \right) \right) < \frac{R}{2} \right\} } \end{aligned}$$

(93)

Let us introduce a variable related to the channel coefficients and the the transmission rate,

$$\begin{aligned} \beta _k=\log (1 + \frac{\rho }{M} \lambda _k)/R, \end{aligned}$$

where $k=1,2,\ldots ,M$. In addition, since the eigenvalues are considered in order, the values of $\beta _k$ are also ordered, given as

$$\begin{aligned} {\mathcal {B}} = \left\{ {\beta |{\beta _{{M}}} \ge \cdots {\beta _2} \ge {\beta _1} \ge 0,\frac{T}{2} - \mathop {\sum }\limits _{k = 1}^{{M}} {\beta _k} > 0} \right\} . \end{aligned}$$

(94)

When $\beta _k$ is substituted for $\lambda _k$ then pdf $p(\lambda _1,\lambda _2,\ldots ,\lambda _{M})$ is then written in terms of $\beta _k$ using Lemma 4. The resultant equation is multiplied with $2^{c(k) \times R}$ to have,

$$\begin{aligned} P_{otg} 2^{-c \left( k \right) R } &\ge K 2^{-c \left( k \right) R} R^q \left( \frac{M}{\rho } \right) ^{MN} \int \limits _{\mathcal {B}} \prod \limits _{i=1}^{q} \left( 2^{\beta _i R} - 1 \right) ^{\left| M -N \right| } \prod \limits _{i<j} \left( 2^{\beta _i R} - 2^{\beta _j R} \right) ^2 2^{\sum \limits _i \beta _i R} \\ & \quad \times e^{-\frac{M}{\rho } \sum \limits _i \left( 2^{\beta _i R} -1 \right) } d\beta \end{aligned}$$

(95)

where $K_0$ is an normalizing factor. We now lookup for a subset region ${{\mathcal {B}}_{\varepsilon _1}} \subset {\mathcal {B}}$ such that

$$\begin{aligned} \beta _{\varepsilon _1} = \left\{ \beta \epsilon {\mathcal {B}}_1 | \varepsilon _1> 0, s.t. \frac{\log \rho }{R}-\beta _q \ge \varepsilon _1, \beta _1>\varepsilon _1, \left| \beta _j - \beta _i \right| > \varepsilon _1, \forall i \ne j \right\} \end{aligned}$$

(96)

The Eq. (95) is valid for regardless of the choice for c(k), so we consider c(k) as an arbitrary function. We will then have

$$\begin{aligned} \begin{aligned} P_{otg} 2^{-c(k)R} \ge K 2^{-c \left( k \right) R} R^q \left( \frac{M}{\rho } \right) ^{MN} e^{\frac{Mq}{\rho }} \int \limits _{\beta _{\varepsilon _1}} 2^{\sum \limits _i \beta _i R} \prod \limits _{i=1}^q \left( 1-2^{-\varepsilon _1 R} \right) ^{\left| M -N \right| } 2^{\beta _i R \left| M-N \right| } \\ \prod \limits _{i<j}^q \left( 1-2^{-\varepsilon _1 R} \right) ^{2} 2^{2\beta _j R} e^{-Mq2^{-\varepsilon _1 R}} d\beta \end{aligned} \end{aligned}$$

(97)

Realizing that $e^{-2^{-\varepsilon _1 R}} \ge \left( 1 - 2^{- \varepsilon _1 R} \right)$ and assigning $f \left( \beta \right) \,\triangleq\, \sum \limits _{i=1}^q \left( \left| M-N \right| + 2i -1 \right) \beta _i$ will lead us to have

$$\begin{aligned} \begin{aligned} P_{otg} 2^{-c(k)R} \ge K 2^{-c \left( k \right) R} R^q \left( \frac{M}{\rho } \right) ^{MN} e^{\frac{Mq}{\rho }} \int \limits _{\beta _{\varepsilon _1}} 2^{\sum \limits _i \beta _i R} \prod \limits _{i=1}^q 2^{\beta _i R \left| M-N \right| } \prod \limits _{i<j}^q 2^{2\beta _j R} \\ \left( 1-2^{-\varepsilon _1 R} \right) ^{M \left( N + q \right) } d\beta \end{aligned} \end{aligned}$$

(98)

Now, we redefine

$$\begin{aligned} f(\beta ^{\sup }) = \sup \limits _{{{\mathcal {B}}_{\varepsilon _1}}} f(\beta ). \end{aligned}$$

Because of continuity of f, for some $\varepsilon _1 > 0$, there exist a neighborhood ${\mathcal {I}}_{\varepsilon _2} \subset {\mathcal {B}}$, in which $f\left( {{\beta _1}, \cdots ,{\beta _{{M}}}} \right) \ge f\left( \beta \right) - { \varepsilon _2}$. So in the intersection region ${\mathcal {I}}_{\varepsilon _2} \cap {\mathcal {B}}_{\varepsilon _1}$ we will have

$$\begin{aligned} \begin{aligned} \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }} \ge \frac{\log Q}{{\log \rho }} - \left( MN \right) + \left( {f\left( {{\beta ^{\sup }}} \right) - c\left( k \right) } \right) \frac{R}{{\log \rho }} \end{aligned} \end{aligned}$$

(99)

where $Q= {K} M^{MN} R^q e^{\frac{Mq}{\rho }} \left( 1-2^{-\varepsilon _1 R} \right) ^{q(q \left| M-N \right| + q(q-1))+Mq} 2^{\left( -\varepsilon _2\right) } \mathrm{Vol} \left\{ {\mathcal {A}}_{\varepsilon _1} \cap I_{\varepsilon _2} \right\}$ is a constant and $\mathrm{Vol} \lbrace {\mathcal {B}}_{\varepsilon _1} \cap {\mathcal {I}}_{\varepsilon _2} \rbrace = \int \limits _{{\mathcal {B}}_{\varepsilon _1} \cap {\mathcal {I}}_{\varepsilon _2}} d\beta$. To obtain $f(\beta ^{\sup })$ we have to partition the operating region accordingly by $\frac{\log \rho }{R}$ under the constraint ${\mathcal {B}}_{\varepsilon _1}$. Then the operating region ${\mathcal {R}}_\delta (k)$ for integer k values and $\delta \ge \varepsilon _1$ an arbitrary positive constant is defined as

$$\begin{aligned} {{\mathcal {R}}_\delta }\left( k \right) \buildrel \Delta \over = \left\{ {\begin{array}{ll} {\frac{1}{\delta }> \frac{{\log \rho }}{R}> \frac{T}{{2k}} + \delta }&{}{k = 0}\\ {\frac{T}{{2k}} - \delta> \frac{{\log \rho }}{R}> \frac{T}{{2\left( {k + 1} \right) }} + \delta }&{}{ 0< k < {M}}\\ {\frac{T}{{2k}} - \delta> \frac{{\log \rho }}{R} > \frac{1}{{2}} + \delta }&{}{ k = {M}} \end{array}} \right. \end{aligned}$$

(100)

It is now required to determine the function $f(\beta ^{\sup })$ in different operating regions, whereas in this work we have three operating regions. When the system is operating in ${\mathcal {R}}_\delta (0)$ region, we have

$$\begin{aligned} f\left( {{\beta ^{\sup }}} \right) = \left( M+N-1 \right) \left( \frac{M+1}{2} \right) \end{aligned}$$

(101)

whose supremum happens at $\beta = (0,\ldots ,\frac{T}{2})$ where $T=M + 1$. The second case is when the system is operating in ${\mathcal {R}}_\delta (k)$ region, we then have

$$\begin{aligned} f\left( {{\beta ^{\sup }}} \right) = \lbrace M + N - \left( 2k+1\right) \rbrace \frac{T}{2} + k\left( k+1\right) \left( \frac{\log \rho }{R} + \varepsilon \right) \end{aligned}$$

(102)

where the supremum happens at

$$\begin{aligned} {\beta = \left( {0, \cdots ,\frac{T}{2} - k\left( {\frac{{\log \rho }}{R} + \varepsilon } \right) ,\overbrace{\frac{{\log \rho }}{R} + \varepsilon , \cdots ,\frac{{\log \rho }}{R} + \varepsilon }^{k\;times}} \right) }. \end{aligned}$$

Finally, the operating region is in ${\mathcal {R}}_\delta (k)$ where $k=M$, whose supremum happens at

$$\begin{aligned} {\beta = \left( {\frac{T}{2} - {M}\left( {\frac{{\log \rho }}{R} + \varepsilon } \right) ,\overbrace{\frac{{\log \rho }}{R} + \varepsilon , \cdots ,\frac{{\log \rho }}{R} + \varepsilon }^{({M} - 1)\;times}} \right) } \end{aligned}$$

for which the $f(\beta ^{\sup })$ value is

$$\begin{aligned} f\left( {{\beta ^{\sup }}} \right) = \lbrace M + N - \left( 2k+1\right) \rbrace \frac{T}{2} + k\left( k+1\right) \left( \frac{\log \rho }{R} + \varepsilon \right) \end{aligned}$$

(103)

By letting the function $f(\beta ^{\sup })$ in different operating regions with assumption $\varepsilon \le \delta$ guarantees that $\frac{{{M} + 1}}{2} - k\left( {\frac{{\log \rho }}{R} + \varepsilon } \right) > 0$ and as ${\mathcal {R}}(k)$ shown in (38). We will then have

$$\begin{aligned} &\mathop {\lim \inf }\limits _{\rho \rightarrow \infty } \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }} \le - \left( MN - k(k+1) \right) \\ &\quad +\left( \left( M+N-(2k+1)\right) \left( \frac{M+1}{2} \right) - c\left( k \right) \right) \mathop {\lim \inf }\limits _{\rho \rightarrow \infty } \frac{R}{{\log \rho }} \end{aligned}$$

(104)

2. Upper Bound    We now turn our interest towards upper bound of the outage probability, given as

$$\begin{aligned} {P_{otg} \le 2P_{1,otg} = 2 \Pr \left\{ \frac{1}{T} \log \left( \prod \limits _{i=1}^q \left( 1 + \frac{\rho }{M} \lambda _i \right) \right) < \frac{R}{2} \right\} } \end{aligned}$$

(105)

We will now consider for change of variable and redefine for $\log \left( {1 + \frac{\rho }{M} {\lambda _k}} \right) /R = {\beta _k}$, where $k = 1,2, \ldots ,{q}$. The ${\beta _k}$s are ordered as like the eigenvalues, given by,

$$\begin{aligned} {\mathcal {B}} = \left\{ {\beta |{\beta _{{M}}} \ge \cdots {\beta _2} \ge {\beta _1} \ge 0,\frac{T}{2} - \mathop {\sum }\limits _{i = 1}^{{M}} {\beta _{k > 0}}} \right\} . \end{aligned}$$

(106)

Using Lemma 4, the pdf of ${\lambda _1} \le {\lambda _2} \le \cdots \le {\lambda _{{q}}}$ is written in terms of $\beta$ and substituted for outage probability after multiplying with ${2^{ - c\left( k \right) R}}$ gives

$$\begin{aligned} \begin{aligned} P_{otg} 2^{-c \left( k \right) R } \le 2 K 2^{-c \left( k \right) R} R^q \left( \frac{M}{\rho } \right) ^{MN} \int \limits _{\mathcal {B}} \prod \limits _{i=1}^{q} \left( 2^{\beta _i R} - 1 \right) ^{\left| M -N \right| } \prod \limits _{i<j} \left( 2^{\beta _i R} - 2^{\beta _j R} \right) ^2 \\ 2^{\sum \limits _i \beta _i R} e^{-\frac{M}{\rho } \sum \limits _i \left( 2^{\beta _i R} -1 \right) } d\beta . \end{aligned} \end{aligned}$$

(107)

Considering $c\left( k \right)$ as an arbitrary function at this point of time, we write this inequality as ${P_{otg}}{2^{ - c\left( k \right) R}} \le B{}_1 + {B_2}$. The functions $B{}_1$ and $B{}_2$ are the subsets of outage probability that are defined as,

$$\begin{aligned}&{{B_1} \buildrel \Delta \over = {2^{ - c\left( k \right) R}}\int \limits _{{{\mathcal {B}}_1}} p(\beta ) d\beta } \nonumber \\&{{B_2} \buildrel \Delta \over = {2^{ - c\left( k \right) R}}\int \limits _{{{\mathcal {B}}_2}} p(\beta ) d\beta } \end{aligned}$$

(108)

where the outage region which is the limit of integration is then defined as

$$\begin{aligned}&{\mathcal {B}}_1 \,\triangleq\, \left\{ \beta \epsilon {\mathcal {B}} \vert \beta _{q} > \frac{\log \rho }{R} + \varepsilon \right\} \nonumber \\&{\mathcal {B}}_2 \,\triangleq\, \left\{ \beta \epsilon {\mathcal {B}} \vert \beta _{q} \le \frac{\log \rho }{R} + \varepsilon \right\} \end{aligned}$$

(109)

With the defined subsets we will have

$$\begin{aligned} \begin{aligned} \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }} \le \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \frac{{\log \left( {1 + {{{B_1}} / {{B_2}}}} \right) }}{{\log \rho }} + \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \frac{{\log \left( {1 + {B_2}} \right) }}{{\log \rho }} \end{aligned} \end{aligned}$$

(110)

Solving for $B_1$ and $B_2$ individually and later substituting in the Eq. (110) yields the result. Now considering $B_1$ for the specified region ${\mathcal {B}}_1$, we get

$$\begin{aligned} \begin{aligned} B_1 \le 2K R^q e^{\frac{M}{\rho }} \rho ^{-MN} 2^{-c\left( k\right) R} \int \limits _{{\mathcal {B}}_1} 2^{f\left( \alpha \right) R} e^{-M2^{-\varepsilon R}} d\beta \end{aligned} \end{aligned}$$

(111)

where $f \left( {{\beta _1},{\beta _2}, \ldots ,{\beta _M}} \right)$ redefined as $( N \beta _1 + (N-1) \sum \limits _{k=2}^M \beta _k ).$ Realizing that ${{\mathcal {B}}_1}$ is a subset of ${B_1}$ which belongs to $\left\{ {\beta |1 \ge {\beta _k} \ge 0,\forall k} \right\}$ and hence $Vol\left\{ {{{\mathcal {B}}_1}} \right\} \le 1.$ This leads to the result

$$\begin{aligned} \begin{aligned} B_1 \le K R^q ( 2^{\varepsilon R} )^{\frac{f_1 - c(k)}{\varepsilon }} e^{-M 2^{\varepsilon R}} e^{\frac{M}{\rho }} \rho ^{-MN} \end{aligned} \end{aligned}$$

(112)

where $f\left( {{{\mathcal {B}}_1}^{\sup }} \right)$ is defined as

$$\begin{aligned} f\left( {{{\mathcal {B}}_1}^{\sup }} \right) \,\triangleq\, \mathop {\sup }\limits _{{{\mathcal {B}}_1}} f\left( \beta \right) . \end{aligned}$$

(113)

Now, we have to look at other subset ${B_2}$ of the Eq. (110) for

$$\begin{aligned} {{\mathcal {B}}_{\varepsilon _1}} \,\triangleq\, \left\{ \beta \epsilon {\mathcal {B}}_2 | \frac{\log \rho }{R} - \beta _q> \varepsilon _1, \beta _1> \varepsilon _1, |\beta _j - \beta _i|>\varepsilon _1, \forall i \ne j \right\} \end{aligned}$$

(114)

and realizing that $e^{-2^{-\varepsilon _1 R}} \ge \left( 1 - 2^{-\varepsilon _1 R} \right)$ to have the expression,

$$\begin{aligned} B_2 \ge 2 K R^q e^{\frac{M q}{\rho }} (1-2^{-\varepsilon _1 R})^{M(N+q)} \rho ^{-MN} 2^{-c(k) R} \int \limits _{{\mathcal {B}}_{\varepsilon _1}} 2^{f(\beta )R} d\beta \end{aligned}$$

(115)

For the function given in Eq. (114), there exists a supremum value, defined as,

$$\begin{aligned} f\left( {{{\mathcal {B}}_2}^{\sup }} \right) \buildrel \Delta \over = \mathop { \sup }\limits _{\beta \epsilon {{\mathcal {B}}_{{ \varepsilon _{_3}}}}} f\left( \beta \right) . \end{aligned}$$

(116)

Then for any $\varepsilon _4>0$, there exists a neighborhood $I_{\varepsilon _4}$ belongs to $\beta ^{\sup }$ such that,

$$\begin{aligned} f\left( {{\beta _1}, \ldots ,{\beta _{{M}}}} \right) \ge f\left( {{\beta ^{\sup }}} \right) - { \varepsilon _4} \end{aligned}$$

(117)

This condition let us to write

$$\begin{aligned} \begin{aligned} B_2 \ge 2 K R^q e^{\frac{M q}{\rho }} ( 1- 2^{-\varepsilon _1 R})^{M(N+q)} \rho ^{-MN} 2^{(f(\beta ^*) - c(k) -\varepsilon _2)R} \mathrm{Vol} \{ {\mathcal {B}}_{\varepsilon _1} \cap I_{\varepsilon _2} \}. \end{aligned} \end{aligned}$$

(118)

From Eqs. (111) and (118) and the fact that $e^{-\frac{M(q-1)}{\rho }} \le 1$ we will have

$$\begin{aligned} \frac{B_1}{B_2} \le \left( 1-(2^{\varepsilon R})^{-\frac{\varepsilon _1}{\varepsilon }}\right) ^{-M(N+q)} (2^{\varepsilon R})^{\frac{f_1 - f (\beta ^*)+\varepsilon _2}{\varepsilon }} e^{-M(2^\varepsilon R)} \mathrm{Vol}^{-1} \lbrace {\mathcal {B}}_{\varepsilon _1} \cap I_{\varepsilon _2} \rbrace . \end{aligned}$$

(119)

This means that

$$\begin{aligned} \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \quad \frac{{\log \left( {1 + {B_1}/{B_2}} \right) }}{{\log \rho }} = 0 \end{aligned}$$

(120)

because the value $2^{\varepsilon R}$ decays exponentially in right-hand side while it grows polynomially with the same variable in Eq. (119). For the second term in Eq. (110), we note that

$$\begin{aligned} \begin{aligned} B_2 \le 2 K R^{q} 2^{f(\beta ^{\sup }) - c(k))R} \rho ^{-MN} \end{aligned} \end{aligned}$$

(121)

is the supremum of $\frac{\log B_2}{\log \rho }$ in the region ${\mathcal {B}}_{\varepsilon _3}$. We will have

$$\begin{aligned} \begin{aligned} \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }} \le \frac{{2K{R^{{M}}}}}{{\log \rho }} - \left( {{M}{N} - {M} + 1} \right) + \left( {f\left( {{\beta ^{\sup }}} \right) - c\left( k \right) } \right) \frac{R}{{\log \rho }} \end{aligned} \end{aligned}$$

(122)

where $f(\beta ^{\sup })$ redefined as

$$\begin{aligned} f(\beta ^{\sup }) \,\triangleq\, \sup \limits _{{\mathcal {B}}_2}^{} f(\beta ). \end{aligned}$$

(123)

To obtain supremum value for Eq. (105) we have to let $f(\beta ^{\sup })$ in three operating regions as mentioned earlier, thus we get

$$\begin{aligned} &\mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \frac{{\log {P_{otg}} - c\left( k \right) R}}{{\log \rho }} \ge - \left( MN - k(k+1) \right) \\ &\quad +\,\left( \left( M+N-(2k+1)\right) \left( \frac{M+1}{2} \right) - c\left( k \right) \right) \mathop {\lim \sup }\limits _{\rho \rightarrow \infty } \frac{R}{{\log \rho }}. \end{aligned}$$

(124)

The Eqs. (104) and (124) are TRT expressions that belong to the operating region given in Eq. (23) for the D-BLAST transmission employing SIC at the receiver and completes the proof.