### SINR for the MRC-OFDM demodulator

Substituting Eq. (7) into Eq. (9), the output signal vector of the pulse blanking can be represented as

$$ {\mathbf{y}^{v}} = {\bar{\mathbf{B}}^{v}}(\mathbf{s} \otimes {\mathbf{h}^{v}}) + {\bar{\mathbf{B}}^{v}}{\mathbf{i}^{v}} + {\bar{\mathbf{B}}^{v}}{\mathbf{n}^{v}},v = 1,2,\ldots,N $$

(14)

where \({\bar {\mathbf {B}}^{v}}{\mathbf {i}^{v}} = {\text {diag}}\left ({\bar {b}_{0}^{v} {i_{0}^{v}},\ldots,\bar {b}_{n}^{v} {i_{n}^{v}},\ldots,\bar {b}_{K - 1}^{v} i_{K - 1}^{v}} \right)\) with the *n*th component represented as

$$ \begin{aligned} \bar{b}_{n}^{v} {i_{n}^{v}} = \left({1 - {b_{n}^{v}}} \right) {b_{n}^{v}} {g_{n}^{v}}, \quad n &= 0,1,\ldots,K - 1,\\ v &= 1,2,\ldots,N \end{aligned} $$

(15)

As \({{b_{n}^{v}}}\) is a Bernoulli RV that takes a value of one or zero, the product \(\left ({1 - {b_{n}}} \right) {b_{n}}\) identically equals to 0, thus \(\bar {b}_{n}^{v} {i_{n}^{v}} = 0\), and then \(\bar {\mathbf {B}} \mathbf {i} = {\mathbf {0}}\). Given all the above, Eq. (14) is further simplified as

$$ {\mathbf{y}^{v}} = \mathbf{s} \otimes {\mathbf{h}^{v}} + {\tilde{\mathbf{i}}^{v}} + {\tilde{\mathbf{n}}^{v}},v = 1,2\ldots.~,N $$

(16)

where \({\tilde {\mathbf {i}}^{v}} = \left ({{{\bar {\mathbf {B}}}^{v}} - \mathbf {I}} \right) \left ({\mathbf {s} \otimes {\mathbf {h}^{v}}} \right)\) denotes the equivalent impulse noise vector with the *n*th component represented as

$$ \tilde{i}_{n}^{v} = - {b_{n}^{v}}\sum\limits_{m = 0}^{K - 1} {{s_{m}} h_{\left({n - m} \right)~\text{mod}~K}^{v}},v = 1,2,\ldots.~,\ $$

(17)

and \({\tilde {\mathbf {n}}^{v}} = {\bar {\mathbf {B}}^{v}} {\mathbf {n}^{v}}\) denotes the complex Gaussian white noise vector in the output of pulse blanking with the *n*th component \(\tilde {n}_{n}^{v}\) represented as

$$ \tilde{n}_{n}^{v} = \left({1 - {b_{n}^{v}}} \right){n_{n}^{v}},v = 1,2,\ldots.~,N $$

(18)

Substituting Eq. (16) into Eq. (10) yields

$$ {\mathbf{Y}^{v}} = \mathbf{S} {\mathbf{H}^{v}} + \mathbf{F} {\tilde{\mathbf{i}}^{v}} + \mathbf{F} {\tilde{\mathbf{n}}^{v}},v = 1,2,\ldots.~,N $$

(19)

Expanding Eq. (19), the *k*th component of **Y**
^{v} can be expressed as

$$ \begin{aligned} {Y_{k}^{v}} &= {S_{k}} {H_{k}^{v}} + \frac{1}{{\sqrt K }}\sum\limits_{n = 0}^{K - 1} {\tilde{i}_{n}^{v}} {e^{- 2\pi j\frac{{k n}}{K}}}\\ &\quad + \frac{1}{{\sqrt K }}\sum\limits_{n = 0}^{K - 1} {\tilde{n}_{n}^{v}} {e^{- 2\pi j\frac{{k \cdot n}}{K}}} \end{aligned} $$

(20)

Substituting Eq. (20) into Eq. (13), the *k*th component of **Y**
_{MRC} can be expressed as

$$ \begin{aligned} {Y_{\text{MRC},k}} =& {S_{k}}\sum\limits_{v = 1}^{N} {{{\left| {{H_{k}^{v}}} \right|}^{2}}}\\ &+ \frac{1}{{\sqrt K }}\sum\limits_{v = 1}^{N} {{{\left({{H_{k}^{v}}} \right)}^ * }\sum\limits_{n = 0}^{K - 1} {\tilde {i_{n}^{v}}} {e^{- 2\pi j\frac{{k n}}{K}}}}\\ &+ \frac{1}{{\sqrt K }}\sum\limits_{v = 1}^{N} {{{\left({{H_{k}^{v}}} \right)}^ *} \sum\limits_{n = 0}^{K - 1} {\tilde {n_{n}^{v}}} {e^{- 2\pi j\frac{{k n}}{K}}}} \end{aligned} $$

(21)

From Eq. (21), *Y*
_{MRC,k
} can be split into two parts, the first containing the desired signal of the *k*th subchannel, denoted by *E*
_{
k
}, and the second containing the noise and the ICI caused by pulse blanking, denoted by *W*
_{
k
}. *Y*
_{MRC,k
} can be further expressed as

$$ {Y_{\text{MRC},k}} = {E_{k}} + {W_{k}} $$

(22)

where

$$ {E_{k}} = {S_{k}}\sum\limits_{v=1}^{N} {{{\left| {{H_{k}^{v}}} \right|}^{2}}} $$

(23)

and

$$ \begin{aligned} {W_{k}} =& \frac{1}{{\sqrt K }}\sum\limits_{v = 1}^{N} {{{\left({{H_{k}^{v}}} \right)}^ * }\sum\limits_{n = 0}^{K - 1} {\tilde {i_{n}^{v}}} {e^{- 2\pi j\frac{{k n}}{K}}}} \\ &+ \frac{1}{{\sqrt K }}\sum\limits_{v = 1}^{N} {{{\left({{H_{k}^{v}}} \right)}^ * }\sum\limits_{n = 0}^{K - 1} {\tilde{n}_{n}^{v}} {e^{- 2\pi j\frac{{k n}}{K}}}} \end{aligned} $$

(24)

The variance of *E*
_{
k
} is given as

$$ {V_{\text{ar}}}\left({{E_{k}}} \right) = {\sigma_{S}^{2}}{\left({\sum\limits_{v = 1}^{N} {{{\left| {{H_{k}^{v}}} \right|}^{2}}}} \right)^{2}} $$

(25)

From the derivation in the Appendix, the variance of *W*
_{
k
} can be calculated as

$$ {V_{\text{ar}}}\left({{W_{k}}} \right) = \left[ {p{\sigma_{S}^{2}} + \left({1 - p} \right){\delta_{n}^{2}}} \right]\sum\limits_{v = 1}^{N} {{{\left| {{H_{k}^{v}}} \right|}^{2}}} $$

(26)

Combining Eqs. (25) and (26), the instantaneous SINR of the *k*th subchannel at the output of MRC is given as

$${} \begin{aligned} {\gamma_{k}} &\buildrel \Delta \over = \frac{{{V_{\text{ar}}}\left({{E_{k}}} \right)}}{{{V_{\text{ar}}}\left({{W_{k}}} \right)}}\\ &= \sum\limits_{v = 1}^{N} {\frac{{{\sigma_{S}^{2}}}}{{p{\sigma_{S}^{2}} + \left({1 - p} \right){\delta_{n}^{2}}}}{{\left| {{H_{k}^{v}}} \right|}^{2}}},\quad k=0,1,\ldots,K-1 \end{aligned} $$

(27)

Let \(\rho \buildrel \Delta \over = {{{\delta _{S}^{2}}} \left /{{\delta _{n}^{2}}}\right.}\) denote the average input SNR per antenna (channel), Eq. (27) becomes

$$ \begin{aligned} {\gamma_{k}} &= \sum\limits_{v = 1}^{N} {\frac{\rho }{{p \rho + \left({1 - p} \right)}}{{\left| {{H_{k}^{v}}} \right|}^{2}}} \\ &= \sum\limits_{v = 1}^{N} {{\gamma_{k}^{v}}},\quad k=0,1,\ldots,K-1 \end{aligned} $$

(28)

where \({\gamma _{k}^{v}} \buildrel \Delta \over = \frac {\rho }{{p \rho + \left ({1 - p} \right)}} {\left | {{H_{k}^{v}}} \right |^{2}}\) denotes the instantaneous SINR for the *k*th subchannel of the *v*th channel at the output of MRC. We assume the independence between different diversity reception branches, then \({{H_{k}^{v}}}\) is independent of \({{H_{k}^{j}}}\) if *j*≠*v*, and therefore \({\gamma _{k}^{v}}\) is independent of \({\gamma _{k}^{j}}\) if *j*≠*v*.

### SER of the MRC-OFDM receiver with pulse blanking over Rayleigh fading channel

For the Rayleigh fading channel, the frequency response of the *k*th subchannel of the *v*th receive branch \({H_{k}^{v}} = \sum \limits _{l = 0}^{{L_{v}} - 1} {h_{_{l}}^{v}{e^{- j2\pi \frac {{kl}}{K}}}} \) is a complex Gaussian RV of mean zeros and variance one, thus \(\left |{H_{k}^{v}}\right |^{2}\) is *χ*
^{2} distributed with 2 degrees of freedom and \(E\left \{ {|{H_{k}^{v}}{|^{2}}} \right \} = 1\).

Considering \({\rho \left /{\left ({p\rho + \left ({1 - p} \right)} \right)}\right.}\) is a constant when the probability of impulsive noise occurrence *p* is given, \({\gamma _{k}^{v}}\) is also *χ*
^{2} distributed with 2 degrees of freedom and mean \(\bar \gamma = {\rho \left /{\left ({p\rho + \left ({1 - p} \right)} \right)}\right.}\). Therefore, the probability density function (PDF) of \({\gamma _{k}^{v}}\) is given by [20]

$${} {{\begin{aligned} p\left({{\gamma_{k}^{v}}} \right) = \frac{1}{{\bar{\gamma} }}{e^{\frac{{ - {\gamma_{k}^{v}}}}{{\bar{\gamma} }}}},{\gamma_{k}^{v}} \ge 0,k = 0,1,\ldots,K - 1,v = 1,2,\ldots,N \end{aligned}}} $$

(29)

The moment generating function (MGF) of \({\gamma _{k}^{v}}\) can be obtained by applying the Laplace transform to \(p\left ({{\gamma _{k}^{v}}} \right)\) with the exponent reversed in sign and is given by [21]

$$ {M_{{\gamma_{k}^{v}}}}\left(s \right) = \frac{1}{{1 - s\bar{\gamma} }},k = 0,1,\ldots,K - 1,v = 1,2,\ldots,N $$

(30)

Note that the MGF of the sum of independent RVs is the product of the MGFs of individual RVs [21]. Using Eqs. (28) and (30), the MGF of the combined SINR with MRC of the *k*th subchannel *γ*
_{
k
} becomes

$$ \begin{aligned} {M_{{\gamma_{k}}}}\left(s \right) &= \prod\limits_{v = 1}^{N} {{M_{{\gamma_{k}^{v}}}}\left(s \right)} \\ &= \frac{1}{{{{\left({1 - s \bar{\gamma}} \right)}^{N}}}},\quad k = 0,\ldots,K - 1 \end{aligned} $$

(31)

Using the result in [21], the SER of the *k*th subchannel for the OFDM receiver with *M*-phase-shift keying (PSK) modulation over Rayleigh fading channel can be expressed as

$$ \text{SER}_{\text{MPSK},k}^{\text{Rayleigh}} = \frac{1}{\pi }\int_{0}^{{{\left({M - 1} \right)\pi} \left/M\right.}} {{M_{{\gamma_{k}}}}\left({\frac{{ - {g_{\text{psk}}}}}{{{{\sin }^{2}}\theta }}} \right)} {\mathrm{d}}\theta $$

(32)

where \({g_{\text {psk}}} = {\sin ^{2}}\left ({{{\mathrm {\pi }} \left /M\right.}} \right)\). For large \(\bar {\gamma } \), the average SER of Eq. (32) behaves as [16]

$$ \text{SER}_{\text{MPSK},k}^{\text{Rayleigh}} \simeq \frac{{{C_{\text{PSK}}}\left({N,M} \right)}}{{{{\bar{\gamma} }^{N}}}} $$

(33)

where

$$ \begin{aligned} {C_{\text{PSK}}}\left({N,M} \right) &= \frac{1}{{{{\left[ {2\sin \left({{\pi \left/M\right.}} \right)} \right]}^{2N}}}}\left[ {\left({ \begin{array}{c} {2N}\\ N \end{array}} \right)\frac{{M - 1}}{M}} \right.\\ &\quad\left. { - \sum\limits_{j = 1}^{N} {\left({ \begin{array}{c} {2N}\\ {N - j} \end{array}} \right){{\left({ - 1} \right)}^{j}}\frac{{\sin \left({{{2\pi j} \left/M\right.}} \right)}}{{\pi j}}}} \right] \end{aligned} $$

(34)

The SER of the *k*th subchannel for the OFDM receiver with *M*-quadrature amplitude modulation (QAM) over Rayleigh fading channel can be expressed as

$${} \begin{aligned} \text{SER}_{\text{MQAM},k}^{{\text{Rayleigh}}} &= \frac{4}{\pi }\left({1 - \frac{1}{{\sqrt M }}} \right)\int_{0}^{{\pi \left/2\right.}} {{M_{{\gamma_{k}}}}\left({ - \frac{{{g_{\text{QAM}}}}}{{{{\sin }^{2}}\theta }}} \right)} {\mathrm{d}}\theta \\ &\quad - \frac{4}{\pi }{\left({1 - \frac{1}{{\sqrt M }}} \right)^{2}}\int_{0}^{{\pi \left/4\right.}} {{M_{{\gamma_{k}}}}\left({ - \frac{{{g_{\text{QAM}}}}}{{{{\sin }^{2}}\theta }}} \right)} {\mathrm{d}}\theta \end{aligned} $$

(35)

where \({g_{\text {QAM}}} = {3 \left /{\left ({2\left ({M - 1} \right)} \right)}\right.}\). For large \(\bar \gamma \), the average SER of Eq. (35) behaves as

$$ \text{SER}_{\text{MQAM},k}^{\text{Rayleigh}} \simeq \frac{{{C_{\text{QAM}}}\left({N,M} \right)}}{{{{\bar{\gamma}}^{N}}}} $$

(36)

where

$${} \begin{aligned} {C_{\text{QAM}}}\left({N,M} \right) &= \frac{1}{{{2^{2N}}}}{\left({\frac{{\sqrt M - 1}}{{\sqrt M }}} \right)^{2}}{g_{\text{QAM}}}^{- N}\\ &\quad\times \left[\vphantom{{ - \frac{4}{\pi }\sum\limits_{j = 1}^{N} {\left({\begin{array}{c} {2N}\\ {N - j} \end{array}} \right)\frac{{{{\left({ - 1} \right)}^{{{\left({j + 1} \right)} \left/2\right.}}}}}{j}}}} {\left({\begin{array}{c} {2N}\\ N \end{array}} \right)\left({\frac{{2\sqrt M }}{{\sqrt M - 1}} - 1} \right)} \right.\\ &\quad\left. { - \frac{4}{\pi }\sum\limits_{j = 1}^{N} {\left({\begin{array}{c} {2N}\\ {N - j} \end{array}} \right)\frac{{{{\left({ - 1} \right)}^{{{\left({j + 1} \right)} \left/2\right.}}}}}{j}}} \right] \end{aligned} $$

(37)

In the SER expressions given in Eqs. (32), (33), (35), and (36), the SER of the *k*th subchannel is independent of the subchannel index, which indicates that pulse blanking has the same effect on the error performance of each subchannel of the MRC-OFDM receiver.

In particular, for the binary PSK (BPSK) modulation, the closed-form expression for the integral in Eq. (32) becomes [16]

$$\begin{array}{@{}rcl@{}} \text{SER}_{\text{BPSK},k}^{\text{Rayleigh}} &=& {\left[ {\frac{1}{2}\left({1 - \mu} \right)} \right]^{N}}\sum\limits_{j = 0}^{N - 1} {\left({ \begin{array}{*{20}{c}} {N - 1 + j}\\ j \end{array}} \right)} \\ &&\times {\left[ {\frac{1}{2}\left({1 + \mu} \right)} \right]^{j}} \end{array} $$

(38)

where \(\mu = \sqrt {{{\bar {\gamma }} \left / {\left ({1 + \bar {\gamma }} \right)}\right.}} \).

The exact expression in Eq. (38) provides insight into a few special cases. (i) When there is no interference, i.e., *p*=0, \(\bar {\gamma }= \rho \), Eq. (38) reduces to the symbol error probability of a conventional *N*th-order space diversity receiver employing MRC. (ii) When there is interference, i.e., *p*≠0, and in the limiting case the input SNR is large, i.e., \(\rho \to \infty \), \(\mathop {\lim }\limits _{\rho \to \infty }\bar \gamma ={1\left / p\right.}\), thus, there will be an error floor for the SER performance curve. Considering a small *p* value (less than 0.1) fulfilling the condition \(\bar {\gamma }\gg 1\), the approximate theoretical expression of the error floor behaves as \({\left ({\frac {p}{4}}\right)^{N}}\left ({ \begin {array}{{c}} {2N-1}\\ N \end {array}} \right)\). Therefore, the error floor decreases proportionally with the *N*th power of the probability of impulsive noise occurrence *p* at an *N*th-order space diversity. For a given *p* value, the error floor can be efficiently reduced as the number of receive antenna increases. The same conclusion can be reached based on the approximate expressions in Eqs. (33) and (36).

### SER of the MRC-OFDM receiver with pulse blanking over Ricean fading channel

For the Ricean fading channel, \({h_{0}^{v}}\) is assumed to be a non-zero mean complex Gaussian RV, i.e., \({h_{0}^{v}} \sim \mathrm {\mathcal {CN}}\left ({u_{v}},\delta _{0}^{{v}^{2}} \right)\), \(\left \{ {{h_{l}^{v}},l = 1,\ldots,{L_{v}} - 1} \right \}\) are assumed to be complex Gaussian RVs, i.e., \({h_{l}^{v}}\sim {\mathrm {\mathcal {CN}}}\left (0,\delta _{l}^{{v}^{2}} \right)\). The channel power is normalized to 1, i.e., \({\left | {{u_{v}}} \right |^{2}} + {\sum \nolimits }_{l = 0}^{L - 1} {{{\left | {{\delta _{l}^{v}}} \right |}^{2}}} = 1\), and the Ricean factor of the *v*th channel is defined as \({K_{v}} \buildrel \Delta \over = {{{{\left | {{u^{v}}} \right |}^{2}}} \left / {{\sum \nolimits }_{l = 0}^{L - 1} {{{\left | {{\delta _{l}^{v}}} \right |}^{2}}} }\right.}\). Hence, the frequency response of the *k*th subchannel \({H_{k}^{v}}\) is distributed according to complex Gaussian with mean *u*
^{v} and variance \({\sum \nolimits }_{l = 0}^{L - 1} {{{\left | {{\delta _{l}^{v}}} \right |}^{2}}} - {\left | {{u^{v}}} \right |^{2}}\). Furthermore, \({\left | {{H_{k}^{v}}} \right |^{2}}\) is noncentral *χ*
^{2} distributed with 2 degrees of freedom and \(E\left \{ {|{H_{k}^{v}}{|^{2}}} \right \} = 1\).

Considering \({\rho \left /{\left ({p\rho +\left ({1-p}\right)}\right)}\right.}\) is a constant when the probability of impulsive noise occurrence *p* is given, \({\gamma _{k}^{v}}\) is also according to noncentral *χ*
^{2} distributed with 2 degrees of freedom with mean \(\bar {\gamma } = {\rho \left / {\left ({p\rho + \left ({1 - p} \right)} \right)}\right.}\). Therefore, the PDF of \({\gamma _{k}^{v}}\) is given by [15]

$$ p\left({{\gamma_{k}^{v}}} \right) = \frac{{\left({1 + {K_{v}}} \right){e^{- \frac{{\left({1 + {K^{v}}} \right){\gamma_{k}^{v}}}}{{\bar{\gamma}}}}}}}{{\bar{\gamma}}}{I_{0}}\left({2\sqrt {\frac{{{K_{v}}\left({1 + {K_{v}}} \right){\gamma_{k}^{v}}}}{{\bar{\gamma}}}}} \right) $$

(39)

where \({I_{0}}\left (\cdot \right)\) is the zero-order modified Bessel function of the first kind. The MGF of \({\gamma _{k}^{v}}\) can be expressed as

$$ {M_{{\gamma_{k}^{v}}}}\left(s \right) = \frac{{1 + {K_{v}}}}{{1 + {K_{v}} - s\bar{\gamma}}}\exp \left({\frac{{{K_{v}}s\bar{\gamma}}}{{1 + {K_{v}} - s\bar{\gamma}}}} \right) $$

(40)

Then, the MGF of the combined SINR can be written as

$$ \begin{aligned} {M_{{\gamma_{k}}}}\left(s \right) &= \prod\limits_{v = 1}^{N} {{M_{{\gamma_{k}^{v}}}}\left(s \right)} \\ &= \prod\limits_{v = 1}^{N} {\left\{ {\frac{{1 + {K_{v}}}}{{1 + {K_{v}} - s\bar{\gamma}}}\exp \left({\frac{{{K_{v}}s\bar{\gamma} }}{{1 + {K_{v}} - s\bar{\gamma}}}} \right)} \right\}} \end{aligned} $$

(41)

Using the result in [21], the SER of the *k*th subchannel for the OFDM receiver with *M*-PSK modulation over Ricean fading channel can be expressed as

$$ \text{SER}_{\text{MPSK},k}^{\text{Ricean}}{=}\frac{1}{\pi }\int_{0}^{\left({M - 1} \right)\pi /M} {{M_{\gamma_{k}^{}}}\left({ - \frac{{{g_{\text{psk}}}}}{{{{\sin }^{2}}\theta }}} \right)} {\mathrm{d}}\theta $$

(42)

For large \(\bar {\gamma }\), the average SER of Eq. (42) behaves as

$$ \begin{aligned} \text{SER}_{\text{MPSK},k}^{\text{Ricean}} \simeq& \frac{{\exp \left({ - {\sum\nolimits}_{v = 1}^{N} {{K_{v}}}} \right){C_{\text{PSK}}}\left({N,M} \right)}}{{{{\bar \gamma }^{N}}}}\\ &\times \prod\limits_{v = 1}^{N} {\left({1 + {K_{v}}} \right)} \end{aligned} $$

(43)

The SER of the *k*th subchannel for the OFDM receiver with *M*-QAM modulation over Ricean fading channel can be expressed as

$${} \begin{aligned} \text{SER}_{\text{MQAM},k}^{\text{Ricean}} &= \frac{{4\left({\sqrt M - 1} \right)}}{{\pi \sqrt M }}\int_{0}^{\pi /2} {{M_{{\gamma_{k}}}}\left({ - \frac{{{g_{\text{QAM}}}}}{{{{\sin }^{2}}\theta }}} \right)} {\mathrm{d}}\theta \\ &\quad- \frac{{4{{\left({\sqrt M - 1} \right)}^{2}}}}{{\pi M}}\int_{0}^{\pi /4} {{M_{{\gamma_{k}}}}\left({ - \frac{{{g_{\text{QAM}}}}}{{{{\sin }^{2}}\theta }}} \right)} {\mathrm{d}}\theta \end{aligned} $$

(44)

For large \(\bar {\gamma }\), the average SER of Eq. (44) behaves as [21]

$$\begin{array}{@{}rcl@{}} \text{SER}_{\text{MQAM},k}^{\text{Ricean}} &\simeq& \frac{{\exp \left({ - {\sum\nolimits}_{v = 1}^{N} {{K_{v}}}} \right){C_{\text{QAM}}}\left({N,M} \right)}}{{{{\bar{\gamma}}^{N}}}}\\ &&\times \prod\limits_{v = 1}^{N} {\left({1 + {K_{v}}} \right)} \end{array} $$

(45)

Also, the SER derived above is only for the *k*th subcarrier. It has to be averaged over all the subcarriers. Comparing Eq. (44) with Eq. (33), Eq. (45), and Eq. (36), we find that for the specific Ricean factors, the same conclusion presented in Section 3.2 can be reached based on the approximate expressions in Eqs. (45) and (46).