Bit error probability analysis of IR-UWB ED-OOK system using cooperative dual-hop DTF strategy

Abstract

The Bit Error Rate (BER) of a single user cooperative Impulse Radio (IR)-Ultrawideband (UWB) communication system employing Energy Detector (ED) receiver with On-Off Keying (OOK) in IEEE 802.15.4a UWB multipath environment is investigated in this paper. Analytical evaluation based on Energy detection principle is performed to derive approximate BER Expressions for various diversity combining cases, namely optimum combining, linear combining and selective combining using cooperative dual-hop Detect and Forward (DTF) relay protocol in presence and absence of Inter-symbol interference (ISI). Numerical results reveal that there is a significant improvement in BER, with increase in number of relay diversity paths and decrease in number of frames N_f. The accuracy and perfection in approximation used in investigation of BER is confirmed with the validation of the analytical BER expressions with that of the simulation results. The analytical and simulation results confirm that among the diversity combining schemes, the performance of optimum combining is better compared with linear combining, which in turn performs better than selective combining.

Appendices

Appendix A: Determination of variance of ${n_{k}^{2}}(t)$

Let u = n_SD(t),n_SR(t),n_RD(t) and v = u². Therefore, the variance of v can be simplified as:

$$ \begin{array}{@{}rcl@{}} {{\sigma}_{v}^{2}} &=& \mathbb{E}[u^{2}]-\left\{\mathbb{E}[u]\right\}^{2} = \mathbb{E}[u^{4}]-\left\{\mathbb{E}[u^{2}]\right\}^{2} \\ &=& 3\left\{{{\sigma}_{u}^{2}}\right\}^{2}-\left\{{{\sigma}_{u}^{2}}\right\}^{2} = 2\left\{{{\sigma}_{u}^{2}}\right\}^{2} = \frac{{N_{0}^{2}}}{2} \end{array} $$

(63)

where $\mathbb {E}[u]=0$ and ${{\sigma }_{u}^{2}}=\frac {N_{0}}{2}$. The term $k \in \left \{1,2,3\right \}$ represents S-D, S-R and R-D link respectively as mentioned already.

Appendix B: Determination of optimum combining factor ζ

• i. Correct Detection: In order to evaluate the value of ζ, we differentiate the SNR expression given by Eq. (54) w.r.t. ζ and equate the result to 0.

  $$ \begin{array}{@{}rcl@{}} \frac{d{\gamma}_{LOC}}{d\zeta} &=& \frac{(A-B)}{C} = 0, \end{array} $$

  (64)

  where $A = 2(s_{SD}+\zeta s_{RD})s_{RD}({\sigma }_{Z_{\text {noise}-SD}}^{2}+\zeta ^{2}{\sigma }_{Z_{\text {noise}-RD}}^{2})$, $B=(s_{SD}+\zeta s_{RD})^{2}2\zeta ({\sigma }_{Z_{\text {noise}-RD}}^{2})$ and $C=({\sigma }_{Z_{\text {noise}-SD}}^{2}+\zeta ^{2}{\sigma }_{Z_{\text {noise}-RD}}^{2})^{2}$.

  $$ \begin{array}{@{}rcl@{}} \frac{d{\gamma}_{LOC}}{d\zeta} &=& 2[(s_{SD}+\zeta s_{RD})s_{RD}({\sigma}_{Z_{\text{noise}-SD}}^{2}\\&&+\zeta^{2}{\sigma}_{Z_{\text{noise}-RD}}^{2})]-[(s_{SD}+\zeta s_{RD})^{2}\\&&\times 2\zeta{\sigma}_{Z_{\text{noise}-RD}}^{2}] = 0. \end{array} $$

  (65)

  Using quadratic rule, we find the roots for ζ.

  $$ \begin{array}{@{}rcl@{}} \zeta &=& \frac{-4({\sigma}_{Z_{\text{noise}-SD}}^{2})s_{RD}^{2}}{-4s_{SD}s_{RD}{\sigma}_{Z_{\text{noise}-RD}}^{2}} = \frac{({\sigma}_{Z_{\text{noise}-SD}}^{2})s_{RD}}{({\sigma}_{Z_{\text{noise}-RD}}^{2})s_{SD}}. \end{array} $$

  (66)

• ii. Incorrect Detection: In order to evaluate the value of ζ, we differentiate the SNR expression given by Eq. (56) w.r.t. ζ and equate the result to 0. Following the same procedure as above, we obtain the value of ζ.

Appendix C: Determination of optimum combining factor ζ in presence of ISI

• i. Correct Detection: To solve the value of ζ, we differentiate the SNR expression given by Eq. (53) w.r.t. ζ and equate the result to 0.

  $$ \begin{array}{@{}rcl@{}} \frac{d{\gamma}_{LOC-ISI}}{d\zeta} &=& \frac{(A_{1}-B_{1})}{C_{1}} = 0, \end{array} $$

  (67)

  where $A_{1}=2(s_{SD}+\zeta s_{RD})s_{RD}({\sigma }^{2}_{Z_{\text {noise}-SD}}+\zeta ^{2}{\sigma }^{2}_{Z_{\text {noise}-RD}}+I_{SD}+\zeta I_{RD})$, $B_{1}=(s_{SD}+\zeta s_{RD})^{2}2\zeta ({\sigma }^{2}_{Z_{\text {noise}-RD}}+I_{RD})$ and $C_{1}=({\sigma }^{2}_{Z_{\text {noise}-SD}}+\zeta ^{2}{\sigma }^{2}_{Z_{\text {noise}-RD}}+I_{SD}+\zeta I_{RD})^{2}$.

  $$ \begin{array}{@{}rcl@{}} \frac{d{\gamma}_{LOC-ISI}}{d\zeta} &=& 2[(s_{SD}+\zeta s_{RD})s_{RD}({\sigma}^{2}_{Z_{\text{noise}-SD}}\\&&+\zeta^{2}{\sigma}^{2}_{Z_{\text{noise}-RD}}+I_{SD}+\zeta I_{RD})]\\&&-[(s_{SD}+\zeta s_{RD})^{2}(2\zeta{\sigma}^{2}_{Z_{\text{noise}-RD}}+I_{RD})] \\&& = 0. \end{array} $$

  (68)

  Using quadratic rule, we find the roots for ζ.

  $$ \begin{array}{@{}rcl@{}} \zeta &=& \frac{-2b_{1}\pm 2\sqrt{{b_{1}^{2}}-a_{1}c_{1}}}{2a_{1}}, \end{array} $$

  (69)

  where $a_{1}=s_{RD}^{2}I_{RD}-2s_{SD}s_{RD}{\sigma }^{2}_{Z_{\text {noise}-RD}}$, $b_{1}=s_{RD}^{2}({\sigma }^{2}_{Z_{\text {noise}-SD}}+I_{SD})-s_{SD}^{2}{\sigma }^{2}_{Z_{\text {noise}-RD}}$ and $c_{1}=2s_{SD}s_{RD}({\sigma }^{2}_{Z_{\text {noise}-SD}}+I_{SD})-s_{SD}^{2}I_{RD}$.

• ii. Incorrect Detection: In order to evaluate the value of ζ, we differentiate the SNR expression given by Eq. (57) w.r.t. ζ and equate the result to 0. Following the same procedure as above, we obtain the value of ζ.