A New Method of Generating Spectral Nulls at the Transmitter in Cognitive Radio

Abstract

In cognitive radio (CR), cognitive users can sense the wholes and white spectrum and generate spectrum notch in the spectrum bands occupied by primary users (PUs) or interference. Thus, the key technology in CR is to control the spectrum shape of the transmitted signal to avoid PUs and interference. In this paper, a new method of shaping the transmitted signal spectrum envelope by spectrum-spread technology is proposed. The proposed method can generate spectral nulls at the band of PUs or interference in the CR environment. Compared to the existing methods generating spectrum nulls, the proposed method can effectively generate spectral nulls to avoid interference or PUs only by designing the pseudo-random code waveform (PCW) based on direct sequence spread spectrum technology. The condition of electromagnetic spectrum occupation is detected by CR technology so as to construct an ideal spectrum template. Based on the spectrum template, we study the design of the baseband waveform. The bit error rate (BER) performance of the proposed method in different sorts of interferences, and the relation between the BER and the spectrum overlap degree (SOD) are derived, of which the concept of SOD is proposed. The expression between BER and SOD shows that BER is proportional to SOD, which shows the criterion to design the PCW. The signal spectrograms in the receiver in presence of tone jamming and BPSK jamming indicate that the proposed scheme can effectively generate spectrum nulls in the frequency band occupied by PUs or interference. Furthermore, the BER versus SNR and BER versus SIR simulation results both in presence of tone jamming and BPSK jamming show that the proposed method has a significantly improved the BER performance by generating spectral nulls to avoid PUs or interferences. Simulation results are carried out to corroborate our theoretical analysis.

Appendices

Appendix 1

$$S\left( t \right){ = }a_{i} \int_{0}^{T} {PN^{2} \left( t \right)dt} + a_{i} \int_{0}^{T} {PN^{2} \left( t \right)\cos 2\omega_{0} tdt}$$

(24)

where

$$\begin{aligned}& \int_{0}^{T} {PN^{2} \left( t \right)\cos 2\omega_{0} tdt} = \int_{ - \infty }^{\infty } {PN^{2} \left( t \right)\cos 2\omega_{0} tdt} \hfill \\ &= \int_{ - \infty }^{\infty } {\left[ {\frac{1}{2\pi }\int_{ - \infty }^{\infty } {\left( {\frac{1}{2\pi }PN\left( {j\omega } \right) \times PN\left( {j\omega } \right)} \right)e^{j\omega t} d\omega } } \right]\cos 2\omega_{0} tdt} \hfill \\& = \frac{1}{4\pi }\int_{ - \infty }^{\infty } {\left( {PN\left( {j\omega } \right) \times PN\left( {j\omega } \right)} \right)d\omega } \cdot \frac{1}{2\pi }\int_{ - \infty }^{\infty } {\left( {e^{{j\left( {\omega + 2\omega_{0} } \right)t}} + e^{{j\left( {\omega - 2\omega_{0} } \right)t}} } \right)d\omega } \hfill \\ &= \frac{1}{4\pi }\int_{ - \infty }^{\infty } {\left( {PN\left( {j\omega } \right) \times PN\left( {j\omega } \right)} \right)\left[ {\delta \left( {\omega + 2\omega_{0} } \right) + \delta \left( {\omega - 2\omega_{0} } \right)} \right]d\omega } \hfill \\ &= \frac{1}{4\pi }\left[ {\left. {PN\left( {j\omega } \right) \times PN\left( {j\omega } \right)} \right|_{{\omega = 2\omega_{0} }} + \left. {PN\left( {j\omega } \right) \times PN\left( {j\omega } \right)} \right|_{{\omega = - 2\omega_{0} }} } \right] \hfill \\ &\approx 0 \hfill \\ \end{aligned}$$

(25)

Therefore we can obtain that

$$S\left( t \right) = a_{i} \int_{0}^{T} {PN^{2} \left( t \right)dt}$$

(26)

Appendix 2

$$E\left[ {N\left( t \right)} \right] = 2\int_{0}^{T} {E\left[ {n\left( t \right)} \right]} E\left[ {PN\left( t \right)\cos \omega_{0} t} \right]dt = 0$$

(27)

$$D\left[ {N\left( t \right)} \right] = E\left[ {N^{2} \left( t \right)} \right]{ - }E^{2} \left[ {N\left( t \right)} \right]{ = }E\left[ {N^{2} \left( t \right)} \right]$$

(28)

$$\begin{aligned} N^{2} \left( t \right) = 4\int_{0}^{T} {n\left( t \right)} PN\left( t \right)\cos \omega_{0} tdt\int_{0}^{\tau } {n\left( \tau \right)} PN\left( \tau \right)\cos \omega_{0} \tau d\tau \hfill \\ = 4\int_{0}^{T} {\int_{0}^{\tau } {n\left( t \right)n\left( \tau \right)} PN\left( t \right)PN\left( \tau \right)\cos \omega_{0} t\cos \omega_{0} \tau dtd\tau } \hfill \\ \end{aligned}$$

(29)

$$\begin{aligned}& E\left[ {N^{2} \left( t \right)} \right] = \hfill \\ &4\int_{0}^{T} {\int_{0}^{\tau } {E\left[ {n\left( t \right)n\left( \tau \right)} \right]} PN\left( t \right)PN\left( \tau \right)\cos \omega_{0} t\cos \omega_{0} \tau dtd\tau } \hfill \\ &= 4\int_{0}^{T} {\int_{0}^{\tau } {\frac{{n_{0} }}{2}\delta \left( {t - \tau } \right)} PN\left( t \right)PN\left( \tau \right)\cos \omega_{0} t\cos \omega_{0} \tau dtd\tau } \hfill \\& = 2n_{0} \int_{0}^{T} {PN^{2} \left( t \right)\cos^{2} \omega_{0} tdt} \hfill \\ &= n_{0} \int_{0}^{T} {PN^{2} \left( t \right)dt} + n_{0} \int_{0}^{T} {PN^{2} \left( t \right)\cos 2\omega_{0} tdt} \hfill \\ &= n_{0} \int_{0}^{T} {PN^{2} \left( t \right)dt} \hfill \\ \end{aligned}$$

(30)

Appendix 3

When there is interference in the communication system, we can obtain thatr(t) ∼ N(I(t) − AE _PN , n ₀ E _PN ) for the case where 0 is sent and r(t) ∼ N(I(t) + AE _PN , n ₀ E _PN ) for the case where 1 is sent. We can obtain that

$$\begin{aligned} P_{e} = \frac{1}{2}\left( {\phi \left( {\frac{I\left( t \right)}{{\sqrt {n_{0} E_{PN} } }} - \sqrt {\frac{{A^{2} E_{PN} }}{{n_{0} }}} } \right) + \phi \left( { - \frac{I\left( t \right)}{{\sqrt {n_{0} E_{PN} } }} - \sqrt {\frac{{A^{2} E_{PN} }}{{n_{0} }}} } \right)} \right) \hfill \\ = \frac{1}{2}\left( {\phi \left( {\frac{I\left( t \right)}{{\sqrt {2n_{0} E_{b} } }} - \sqrt {\frac{{2E_{b} }}{{n_{0} }}} } \right) + \phi \left( { - \frac{I\left( t \right)}{{\sqrt {2n_{0} E_{b} } }} - \sqrt {\frac{{2E_{b} }}{{n_{0} }}} } \right)} \right) \hfill \\ = \frac{1}{2}\left( {\phi \left( {\sqrt {\frac{{2E_{b} }}{{n_{0} }}} \left( {\frac{I\left( t \right)}{{2E_{b} }} - 1} \right)} \right) + \phi \left( {\sqrt {\frac{{2E_{b} }}{{n_{0} }}} \left( { - \frac{I\left( t \right)}{{2E_{b} }} - 1} \right)} \right)} \right) \hfill \\ \end{aligned}$$

(31)

Appendix 4

$$\begin{aligned} \frac{I\left( t \right)}{\alpha } = 2\int_{ - \infty }^{\infty } {PN\left( t \right)\cos \left( {\omega_{I} t + \theta } \right)\cos \omega_{0} tdt} \hfill \\ = \int_{ - \infty }^{\infty } {PN\left( t \right)\cos \left( {\left( {\omega_{I} - \omega_{0} } \right)t + \theta } \right)dt} + \int_{ - \infty }^{\infty } {PN\left( t \right)\cos \left( {\left( {\omega_{I} + \omega_{0} } \right)t + \theta } \right)dt} \hfill \\ \end{aligned}$$

(32)

Let Δω = ω _I  − ω ₀, then

$$\begin{aligned} &\int_{ - \infty }^{\infty } {PN\left( t \right)\cos \left( {\left( {\omega_{I} - \omega_{0} } \right)t + \theta } \right)dt} \hfill \\ &= \int_{ - \infty }^{\infty } {PN\left( t \right)\cos \left( {\varDelta \omega t + \theta } \right)dt} \hfill \\ &= \frac{1}{2}\int_{ - \infty }^{\infty } {\left[ {\frac{1}{2\pi }\int_{ - \infty }^{\infty } {PN\left( {j\omega } \right)e^{j\omega t} d\omega } } \right]\left( {e^{j\theta } e^{j\varDelta \omega t} + e^{ - j\theta } e^{ - j\varDelta \omega t} } \right)dt} \hfill \\ &= \frac{1}{2}\int_{ - \infty }^{\infty } {PN\left( {j\omega } \right)d\omega } \cdot \frac{1}{2\pi }\int_{ - \infty }^{\infty } {\left( {e^{j\theta } e^{{j\left( {\omega + \varDelta \omega } \right)t}} + e^{ - j\theta } e^{{j\left( {\omega - \varDelta \omega } \right)t}} } \right)dt} \hfill \\ &= \frac{1}{2}\int_{ - \infty }^{\infty } {PN\left( {j\omega } \right)\left[ {e^{j\theta } \delta \left( {\omega + \varDelta \omega } \right) + e^{ - j\theta } \delta \left( {\omega - \varDelta \omega } \right)} \right]d\omega } \hfill \\ &= \frac{1}{2}\left[ {e^{ - j\theta } PN\left( {j\varDelta \omega } \right) + e^{j\theta } PN\left( { - j\varDelta \omega } \right)} \right] \hfill \\ &= {\text{Real}}\left\{ {e^{ - j\theta } PN\left( {j\varDelta \omega } \right)} \right\} \hfill \\ \end{aligned}$$

(33)

Similarly we can obtain that

$$\int_{ - \infty }^{\infty } {PN\left( t \right)\cos \left( {\left( {\omega_{I} { + }\omega_{0} } \right)t + \theta } \right)dt} = \text{Re} al\left\{ {e^{ - j\theta } PN\left( {j\left( {\omega_{I} { + }\omega_{0} } \right)} \right)} \right\} \approx 0$$

(34)

Therefore, we can obtain that

$$I\left( t \right) = \alpha \cdot Real\left\{ {e^{ - j\theta } PN\left( {j\varDelta \omega } \right)} \right\}$$

(35)

Appendix 5

$$\begin{aligned} I\left( t \right) = 2\int_{ - \infty }^{\infty } {PN\left( t \right)i\left( t \right)\cos \omega_{0} tdt} \hfill \\ = \int_{ - \infty }^{\infty } {\left( {\frac{1}{2\pi }\int_{ - \infty }^{\infty } {PN\left( {j\omega } \right)e^{j\omega t} d\omega_{1} } } \right)\left( {\frac{1}{2\pi }\int_{ - \infty }^{\infty } {I\left( {j\omega_{I} } \right)e^{{j\omega_{I} t}} d\omega_{I} } } \right)\left( {e^{{j\omega_{0} t}} + e^{{ - j\omega_{0} t}} } \right)dt} \hfill \\ = \frac{1}{2\pi }\int_{ - \infty }^{\infty } {\int_{ - \infty }^{\infty } {PN\left( {j\omega } \right)} I\left( {j\omega_{I} } \right)d\omega d\omega_{I} } \cdot \frac{1}{2\pi }\int_{ - \infty }^{\infty } {\left( {e^{{j\left( {\omega + \omega_{I} + \omega_{0} } \right)t}} + e^{{ - j\left( {\omega + \omega_{I} - \omega_{0} } \right)t}} } \right)dt} \hfill \\ = \frac{1}{2\pi }\int_{ - \infty }^{\infty } {\int_{ - \infty }^{\infty } {PN\left( {j\omega } \right)} I\left( {j\omega_{I} } \right)d\omega d\omega_{I} } \left[ {\delta \left( {\omega + \omega_{I} + \omega_{0} } \right) + \delta \left( {\omega + \omega_{I} - \omega_{0} } \right)} \right] \hfill \\ = \frac{1}{2\pi }\int_{ - \infty }^{\infty } {PN\left( {j\omega } \right)d\omega } \int_{ - \infty }^{\infty } {I\left( {j\omega_{I} } \right)\left[ {\delta \left( {\omega + \omega_{I} + \omega_{0} } \right) + \delta \left( {\omega + \omega_{I} - \omega_{0} } \right)} \right]d\omega_{I} } \hfill \\ = \frac{1}{2\pi }\int_{ - \infty }^{\infty } {PN\left( {j\omega } \right)\left[ {I\left( { - j\left( {\omega + \omega_{0} } \right)} \right) + I\left( { - j\left( {\omega - \omega_{0} } \right)} \right)} \right]d\omega } \hfill \\ = \frac{1}{2\pi }\int_{ - \infty }^{\infty } {PN\left( {j\omega } \right)\left[ {I\left( {j\left( {\omega + \omega_{0} } \right)} \right) + I\left( {j\left( {\omega - \omega_{0} } \right)} \right)} \right]^{ * } d\omega } \hfill \\ \end{aligned}$$

(36)

where denotes complex conjugate.

Appendix 6

When we analyze I(t), it shows the correlation value of PN(t) cos ω ₀ t and i(t). For real signal i(t), when i(t) = PN(t) cos ω ₀ t, I(t) obtains the maximum value

$$I_{\hbox{max} } \left( t \right) = 2\int_{T} {PN^{2} \left( t \right)\cos^{2} \omega_{0} tdt} = \int_{T} {PN^{2} \left( t \right)dt} = E_{PN} = 2E_{b}$$

(37)

Then $\frac{I\left( t \right)}{{2E_{b} }}$ reaches the maximum value 1. When $i\left( t \right) = { - }PN\left( t \right)\cos \omega_{0} t$, I(t) obtains the minimum value and

$$I_{\hbox{min} } \left( t \right) = - 2\int_{T} {PN^{2} \left( t \right)\cos^{2} \omega_{0} tdt} = - E_{PN} = - 2E_{b}$$

(38)

Then $\frac{I\left( t \right)}{{2E_{b} }}$ attains the minimum value −1.

Appendix 7

$$\begin{aligned} P_{e} = \frac{1}{2}\left( {\phi \left( {\sqrt {\frac{{2E_{b} }}{{n_{0} }}} \left( {SOD - 1} \right)} \right) + \phi \left( {\sqrt {\frac{{2E_{b} }}{{n_{0} }}} \left( { - SOD - 1} \right)} \right)} \right) \hfill \\ { = }\frac{1}{4}erfc\left( {\sqrt {{{E_{b} } \mathord{\left/ {\vphantom {{E_{b} } {n_{0} }}} \right. \kern-0pt} {n_{0} }}} \left( {1 - SOD} \right)} \right) + \frac{1}{4}erfc\left( {\sqrt {{{E_{b} } \mathord{\left/ {\vphantom {{E_{b} } {n_{0} }}} \right. \kern-0pt} {n_{0} }}} \left( {1 + SOD} \right)} \right) \hfill \\ \end{aligned}$$

(39)

where $erfc\left( x \right) = \frac{1}{\sqrt 2 }\int_{x}^{\infty } {\exp \left( { - {\text{z}}^{2} } \right){\text{dz}}}$.

For simplify and without loss of generality, set k equal to $\sqrt {{{E_{b} } \mathord{\left/ {\vphantom {{E_{b} } {n_{0} }}} \right. \kern-0pt} {n_{0} }}}$ and (39) can be rewritten as

$$\begin{aligned} P_{e} = erfc\left( {k\left( {1 - SOD} \right)} \right) + erfc\left( {k\left( {1 + SOD} \right)} \right) \hfill \\ = 1 - erf\left( {k\left( {1 - SOD} \right)} \right) - erf\left( {k\left( {1 + SOD} \right)} \right) \hfill \\ \end{aligned}$$

(40)

Take the derivative of (40) we can achieve that

$$\begin{aligned} \frac{{{\text{d}}P_{e} }}{{{\text{d}}SOD}} = \frac{2k}{\sqrt \pi }e^{{ - k^{2} \left( {1 - SOD} \right)^{2} }} - \frac{2k}{\sqrt \pi }e^{{ - k^{2} \left( {1 + SOD} \right)^{2} }} \hfill \\ = \frac{2k}{\sqrt \pi }\left[ {e^{{ - k^{2} \left( {1 - SOD} \right)^{2} }} - e^{{ - k^{2} \left( {1 + SOD} \right)^{2} }} } \right] \hfill \\ \end{aligned}$$

(41)

(41) shows that when SOD > 0, $\frac{{{\text{d}}P_{e} }}{{{\text{d}}SOD}} > 0$ ;when SOD = 0, $\frac{{{\text{d}}P_{e} }}{{{\text{d}}SOD}} = 0$ ;when SOD ≤ 0, $\frac{{{\text{d}}P_{e} }}{{{\text{d}}SOD}} < 0$. Thus, P _e can get the minimal value when SOD = 0.P _e is proportional to SOD when SOD > 0.