Spectral efficient pulse shape design for UWB communication with reduced ringing effect and performance evaluation for IEEE 802.15.4a channel

Abstract

Ultra-Wideband (UWB) faces some regulations due to the interference issues with the co-existing narrowband communication systems, due to which the effective utilization of bandwidth is not possible that causes lower data rate of transmission. We propose a pulse shaping method which can curtail the interference with the co-existing band of communication and have an ability to depress the ringing oscillation. The modified pulse shape has an ability to control over the transmitted power spectral density and offers improved antenna power resolution. To obtain the specific results, the seventh derivative of Gaussian pulse is used as a basic pulse. This pulse is spectrally modified by windowing with Gaussian window to achieve the desired performance. The analysis is not only limited to spectral management but also inculcate the performance estimation of the resultant pulse in multi-user scenario for indoor multipath channel IEEE 802.15.4a. We have also analysed the modified pulse to prove its ability in the location accuracy improvement for high-end application of UWB communication.

Appendix

The Eq. (1-b) is representing the seventh order derivative of Gaussian pulse given by

$$y\left( t \right) = A*exp\left( { - \frac{{t^{2} }}{{2 \cdot \sigma_{1}^{2} }}} \right)$$

(34)

where A and t are the normalized amplitude and time respectively.$\sigma_{1}$ is the standard deviation or PSF. The equation is derived with the help of computational method and verified using MATLAB simulations.

Equation (3) is a basic normalized Gaussian window function used by mathematicians with A = 1 and PSF as $\sigma_{2}$.

Equations (2) and (4) have been obtained as follows,

$$f\left( t \right) = exp\left\{ { - \frac{{\pi \cdot t^{2} }}{{2 \cdot \sigma_{1}^{2} }}} \right\} = exp\left\{ { - at^{2} } \right\}$$

(35)

Where, $a = \frac{\pi }{{2 \cdot \sigma_{1}^{2} }}$. Now, after differentiating the above Eq. (8.2) in time domain, we obtain

$$\frac{d}{dt}f\left( t \right) = - 2 \cdot a.t \cdot f\left( t \right)$$

(36)

Now, consider the properties of continuous time Fourier transform, we have

$$j\omega F\left( \omega \right) = - \frac{2a}{j}\frac{d}{d\omega }F\left( \omega \right)$$

(37)

The solution of differential Eq. (8.4) for $F\left( \omega \right)$ will be

$$F\left( \omega \right) = K \cdot exp\left( { - \frac{{\omega^{2} }}{4a}} \right)$$

(38)

where $\omega$ is the angular frequency associated with frequency $\omega = 2\pi f$. The value of K is obtained as under in Eq. (8.6)

$$F\left( 0 \right) = K = \mathop \smallint \limits_{ - \infty }^{\infty } exp\left( { - at^{2} } \right)dt = 2\mathop \smallint \limits_{0}^{\infty } exp\left( { - at^{2} } \right)dt = \frac{2}{\sqrt a }\mathop \smallint \limits_{0}^{\infty } exp\left( { - \rho^{2} } \right)d\rho = \sqrt {\frac{\pi }{a}}$$

(39)

where error function $\frac{2}{\sqrt \pi }*\mathop \smallint \limits_{0}^{\infty } exp\left( { - \rho^{2} } \right)d\rho = 1$.

Now, the solution and the derivation of Eq. (4), can be expressed as Eq. (8.7)

$$F\left( f \right) = \sqrt 2 \sigma_{1} exp\left( {\left( { - \frac{{\left( {2\pi f} \right)^{2} \sigma_{1}^{2} }}{2\pi }} \right)} \right)$$

(40)