# An Optimization Of Gaussian Uwb Pulses Biology Essay

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UWB is a new interesting tehnology for wireless communications. It can replace traditionally carrier-based radio transmission by pulse-based transmission using ultrawide band frequency but at a very low energy. An important aspect of research in this domain is to find a pulse with an optimal shape, whose power spectral density respect and best fit emission limitation mask imposed by FCC.

In this paper we review common used Gaussian Pulse and his derivatives and the influence of shape factor, finding an optimal specific value for each derivatives. Next, we search to obtain possible better pulse as linear combinations of Gaussian derivatives. Older study refer in one case of same shape factor for all derivatives and in other case of higher factor for first derivative and smaller factor shape for next derivatives.

Our new idea is to use Gaussian derivatives but each with its specific optimal shapes factor and to use an "trial and error" algorithm to obtain a linear combination pulse with a more better performance.

Index Terms - Gaussian Monocycle; Shape factor; Trial and error; UWB; Wireless LAN

## INTRODUCTION

UWB (Ultra-Wide-Band) wireless transmission is based on impulse radio and can provide a very high data rates over short distances. Its traditional application were in non-cooperative radar. UWB device by definition has a bandwith egual or greater than 20% of the center frequency or a bandwith egual or more than 500Mhz. Since FCC (Federal Communications Commision) authorized in 2002 the unlicensed use in the domain 3.1- 10. 6 GHz, UWB become very interesting for commercial development. High data rate UWB can enable wireless monitors, efficient transfer of data between computer in a Personal Wireless Area Network, from digital camcorders, transfer of files among cell phone and home multimedia devices, and other radio data communication over short distances.

Traditional wireless technologies use radio sine waves that provide "continual" transmission at a specific frequency. UWB radio system is associated with its impulse-based, carrier-free, time-domain radio system format that transmits very short UWB pulse signal trains (sub-nanosecond pulse width) without using any continuous sinusoidal wave carriers. For pulse-based UWB system, an extremely short pulse spreads its signal power over a very wide frequency spectrum (3.1GH-10.6GHz) where the duty cycle of UWB pulse train can be as low as 1%. The pulses are emitted in a rhythm unique to each transmitter. The receiver must know the transmitter's rhythm signature or pulse sequence to "know how to listen" for the data being transmitted.[5].

Study of shape of the base pulse is fundamental becouse on it depend the performances of an UWB system like efficiently use of permitted emission power, coexistence with other radio communication systems and a simple circuit implementation.

One of the fundamental challenge is to maximize the radiated energy of the pulse while the spectral power density complying with the spectral mask FCC.

The FCC reglemented use of UWB devices respecting emission limit values as depicted in Fig.1 [4]. Due to the extremely low emission levels currently allowed comparable with unnintended emission (FCC Part 15) UWB systems tend to be for short-range and indoors applications. UWB operates best over short distance of about 2-3 meters delivering data speeds of 480 Mbps. As distance increases, speed decreases, but at 10 meters still reach or exceed 100 Mbps.

Fig.1. UWB indoor emission mask reglemented by FCC.

Since the ultra-short pulses are relative easy to generate only with analog components, the Gaussian Monocycle and his derivatives is common used for UWB.

In this paper, we study the spectral properties of UWB pulses. Section II analyzes the power spectral density of the Gaussian monocycle and then in Section III extend the study to higher-order derivatives of the Gaussian pulse and about the influence of shape factor Ïƒ and his optimal values.

In Section IV, we study a new types of pulses obtained by linear combinations of Gaussian pulses and an algorithm to find optimal coefficients. We propose a new set of based pulses, having shape factor Ïƒn specific for every derivatives. The conclusion about performance of a combination pulses obtained is presented in Section V.

## GAUSSIAN monocycle

By far the most popular pulse shapes discussed in UWB communication literature are the Gaussian pulse and its derivatives, as they are easy to describe and work with.

Basic Gaussian pulse is described analytically as:

If a Gaussian pulse is transmitted, due to the derivative characteristics of the antenna, the output of the transmitter antenna can be modeled by the first derivative of the Gaussian pulse [1]. Therefore, the pulse radiated is given by first derivative of Gaussian pulse, called monocycle :

For a shape factor Ïƒ =0.1ns the waveform of the pulse in presented in Fig.2 , and the corresponding spectrum in Fig. 3

Fig.2 Waveform of Gaussian monocycle

Fig.3 Power Spectral Density of Gaussian monocycle

Strictly speaking, the duration of the Gaussian pulse and all of its derivatives is infinite. Here, we define the pulse width, Tp, as the interval in which 99.99% of the energy of the pulse is contained. Using this definition, it can be shown that Tp â‰ˆ 7Ïƒ for the first derivative of the Gaussian pulse.

The posibillity for tuning PSD spectrum in order to respect and fit the mask FCC is to choose the shape factor Ïƒ. We observe than decreasing the value of Ïƒ in time domain the duration of pulse Tp is shorter, and leads in frequency domain the spectrum to migrate to higher frequency. For example, when value of Ïƒ is 0.12 ns, Tp=0.84 ns and frequency at maxim PSD is f_peak=1.24 Ghz (continuous curve in Fig.4) ; when Ïƒ decreases at 0.080 ns, Tp become shorter, 0.56 ns and f_peak = 2Ghz, higher (dashed curve) ; for Ïƒ=0.04 ns, Tp=0.28ns and f_peak move to high frequency, f_peak=4Ghz (dotted curve).

Fig.4. PSD of Gaussian monocycle for three Ïƒ values

As we see, it is clear that the PSD of the first derivative pulse doesn't meet the FCC equirement no matter what value of the pulse width is used. Therefore, a new pulse shape must be found that satisfies the FCC emission requirements. One possibility is to shift the center frequency and adjust the bandwidth so that the requirements are met. This could be done by modulating the monocycle with a sinusoid to shift the center frequency and by varying the values of Ïƒ.

Impulse UWB, however, is a carrierless system; modulation will increase the cost and complexity. Therefore, alternative approaches are required for obtaining a pulse shape which satisfies the FCC mask.

In the time domain, the high-order derivatives of the Gaussian pulse resemble sinusoids modulated by a Gaussian pulse-shaped envelope. As the order of the derivative increases, the number of zero crossings in time also increases; more zero crossings in the same pulse width correspond to a higher "carrier" frequency sinusoid modulated by an equivalent Gaussian envelope.

These observations lead to considering higher-order derivatives of the Gaussian pulse as candidates for UWB transmission.[2]

## HIGHER ORDER DERIVATIVES

We investigate in the folow the pulses as derivatives of the basic gaussian pulse.

Ecuation of n-derivative pulse is done by:

In time domain, we observe that the duration of pulse remain same for different order derivatives, and we can consider Tp = 10Ïƒ . Waveforms for this pulses and respectively theirs PSD is presented below in Fig.5 and 6.

Fig.5 Waveforms for 5,7, and 12th derivative of

Gaussian pulses

Fig.6 PSD for 5,7,and 12th derivative Gaussian pulse

Interesting, in frecvency domain Fourier transforms of those pulses has a relative simple expression, and spectrum has a amplitude :

We can study the spectrum by easy compute the

frecvency peak and the bandwith. Frecvency peak

of spectrum is done by :

For every derivative, we choose a value for Ïƒ to obtain a pulse that matches the FCC's PSD mask as closely as possible. For example, in Fig.7 is presented results of simulations for 5th derivative Gaussian pulses with Ïƒ having

differents values. The bold-plotted curve is the result for optimal value for Ïƒ5=0.051 ns.

Fig.7 Optimization of Ïƒ for 5th derivative pulse

In Table 1, we summarize the optimal parameter Ïƒn , the peak emission frequency, and the 10dB bandwidth obtained for first 15 order derivatives of Gaussian pulses :

Table 1. Optimal values for pulses gaussian derivatives

n-order

Ïƒn

[ns]

fM

[Ghz]

B10dB

[Ghz]

1

0.033

4.79

7.50

2

0.039

5.78

7.50

3

0.044

6.34

7.40

4

0.047

6.72

7.07

5

0.051

7.01

6.64

6

0.053

7.23

6.19

7

0.057

7.42

5.59

8

0.060

7.57

5.67

9

0.062

7.70

5.48

10

0.064

7.81

5.24

11

0.067

7.90

5.08

12

0.069

8.01

4.94

13

0.071

8.10

4.79

14

0.073

8.18

4.66

15

0.075

8.25

4.54

These results show that the pulse width will be less than 1 nanosecond for all cases, and the 10dB bandwidth is 4.5 GHz or greater. The maximum PSD can be controlled by changing the value of the amplitude A of the pulse. [9].

## OPTIMAL COMBINATION OF

## GAUSSIAN PULSES

For obtain a better performance UWB pulse ,we intend to study linear combinations of gaussian derivatives pulses. Let's to consider a base pulses with first 15 gaussian derivatives pulses each with a individual shape factor.

The combination pulse has this expression :

The problem is to find the best set of coefficients S={cn } than the resulting combination pulse respect and best fit FCC requirements [1]. We propose a computer-based method by means of a giving a random sets of coefficients and testing with a trial-and-error procedure described as follows:

Step 1. Initialize the random number generator.

Step 2. Generate a random set of coefficients S.

Step 3.Check if the PSD of the linear combination

obtained with coefficients and base pulses

satisfies the emission limits.

Step 4. If the emission limits are not meeting,

go to step 2 and generate another combination.

Step 5. If the emission mask are meeting and this

is first set verifying the limits, initiate the

optimal set BS=S.

Step 6. If the emission mask are meeting and

already exist an optimal set, compare actual valid

set S with optimal set BS. If S have a better fit of

a mask, i.e. the sum of PSD(exprimed in mW/Hz)

of all frecvency is greater, redefine BS=S.

Step 7. Repeat this procedure going to step 1 for

some number of "trial" cycles, obtaining new

possible improved pulse.

Step 8. After that number of independent random

searches (becouse in step 1 random generator is

reinitialized) ,algoritm stops and current BS is the

optimal found.

If we rule this algorith for a sufficiently big number of "trials", the results converge and we obtain a best result possible. Ours results is obtaining by rule the algorithm with 1000 number of independent "trials".

We running this algorithm for three case, in function of choises of set of shape factors Ïƒn of each based pulses for combination.

A. Case of same factor shape Ïƒ for all derivatives.

In that case, we consider the pulses derivatives having same shape factor Ïƒ. The result for values of Ïƒn=0.2ns is

presented in Fig.8

Fig.8 PSD of optimal combination for same Ïƒn=0.2ns

As observe, FCC limits is respected at all frequency, but fitt of the mask is not better, only to about 4 GHz.

B. Case of differens factosr shape.

Improved performance can be achieved by adopting different values for the different derivatives.

Consider a second set of a values characterized by a higher value of Ïƒ (0.42 ns) for the first derivative and smaller values (0.08 ns) for the higher derivatives.[1]

The new values improve performance of the trial-and-error procedure, leading to a PSD that is quite close to the target for frequencies up to about 8 GHz, as is depicted in Fig.9.

Fig.9 PSD of optimal combination for Ïƒ1=0.42ns and Ïƒ2-15=0.08ns

C. Case of optimized factosr shape

Regarding the discution from Section 2 , we have the idea to consider Gaussian derivative pulses having its specific optimized factor shape Ïƒn conform Table 1.The PSD of that optimized pulses is depicted below in Fig.10 :

Fig.10 Ïƒ-optimal Gaussian derivatives pulses

With those based pulses, applying optimization combination algorithm will result a pulse combination with PSD presented n Fig.11.

Fig.11 Optimal Pulse Combination

As we see, this combination pulse have very good properties in fitting the mask and therefor using maximum allowed emission power.

For a objective comparison and correct performance evalation of this pulse, in every figure is displaying the parameter used in algorithm "trial and error", PS, what is "Spectral Power" obtained summarizing numerical values

of PSD(exprimed in mW/Hz) at all sampled frequency .

The value obtained here is the best , PS =2.69âˆ™10Â-4 mW. In case of same values for Ïƒ obtained value is poor PS=7.5âˆ™10Â-6 mW and in case of higher Ïƒ for first derivative and smaller for higher order derivatives, PS is medium PS=1.45âˆ™10Â-5 mW.

## conclusion

UWB techology is new and subject to more improvements. Signal pulses by order of ns is easy to generate only it is simple and with an analogue circuit. Gaussian monocycle and his derivatives is common used waveform for pulses in UWB. It is easy to generate, but don't have ehough good performance. One of the problem is to obtain an PSD that better fit limitted emission mask. Higher order derivatives has been studied and proving better performance ,but not enough. For indoor application fifth-order derivatives is currently a choise for implementation.

Now, we try an algorithm to obtain a pulse as a liniar combination of first 15 derivatives gaussian pulse. Using this method with same shape factor for all derivatives, and next an implementation with small values for monocycle and high value for higher derivatives leads to obtain pulses with PSD respecting and some improved fitting of the mask, but not enough good above 7 Ghz.

We propose an idea to use as base pulses for combination Gaussian derivatives with Ïƒ values individually optimized by critery to obtain maximum bandwith. With this pulses, the optimal combination obtained by algorithm "trial and error" lead to a pulse with very goog performance to respect and better fit the mask. The diagram ilustrates than this pulse have a PSD well approximating the mask at all frequency, therefore efficiently use of available bandwith and power.

This result from the diagram is argumented by the computed parameter PS (Spectral Power) , as the sum of values of power spectral density, the combination pulse having in this case the best value.

Results and simulations obtained by implementation of algorithm in MatLab is interesting and subject to further discutions and implementations.