**On the Performance of Ultra Wideband Radio in Stochastic Tapped Delay
Line Model of the Ultra Wideband Channel**

Kazi M. Ahmed, Mohammad Upal Mahfuz\*, Rabindra Ghimire, and Mohammad E.
R. Khan

Telecommunications Program, Asian Institute of Technology, PO Box: 4,

Khlong Luang, Pathumthani 12120, Thailand.

Email\*: [st100136@ait.ac.th](mailto:kahmed@ait.ac.th)

*Abstract*---This paper demonstrates the performance of Ultra Wideband
(UWB) systems in the Stochastic Tapped Delay Line (STDL) model of the
UWB channel with RAKE receiver. System models are based on Direct
Sequence Code Division Multiple Access (DS-CDMA) principle. The system
has been studied in presence of multiple user interference and Gaussian
noise. During the analysis, various types of narrow, sub-nanosecond
pulses with same power were used and extensive simulations were run in
3.1 to 10.6 GHz frequency band. The simulation results show that pulse
shape has noticeable impact on the performance of the system. The
performance of the pulse waveform based on third derivative of Gaussian
pulse has proved to be better than other pulses that were used. With a
thorough analytical approach, spectral behavior of higher order Gaussian
pulses has been explained. Finally, it is concluded that the third
derivative of Gaussian pulse is the most suitable pulse shape for
fulfilling the FCC regulated power spectral diagram (PSD) has also been
illustrated and in multi-user interference environment.

*Index Terms*---Ultra wideband, channel model, pulse shape.

I.  [Introduction]{.smallcaps}

Ultra Wideband Impulse Radio (UWB-IR) is currently receiving a great
deal of global attention because of its ability to provide higher data
rate with low cost and relatively low power consumption. UWB radio
communicates with baseband pulses of very short duration on the order of
tenth of a nanosecond. According to the regulation of Federal Commission
of Communications (FCC), a signal is defined as UWB signal if it has a
$- \text{10}$dB fractional bandwidth, $F_{\text{BW}}$ greater than (or
equal to) 0.20 or it occupies at least 500MHz of the spectrum \[1\]
expressed as

$$F_{\text{BW}} = 2\frac{f_{H} - f_{L}}{f_{H} + f_{L}} \geq 0\text{.}\text{20}$$

where *f~H~* and *f~L~* correspond to the --10 dB bandwidths. FCC has
also regulated the spectral shape and maximum power spectral density ($$
-41.3 dBm/MHz) of the UWB radiation in order to limit the interference
with other communication systems like UMTS or WLAN \[2\]. Although UWB
impulse radio can be termed as 'Spread Spectrum' technique because of
its extremely large operational bandwidth (7.5 GHz), the main difference
is that conventional spread spectrum techniques utilize carriers for
transmitting information whereas UWB radio transmits information without
carrier. The waveforms that are used for UWB radio are very short in
duration, causing their energy to be spread across the frequency
spectrum. With UWB signals the dense multipath can be resolved, allowing
a rake receiver for signal demodulation \[3\].

In order to analyze the potential of UWB systems, models describing the
UWB propagation channel must be properly known. The channel models help
analyze system performance and make design trades. In \[4\] the
Statistical analysis of indoor radio propagation and the modified
Poisson process or (Δ - K) model for path arrival of UWB system was
presented. Examination of the Spread Spectrum (SS) principle with time
hopping technique in the presence of wideband interference with AWGN
channel has been considered \[5\]. The modified Poisson process or Δ-K
model of the channel was used to study the UWB system for various M-PAM
signals \[6\], but the channel model was not suitable for the UWB system
because the frequency band used for measurement was in the range from
900 MHz to 1300 MHz. The effectiveness of multiuser detection for UWB
using DS-CDMA was shown in \[7\]. A channel model based on measurements
made in the indoor environment was presented in \[8\]. The model
presented in \[8\] was formulated as a stochastic tapped delay line
(STDL) model of the UWB indoor channel.

In this paper, to develop the UWB system model, DS-CDMA spreading
approach, Stochastic Tapped Delay Line (STDL) model of the channel \[8\]
and RAKE receiver in presence of multiple user interference are
considered. The Bit Error Rate (BER) performance of DS-CDMA systems in
UWB using STDL model and RAKE receiver with multiple user interference
is shown. The effect of pulse shapes on channel characteristics has also
been indicated. A thorough analytical explanation supporting the fact
that third derivative of Gaussian pulse is the most suitable pulse shape
for simultaneously fulfilling the FCC regulated power spectral density
(PSD) has also been illustrated. Finally, on the basis of extensive
simulations of BER performance and fulfillment of FCC power emission
regulation, a conclusion has been drawn to choose the third derivative
of Gaussian pulse as the best pulse shape for UWB radio communication.

The paper is organized as follows: Section II provides with a brief
description of the concept of DS-CDMA based multi-user interference. The
discussion is followed by Section III describing the complete STDL
system model under investigation. Simulation environment and
corresponding results have been explained in Section IV. Finally,
Section V concludes the paper.

# Concept of Multi-user Interference

Multiple User Interference (MUI) is modeled as signals coming from other
UWB users having the same basic signal characteristics as those of the
user of interest, but using different spreading codes. Assuming a total
of $\text{Nu}$users, the received signal $r(t)$can be written as \[9\]

$$r(t) = s^{(1)}(t) \otimes h^{(1)}(t) + i(t) + n(t)$$

where $s^{(1)}(t)$ is the signal of interest at the output of the
transmitting antenna, $h^{(1)}(t)$ represents the impulse response of
the channel corresponding to the user of interest, *n*(*t*) is the
Gaussian noise and *i*(*t*) is the interference coming from the other
$(\text{Nu} - 1)$users. Here the interference, $i(t)$can be defined as

$$i(t) = \sum_{n = 2}^{\text{Nu}}{s^{(n)}(t - \tau_{n}) \otimes h^{(n)}(t)}$$

where $h^{(n)}(t)$ for
$n = 2,3,\text{.}\text{.}\text{.}\text{.}\text{.}\text{Nu}$are the
impulse responses of the channels corresponding to the
$(\text{Nu} - 1)$interfering users and $\tau_{n}$ is the delay of the
arrival time of the n^th^ user from the user of interest i.e.
$s^{(1)}(t)$ for which $\tau_{1} = 0$. Assuming DS-CDMA as the used
multiple access technique, $s^{(n)}(t)$can be written as \[9\]$$

$$s^{(n)}(t) = \sum_{k = - \infty}^{+ \infty}{\sum_{j = 1}^{\text{Np}}{w_{d_{k}^{(n)}}(t - \text{kT}_{d} - \text{jT}_{c})}(c_{p})^{(n)}}$$

where $w(t)$is the used pulse waveform, $(c_{p})_{j}$is the *j*-th chip
of the Pseudo-random Noise (PN) code, generated by using bipolar PN
sequences with values $\left\{ + 1, - 1 \right\}$, $d_{k}$is the *k*-th
data bit chosen which, in Pulse Shape Mmodulation (PSM) case, also
defines the pulse waveform to be transmitted, $\text{Np}$represents the
number of pulses per data bit, $\text{Tc}$is the chip length and
$\text{Td}$is the data bit length defined as $T_{d} = N_{p}T_{c}$. Each
user experiences the channel with a different channel realization so
that the channel impulse response

$h^{(n)}(t) \neq h^{(m)}(t)$ where
$n,m = 1,2,\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{Nu},n \neq m$

Each MUI user has the same power of the user of interest. Interfering
users are considered either synchronous or asynchronous with the user of
interest. The 'synchronous' condition refers to 'frame synchronization'
where the receiver is synchronized with all other users and can receive
all users' data at the same time instant. On the other hand, in the
asynchronous case, the interfering signals arrive at the receiver with
resolution of a time sample. The asynchronism is performed between every
interfering user and the user of interest, and also among the
interfering users.

# System Model

In this research study higher order derivatives of Gaussian and Rayleigh
pulses have been used with STDL channel model. STDL channel model is a
derivative of S-V model. STDL model has been chosen for channel
representation because S-V channel has been very popular for indoor
wireless channel modeling and many related studies have been based on
this channel \[9\]. The time domain representations of some pulses are
given in \[10\]. The overall block diagram of the proposed system is
then shown in Figure 1.

A.  *Sub-nanosecond mono-pulses*

The basic pulses used in simulation of this study are the n^th^ order
derivatives of Gaussian pulse expressed as

$$\Omega_{G}^{n}(t) = \frac{d^{n}}{\text{dt}^{n}}\left( \frac{A}{\sqrt{2\pi}\sigma}e^{- \left( \frac{t^{2}}{2\sigma^{2}} \right)} \right)$$

where the basic waveform of Gaussian pulse is given by

$$\Omega_{\text{oG}}(t) = \frac{A}{\sqrt{2\pi}\sigma}e^{- \left( \frac{t^{2}}{2\sigma^{2}} \right)}$$

#  {#section .Reference-Head}

In addition to above, the channel effects using Rayleigh pulse as
expressed below are also compared.

$$\Omega_{\text{oR}}(t,\sigma) = \frac{t}{\sigma^{2}}\text{exp}( - \frac{t^{2}}{2}\sigma^{2})$$

Truncated *Sinc Pulse* and *Dual Gaussian Monopulse* were also used for
simulation.

B.  *Transmitter Section*

In the transmitter section Direct Spread CDMA spreading approach has
been used. The equiprobable binary bit stream
$\left\{ b_{j}(i) \right\} \in \left\{ 1, - 1 \right\}$is first
multiplied by the spreading code, $c_{j}(k)$, named Walsh Hadamard Code
of length 16. This DS-CDMA spreading is then followed by BPSK modulator
where the spread output is BPSK modulated by a signature waveform,
$\Omega_{c}(t)$. The modulated signal for the j^th^ user is represented
as \[11\]

$$x_{j}(t) = \sum_{i = - \infty}^{+ \infty}{\sum_{k = 1}^{N}{\Omega_{c}(t - \text{iT}_{s}} - \text{kT}_{c})c_{j}(k)b_{j}(i)}$$

where, *b~j~*(i) and *c~j~(k)* represent the *i^th^* information bit and
the *k^th^* chip of the spreading code with length, *N* and chip
duration, *T~c~* respectively for the *j*-th user. *T~s~* is the symbol
duration and$\Omega$*~c~(t)* is the chip pulse waveform. The radiated
waveforms for Rayleigh Pulse and Third derivative of Gaussian pulse are
shown in Figure 2(a) and 2(b).

## Channel Model

The transmitted BPSK modulated signals,$x_{j}(t)$are then passed through
the STDL propagation channel model, which is based on measurements made
in a typical office environment for UWB radio channel \[8\]. The model
characterizes the shape of the power-delay profile (PDP) in terms of
path gains,$G_{K}$ and delays,$\tau_{K}$ of multipath components. The
delay axis was adjusted in such a way that the delay bin of the first
quasi Line of Sight (LOS) path began at time instant τ = 0. The path
resolution of the considered system was $\Delta\tau = 2\text{ns}$.
Before the signal is received by the receiver section, thermal
noise,$n(t)$and interference from narrow-band transmitters,$n_{f}(t)$get
added to it. The model represented the statistics of the path gains and
its dependence on the delays.

The large-scale and the small-scale fading statistics were also
distinguished. The Power Delay Profile (PDP) at one location and that
averaged over all locations within the measurement area were referred to
as 'Local PDP' and 'Small Scale Averaged PDP (SSA-PDP)' respectively.
Large scale fading was investigated by considering the SSA-PDPs of
different locations, i.e. after having removed partially the small scale
statistics by spatially averaging the PDPs over the locations within one
measurement area. The 'global' parameters, referring to large scale and
small scale fading effect of the channel respectively, have been shown
in Table 1.

For large scale fading, the SSA-PDP exhibits an exponentially decay
function of excess delay. The power ratio, $r$and the decay constant,
$\varepsilon$, which vary from location to location, were treated as
stochastic variables. Since there were limited values of$r$and
$\varepsilon$, they were modeled as Log-Normal variables \[8\] as shown
in Table 1. The path loss (PL) was defined as a function of distance
also as shown in Table 1 \[8\]. The total average energy
gain,${\overline{G}}_{\text{tot}}$ can be calculated by integrating the
SSA-PDP of each place over all delay bins. The
${\overline{G}}_{\text{tot}}$is also log-normally distributed about path
loss (PL) and a standard deviation of 4.3 \[8\]. The average PDP is
specified according to \[8\]

$$\overline{g}(\tau) = \frac{{\overline{G}}_{\text{tot}}}{1 + r\frac{1}{1 - e^{( - \frac{\Delta\tau}{\varepsilon})}}}\begin{array}{r}
\left\{ \delta(\tau - \tau_{1}) + \middle| \right\}
\end{array}\left\{ \right\}$$

On the other hand, small-scale fading characterizes the changes in the
received signal when the receiver position changes only by a small
fraction of the distance between transmitter and receiver. This was
derived by considering the deviations of all the local PDP from the
respective SSA-PDP. The 'local' parameters, referring to small scale
fading effect, have also been shown in Table 1. The local path gain
values,$G_{k}$are the superposition of large and small scale statistics
and are Gamma distributed random variables with
mean,${\overline{G}}_{k}$ and parameters, $m_{k}$. The values of
$m_{k}$are independent truncated Gaussian random variables with
parameters dependent on the delay,$\tau_{k}$and
mean,${\overline{G}}_{k} = \overline{g}(\tau_{k})$\[8\].

The received signals are the sum of replicas of the transmitted signals.
The received signal is, therefore, expressed as

$$r(t) = \sum_{i = 1}^{N_{\text{path}}}{(c_{i}x(t - \tau_{i}) + n(t)} + n_{f}(t))$$

where, n(t) is zero mean AWGN and $n_{f}(t)$ is the interference signal.

## Receiver Section

For the simulation of this study RAKE receiver with a maximum of 12 arms
are used. First arm is locked to the first multipath component, *m~1~*.
Multipath component, *m~2~* arrive τ~1~ time units later than *m~1~* and
is captured by second tap and so on. The signals in the correlators are
despread by Walsh Hadamard Code of length 16. All decision statistics
are weighted by a weighting factor,$\alpha$ to form overall decision
statistics. The signals are then integrated over the entire period the
entire period. The integrated signal is then compared with the
appropriate threshold value to receive the better estimate of the
transmitted signal.

Table 1: Expressions of global and local parameters

+-----------------------------------------------------------------------+
| [Global Parameters:]{.underline}                                      |
|                                                                       |
| $$\text{Path Loss } = \left\{ \begin{matrix}                          |
| \text{20}\text{.}4\                                                   |
| text{.}\text{log}_{\text{10}}(\frac{d}{d_{0}}) & d \leq \text{11}m \\ |
|  - \text{56} + \tex                                                   |
| t{74}\text{.}\text{log}_{\text{10}}(\frac{d}{d_{0}}) & d > \text{11}m |
| \end{matrix} \right.\ $$                                              |
|                                                                       |
| $$\text{Decay Constant   }\varepsilo                                  |
| n\text{\textasciitilde}L_{N}(\text{16}\text{.}1;1\text{.}\text{27})$$ |
|                                                                       |
| $$\text{Shadowing }{\overline{G                                       |
| }}_{\text{tot}}\text{\textasciitilde}L_{N}( - \text{PL};4\text{.}3)$$ |
|                                                                       |
| $$\text{Power Ratio  }r\text{\textasciitilde}L_{N}( - 4;3)$$          |
+-----------------------------------------------------------------------+
| [Local Parameters:]{.underline}                                       |
|                                                                       |
| $$\text{Energy                                                        |
| Gains  }G_{k}\text{\textasciitilde}\Gamma({\overline{G}}_{k};m_{k})$$ |
|                                                                       |
| $$\text{m Values  }\begin{array}{r}                                   |
| \left\{ m_{k}\text{\textasciitilde}T_{N}(\mu_{m}(\t                   |
| au_{k});\sigma_{m}^{2}(\tau_{k})) \middle| \right.\ \left\{ \mu_{m}(\ |
| tau_{k}) = 3\text{.}5 - \frac{\tau_{k}}{\text{73}} \middle| \right.\  |
| \end{array}$$                                                         |
+-----------------------------------------------------------------------+

[IV. Results]{.smallcaps}

In this section, results based on the simulation of performance of an
ultra wideband system using DS-CDMA multiple access technique,
Stochastic Tapped Delay Line (STDL) channel model and RAKE receiver with
individual cases of 8 and 12 Rake arms and various types of
sub-nanosecond pulses as well as multiple user interference (MUI) have
been presented. The chip rate is 320 Mchip/s with spreading length of 16
(Walsh Hadamard code). The arrival of multiple users is assumed to be
Gaussian in nature and exist over the entire period. Performance of
various sub-nanosecond pulses in STDL model has been discussed in the
next sub-section which is followed by a discussion on results of system
performance in the presence of multiple user interference.

## The effect of pulse shapes on channel model

Figures 3.a and 3.b show the results of simulation for different pulse
shapes with a number of Rake receiver taps equal to 8 and 12 at a
distance of 5m with STDL model of the channel and in the presence of
white Gaussian noise. For each case, synchronization and channel
estimation are assumed to be perfect. The results show that the pulse
shape has noticeable impact on the performance of the system. It has
been found in simulation that the performance of third derivative of
Gaussian pulse is better than any other pulses.

For the Bit Error Rate (BER) of 10^-4^, using RAKE arms equal to 8 at a
distance of 5m as shown in Figure 3.a, the third derivative of Gaussian
pulse has 3.5 dB *E~b~/No* gain over that value of Gaussian monopulse,
whereas the fifth derivative of Gaussian pulse has almost the same
*E~b~/No* ratio as that of Gaussian pulse. Similarly, for the BER of
10^-4^, increasing the number of RAKE arms to 12, as shown in Figure 3.b
at a distance of 5m the third derivative of Gaussian pulse has 2.9 dB
*E~b~/No* gain over that value of Gaussian monopulse, whereas the fifth
derivative of Gaussian pulse needs 0.5 dB *E~b~/No* more than what is
required by the Gaussian pulse.

Pulse shape has noticeable impact because in the multipath environment
there is more chance that the pulses will arrive with different delay,
and the arrived components are out of phase with each other. In UWB
systems, baseband pulses and BPSK modulation schemes have been used for
simulation. If the arrived components are out of phase, in such case the
resulting output of the receiver depends on pulse shape. A comparison
between performances shown in Figure 3.a and Figure 3.b also indicates a
performance improvement in multipath fading environment as a result of
using an increased number of RAKE arms.

## The effect of Multiuser Interference on System Performance

Figures 4.a, 4.b and 4.c show system performances in presence of
multiple user interference at a distance of 5m. As shown in the above
figures, it is found that degradation in the performance of third
derivative of Gaussian pulse is less than the degradations of Rayleigh
pulse and second derivative of Rayleigh pulse in the presence of
multiple user interference. Simulations at a distance of 7m without
interference and in the presence of interference have also been
performed and the results have been shown in Figures 5.a, 5.b and 5.c.
At a distance of 7m the degradation in the performance is more if the
number of interferences is increased. At the distance of 7 meters,
performance of third derivative of Gaussian pulse is also better than
other sub-nanosecond pulses. The results at a distance of 10m and 12m
without interference are shown in Figure 6 and in Figure 7 respectively.
The result showed that third derivative of Gaussian pulse shows better
performance all other pulses.

[V. Validation into FCC Power Mask Requirement]{.smallcaps}

In this section, an analytical approach has been shown for the
validation that the third derivative of Gaussian pulse suits the FCC
power emission requirement the best among all possible pulse shapes,
especially when the wireless channel is impaired with the presence of
multiple paths and Multi-user Interference (MUI). According to Part 15
of FCC Regulation on UWB emission limits in the form of a spectral mask
for indoor and outdoor systems \[12\], UWB radios can emit a peak
transmit power of $- \text{41}\text{.}3$dBm/MHz in the frequency band
from 3.1 GHz to 10.6 GHz. Outside this band, the Power Spectral Density
(PSD) must be decreased in order not to interfere with other wireless
communication systems operating in that frequency bands. From 0.96 GHz
to 1.61 GHz range, the reduction in admissible transmitted power is
necessary to protect GPS transmissions. Also, to protect PCS
transmission for outdoor systems in the band from 1.99 GHz to 3.1 GHz,
the required backoff to be ensured is 20 dB for outdoor systems.

The PSD of the first derivative Gaussian pulse does not meet the FCC
power emission requirement regardless of the pulsewidth being used. On
the other hand, as indicated in Section 4, it has been found by
simulation in our experiment that higher order derivatives of Gaussian
pulse have significant $\frac{E_{b}}{N_{o}}$ gain for the same BER value
over all other pulses used. Therefore, an analytical investigation has
been made to research into the PSD of higher order derivatives of
Gaussian pulses, more specifically for third derivative of Gaussian
pulse. In Figure 8, the normalized PSD has been drawn for first to fifth
order derivatives of Gaussian pulse for a chosen value of σ. The value
of σ has been calculated according to Bisection method by numerical
analysis \[14\]. The normalization factor is the peak value allowed by
the FCC, -41 dBm/MHz. It is clear from the figure that increasing order
of derivative increases the center frequency and thus help to suit the
FCC power emission regulation, while at the same time reduces the
relative bandwidth \[14\]. Shifting the center (peak) frequency and
adjusting the bandwidth so that the FCC requirements are met could be
done by modulating the monocycle with a sinusoid by varying the values
of σ. But since Impulse radio Ultra Wideband (IR-UWB) systems are
carrierless systems, modulation will increase the cost and complexity in
design. After an analytical explanation described in Appendix A, it can
be concluded that higher order derivatives have a clear impact in
fulfilling the FCC PSD mask for UWB transmission and more specifically,
third order derivative of Gaussian pulse meets the FCC requirement quite
well the 3.1 GHz -- 10.6 GHz region while providing a good trade-off
between PSD and operational bandwidth. The 3dB bandwidths of higher
order Gausian pulses have been calculated and shown in Figure 9 \[14\].
As order of derivative increases, the relative bandwidth decreases. This
point is important for choosing the third order derivative, since its
relative bandwidth is much larger than 4^th^ or 5^th^ order derivatives.
Also, it has been found from simulation that for the same BER value to
be ensured the required $\frac{E_{b}}{N_{o}}$value for the third
derivative of the Gaussian pulse is the minimum among all. A more
in-depth research is necessary to shape the third derivative of Gaussian
pulse so that it fits the FCC PSD requirement more accurately in the
0.96 GHz-1.61 GHz and 1.99 GHz-3.1 GHz ranges. The effectiveness of
matched filter in this case can also be investigated.

# Conclusions

At present, Ultra Wideband (UWB) radio is an emerging technology
especially in short distance indoor communications. In this study, the
performance in the form of Bit Error Rate (BER) in MUI environment for
Ultra Wideband radio has been described. Stochastic Tapped Delay Line
(STDL) model derived basically from Saleh-Valenzuela model and accepted
as 'modified S-V model' by IEEE 802.15.3a study group, has been chosen
here to investigate the system performance for various sub-nanosecond
pulses and in the presence of multi-user interference. DS-CDMA, a
comparatively less complicated and cheaper multiple access technique
currently available, is used in the transmitter section. Till date, the
most currently adopted and widely accepted pulse shape used for UWB
indoor communication is modeled as the second derivative of Gaussian
function. \[13\]. Experiments are still going on so that pulse shapes
meet the FCC requirement more accurately. In our experiment, it has been
found that third derivative of Gaussian pulse shows the best performance
among all pulses used even in the presence of MUI environment. Gausssian
pulse and its derivatives have been given more importance because they
can be generated at the pulse generator at the easiest way. In our
simulation, the best results have been obtained for the third derivative
of the Gaussian pulse, where considering the antenna effects is referred
as future work of the current research. Through spectral analysis it has
also been proved that third derivative of Gaussian pulse fits the FCC
power emission mask quite effectively in the 3.1 GHz -- 10.6 GHz range
where UWB devices are supposed to be used in indoor communications.

# References {#references .Reference-Head}

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2.  [D. Barras, F. Ellinger, H. Jäckel, \"A comparison between
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    2002](http://www.ife.ee.ethz.ch/~ellinger/Homepage/UWB2002.pdf)

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**\[**14\] Hongsan Sheng, P. Orlik, A. M. Haimovich, L. J. Cimini, Jr.
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# Appendix: A {#appendix-a .Reference-Head}

Using the expression described in Section 3.A the n-th order derivative
of Gausssian pulse can be obtained recursively from

$$\Omega_{G}^{n}(t) = - \frac{n - 1}{\sigma^{2}}\Omega^{(n - 2)}(t) - \frac{t}{\sigma^{2}}\Omega^{(n - 1)}(t)$$

# The Fourier transform of the n-th derivative of Gaussian pulse is  {#the-fourier-transform-of-the-n-th-derivative-of-gaussian-pulse-is .unnumbered}

$$\Omega_{n}(f) = A(j2\pi f)^{n}e^{- \frac{(2\pi f\sigma)^{2}}{2}}$$

from where the amplitude spectrum can be considered as

$$\left| \Omega_{n}(f) \right| = A(2\pi f)^{n}e^{- \frac{(2\pi f\sigma)^{2}}{2}}$$

The peak emission frequency, $f_{M}$can be determined by equating
$\frac{d\left| \Omega_{n}(f) \right|}{\text{df}}$to zero. The value of
$f_{M}$and the corresponding $\left| \Omega_{n}(f_{M}) \right|$are found
as

$f_{M} = \frac{\sqrt{n}}{2\text{πσ}}$ and
$$$\left| \Omega_{n}(f_{M}) \right| = A\left( \frac{\sqrt{n}}{\sigma} \right)^{n}e^{\left( - \frac{n}{2} \right)}$

The normalized PSD, $\left| P_{n}(f) \right|$ can be defined as

$$\left| P_{n}(f) \right| \equiv \frac{\left| \Omega_{n}(f) \right|^{2}}{\left| \Omega_{n}(f_{M}) \right|^{2}} = \frac{(2\pi f\sigma)^{2n}e^{- (2\pi f\sigma)^{2}}}{n^{n}e^{( - n)}}$$

which has a peak value of 1 (0 dB). Considering the n-th derivative of
Gaussian pulse as the transmitted pulse of UWB transmission where
$A_{\text{max}}$ is the peak power spectral density that has been set as
limit by FCC then the PSD of the transmitted signal can be expressed as

$$\left| P_{n}(f) \right| \equiv A_{\text{max}}\left| P_{n}(f) \right| = \frac{A_{\text{max}}(2\pi f\sigma)^{2n}e^{\left\{ - (2\pi f\sigma)^{2} \right\}}}{n^{n}e^{( - n)}}$$

$$It has also been found from the expression of $f_{M}$that $f_{M}$
varies proportionally with $\sqrt{n}$ for a given decaying factor,
$\sigma$. Gaussian derivatives of higher order are characterized by
higher peak frequencies. Thus, differentiation is a way to move energy
to higher frequency bands.

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