### 2.1. Channel modeling

The following two steps are used to model the multipath radio channel.

#### (1) Frequency responses for sinusoidal waves using the SBR/Image technique

The SBR/Image method can deal with high-frequency radio wave propagation in the complex indoor environment [7, 8]. It conceptually assumes that many triangular ray tubes are shot from the transmitting antenna (TX), and each ray tube, bouncing and penetrating in the environment is traced in the indoor multipath channel. If the receiving antenna (RX) is within a ray tube, the ray tube will produce image contributions to the received field at the RX, and the corresponding equivalent source (image) can be determined. By summing all contributions of these images, we can obtain the total received field at the RX. In the real environment, external noise in the channel propagation will be considered. The depolarization yielded by multiple reflections, refraction, and first-order diffraction, is also taken into account in our simulations. Note that the different values of dielectric constant and conductivity of materials for different frequencies are carefully considered in channel modeling.

Using ray-tracing techniques to predict channel characteristic is effective and fast [7–9]. Thus, a ray-tracing channel model is used to calculate the channel matrix of the UWB system. The flow chart of the ray-tracing process is shown in Figure 1. It conceptually assumes that many triangular ray tubes (not rays) are shot from a transmitter. Here, the triangular ray tubes whose vertexes are on a sphere are determined by the following method. First, we construct an icosahedron which is made of 20 identical equilateral triangles. Then, each triangle of the icosahedron is tessellated into many smaller equilateral triangles. Finally, these small triangles are projected onto the sphere and each ray tube whose vertexes are determined by the small equilateral triangle is constructed.

For each ray tube bouncing and penetrating in the environment, we check whether reflection and penetration times of the ray tube are larger than the number of maximum reflection *N*_{ref} and maximum penetration *N*_{pen}, respectively. If not, we check whether the receiver falls within the reflected ray tube. If yes, the contribution of the ray tube to the receiver can be assumed to be emitted from an equivalent image source. In other words, a specular ray going to receiver is assumed to exist in this tube and this ray can be thought as launched from an image source. Moreover, the field diffracted from illuminated wedges of the objects in the environment is calculated by uniform theory of diffraction [10]. Note that only first diffraction is considered in this article, because the contribution of second diffraction is very small in the analysis.

By using these images and received fields, the channel frequency response can be obtained as follows:

H\left(f\right)=\sum _{p=1}^{{N}_{\mathsf{\text{p}}}}{a}_{p}\left(f\right){e}^{j{\theta}_{p}\left(f\right)}

(1)

where *p* is the path index, *N*_{p} is the number of paths, *f* is the frequency of sinusoidal wave, θ_{
p
}(*f*) is the *p* th phase shift, and *a*_{
p
}(*f*) is the *p* th amplitude. Note that the channel frequency response of UWB systems can be calculated by Equation (1) in the frequency range of UWB for both desired and interference signals.

#### (2) Inverse Fast Fourier Transform (IFFT) and Hermitian Processing

The frequency response can be transformed to the time domain by using the inverse Fourier transform with the Hermitian signal processing [11]. By using the Hermitian processing, the pass-band signal is obtained with zero padding from the lowest frequency down to direct current (DC), taking the conjugate of the signal, and reflecting it to the negative frequencies. The result is then transformed to the time domain using IFFT [12]. Since the signal spectrum is symmetric around DC, the resultant doubled sideband spectrum corresponds to a real signal in the time domain.

The equation used to model the multipath radio channel is a linear filter with an equivalent impulse response given by

{h}_{b}\left(t\right)=\sum _{l=1}^{N}{\alpha}_{l}\delta \left(t-{\tau}_{l}\right)

(2)

where *l* is the path index, *α*_{
l
} is the amplitude of *l* th path and *τ*_{
l
} is the time delay of the *l* th path. δ(.) is the Dirac delta function [13]. The goal of channel modeling is to determine the *α*_{
l
} and *τ*_{
l
} for a transmitter-receiver location in the system. The impulse response function of the station for a transmitter-receiver location is computed by the following two steps: step one is the obtaining of frequency response for sinusoidal waves by the SBR/Image technique and step two is the use of IFFT and Hermitian processing [14].

The SBR/Image method can deal with high-frequency radio wave propagation in the complex outdoor environment. It conceptually assumes that many ray tubes are short from the transmitting antenna (TX) and each ray tube bouncing and penetrating in the environment is traced. The first-order wedge diffraction is included, and the diffracted rays are attributed to the corresponding image. A frequency response is transformed to the time domain by using inverse Fourier transform with Hermitian signal processing. Using Hermitian processing, the pass-band signal is obtained with zero padding from the lowest frequency down to DC, taking the conjugate of the signal, and reflecting it to the negative frequencies. The result is then transformed to the time domain by using IFFT. Since the signal spectrum is symmetric around DC, the resultant doubled-side spectrum corresponds to a real signal in the time domain.

### 2.2. System block diagram

The diagram of transmitted waveform is shown in Figure 2. The transmitted UWB pulse stream can be expressed as [15–24]:

x\left(t\right)=\sqrt{{E}_{\mathsf{\text{TX}}}}\sum _{n=0}^{\infty}p\left(t-n{T}_{\mathsf{\text{d}}}\right){d}_{n}

(3)

where *E*_{TX} is the average transmitted energy symbol and *p*(*t*) is the second derivative of the Gaussian waveform. *T*_{d} is the duration of the transmitted signal. The binary PAM symbol *d*_{
n
} ∈{ ± 1} is assumed to be independent, identically distributed (i.i.d.). The waveform *p*(*t*) has an ultra-short duration *T*_{
p
} usually of the order of nanoseconds and is usually much smaller than *T*_{d}. The waveform *p*(*t*) can be described by the following expression:

p\left(t\right)=\frac{{d}^{2}}{d{t}^{2}}\left(\frac{1}{\sqrt{2\pi}\sigma}{e}^{\frac{-{t}^{2}}{2{\sigma}^{2}}}\right)

(4)

where *t* and *σ* are time and standard deviation of the Gaussian wave, respectively.

The average transmitted energy symbol *E*_{TX} can be expressed as

{E}_{\mathsf{\text{TX}}}={\int}_{0}^{{T}_{\text{d}}}{p}^{2}\left(t\right)dt

(5)

Block diagram of the simulated communication system is shown in Figure 3. The received signal after the bandpass filter *r*(*t*) can be expressed as follows:

r\left(t\right)=\left[x\left(t\right)\otimes {h}_{b}\left(t\right)\right]+n\left(t\right)

(6)

where *x*(*t*) is the transmitted signal and *h*_{
b
}(*t*) is the impulse response of the UWB channel, *n*(*t*) is the white Gaussian noise with zero mean and power spectral density *N*_{0}/2 w/Hz.

The correlation receiver samples the received signal at the symbol rate with a delay given by

q\left(t\right)=p\left(t-{\tau}_{1}-n{T}_{\mathsf{\text{d}}}\right)

(7)

where *τ*_{1} is the delay time of the first received wave.

The output of the correlator is

{Z}_{n}={\int}_{0}^{{T}_{\mathsf{\text{d}}}}\left\{\left[\sqrt{{E}_{\mathsf{\text{TX}}}}\sum _{n=0}^{\infty}p\left(t-n{T}_{\mathsf{\text{d}}}\right){d}_{n}\right]\otimes {h}_{b}\left(t\right)\right\}\cdot q\left(t\right)dt+{\int}_{0}^{{T}_{\mathsf{\text{d}}}}n\left(t\right)q\left(t\right)dt={V}_{n}+\eta

(8)

It can be shown that the noise components *η* of Equation (8) are uncorrelated Gaussian variables with zero mean. The variance of the output noise *η* is

{\sigma}^{2}=\frac{{N}_{0}}{2}{E}_{\mathsf{\text{TX}}}

(9)

The average probability of error is thus expressed by:

{P}_{e}[\left(\right)close="|">Z\left(t=n{T}_{\mathsf{\text{d}}}\right){\stackrel{\rightharpoonup}{d}}_{n}]\n =\n \n \n 1\n \n \n 2\n \n \n e\n r\n f\n c\n \n \n \n \n V\n \n (\n \n t\n =\n n\n \n \n T\n \n \n \n d\n \n \n \n \n )\n \n \n \n \n \n 2\n \n \n \sigma \n \n \n \u22c5\n \n (\n \n \n \n d\n \n \n n\n \n \n \n )\n \n \n \n

(10)

where erfc\left(x\right)=2/\sqrt{\pi}{\int}_{x}^{\infty}{e}^{-{y}^{2}}dy is the complementary error function and \left\{{\stackrel{\rightharpoonup}{d}}_{n}\right\}=\left\{{d}_{0},{d}_{1},...,{d}_{N}\right\} is the binary sequence. Finally, the BER for the BPAM IR UWB system can be expressed as

\mathsf{\text{BER}}=\sum _{n=0}^{N}P\left({\stackrel{\rightharpoonup}{d}}_{n}\right)\cdot \frac{1}{2}erfc\left[\frac{V\left(t=n{T}_{\mathsf{\text{d}}}\right)}{\sqrt{2}\sigma}\cdot \left({d}_{n}\right)\right]

(11)

### 2.3. RMS delay spread and mean excess delay

In order to compare different multipath channels and develop general design guidelines for wireless systems, parameters which grossly quantify the multipath channel are used. The mean excess delay spread and root mean square (RMS) delay spread are parameters of the multipath channel that can be determined from a power delay profile or channel impulse response. However, the time dispersive properties of wideband multipath channels are most commonly quantified by their mean excess delay and RMS delay spread. The mean excess delay is defined as

\stackrel{\u0304}{\tau}=\frac{\sum _{k}{\beta}_{k}^{2}{\tau}_{k}}{\sum _{k}{\beta}_{k}^{2}}=\frac{\sum _{k}P\left({\tau}_{k}\right){\tau}_{k}}{\sum _{k}P\left({\tau}_{k}\right)}

(12)

The RMS delay spread is the square root of the second central moment of the power delay profile and is defined as

{\sigma}_{\tau}=\sqrt{\overline{{\tau}^{2}}-{\left(\overline{\tau}\right)}^{2}}

(13)

where

{\stackrel{\u0304}{\tau}}^{2}=\frac{\sum _{k}{\beta}_{k}^{2}{{\tau}_{k}}^{2}}{\sum _{k}{\beta}_{k}^{2}}=\frac{\sum _{k}P\left({\tau}_{k}\right){{\tau}_{k}}^{2}}{\sum _{k}P\left({\tau}_{k}\right)}

(14)

### 2.4. Rake receiver techniques

Signal reception in a multipath fading channel can be enhanced by a diversity technique using the rake receiver. Rake receivers combine different signal components that have propagated through different paths in the channel. This is a time diversity technique. The combination of different signal components will increase the signal-to-noise ratio (SNR), thus improving the reception performance. There are three major types of rake receivers to be considered here, i.e., I-rake, S-rake, and P-rake. First, I-rake is an ideal rake receiver that captures all of the received signal power by having numbers of fingers equal to the number of multipath components. Using the I-rake, an infinite number of correlators are required to distinguish infinite multipath components. This is difficult to carry out in reality. Second, S-rake is a selective rake receiver which selects *L* multipath components having largest signal amplitudes. The S-rake receiver has much less complexity than the A-rake. Third, P-rake is similar to S-rake. The principle is that the first multipath component will typically be strongest and contain the most received signal power. But, the disadvantage is that it may not correctly choose the strongest multipath components. In this article, we shall take the S-rake receiver. We will evaluate system performance using *L* as a parameter (*L* = 1, 2, 5, and 8).