UWB Propagation Channels

Channel is the medium in which a signal propagates from the transmitter to the receiver. A good understanding of the propagation phenomena allows to model the propagation channels adapted to the constraints of UWB. A signal propagating in the free space is only affected by the attenuation and a delay which is a function of the distance between the transmitter and the receiver. However, antenna gains and antenna efficiencies make it frequency dependent as well. The path gain in the free space is given by the formula as under:

G_pr = P_R

P_T =G_{T X}(f)η_{T X}(f)G_RX(f)η_RX(f) c

4πdf 2

(1.9) wherePRandPT are the received and the transmitted powers respectively,GT X(f) and G_RX(f) are the frequency dependent antenna gains for the transmitter and the receiver respectively,η_{T X}(f) andη_RX(f) are the antenna efficiencies of the transmitter and the receiver respectively, f is the frequency of the transmitted signal, and d is the distance between the transmitter and the receiver. This equation is normally interpreted such that the path gain decreases with the square of frequency. However, the above formulation also shows us that this is valid only if the antenna gains are constant over frequency [24]. The antenna area, A_RX, is also related to the antenna gain and can be written as:

A_RX = G_RX 4π

c f

2

(1.10) In real environments, there are multiple obstacles which disturb the signal prop-agation. The received signal is a recombination of multiple waves, which have either been attenuated or experienced different phase shifts and reach the receiver with a delay corresponding to the path they have traversed. The presence of multi-paths in the propagation channel can lead to significant distortions in the received signal.

Furthermore, especially in the indoor environment (which is often the propagation environment for UWB signals), a line of sight (LOS) path is not always available.

Therefore, the signals propagating in the real environment undergo different phenom-ena such as reflection, transmission, diffraction, diffusion and wave guide effect etc.

objects, then resulting propagation process is also dependent on the frequency as the dielectric properties of most materials show strong variations over the frequency range of the interest for UWB systems. The diffraction and the scattering on the rough surfaces are other properties which show strong dependence on the frequency [24].

Along with the frequency, the propagation is also affected largely by the distance between the transmitter and the receiver. One of the most important propagation parameter which is a function of the distance is the path loss or the attenuation caused by the propagation to a certain distance d. Different obstacles present in the channel further attenuate the transmitted wave. Generally, the attenuation of the wave is a function of distance d between the transmitter and the receiver which is characterized by the coefficient of the propagation loss, α. The received power decreases in proportion tod^−α. The value ofαin the free space is 2, whereas it varies between 2 and 5 in a non line of sight (NLOS) configuration. In LOS configuration, in the presence of wave guide effects,α is usually less than 2.

1.5.1 UWB Channel Characterization Parameters

Different parameters affect the channel in different ways. The transmitted signal propagating in a given propagation channel can experience constructive or destructive interference, frequency selectivity and Doppler effect etc. depending upon the channel and its variations. Considering a narrow band carrier based signal, the constructive and destructive interference are caused by overlapping of two waves received with certain delay in time. If the received signals are in phase then the amplitude will add up and an enforced signal will be formed, whereas if the received signals are out of phase, then the amplitude will be nearly canceled out resulting in a weaker signal. The phase shift between the two signals depends upon the wavelength of the signal and the difference in the path followed by the signals. In case of mobile displacement, the phase rotation of each path leads to a succession of maxima and minima which represents a signal with fast fading. When this phenomenon applies to a large number of multi-paths, the received signal appears as a random process. As the signal covers a narrow band, we can suppose that all the frequency components of the signal experience similar phase shifts and the power attenuation of all the frequency components is constant throughout the considered band. This phenomenon is also termed as flat fading.

If however, the transmitted signal has a large bandwidth, diverse frequency compo-nents are attenuated in different ways. The signal thus undergoes a frequency selective fading and therefore, the received power varies with the frequency. The bandwidth over which the spectral components of the signal are affected in the same way is called coherence bandwidth or correlation bandwidth. In the time domain, frequency selec-tivity results in a reception of different signals with delays in the order of nanoseconds.

Depending on the bandwidth of the signal, these echoes overlap and thus cause sig-nificant attenuation of the signal. For signals with very wide spectrum, for example

UWB signals, the resolution of multi-path becomes very low and thus limits the in-terference between the different versions of the delayed signal. In this case, the fading power is less important. Some advanced techniques for the reception of the signals such as channel equalization can be applied to maximize the received energy present in the multi-paths [25, 26, 27]. Therefore, we can say that the large bandwidth is responsible for the temporal spread of the transmitted signal. The knowledge of this dispersion is necessary for calibrating the communication systems and avoiding the problems of inter-symbol interference (ISI).

The phenomenon of constructive or destructive interference shows that the prop-erties of the radio propagation channel may differ significantly when the receiving antenna is positioned at different locations. Therefore, the behavior of the propa-gation channel becomes of special interest when the transmit antenna, the receiving antenna (or both), are moving. The Doppler effect corresponds to the frequency shift introduced in the electromagnetic signal caused by the variation of the path.

1.5.2 Channel Modeling and Practical Constraints

In the rest of the chapter, we will use the term ‘narrow band’ for all the signals which are not UWB. The channel impulse response (CIR) of narrow band signals can be modeled as the sum of different multi path components (MPCs) [28]. The channel model will only be deterministic if all of the echoes of the received signal are resolvable. However, because of the narrow bandwidth, not all of the MPCs are resolvable. Therefore, the CIR can be written as:

h(t, τ) = where N is the number of MPCs, while N⁰ is the number of resolvable MPCs where one resolvable MPC consists of k physical MPCs, c_i is the resultant amplitude of the resolvable multi-path components. In the case of UWB systems, CIR differs from the narrow band propagation channel in many respects:

• Due to the fine temporal resolution, the number of physical MPCs which form one resolvable MPC is lower

• Each of the MPCs is subjected to some distortions due to the frequency depen-dent effects

From [24], the CIR for a UWB channel can be written as:

h(t, τ) = XN

i=1

a_i(t)χ_i(t, τ)⊗δ(t−τ_i) (1.12) where χ_i(t, τ) denotes the (time-averaging) distortion of the i_th echo due to the frequency selectivity of the interactions with the environment. No matter how wide

long as the system is band limited, any deterministic CIR can be represented by a tapped delay line model implying that the distortion due to the frequency has not fundamentally changed the description method. It is assumed that the tap spacing is at least as dense as required by the Nyquist sampling theorem (time sampling must be carried out at a rate of at least twice the highest spectral frequency). On the other hand, the number of taps that is required to represent the impulse response can increase due to the pulse distortion.

Depending upon the bandwidth of the UWB signal, the UWB channels can be categorized as sparse and dense. The sparse channel is the one in which the arrival time of certain MPCs is larger than the inverse of the bandwidth of the channel.

Therefore, every resolvable MPC might not carry significant amount of energy. On the other hand, in thedense channel, the inter arrival time of the MPCs is smaller than the resolvable bandwidth. This significantly impacts the design of Rake receivers [24].

Dense orsparse power delay profiles (PDP) depend on the considered bandwidth and the considered environment. Channels with larger bandwidths are more likely to be sparse than the channels with lesser bandwidths. However, large number of reflecting and diffracting objects in the propagation environment will lead to dense channels even for extremely large bandwidths. For instance, dense channels are observed for a large bandwidth of 7.5 GHz in an industrial environment [29], while residential environments [30] show sparse behavior at that bandwidth. More information on UWB channels can be found in [24].