Section 1.3. The I-UWB System Model

1.3. The I-UWB System Model

1.3.1. Overview of the I-UWB System

This section presents the overall system model and notation convention that will be used throughout the book. The basic model for an unmodulated I-UWB pulse train was given in (1.1) and is repeated here

Equation 1.3

Note that we have assumed that the pulse is not distorted by the channel. Thus, p(t) is the pulse that would be observed by the receiver in a distortionless and noiseless channel with infinite SNR. However, this is not necessarily equal to the pulse generated by the pulser circuitry, nor is it necessarily equal to the pulse launched by the transmitter. This is a unique feature of UWB systems and arises from the fact that the transmit and receive antennas can, and often do, distort the pulse shape.

Thus, in our system model we assume that the antenna-induced distortion is included in the received pulse p(t). Recall also that the antenna-induced distortion can change with elevation and azimuth angles. Thus, we assume that such effects will ultimately be included in the assumed channel model.

1.3.2. Pulse Shapes

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

Equation 1.4

where s is the standard deviation of the Gaussian pulse in seconds, and m is the location in time for the midpoint of the Gaussian pulse in seconds. Note that the pulse width, t_p, is related to the standard deviation as t_p = 2ps. An example is plotted in Figure 1.4(a). The first derivative of a Gaussian pulse is also a commonly used analytical pulse shape, due to the fact that a UWB antenna may differentiate the generated pulse (assumed to be Gaussian) with respect to time,^[3] leading to the following pulse shape

^[3] Certain types of antennas, such as the Bicone antenna, can be configured to prevent differentiation of the generated pulse.

Equation 1.5

where k is a constant that determines the pulse width, and we have assumed m = 0. A third model uses the second derivative of a Gaussian pulse or

Figure 1.4. Example UWB Pulses.

These three pulse types are plotted in Figure 1.4. The time axis is arbitrary and depends on the values assumed above for k and s. We should also note that the current FCC rules make UWB transmission most practical in the 3.110.6 GHz band. As a result, the preceding pulse shapes may not be useful for commercial systems. Instead, the Gaussian modulated sinusoidal pulse is more practical. Specifically, the pulse shape

Equation 1.6

where f_c is the desired center frequency for the pulse. An example plot of this pulse is given in Figure 1.5.

Figure 1.5. Example of a Sinusoidal Gaussian Pulse.

1.3.3. Modulation Schemes

I-UWB systems allow for several modulation schemes, including Pulse Position Modulation (PPM) and Pulse Amplitude Modulation (PAM).^[4] A detailed discussion of modulation schemes will be presented in Chapter 5, "Transmitter Design." We introduce them here in order to establish the notation and system model used throughout this book. The transmit signal in the case of amplitude modulation is represented by

^[4] Pulse Amplitude Modulation (PAM) is also referred to as Pulse Position Modulation (PPM), or BPSK in the literature. We have chosen to use the generic PAM, as it allows for a more general discussion of UWB modulation. PPM thus becomes a specal case of PAM (2-PAM).

Equation 1.7

where A_i =d_i(t) now represents the amplitude of the i^th pulse, which is dependent on the data d_i(t) and the specific modulation scheme. A pulse position modulation scheme is represented by

Equation 1.8

where d_i(t) is the time modulation based on the information and d is the base time increment.

As an example, let d_i(t) be anantipodal binary bit stream consisting of +1's and 1's. The transmitted PAM signal will consist of a stream of positive and negative pulses (Figure 1.6a). The transmitted PPM signal will consist of pulses that are shifted either slightly before or slightly after their ideal positions in a regularly spaced pulse train (Figure 1.6b).

Figure 1.6. Example of Modulated UWB Signals Using the Data Sequence {1-1 1 1-1}.

A key characteristic of UWB systems is their low power spectral density. The desire for low power spectral density (PSD) impacts the system model in two distinct ways. First, the pulse rate is often higher than the data rate. In other words, to obtain sufficient energy per symbol while maintaining sufficiently low PSD, multiple pulses will be associated with a single symbol. In this case the received signal is represented by

Equation 1.9

where N_s is the number of pulses per symbol, and the symbol rate is . For PPM systems where multiple pulses per symbol are used, the received signal is represented as

Equation 1.10

The received signal is then modeled as

Equation 1.11

where h(t) represents the channel that possibly distorts the transmit signal and is assumed to have unit average energy and represents the convolution operation. That is, we scale out all gross attenuation effects and include them in the noise power. The noise is assumed to be Additive White Gaussian Noise (AWGN) with power , where SNR is the average signal-to-noise ratio.^[5] In the case where the channel is modeled using a Finite Impulse Response (FIR) filter approach (Chapter 3), the channel is modeled as

^[5] Unfortunately, narrowband interferers located within the bandwidth of the I-UWB signal invalidate the AWGN assumption. For the sake of simplicity, however, we will adhere to the assumption of AWGN in order to provide a means to enable some selected comparison of I-UWB with traditional narrowband communcation systems.

Equation 1.12

where b_i is the amplitude and sign of the i^th path, t_i is the relative delay of the i^th path, d(t) is an impulse function, and N_p is the number of paths. Note that in general we will model these parameters as random variables, as discussed in Chapter 3.

1.3.4. Multiple Access Schemes

When a system has multiple users, we represent the transmit signal for user k as s_k(t). The total received signal is given by

Equation 1.13

where h^(k) (t) is the channel impulse response between the k^th user and the receiver, K is the total number of users considered, and n(t) is AWGN.

In multiuser systems there are many forms of multiple access that will be discussed in Chapter 5. The signal model for TDMA-based (or random access methods) will not differ from that presented here. However, CDMA systems will require additional notation. When pseudorandom amplitude modulation is used to distinguish users, we represent the signal from the k^th user as

Equation 1.14

where and c_i is the pseudorandom code value for the i^th pulse. This value can repeat every symbol (short codes) so that c_i =c_i+Ns, or it can repeat at some much longer interval (long codes).

Pseudorandom codes can also be applied to PPM schemes, and are often referred to as time hopping or pseudorandom dithering. In this case the transmit signal of the k^th user is

Equation 1.15

where the time hopping is accomplished with the sequence , and T_c is the fundamental hopping granularity.

1.3.5. Receiver Decision Statistic

The receiver estimates the most likely transmitted data symbol by using a decision statistic that is a function of the received signal

Equation 1.16

where Z represents the output of the receiver, and the function f (Z) depends on the modulation scheme and receiver structure, as discussed in Chapter 6, "Receiver Design Principles," and Chapter 7, "On the Coexistence of UWB and Narrowband Radio Systems." Additionally, in diversity systems with multiple receiver branches (such as multiple antenna structures or a RAKE receiver) the decision statistic Z will be the sum of several statistics

Equation 1.17

where L is the number of diversity branches, either in time or space, and z_i is the statistic calculated per diversity branch.