Section 5.3. I-UWB Transmitters

5.3. I-UWB Transmitters

In this section, signal models (and system models where needed) that capture all the different kinds of modulation techniques proposed to date for I-UWB are presented. These models include time-hopped pulse position modulation (TH-PPM), time-hopped antipodal pulse amplitude modulation (TH-A-PAM), optical orthogonal coded pulse position modulation (OOC-PPM), direct sequence spread spectrum modulation (DS), transmitted reference (TR), and pilot waveform assisted modulation (PWAM) .

5.3.1. TH-PPM and TH-(A-PAM) UWB Signals

A typical TH-PPM modulated UWB signal can be modeled as [20]

Equation 5.16

The TH-(A-PAM) modulated UWB signal can be modeled as [20]

Equation 5.17

where s^(k) (t) is the random process describing the signal transmitted by the k^th user and w (t) is the signal pulse normalized so that The energy of the transmitted waveform is E_w. The pseudorandom time hopping code k provides an additional shift in order to reduce the effect of collisions in multiple access schemes. The binary information stream is transmitted using a PPM modulation format introducing an additional shift, d, that is used to distinguish between pulses carrying bit 0 and bit 1. for TH-A-PAM. Each information bit is transmitted using N_s consecutive pulses, and the resulting bit rate is R_b = (N_sT_f)^-1. T_c is the chip duration.

5.3.2. OOC-PPM UWB Signals

Optical orthogonal codes are families of binary sequences typically employed to provide multiuser capabilities to "positive systems," which are systems in which the transmitted waveforms cannot be summed to obtain a zero. Refer to [20] for details. These codes are characterized by four parameters, F, K, l_a, and l_c that symbolize respectively the code length, the number of '1's in the code words, and the maximum out of phase autocorrelation and cross correlation values. If , k is the code word assigned to user

5.3.3. DS-UWB Signals

A DS-UWB modulated signal can be modeled as

Equation 5.19

where is the spread spectrum code assigned to user k, and is the data bit of user k.

5.3.4. Transmitter Reference (TR) UWB System

TR systems were first proposed in the 1920s [21]. In a TR system, a pair of unmodulated and modulated signals is transmitted, and the former is employed to demodulate the latter. The receiver for this transmitter can capture the entire signal energy for a slowly varying channel without requiring channel estimation. Another potentially attractive feature of UWB autocorrelation receivers is their relative robustness to synchronization problems [22]. However, fundamental system weaknesses, such as bandwidth inefficiency and high noise vulnerability, coupled with the advent of stored reference and matched filter implementations in the 1950s and 1960s, largely diminished research interest in TR schemes [23]. However, research in UWB autocorrelation receivers has been relatively active in the last two years. A delay hopped, TR communications system was recently built by the research and development center of GE. Experiments show the viability of such a system in an indoor multipath environment [22, 24]. Giannakis, et al., [25, 26], introduced a general pilot waveform assisted modulation (PWAM) scheme, which subsumes TR as a special case. The values of the system's parameters were derived to minimize the channel's mean square error (MSE) and maximize the average capacity. The circumstances under which the UWB autocorrelation-TR system is optimal were also analyzed. We briefly describe the TR and the PWAM I-UWB transmitters here.

TR System Model

The TR system described in [21] and [27] employs binary pulse position modulation (PPM). The transmitted signals consist of N_p UWB pulses, p (t), of duration T_p and energy E_p. The waveforms are comprised of unmodulated pulses interleaved with PPM-modulated pulses. Time hopping is not considered in this model. The signals s₀ (t) and s₁ (t) are equally likely, and are transmitted on t [0,N_p,T_f], where T_f >> T_p). The signals can be written as

Equation 5.20

where t_p is the delay associated with PPM. Moreover, the authors of [21] and [27] assume e_0,i = i mod 2 and e_1,i = 1 - e_0,i.

PWAM Optimization

In [25] and [26], Giannakis, et al., described a general PWAM scheme and derived conditions for which the scheme operates optimally. In these works, minimum channel MSE and maximum average capacity are the two optimality measures. The system model can be viewed as a generalization of the model proposed by Stark, et al., [21], [27]. Every binary symbol is shaped by p(t) and is transmitted repeatedly over N_f consecutive frames, each of duration T_f seconds. The channel is assumed to be static over a burst of duration seconds. Each burst includes up to symbols that are either pilot or information bearing. Pilot waveforms are used at the receiver to form a channel estimate. During each burst, N_s distinct information symbols are sent, corresponding to waveforms. The number of pilot waveforms is thus . The power of the pilot waveform is denoted by P (n_p); likewise, the power of the data waveform is denoted by P (n_d).

The following four constraints must be observed to ensure that the system is optimal:

1. Given the total number of pilot waveforms per burst, , and the total energy assigned to pilot waveforms, e_p, equi-probable pilot power waveforms minimize the channel MSE.

2. Given the total data energy, e_d, and the total pilot energy, e_p, equi-powered information symbols maximize the average capacity, C.

3. Given the total data energy, e_d, the total pilot energy, e_p, and the number of waveforms per burst , the number of pilot waveforms that maximizes the average capacity is given by , defined by: , where

   

   
   

4. With fixed burst size N, number of information symbols per burst N_s, and total transmission energy e, the optimal energy allocation factor, , that maximizes the average capacity is

   Equation 5.21

   

   
   

For a proof of these claims, the reader is referred to [26]. Note that for N = 2, we get N_p = N_s = 1 and for optimal PWAM. This corresponds to a system where the transmitted waveforms are split evenly between pilot and data waveforms of equal power. This resulting PWAM system turns out to be equivalent to the TR autocorrelation system described by Stark. (Note that one of Stark's TR models assumes the channel to be time invariant over 2N_pT_f seconds, which corresponds to N = 2.)