Section 7.5. Interference Analysis: UWB on UWB

7.5. Interference Analysis: UWB on UWB

Under appropriate conditions, properly designed time-hopping and DS-CDMA type spreading codes can mitigate interference due to multiple users; however, in the presence of asynchrony, performance may degrade significantly. If the duty cycle is small or equivalently the load is light, that is, the number of active users N_u « N_f, random time hopping results in nearly orthogonal (or collision-free) low-rate multiple users.

A transmitted symbol corresponding to N_f pulses per bit occupies a total of N_fN_c chip slots. In the traditional TH scenario, exactly one chip slot per frame (consisting of N_c chip slots) is non-zero. Generalizing this, we let a_i, i = 0, ..., N_fN_c 1, denote the length N_fN_c spreading code. Rather than insisting that N_f chips be non-zero, we will require that the expected number of non-zero chips be aN_fN_c per symbol. Thus, we model a_i as i.i.d. random variables, taking on values 1, 0,1 with probability mass function (PMF)

Equation 7.46

where . The PMF is depicted in Figure 7.18. In a given chip slot, a chip (or pulse) occurs with probability 2a. If the chip occurs, it takes on values ±1 with equal probability. The case a = 1/2 corresponds to conventional non-episodic DS spreading. The duty cycle is 1/2a. If the processing gain N_f is fixed (as is the usual case), the average symbol duration is N_f/2a chip slots or N_fN_cT_c/2a s.

Figure 7.18. A Ternary Random Process Model (Pdf Shown at Bottom) May Be Used to Model Episodic Bipolar Pulsed Transmission. Cases with a < 0.5 (Top) Correspond to Episodic Transmission (Off-Times Between Pulses), Whereas a = 0.5 (Middle) Corresponds to a Conventional Binary Random Process Model (No Off-Times Between Pulses).

This ternary PMF is very useful in capturing the effect of multiple access interference in a lightly loaded system. Specifically, if the user of interest is k, the receiver synchronizes itself to the k-th user's (random) hopping sequence, which it is assumed to know. Within any of the N_f active chips in a symbol, the receiver will see interference from one or more users. With N_u denoting the number of active users, the probability that any one given user causes a chip collision is 1/2a. The per-chip collision probability is then (N_u 1)/2a. The duty factor 1/2a effectively reduces the interference seen by the receiver.

Let k = 0 denote the user of interest, with N_u being the total number of users. A Chernoff bound on the BER was derived in [39], and is given by

Equation 7.47

where E₀ is the desired user's energy per pulse, and I_o denotes the effective interference,

Equation 7.48

Note that the effective MUI is reduced by the duty cycle factor, 2a. In [39], this approach was used to obtain bounds on BER when training is used to obtain estimates of a random multipath UWB fading channel, which is then used in a RAKE receiver and in the presence of MUI.

Example 7.7

[39] We simulated a 10-user system, with processing gain N_f = 128. The desired user's SNR, SNR₁ = 10log ₁₀(E₁/s²_v) was varied, while the remaining 9 interfering users have SNRs fixed at 0 dB. We implemented a matched filter receiver with threshold at zero. The interfering bits were generated according to the ternary PMF to simulate the effects of random time hopping codes. Figure 7.19 shows simulation curves and the Chernoff bound for two values of a. The bound becomes increasingly tight for meaningful BER's ( 10a = 0.003, the performance is essentially that of the MUI-free case. In AWGN, the single user theoretical BER is given by . This bound is also shown in Figure 7.19, virtually coinciding with the a = 0.003 simulation.

Figure 7.19. Chernoff Upper Bound on the BER of a UWB System Employing a Single-User RAKE Receiver, in the Presence of Multi-User Interference, and Compared with Simulations. All Users Have Processing Gain n = 128 Pulses/Bit. Interfering Users Have SNR = 0 dB, while the Desired User Per-Pulse SNR is Varied over [20, 8] dB. The Bounds are Increasingly Tight as the SNR Increases.

Here, the time-hopping pattern was considered to be random and the modulation A-PAM. With the hop pattern modeled as deterministic, BER expressions have been derived for PPM in [45] assuming that the MUI is Gaussian, and in [38] without this assumption.