Section 7.3. Aggregate UWB Interference Modeling

7.3. Aggregate UWB Interference Modeling

In the preceding, we derived detailed expressions for the BER of an NB receiver when subjected to interference from a single UWB emitter. In this section, we consider the aggregate effect due to multiple interferers. We derive a pdf for the asymptotic aggregate interference, which would be useful in a link budget analysis.

7.3.1. Received Power

Although narrowband propagation models do not readily apply to the UWB signal (see, for example, [49-51], the propagation channel is well-modeled as linear, and the narrowband propagation model can be applied because the NB receiver only sees a narrowband interfering signal.

A gross characterization of the geometric path loss is given by

Equation 7.14

where P_R denotes the received power at distance d from the source, b is the path loss exponent (typically between 2 and 6), and P_o is the power measured at distance d_o from the transmitter.

Consider a single UWB signal, with the Gaussian monocyle pulse, p(t). As we saw earlier, the peak of the UWB signal spectrum is at f =f_o. Assuming P (f) is approximately flat over the NB signal bandwidth, [f_cWf_c +W], with mean power P²(f_c) given by (7.3), the interference power due to the UWB pulse is easily found to be

Equation 7.15

where P_R is the total received power from the UWB source, and h = 2W/f_c« 1 for the typical NB receiver. Figure 7.9 shows the normalized interference versus the frequency ratio f_c/f_o. As expected, the interference is worst when the spectral peak of the UWB pulse matches the center frequency of the NB signal and decreases only slowly as the UWB peak shifts away from the center of the NB signal. Comparing the previous equation with (7.3), we note that this normalized interference is not the PSD of the UWB signal.

Figure 7.9. Normalized Interference Versus Normalized Frequency for a Single UWB Interferer into a Victim NB System. The Interference Is Worst When the UWB Spectral Peak Matches the NB Victim Center Frequency.

In order to assess the aggregate effect of a number of radios, we consider the following scenario. But it should be noted that the following analysis is not confined to UWB signals. We assume that the receiver uses an omni-directional antenna. The transmitters all have the same transmit power and are uniformly distributed, with density r per square unit, over a concentric ring centered at the receiver with inner and outer radii R_min and R_max. Figure 7.10 illustrates this. The uniform distribution implies that the number of transmitters in an area A is Poisson distributed with rate parameter l =rA. The inner ring represents a zone of exclusion and models the minimum separation that one might expect between the radios. The transmitted signals are assumed to be independent of each other. Additionally, it is assumed (although not strictly needed) that all transmitters use the same modulation format. The assumption of equal transmit powers is reasonable in a scheme where power control is difficult. The receiver's basic operation would be to project the received signals onto the signal space, which describes the signal of interest. For example, in a conventional M-ary narrow-band system, this projection would consist of a band-pass filter (BPF), followed by down-conversion to baseband, low-pass filtering to get rid of the double frequency terms, and matched filtering with a bank of filters corresponding to the M waveforms.

Figure 7.10. Physical Layout for UWB Aggregate Interference Modeling. The Victim Receiver Is at the Center of the Circle, with Interferers Uniformly Distributed in the Annulus with Inner Radius R_min and Outer Radius R_max.

Mean Received Power

The UWB emitters are randomly located and asynchronous. As such, their complex baseband signals (after front-end processing in the NB receiver) will not be in phase. Because these signals are independent, their powers will add at the receiver. The number of transmitters and their locations, are random; hence, the total received power is also a random variable. With P_o denoting received power at reference distance d_o, as described previously, the expected total power received by the NB radio at the origin from all the transmitters within the annulus is given by

Equation 7.16

where we recall that b > 2. In the last equation, the first term describes the received power due to a transmitter at a distance R_min and the second term describes the aggregate effect due to all the transmitters. Note that because r has dimensions of 1/area, the second term is dimensionless. The term pR²_minr represents the effective number of sources within a circle of radius R_min; the denominator is related to the path loss exponent. Thus, the aggregate effect is represented by a single transmitter at the minimum distance, R_min, whose received power at the reference point is P^'₀, given by

Equation 7.17

We next consider the moment generating function (MGF).

MGF for Received Power

We just evaluated the expected value of the total received power, P_R. One way to characterize the random variable (r.v.) P_R is to evaluate its moments. Following the steps leading to (7.16), we obtain for m real,

Equation 7.18

where we have assumed that bm > 2. When m = 1, we recover the result in (7.16). Note that the limiting moments do not exist if bm < 2, that is, the lower-order moments, m < 2/b do not exist. This implies that the pdf is very peaked at the origin.

The variance of the received power is given by^[6]

^[6] An apparently convenient representation is obtained if the reference distance is also at the edge of the exclusion zone, that is, d_o =R_min. In this case, we have

Unfortunately, this masks the effect of shrinking the exclusion zone, that is, this representation cannot be used in the limit R_min 0. Hence, we will not pursue this approach.

Equation 7.19

For b 2 (corresponding to some reported indoor applications with LOS), some of the lower-order integer moments may not exist. For

Unfortunately this does not lead to a closed-form expression for the pdf.^[7]

^[7] The pdf has k-th order moment, 1/(kb2).

7.3.2. Asymptotic Pdf of Aggregate Noise

We can conjecture that the aggregate received signal is heavy-tailed. The rationale is that the aggregate of impulsive processes must be impulsive. We establish this analytically.

In the preceding, we assumed that the transmitters are Poisson distributed over the plane. Let l denote the Poisson rate parameter and n denote the dimension parameter. Thus n = 2 for sources in the plane, n = 3 for sources in a volume. If l denotes the rate of a n-D homogeneous point process (PP), with points located at distances r_i from the origin, represent occurrence times on the line with rates pl (n=2), and 4pl/3 (n=3) [52].

Theorem 1 [53, Theorem 1.4.5]. Let {Wi}, and {G_i} be independent sequences of random variables, where G_i is the sequence of arrival times of a unit rate Poisson process. If the W_i are positive valued with E{|W|^a} < , for some (0, 1), we can conclude as a special case of [53, Theorem 1.4.5] that the random variable

Equation 7.21

converges almost surely to a positive-valued alpha-stable r.u., with index a, and dispersion parameter s^a =E|W|^aG(2a)cos(pa/2)/(1a) where G(.) is the Gamma function.

The second characteristic function of the r.v. Z is ([53, p. 7])

Equation 7.22

With r_i denoting the distances of the Poisson distributed interferers, the total normalized received power is

Equation 7.23

The positive-valued random variables W_i represent the effects of shadowing and/or fading, assumed to be independent from transmitter to transmitter. The r.v.s rⁿ_i are Poisson distributed over [0, ). Applying Theorem 1, we conclude that the normalized aggregate received power converges to a positive stable distribution with characteristic exponent or index a =n/b. Because 2 < b < 6 typically, we have 1/3 < a < 1 for sources in the plane^[8]. Note that these pdfs are heavier-tailed than the Cauchy.

^[8] For sources in a volume, n = 3, we have 3/4< a < 3/2. The r.v. Z converges to a stable distribution with index a = 3/b, and Theorem 1 has to be modified slightly.

Although a closed-form expression (CFE) for the characteristic function of Z is given in (7.22), a CFE for the pdf exists only when a = 1/2, yielding the Levy distribution whose pdf is [53, p. 10]

and cdf is

where Q(z) is the tail probability of the standard Gaussian random variable. Recall that if u is a zero-mean unit variance Gaussian r.v., then z can be generated as s/u².

Although the pdf does not exist in closed form for other values of a, a series expansion may be found in [54, Sec XVII.6],

Equation 7.24

for 0 < a < 1 and z > 0.

From the series expansion, we note that the tails of the pdf decay algebraically as 1/z^1+a; as a consequence, E{Z^m} < , only for m < a. Because a < 1, the mean is not defined, and neither is the variance.

The preceding analysis is an asymptotic one because we assumed that R_min = 0 (that is, there is no exclusion zone) and that R_max = . In effect, the pdf becomes heavy-tailed because the interferer is allowed to come arbitrarily close to the victim receiver. If an exclusion zone R_min is small. Simulations not shown here indicate that even with 1020 interferers, the stable model is a very good approximation. Practically, the implication is that the aggregate signal is strongly non-Gaussian. We note that recent reports [14, 15] describe the signal due to a single UWB source at an NB receiver as being much heavier-tailed than the Gaussian. The index a quantifies this heavy-tailedness: As a decreases (that is, as decay exponent b increases), the pdf becomes more heavy-tailed and more peaked at the origin. If the aggregate power is small enough, this can be lumped with the Gaussian thermal noise. Optimal detection in non-Gaussian noise involves non-linear pre-processing to eliminate the deleterious effects of the heavy-tailed noise. Robust algorithms for this are described in [5557] and references therein.

It is of interest to compare the pdf of the aggregate interference with that of a single interferer. To this end, we derive the pdf of the location of the nearest interferer. We follow the approach in [41]. The probability that there is no interferer within distance d of the receiver is also the probability that the nearest interferer is distance d away. Because the sources are Poisson distributed with rate r, we have

The received power from a source at distance d is P_R = P_o(d_o/d)^b, where we recall that P_o and d_o are reference values. Thus Pr(P_R < z) is the probability that there are no interferers within distance d = d_o[z/P_o]^1/b, yielding

where we used a = 2/b. For z large, the pdf behaves as 1/z^1+a and agrees with the alpha-stable pdf derived for the aggregate interference. In other words, if the scenario is interference limited, the pdf of the interference is adequately described by the nearest interferer that, under the equal transmitted power assumption, dominates the received power.

7.3.3. Amplitudes: Aggregate Pdf

We now derive the pdf for the complex amplitude due to the interferers. The received signal passes through the NB receiver's filters; it is converted to complex baseband, and correlated with a bank of M matched filters. Let I_k denote the vector output of the MF bank due to the k-th interferer. Random vector I_k is well-modeled as circularly symmetric zero-mean complex Gaussian. The sources are assumed to be Poisson distributed. The development in [52] is applicable here. The decision variable is now a vector,

Equation 7.25

with I being spherically symmetric and alpha-stable, again with index a = 2/b. Note that z is a M-dimensional complex vector due to the M-ary receiver. The joint second characteristic function of this Z is given by

Equation 7.26

7.3.4. Bernoulli and Poisson Models

In a TH-UWB system, a pulse occupies one of T_h chip slots in each frame. The probability of slot occupancy in a given frame is p = 1/T_h. If we relax the requirement of one pulse per frame (see the discussion of episodic signaling in Section 7.5) and consider time in units of chip-slot widths, the sequence of pulses constitutes a Bernoulli random process, with probability of occurrence p = 1/T_h. In [25], we had considered this model briefly. If p is small, the normalized kurtosis,

Equation 7.27

indicating that the interference is strongly non-Gaussian. This non-Gaussianity can be exploited to estimate the frequency-selective channel encountered by the UWB signal. The k-th order normalized cumulants are roughly p^{1 k/2}.

A limiting case is the Poisson process model. Let t_k denote the arrival times of the UWB pulses at a receiver. Given that the UWB signals are time-hopped, episodic, and asynchronous, it is reasonable to model the pulse times as the arrival times of a Poisson process. The effects of multipaths can be included in this model. To facilitate analysis, let us assume that the delays and amplitudes associated with each path are independent of one another, and identically distributed. Let l(t) denote the rate (pulses per second) of this inhomogeneous Poisson process (IPP), t_k the arrival times, a_k the amplitudes, and p(

Such a model is called a marked point process (MPP) with multiplicative marks. The marks are the random variables, the a_ks. This is a special case of an MPP that we had considered in [58]. More recently, an unmarked homogeneous point process (HPP) model (a_k 1) was considered in [60].

In an NB receiver, the signal x(t) would be filtered. If the bandwidth of the NB system is small relative to that of x(t), it is reasonable to model the FT's of the individual pulses p_k(t) to be flat over the NB system's band-pass region, [f_c W, f_c + W]. Thus, the effective signal can be represented as

Equation 7.29

where h_nb(t) denotes the NB receive filter. The cumulants of this process are given by [58, eq. (9)] [59]

Equation 7.30

For a HPP, l(t)

If h(t) is band-pass, H(0) = 0, and E{x_f (t)} = 0. The variance is given by

Equation 7.32

for a receive filter with unit energy. The kurtosis is given by

Equation 7.33

This in turn yields the power at the output of an envelope detector

Equation 7.34

which follows readily from for an ideal bandpass filter that has support over . For an unmarked PP, according to [60], these results are validated by experimental results reported in [12]. The process x_f (t) is asymptotically Gaussian, and results on rates of convergence may be found in [61] [62]. When the pulse rate is small, the UWB interference appears as impulse noise and degrades the BER of the NB system. When the arrival rate is large, the Gaussian approximation holds and all NB bits are impacted about the same. Effectively, the noise floor is raised. If the modulation is not zero-mean (such as OOK or PPM without time-hopping), additional terms will appear on the right-hand side of (7.34) corresponding to lines in the spectrum of the interferer [41].

7.3.5. Simulation Examples

Example 7.4

Here we consider the aggregate effect of Poisson distributed sources. The UWB source signal strength is adjusted to ensure a BER of BER_u at the UWB receiver at a distance d_u away. Here, d_u represents the maximum required range of the UWB radios. BPSK modulation is used. N_f pulses are transmitted per UWB bit; one pulse per frame. The pulse width is T_p, the frame length is T_f = N_cT_p. The pulses may be time-hopped by using symbol periodic or long-code hopping sequences. P_R denotes the received power per pulse, so that the received energy per UWB bit is E_b = P_RT_pN_f. Assume that P_R is such that the target BER is achieved in AWGN. The AWGN noise level is assumed to be identical at both the UWB and NB victim receiver (VR), this is a reasonable assumption (difference in noise power can be accommodated, but this facilitates the analysis). Let N_o/2 denote the two-sided noise power-spectral density. With P_o denoting received power at reference distance d_o from the UWB source, the received power at the UWB receiver is P_R,u = P_o(d_o/d_u)^b, where b 2 is the path loss exponent. Let P_R,n = P_o(d_o/d_n)^b. Because the UWB pulse shape does not play a critical role in the interference analysis, assume that it is rectangular; hence E_b = T_pN_fA², where A is the received pulse amplitude, that is, P_u = A². Hence,^[9] E_b/N_o = erfcinv² (2BER _u) and (2BER_u). So this is the received UWB power at distance d_u. Hence, at distance d_n, the received power is P_u,n = P_u(d_u/d_n)^b. The power due to the interference after the NB receive filtering is approximately given by P_i = P_u,nB_u/B_n = P_u(d_u/d_n)^bB_u/B_n, where B_u and B_n denote the bandwidths of the UWB and NB signals. Let the NB receiver be at the origin with UWB interferers Poisson distributed a minimum distance R_min beyond the victim receiver (VR). The squared distances from VR, d²_u,k, are now Poisson arrival times with rate pr. The received powers are now P_u(d_u/d_u,k)^b, before the NB receive filter.

We consider a scenario with UWB parameters d_u = 20m, BER_u = 0.001, T_p = 2ns, that is, B_u = 1 GHz, and N_f = 100, N_c = 100 (1% duty-cycle), and various values of r = 10^k, k = 2 : 6. r = 10⁶ corresponds to a UWB density of 1 radio per square km. The NB-BPSK signal BW was fixed at 50 KHz, and the path loss parameter was b = 3.5. We consider two values of the exclusion radius, d_min = 20, and d_min = 50, with d_max = 1000m. The aggregate interference can be well-modeled as Gaussian, taking into consideration the large number of interferers and the effects of the NB front end filter. Figure 7.11 shows the BER curve for the victim receiver, for d_min = 20m (left panel), and d_min = 50m (right panel). The curves are parameterized by r. Also shown is the theoretical AWGN BER curve. As expected, the degradation is small if UWB density is low, the UWB duty cycle is low, the UWB radio range is short, or the exclusion radius is large.

Figure 7.11. Effect of Aggregate UWB Interference on an NB Receiver, with Curves Parameterized by the Density of the UWB Interferers.

Figure 7.11 shows BER versus SNR; the curves correspond to different values of the density of interferers.

^[9] In the following, x = erfcinv(y) denotes the unique inverse solution to y = erfc(x) .