Section 7.4. Interference Analysis: NB on UWB

7.4. Interference Analysis: NB on UWB

In the frequency range above 3.1 GHz, the FCC mask restricts the radiated power to 41.25 dBm/MHz, with a 10 dB bandwidth of 7.5 GHz. In the 2.4 GHz ISM and 5 GHz NII bands, allowed emissions are 40+ dB higher per MHz. With a minimum required bandwidth of 500 MHz, the UWB receiver will see multiple NB waveforms. The range below 1 GHz, of course, is densely occupied by licensed NB systems (AM, FM radios, TV stations, and so on). The wide-bandwidth of UWB systems permits large processing gains. However, the individual pulses have very little energy; as such, NBI signals, which may be expected to be 1012 dB above the noise floor, may cause severe degradation in performance.

The impact of NBI on single-user systems is studied in [28], [29], [35], [37]. A DSCDMA UWB system is shown to resist NBI from IEEE 802.11a OFDM signals [36], where the NBI was modeled as Gaussian noise, as was the MAI. Use of a maximal ratio combiner (MRC) can suppress MAI but not NBI [34]. BER expressions were derived in [38] for a multi-user TH system operating over a pure AWGN channel (that is, no NBI), which could provide a benchmark to study the performance of NBI suppression schemes.

Consider the time-hopped UWB waveform described in (7.10). Assume PAM modulation so that D = 0. Then, the transmitted waveform during the j-th UWB symbol is

Equation 7.35

The received signal during the j-th UWB symbol interval is

Equation 7.36

where w_j(t) denotes AWGN with two-sided psd N_o/2, and i_j(t) is interference.

In AWGN, the optimal receiver is a MF, matched to the symbol waveform,

Equation 7.37

Equivalently, it does the following: it time gates the received waveform, followed by matched filtering with the pulse shape p(t), and then despreading with the spreading code ; that is, the decision statistic is

Equation 7.38

Equation 7.39

where S_j, W_j and I_j denote signal, noise, and interference components.

Because the pulse p(t) has unit energy, and {a_k} is a ±1 sequence, we readily obtain

The noise term W_j is zero-mean Gaussian with variance N_fN_o/2.

The interference term i_j(t) represents the aggregate effect of modulated or un-modulated narrow-band signals. Let f, , and denote the carrier frequency, rate, and power of the -th NBI. The received signal component due to the -th interferer can be written as

Equation 7.40

where

• , is the n-th transmitted symbol of the -th interferer; the symbol set is normalized so that

• g(

• q and t denote the random carrier phase and delay. Phase q ~ p) and t ~ U[0, ).

Because the UWB signal is time-hopped, there is a random time-offset between successive pulses of the same bit. Hence, the starting phase of the NBI carrier changes randomly from the time associated with the beginning of one pulse to that of the next. Given a pulse duration T_p, and an NB symbol duration T = 1/, the probability of a symbol transition within an UWB pulse is given by T. For a UWB pulse of duration 2 ns and a NB signal with a rate of 50 KHz, this probability is 2 -th NBI to the decision statistic

where ,k,l denotes the symbol modulating the th NBI during the j-th UWB bit, with e_j,k, denoting the corresponding delay. The phase term f_j,k, incorporates both the unknown NBI carrier phase as well as the random time shift between pulses due to time hopping. Given that a constant over the duration of the pulse, but this amplitude varies randomly over {j, k, }. For a rectangular pulse shape, = 1. Recall that [10] and P(f) its FT; hence, we have

^[10] For simplicity, we do not consider partial response signaling.

Equation 7.42

For a symmetric pulse shape, P(f) = P(f) is real-valued and

Equation 7.43

The phase variables, the fs, are random variables and are well modeled as being independent (across j, k, l) and uniform over [0, 2p). If q ~ U[0, 2p], we have for m 0, e^jmq} = 0, and Var{e^jmq} = 1 and phase f are mutually independent, we have E{I_j,} = 0 where the expectation is taken with respect to the fs and bs. Next, we compute the variance:

If N_f is large, we can approximate I_j, as a zero-mean Gaussian r.v. with variance . Because the interferers are modeled as independent, their variances will add.

Consequently, we can write the decision statistic as

Equation 7.44

where h_j is zero-mean Gaussian with variance . The BER is then given by

Equation 7.45

where SNR_p := E_p/(N_o/2) is the per-pulse SNR, and SNR := /(We can derive (7.45) in a more heuristic, but insightful, way as follows. Assume that the UWB pulse has a flat FT so that , , and that the PSD of the NBI is flat over , where f_u and f_c are the center frequencies, and W, B are the bandwidths of the UWB and NBI signals. Assume that the support of the NBI PSD is contained within that of the UWB PSD. Then a simple calculation yields the interference power as . With the interference modeled as Gaussian, this leads to (7.45).

As a specific example, consider the Gaussian pulse of Example 7.1, p(t) = exp(t²/2s²_u). The FT of this pulse peaks at f = 0; the FCC specifications require that the PSD be down by 34 dB at f = 0.96 GHz. To ensure this, we need s_u (½)10⁹. The time domain signal and its FT are shown in Figure 7.12 for s = (½)10⁹. The pulse width is about 1012 ns, and the 3 dB and 10 dB bandwidths are 0.28 GHz and 0.52 GHz.

Figure 7.12. The Gaussian UWB Pulse, and its Power Spectrum.

Figure 7.13 shows signal spectra using the Gaussian pulse and the Raised Cosine Pulse (RCF) for different values of symbol period T and excess bandwidth parameter b. These choices of the RCF meet the FCC specs at 0.96 GHz, while utilizing the spectrum more efficiently (ability to transmit more energy). Note also that the Gaussian pulse extends over 1216 ns, the RCF pulse extends over typically 68 symbol periods, so that the RCF pulses have shorter duration. Finally, if the channel has no delay spread, the RCF pulse is desirable because it satisfies the Nyquist criterion.

Figure 7.13. A Comparison of the Power Spectra of Gaussian and RCF Pulses.

Example 7.5

We simulate a pulsed UWB system, with AWGN and narrowband interference (NBI). The UWB signal is the baseband Gaussian pulse p(t) = A exp(t²/2s²), where A is an amplitude factor. Parameter s was set to 2 x 10⁹ so that the pulse bandwidth is approximately 250 MHz; see Figure 7.12. The UWB processing gain, or N_f the number of pulses per bit, is varied in the set {1, 32, 128, 512}. TH-UWB was assumed so that the NBI carrier phase changes randomly from pulse to pulse. The NBI interferer has a carrier frequency f_c = 100 MHz and is modulated by a BPSK sequence at rate 50 Kbps and has a constant interference-to-noise ratio of 10 dB. Figure 7.14 shows BER versus SNR withcurves parameterized by N_f. The solid lines are the theoretical AWGN curves (without NBI) and the symbols denote simulation results. A degradation of about 1 dB is seen in the curves, indicating good resistance to NBI.

Figure 7.14. BER Performance of a Time-Hopped (TH) UWB System with N_f Pulses per Bit, in the Presence of Modulated NB Interference (NBI).

Example 7.6

We simulate a pulsed UWB system, with AWGN and narrowband interference (NBI).

The UWB signal is the baseband Gaussian pulse p(t) = A exp(t²/2s²), where A is an amplitude factor. Parameter s was set to so that the pulse bandwidth is approximately 1 GHz; see Figure 7.12. The UWB processing gain, or N_f the number of pulses per bit, is varied: 4^k, k = 0 : 1 : 4. In TH-UWB, the pulse is duty-cycled T_f T_p and the T_f = T_c = T_p, but an off-time is assumed between bits. The phase of the NBI carrier changes randomly from bit to bit, but not from pulse to pulse within a bit. The random phase-change from bit to bit is a reasonable model when the NBI and UWB rates are not harmonically coupled, even if there is no off-time between bits. Binary PAM was used so that b_j = ±1 and D = 0. The MF receiver is time synchronized and perfect time gating is assumed so that the TH code and spreading code details are not relevant in the absence of NBI. Thus, without NBI, the system performance is equivalent to BPSK-DS, with processing gain N_f. In this example, note that SNR is defined per pulse and is not the SNR per bit, which is N_f times larger. A sampling frequency of 2 GHz was used to digitally implement the MF receiver. This example consists of three parts described here.

1. The NBI consisted of two pure tone interferers with frequencies 400 MHz and 600 MHz. In Figure 7.15, we plot BER versus SNR for both the TH-UWB (solid curves) and DS-UWB (dashed lines). The interference-to-noise ratio (INR) was varied: 5 dB, 10 dB, and 20 dB, and corresponds to the left, middle, and right panels of the figure. The curves are parameterized by processing gain, N_f, the number of pulses per bit, which increases from right to left, 2^k, k = 0, ..., 4, corresponding to markers circles, plus sign, diamond, pentagram and hexagram. The square markers on the dotted lines denote the theoretical AWGN results in the absence of NBI. The UWB pulse duration was 21 ns, and we can verify that in the DS-UWB case, the NBI interferers did not have an integer number of cycles per bit duration; the fractional cycles were either 0.4 or 0.6 for the parameters used in this example. For DS-UWB, the impact of the NBI is minimal when the INR is small or the processing gain (PG) is large, as may be expected. In contrast, the performance of TH-UWB is not as good, but it does provide some NBI resistance (within 3 dB of AWGN bound in the presence of two 5-dB interferers). With TH-UWB, the phase of the NBI changes randomly from pulse to pulse. Equivalently, the unmodulated carrier is now modulated by a random phase sequence. This, in turn, spreads the effective bandwidth of the interferer and its impact on the UWB receiver. A conclusion from this example is that DS-UWB offers better resistance to unmodulated NBI than does TH-UWB.

   Figure 7.15. BER Performance of a Pulsed UWB System with AWGN and NB Interference Versus Per-Pulse SNR. The Panels Correspond to Different INR Ratios, with Curves Parameterized by Processing Gain N_f. Solid Curves Are for TH-UWB, and Dashed for DS-UWB. Example 7.6A Corresponds to Unmodulated NBI (that is, the NBI = Sinusoids).

   
   
   

2. The NBI was now modulated by a BPSK sequence with a bit rate of 50 Kbps. Because the NBI symbol duration of 20ms is much larger than that of the DS-symbol period with maximum PG, 21 x 10⁹ x 256 = 5.4ms, it was assumed that the NBI bit remained constant across each pulse in TH-UWB and across each bit in DS-UWB (we relax this assumption in part (C)). Notice that the performance of the two systems is now virtually identical. Both offer some resistance to modulated NBI, which increases with PG; see Figure 7.16. Note also that the performance of TH-UWB is about the same in both the modulated and unmodulated cases because the random phase change essentially incorporates phase changes due to the modulation.

   Figure 7.16. BER Performance of a Pulsed UWB System with AWGN and NB Interference Versus Per-Pulse SNR, as in Figure 7.15. For Example 7.6B, the NBI Is Now BPSK, whose Bit Boundaries Do Not Occur During a UWB Symbol.

   
   
   

3. The setting is now the same as in part (B), except that now the NBI symbol may change with probability T_p/T_nbi over a TH-UWB pulse, or with probability T_pN_f/T_nbi over a DS-UWB bit. Results are shown in Figure 7.17 and are similiar to those in Figure 7.16.

   Figure 7.17. BER Performance of a Pulsed UWB System with AWGN and NB Interference Versus Per-Pulse SNR, as in Figures 7.15 and 7.16. For Example 7.6C, the NBI is BPSK, and Bit Boundaries May Randomly Occur During a UWB Symbol.