Section 7.2. Interference of UWB on NB: Waveform Analysis

7.2. Interference of UWB on NB: Waveform Analysis

Our aim in this section is to derive an expression for the degradation in BER due to the UWB emitters. If the channel is noise-limited, the effect of the interference is essentially to raise the noise floor. But when the channel is interference limited, a more detailed analysis is required. We will consider the effect of one or more UWB radios on an NB radio. We do this in two steps: (a) detailed waveform analysis, corresponding to a single interferer; and (b) gross power analysis (discussed in the next section).

At the end of the linear receiver processing, conceptually we have a simple M-ary detection problem:

Equation 7.1

where z is the decision variable at the output of the coherent detector, s_k, k {1, ..., M-ary alphabet, g denotes AWGN, and I denotes the effective interference. However, we do not have an expression for the pdf of I. Given that the BER is limited by the worst case, we need to pay more attention to the received waveform itself. Before proceeding with the analysis, we will describe the UWB signal model.

7.2.1. UWB Pulse Model

A popular choice for the basic UWB pulse is the second-derivative of the Gaussian pulse, which takes into account the differentiation effects of the antennas. This also leads to a symmetric pulse with an effectively limited temporal duration. We will see that the details of the waveform p(t) are not critical to the analysis. Let p(t) denote the Gaussian monocycle, [45, 46]^[3]

^[3] Please see Chapter 1 for details.

Equation 7.2

whose Fourier transform (FT) is given by,

Equation 7.3

which peaks at f = f_o. The energy in this pulse is . Conceptually, a pulse-train (which may be dithered to reduce spectral lines, to accommodate user codes, to represent data via PPM, and so on) is convolved with the pulse-shape, so that the power spectrum of the transmitted UWB signal is essentially given by |P (f)|² [47]. The effect of dithering makes the modulation zero-mean and removes periodicities in the data, so that the PSD has no spectral lines. The pulse p(t) and its 'PSD' |P(f)|² are shown in Figure 7.1; notice that the PSD is skewed. The bandwidth of the transmitted signal is given in Table 7.1. To a first-order approximation, the bandwidth is 2f_o, and is centered at f_o. Assuming that the peak of P(f) is normalized so that the FCC EIRP restriction is met, we notice that this waveform does not meet the spectral mask. Of course, the FCC specs can be met by adequately lowering the power, but at the expense of significantly reduced spectral efficiency. Various techniques have been proposed to shape the monocycle to meet the FCC mask [43] [44]. However, as we shall see, the choice of the waveform is not critical. Hence, we will use the Gaussian monocycle as an illustrative waveform.

Figure 7.1. The Second-Derivative-of-Gaussian UWB Pulse, and its Spectrum, with f_o = 2 GHz.

Table 7.1. Bandwidth of UWB Signal using Gaussian Monocycle: Frequencies Normalized by f_o.

Attenuation	Low f	High f	Bandwidth
3 dB	0.6169	1.4415	0.8246
20 dB	0.1955	2.2113	2.0158
40 dB	0.0607	2.7638	2.7031

We note from Figure 7.1 that the UWB PSD will be essentially constant over the bandwidth of a typical NB signal. As specific examples, note that the channel spacing or signal bandwidth is 30 KHz in AMPS, IS-54 and IS-136, 200 KHz in the 900 MHz GSM, 2.36 MHz in IS-95, and 5 to 20 MHz in W-CDMA and cdma2000, which are in the proposed 3G IMT-2000 standards. AMPS/IS-54/136/95 operate in the 900 MHz and 1.9 GHz bands; IMT-2000 is also expected to operate in the 1.9 GHz band. As such, these systems will neither interfere nor suffer interference from UWB systems operating in the 3.1-10.6 GHz band. Interference will come predominantly from the 5 GHz UNII bands, 802.11 WLANs that will operate at 5.3 GHz, and from the proposed 4G systems that will operate above 5GHz. In Japan, there are also reserved radio astronomy bands in the 3260-67 MHz and 3332-39 and 3345.8-3352.5 MHz.

Even over a 20 MHz range, the UWB spectrum is essentially flat; this will certainly be true with more carefully designed pulse shapes such as those proposed in [43] [44]. Consequently, we expect that the effect of the UWB signal is essentially to raise the noise floor. If P_R denotes the total received power from the UWB signal, then the noise floor will be raised by approximately P_RB_nb/B_uwb, where B_nb and B_uwb denote the bandwidths of the NB and UWB signals.

For a typical conventional wideband signal with a 20 MHz bandwidth, centered at f_c = 6.85 GHz, which is the center of the UWB band, and a UWB signal with the minimum 500 MHz bandwidth, we have B_nb/B_uwb = 0.04. For a more typical NB signal with a 30 KHz bandwidth, B_nb/B_uwb = 6 x 10⁵ « 1. As with any spread-spectrum type signal, its impact on an NB signal decreases as its bandwidth increases (assuming constant power).

As far as the signal waveform is concerned, the UWB pulse appears very impulsive if B_nb is small; hence, we should expect to see the impulse response of the IF band-pass filter. Depending upon the precise ratios, some ringing may also be evident. Additionally, the power amplifier may be driven to saturation. We first consider the impact of a continuously pulsed UWB signal and then study the effects of an episodic pulse train (that is, a train of pulses with variable pulse spacing).

7.2.2. Effect of NB Receive Filter

In a typical NB radio, using digital modulation, the same symmetric baseband pulse-shape is used at both the transmitter and the receiver, namely the root-raised cosine filter (RRCF) with transfer function H_rrcf(f) with nominal bandwidth W, roll-off factor or excess bandwidth parameter denoted by g (0g 1), and overall bandwidth g) [48, p. 560].

Taking into account the fact that the front-end band-pass filter and the low-pass filter following the demodulator typically have larger bandwidth than H_rrcf(f), we note the overall filter for the UWB impulse-train is given by,

Equation 7.4

where P(f) is the FT of the Gaussian monocycle [c.f., eqn (7.3)], , and f_c is the carrier frequency for the NB system. If the bandwidth of H_rrcf(f) is very small, then P (f) is approximately constant over the range [f_cW (1 +g), f_c +W (1 +g)], so that G(f)f_c)H_rrcf (f).

7.2.3. BER Analysis

We first consider the effect of an isolated UWB pulse on the performance of an NB receiver. Later we consider the impact of a train of UWB pulses, including effects due to time-hopping.

Let x(t) denote the received signal, after downconversion to baseband. Consider the usual matched filter (MF) plus threshold receiver for BPSK signaling, which is optimal for the AWGN channel. In the k-th symbol interval, the received signal is

where s(t) is the unit energy signal waveform with duration T (corresponding to H_rrcf (f)), E_b is the energy per bit, b_k {-1, 1} is the unknown bit,

Here v represents the 'noise' term and is zero-mean Gaussian, with variance N_o/2. The term u() represents the interference. Assuming that the interfering pulse t) is completely contained within the symbol period and has a relative delay of , we have

where is defined in (7.4), and E_p is the energy in the received UWB pulse. For an NB S(f), is essentially constant over the bandwidth of S(f), so that

Equation 7.6

We illustrate this via an example.

Example 7.1

The NB modulation is BPSK; pulse shaping is with a RRCF with excess bandwidth parameter g = 0.5, bit rate of 50 Kbps (symbol interval T_s = 20ms) and carrier frequency of 500 MHz. The UWB pulse is the Gaussian pulse shape h(t) = exp(t²/2s²), where s was chosen so that the 3-dB point is at 400 MHz. This waveform is somewhat different from that described in (7.2), but the detailed waveform is not important. Figure 7.2 shows the response of the RRCP matched filter to a single UWB pulse. The circles indicate the RRCP itself, indicating that the MF response to a single UWB pulse is a scaled version of the RRCP.

Figure 7.2. The Response of an NB System Matched Filter with UWB Pulse Input. The UWB Pulse Appears as an Impulse to the NB System, Whose Output Closely Approximates the NB System Impulse Response.

Define the SNR impairment factor,

Equation 7.7

Conditioned on a given time offset , the interference term, ) in (7.6) acts as a fixed bias in the decision statistic of (7.5). Conditioned on the interference, the decision variable and variance N_o/2; see Figure 7.3.

Figure 7.3. UWB Single Pulse Interference in a BPSK NB Receiver Acts to Create a Bias in the Bit Decision Variable.

The MF receiver uses a threshold of zero and the bit-error rate (BER) is then given by

Equation 7.8

In deriving (7.8), we assumed NB BPSK modulation for simplicity, but clearly the result is readily generalized to other modulation formats such as MPSK and QAM. Figure 7.4 shows BER as a function of the impairment factor d for nominal SNRs (E_b/N_o) of 4, 6, and 8 dB. Figure 7.5 shows the extra SNR^[4] required to keep the BER constant at target BERs of 10^-2, 10^-3, and 10^-4. The performance loss is worse with higher SNR simply because the standard BER curve is steeper at higher SNR. But, for a fixed d, the required extra SNR does not change significantly with the target BER.

^[4] Extra SNR is defined as the difference between the SNR required to attain target BER in the presence of interference and that required in the pure AWGN case.

Figure 7.4. The BER Versus UWB Impairment Factor d, with Curves Parameterized by the SNR =E_b/N₀ (Bit Energy to Thermal Noise Ratio).

Figure 7.5. The Excess SNR (E_b/N₀) Required to Maintain a Target BER as the UWB Interference Increases (that is, as the Impairment Factor d Increases).

From (7.7) and Figure 7.4, we see that performance degrades as d increases. For a RRCF s(t), it is easy to verify that max _t , where W is the nominal bandwidth. In Section 7.2.1, we saw that the Gaussian monocycle has an approximate bandwidth centered at f =f_o. To a first order approximation, we then have , and from (7.7), we obtain . For W = 50KHz, f_c = 2GHz, . To obtain d = 0.1, we need E_p/E_b = 300; from Figure 7.5, we see that a 3dB SNR impairment occurs when d = 0.33; this corresponds to a ratio E_p/E_b 3300. As expected, significant degradation occurs only when the UWB pulse has large energy because only a small fraction of the UWB energy is passed by the IF filter. A more serious problem is saturation of the power amplifier (PA) due to the large peak power in the UWB pulse. Depending upon its hysteresis characteristics, the PA may need a long recovery time, during which period the received signal is distorted. This suggests the need for high-speed analog blanking circuits in order to cope with dynamic range issues with high-rate A/D convertors. Peak power issues also arise from low duty cycles, that is, when the off time between pulses is long. In this case, the effect of the interference is to wipe out occasional bits, which may be severe depending upon the strength of the coding used. In general, robust signal-processing techniques are required throughout the receiver chain.

If the average UWB pulse rate is significantly less than the NB symbol rate (approximately the NB filter bandwidth), the filter response will settle down between individual pulses and the individual responses will be seen. In a typical NB receiver, RRCF filters will be used and the impulse response duration is approximately six symbol periods, so that a total of about seven symbols will be affected. This situation corresponds to impulsive or burst noise, and bit errors will occur in bursts. If the UWB pulse rate is large, that is, there are multiple pulses per bit, the responses will overlap. The effective interference then depends upon the pulse timings. In the extreme case, this could lead to a (DC) bias in the MF output. Example 7.2 illustrates this.

Example 7.2

The NB system is similar to that used in Example 7.1, within a bit rate of 100 Kbps. The individual UWB pulse parameters are also the same, with a pulse-width of 2ns. The initial pulse has a random offset with respect to (wrt) the time origin. The next pulse occurs U slots later, where U is randomly picked from the set [1, N_c] and the slot duration is the same as the pulse width, T_p. A total of N_f random hop times were generated for each of the four curves shown in Figure 7.6.

Figure 7.6. The Response of an NB System Matched Filter to a UWB Unipolar PPM Input, for Different Random UWB Pulse Rates.

The duty cycle is 1/N_c so that the average pulse rate is 1/T_pN_c, and the average number of pulses per NB symbol is T_b/T_pN_c, where T_b is the average bit duration. Curves corresponding to 5, 50, and 500 pulses per NB symbol are shown in Figure 7.6. In the last case, shown by the '*' curve, the response is essentially constant. In the first case, depicted by the '+' and 'o' lines, the RRCF impulse response is recovered. The markers denote the nominal sampling points. In the intermediate case, shown by the 'x' curve, note that the sampling phase becomes critical. Except in rare cases, the sampling phase (or time offset) is a random variable, and the average BER must be computed by averaging across this phase. In this example, a separate spreading sequence was not applied to the time-hopped pulses, so that they all had the same polarity as the data (+1 here). This example is also representative of uncoded PPM. If the number of pulses is large and bipolar spreading codes are used, the interference would tend to look Gaussian.

The BER in (7.8) is a function of the timing delay ; Figure 7.7 displays the average BER (averaged over , or equivalently over . The curves are parameterized by nominal SNRs (E_b/N_o) corresponding to target BER's of 10^-2, 10^-3, and 10^-4. Note that as d becomes large, the channel becomes interference limited. Because the basic receiver considered here ignores the interference, the BER goes to 0.5.

Figure 7.7. The BER of a Victim NB Receiver, as a Function of the Interference-to-Signal Ratio (ISR), and Averaged over Random Timing Offsets. The Curves are Parameterized by Nominal BER. For ISR Above 10 dB, the Receiver Becomes Interference Limited.

If the UWB pulse rate, R_p, is very small, not every NB symbol is affected by the interference. With T_s denoting the NB symbol duration, the probability that interference is present is given by p_c =T_sR_p; the average BER is then given by

Equation 7.9

7.2.4. Time-Hopped Case

Many UWB radios use time-hopping to accommodate multiple users and randomize collisions in the asynchronous case. Time-hopping prevents complete collision of pulse trains when users are synchronized. Because the individual UWB pulse may have little energy, multiple pulses are transmitted for each data symbol. A DSCDMA type spreading code may be used with these multiple pulses. A data symbol is conveyed by N_f data modulated pulses, each with duration T_p. Each pulse is transmitted in a separate frame which has duration T_f =N_cT_c, where T_c T_s,u =N_fT_f =N_fN_cT_c, so that the symbol rate is R_s = 1/(N_fN_cT_c). In addition to facilitating multiple access, TH also provides some LPD capabilities. TH shifts the position of the UWB pulse from frame to frame. Let c_k [0_c1] denote the chip position of the pulse in the k-th frame^[5]. Let a_i, 0 _f, denote a binary spreading code that may be used with or instead of the TH code, and b_j,u denote the j-th binary bit in the UWB stream. Finally, let D denote the time shift if PPM is used, and d_i := (b_i,u1)/2. Then the transmitted waveform during the j-th UWB symbol interval is given by [45, 46]

^[5] We have implicitly assumed symbol periodic TH code; this is not necessary, and the analysis carries over to long codes.

Equation 7.10

where E_p denotes the energy per pulse, so that the energy per UWB bit is N_fE_p. The decision statistic can now be written as (c.f., eq. (7.5))

Equation 7.11

where the interference term is now

Equation 7.12

where t_d is propagation delay, and the sum over (j,) includes all UWB pulses seen within the d is now

Equation 7.13

It is important to note that the BER can vary from symbol to symbol and that this BER must be averaged across the UWB symbols, b_j,u, the hop codes c, the spreading codes t_d [26].

7.2.5. Simulation Results

The following examples demonstrate the effects of interference from multiple UWB emitters on an NB BPSK receiver.

Example 7.3

The UWB pulse was modeled as a pure Gaussian pulse, p(t) = exp(t²/2s²). This waveform is somewhat different from that described in (7.2), but the detailed waveform is not important, as seen earlier. Parameter s 1_uwb was adjusted so that the spectral energy is largely confined to the bandwidth [1, 1] GHz. The pulsewidth is about 2 ns. A sampling rate of 2ms was used to implement the NB receiver simulation.

The BPSK signal had a rate of 50 Kbps, that is, T_b =40ms, and carrier frequency f_c = 500 MHz. The receiver bandwidth is extremely narrowband with respect to the UWB pulse bandwidth, with an approximate fractional bandwidth of (50x 10³/10⁹)x 100 = 0.005%. A RRCF with g = 0.5 (50% excess bandwidth) was used.

BER estimates, based on 10⁵ BPSK symbols, are shown in Figure 7.8. Three cases are shown: AWGN only, AWGN plus a single UWB interferer, and AWGN plus 10 UWB interferers. Each UWB interferer is assumed to have random non-overlapping pulse times, and the pulses are randomly bipolar. The UWB pulse rate is taken to be one fourth of the BPSK bit rate; thus, the average spacing between the pulses is 160ms which corresponds to a very small duty cycle, 2x10^-9/160x10^-6 = 1.25 x 10^-5. The pulse energy was set to E_p = 10⁴, the energy per bit to E_b = 1, and the AWGN variance was varied to obtain the target SNR. Due to the very small amount of energy that passes the IF filter of the BPSK system, significantly large pulse energy is required to degrade performance.

Figure 7.8. The BER Performance of a Victim NB Receiver in the Presence of One and Ten UWB Interferers. The Significant Level of Interference only Comes About Due to the Very High Power UWB Pulse Levels in This Example.