The Importance of Fractional Bandwidth in Ultra-Wideband Pulse Design. Matthew Welborn and ... systems. The design of high-speed wireless networks for.
The Importance of Fractional Bandwidth in Ultra-Wideband Pulse Design Matthew Welborn and John McCorkle XtremeSpectrum, Inc. 8133 Leesburg Pike, Suite 700 Vienna, VA 22182
short review of typical UWB signal characteristics. Sections III and IV describe the effects of wide fractional bandwidth on fading due to multipath and section V relates these results to the concept of channel coherence bandwidth. Section VI presents measurements of an actual system demonstrating the variation of received signal strength due to multipath interference.
Abstract - Ultra-wideband (UWB) communications
systems provide extremely high data rates while using low transmit power relative to narrowband systems. Recent research concludes that such systems have characteristics that make them appropriate for highspeed wireless network applications. One of the distinguishing aspects of the UWB signal is an extremely high fractional bandwidth when compared with conventional narrowband systems. This paper describes how a large fractional bandwidth leads to lower worst-case fading in the presence of multipath for UWB communications systems, thereby providing a significant advantage for a low-power wireless systems. Also included are measurements made using an actual UWB communications systems showing the magnitude of signal strength variations due to multipath interference. I.
II. CHARACTERISTICS OF UWB SYSTEMS In general, UWB systems use signals that are based on trains of short duration pulses formed using a single basic pulse shape. For such a signal, the interval between individual pulses can be uniform or variable, and there are a number of different methods that can be used for modulating the pulse train with data for communications. One common characteristic, however, is that the pulse train is transmitted without translation to a higher carrier frequency, and so UWB is sometimes also termed “carrier-less radio”. In other words, a UWB system drives its antenna directly with a baseband signal.
INTRODUCTION
This paper presents an analysis of the role of fractional bandwidth1 in providing robust performance in multipath environments for ultra-wideband communications systems. The design of high-speed wireless networks for indoor environments presents the system designer with a number of very challenging requirements. This system must provide low power consumption (to support mobility battery operation for mobility), low implementation complexity and cost, while still providing high performance and data rates in an indoor environment [3]. The key characteristic of such indoor wireless channels is multipath/fading due to multiple scatters. In particular, indoor channels often result in values for multipath delay spread is shorter than in outdoor environments, and therefore relatively wide channel coherence bandwidths [6].
Another important point common to UWB systems is that the individual pulses are very short in duration, typically much shorter than the interval corresponding to a single bit. We can represent a general UWB pulse train signal as a sum of pulses shifted in time: ∞
s( t ) =
a k p( t − t k )
( 1)
k = −∞
Here s(t) is the modulated UWB signal, p(t) is the basic pulse shape, and ak and tk are the amplitude weight and time offset for the k pulse. Because of the short duration of the pulses, the spectrum of a typical UWB signal can be several gigahertz or more in bandwidth. As with any pulse-amplitude modulation (PAM) system, we can easily derive the PSD of the transmitted signal under the assumption that the data symbols are random (independent and identically distributed) and that the pulses are uniformly spaced in time (i.e tk = kTs, where Ts is one symbol interval). In such a case, the PSD is [7]:
For n UWB system, one of the most important design considerations is the choice of the fundamental pulse shape used to generate the UWB signal. In this paper, we analyze the effect of one aspect of this signal design, the signal fractional bandwidth, on the performance of the system in a multipath environment. Section II provides a
S ( f ) = FT {s( t )} =
1
The definition of fractional bandwidth is the ratio of signal bandwidth to center frequency, or more specifically, 2(fHfL)/(fH+fL). Here fH and fL are the are the upper and lower band edges, measured at some fixed level (e.g. -10 or -20 dB) relative to the peak level of the signal.
σ2 Ts
2
P( f ) +
µ2 Ts
2
P( f )
∞
2
k = −∞
P
k k δ f− T T
(2)
0-7803-7400-2/02/$17.00 © 2002 IEEE
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Here σ2 and µ are the variance and mean, respectively, of the symbol sequence ak and δ(t) is a unit impulse or Dirac delta function. The first term on the right-hand side of (2) represents the continuous portion of the PSD and the second terms shows the presence of spectral lines in the transmit signal [2]. If we further assume that the data symbols values are equi-probable and have zero mean, then the expectation E{ak}=µ is zero and the spectral lines vanish as (2) simplifies to:
S( f ) =
σ2 Ts
P( f )
2
1
0 0
1
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10 10
Volts Volts
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dBdB
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1 1 55 4 3 2 1 0 1 2 3 4 5 4 3 2 1 0 1 2 3 4 5 nanoseconds nanoseconds a a
40 0 40 0
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0.2 0.2
0.3 0.3 Hz Hz
0.4 0.4
0.5 0.5
Figure 1: A Gaussian monopulse, p1(t), and its Fourier transform, P1(f).
(3)
In either of the above equations, whether there are spectral lines present or not, it is clear that the PSD of the UWB signal is primarily determined by the spectrum of the pulse, P(f). For this reason, we see that in such a UWB system, the fundamental relationship between the time-domain pulse shape, its Fourier transform and the PSD of the modulated signal is much the same as for a narrowband PAM signal.
The time and frequency domain equations for this pulse are given here:
p1 ( t ) =
A e te tp
−
1 t 2 tp
2
(4)
and
At this point, we are ready to further analyze how the shape and spectrum of the UWB pulse will directly influence not just the PSD of the transmitted signal, but also the performance of the UWB system itself in a multipath environment. In particular, we will show that it is the fractional bandwidth of the UWB pulse that provides much of the superior performance for a UWB communications system.
P1 (ω ) = A 2π e t p ω e
−1 tpω 2
(
)2
(5)
This basic pulse is used to derive an entire class of pulses by taking successive derivatives of the timedomain waveform. We will refer to these different pulses by using subscripts:
2
dn A e p n +1 ( t ) = n te tp dt
III. WIDE FRACTIONAL BANDWIDTH In anticipation of regulating UWB systems for widespread commercial use, the FCC has proposed that UWB systems be permitted to operate on an unlicensed basis at extremely low transmit power levels under the guidelines of Part 15 of the Code of Federal Regulations. As part of this proposal the FCC has tentatively defined UWB systems as those that have bandwidths exceeding 25% of their center frequency, or 1.5 GHz, whichever is less. At the time of this writing, formal proceedings to evaluate this proposal are ongoing.2
1 t − 2 tp
(6)
These pulses are shown in Figure 2 for values of n up to n=20. At this point it is helpful to note a few properties of the different pulses in this class. First, from the time domain plots it is clear that as the order of the derivative increases, the number of zero crossings in the pulse also increases. As the pulse order increases, the pulses begin to resemble sinusoids modulated by a Gaussian pulse-shaped envelope. From this we would suspect that the fractional bandwidth for each of the successive pulses decreases, since more zeros crossings in the same pulse width would correspond to a higher “carrier” frequency sinusoid modulated by an equivalent Gaussian envelope. This suspicion is correct. In Figure 3, we see a plot that shows the fractional bandwidth of the different pulses as a function of the measurement point (i.e. how far down from the peak of the spectrum we choose to measure the signal bandwidth).
To understand the effect of the fractional bandwidth on the performance of the UWB system, we now present a class of pulses that can be used to produce waveforms with a wide range of fractional bandwidths. Using this class of pulses, we then analyze the effect of a wide fractional bandwidth on the worst-case severity of fades caused by multipath interference. This class of pulses is based on the Gaussian monopulse shown in Figure 1. 2
A discussion of the UWB proceedings is beyond the scope of this paper. Specific proposals and a discussion of relevant issues can be found in the Notice for Proposed Rulemaking for these proceedings, FCC document number 00-163, and other public comments are available under FCC docket number 98-153.
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8 8
The worst-case for this fading phenomenon with two component pulses occurs when the components have equal amplitudes. In such a case, the waveforms (pulses in this case) will be added together with a relative time offset equal to the path delay difference between the two paths. Because a UWB system is baseband, the two received pulse can have either the same or opposite polarity.
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1 10.5 0.50
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-1 -1 -1 -0.5-1 -0.5 -1 -0.5 -0.5 -1 0.5 10.5 -1 -0.5 0 0.5 1 -0.5 1 -0.5 0 1 -0.5 0 0.5 1 -1 1 -0.5 0 0.5 1 -1 1 -0.5 0 0.5 1 -1 1-1 -0.5 0 0.5 1 -1 -1 -1 -1 -1 -1 -1 -1 0.5 0.5-0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 0.5 -0.5 0 0.5 1 -10.5-0.5 0 0.5 1 -1 -1 -0.5 0 0.5 1 -1 0.5 0.5 -1 -0.5 0 0.5 1 1 1 1 1 1 1 0 0 0 0 0 0 0.5 0.5 0.5 0.5 0.5 0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -0.5 -1 -0.5 0 0.5 -0.5 -0.5 1 -1 -0.5 0 0.5 1 -0.5 -0.5 -0.5 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0.5 1 -11 -0.5 0
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For the same-polarity case, the result seen at the output of the correlator will depend on the path delay between the two pulses, and will vary from double the output (for zero path delay difference) to two completely resolved pulses (for path delay greater than the pulse width). For intermediate values of pulse delay (between zero and the pulse width), the resulting correlator output for the combined pulses will depend on the shape of the pulse and the path difference. Figure 4 shows the correlator output relative to a single pulse for two different pulse shapes: one with high fractional bandwidth (pH) and one with low fractional bandwidth (pL). The bottom axis shows the path difference between the pulses and the vertical axis shows that the resulting normalized correlation value ranges from 2 to something less than one. For pH , we see that the minimum output value is around 0.49 (-6.2 dB), while for pL the minimum normalized output value is about 0.024 (-32 dB). The implication here is that for same-polarity pulses, the worst possible fade due to two equal-amplitude multipath components is much worse for the pulse with the lower fractional bandwidth.
0.5 1
Figure 2: Time domain plots for the derivatives of the Gaussian monopulse. 0.5 1
0.5 1
0.5 1
0.5 1
-2 -4
dB Down from Peak
-6
Derivative Order
-8
20
3
2
1
-10 -12 -14 -16 -18 -20
0.2
1.6 1.4 1.2 1 0.8 0.6 0.4 Fractional Bandwidth At Left-Axis Measurement Points
1.8
2
Figure 3: Plot of fractional bandwidth versus measurement point (in dB) for the monopulse derivatives.
2
In this plot we see that the fractional bandwidth, measured at the 10 dB down points, ranges from about 170% for p1(t) to less than 50% for p20(t).
Fractional BW = 0.25
Correlator Amplitude
1.5
IV. PERFORMANCE IN A MULTIPATH ENVIRONMENT In a multipath environment, the reception of multiple received components scattering off different objects can result in either constructive or destructive interference. The different path lengths of the multiple components will cause them to arrive with different path delays, thus potentially out of phase. To understand the role of fractional bandwidth in the system performance in a high multipath environment, we will use a simplified analysis that determines the worst-case destructive interference occurring when two signal components arrive at the receiver with different time delays. For this analysis, we assume that the two received components are added in the receiver and we compare the resulting output of a correlator that is matching the received signal against the transmitted pulse shape.
1
0.5
Fractional BW = 2.0
0
0
0.1
0.2
0.4 0.3 Path Delay Differential (ns)
0.5
0.6
Figure 4: Plot of correlation value versus pulse differential path delay for pulses with the same polarity. This effect is more completely shown in Figure 5, where we plot the worst-case fade for two equalamplitude components as a function of fractional bandwidth. This plot shows that the worst-case fade can be significantly worse for the pulses with lower fractional bandwidth. For the case of two pulses with unequal amplitudes (i.e. one larger that the other), the curves
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variation from the nominal single-pulse correlator output, and thus more robust performance.
equivalent to those in Figure 4 would still be centered around the value of one, but would have smaller amplitudes variations, so we would see that the equal amplitude case is indeed the “worst-case”. The conclusion here is that a high fractional bandwidth pulse will allow the system to experience less severe fading in a multipath environment. From a system design point of view, this means that the system will require a lower margin for multipath fading in the link budget and can therefore provide more robust performance with less transmit power.
V. RELATIONSHIPS OF THESE RESULTS TO COHERENCE BANDWIDTH These results showing the advantage of wide fractional bandwidth can also be understood in the context of the well-known relationship between signal bandwidth and channel coherence bandwidth. This relationship is based on the assumption that the channel can be characterized as having a characteristic coherence bandwidth over which signals are likely to experience similar fading. This coherence bandwidth, Bc, is a statistical measure of the range of frequencies over which the channel appears “flat”, that is, over which spectral components experience equal amplitude fading. A signal is said to experience “flat fading” when its signal bandwidth, Bs, is less than the coherence bandwidth of the channel, BsBc, the signal is said to experience “frequency selective” fading. In this case, the presence of deep multipath fades is unlikely to affect the signal over its entire bandwidth and the overall system is therefore likely to experience much less severe fades in received signal energy than for flat fading [1].
1.6
Relative Bandwidth at –10 dBp
Figure 5: Plot of worst-case fade versus fractional bandwidth for same polarity pulses. 2
When we compared the different pulses p1 through p20 in Figure 2, we showed them for equal center frequencies (roughly corresponding to equal carrier frequencies when we considered them as sinusoids modulated by Gaussian envelopes). For these pulses, the increasing order of the derivative resulted in wider overall pulse widths for each successive pulse, and corresponding narrower bandwidths about the same center frequency. In light of the coherence bandwidth concept described above, then, we see that for a given coherence bandwidth, the shift between flat fading (narrow bandwidth pulses) and frequency selective fading (wider bandwidth pulses) is a gradual shift. This is quantitatively shown in Figure 5 where we plotted worstcase fade depth versus fractional bandwidth.
1.5
Correlator Amplitude
Fractional BW = 2.0
1
Fractional BW = 0.25 0.5
0
0
0.1
0.2
0.4 0.3 Path Delay Differential (ns)
0.5
0.6
Figure 6: Plot of correlation value versus pulse differential path delay for pulses with opposite polarity.
The result of this effect is thus seen to match our existing understanding of the coherence bandwidth concept. The pulses with wide fractional bandwidth are seen to have much less severe worst-case fading than those with lower fractional bandwidth. The implications of this fact on system level design for wireless networks is that the system need not have as large a margin to compensate for signal fading due to multipath for a given environment since the severity of the expected fading is much less.
For two pulses with opposite polarity and simultaneous arrival, there would clearly be complete cancellation, regardless of the pulse shape. There would be no remaining signal for the correlator to detect and the MF output would be zero (negative infinity dB relative to the single-pulse output). For non-zero path delay differences, however, we can again see in Figure 6 that the pulse with wider fractional bandwidth will provide less extreme
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VI. MEASURED RESULTS IN A REAL-WORLD CHANNEL
UWB
The analytical results above are supported by measurements made using an actual UWB system. This system was operated at 50 million pulses per second using a pulse shape that roughly corresponded to p1(t). The system was used to measure propagation loss in an indoor office environment with offices, hallways and partitions, such as might be experience by an indoor wireless network. The transmitter-receiver distance was slowly increased and received power (as indicated by correlator output) was measured at two-foot intervals. Figure 7 shows an example of a plot of the received signal strength versus separation distance. In this case, measurements were made in a hallway with structural columns and a metal bookcase. The particular measurements shown in Figure 7 clearly indicate that the measured receive power was influenced by multipath propagation, because the measured path loss values deviated from the expected curve for free-space path loss. At various points the received power is either greater or less than expected, indicating constructive or destructive multipath interference. The variation of the measurements relative to the free-space curve closely match those predicted by the curves in Figure 5, where the correlator output power is plotted as a function of the differential path delay for two multipath components. Also shown is a curve for 1/R3.5 propagation loss, indicating that the path loss at no time was as indicated y that curve. These results are also consistent with measurements made in previous work on wideband channels reported in [4], where relatively low levels of multipath fading were measured.
1
R2
1
R3.5
Figure 7: Received signal energy versus transmitterreceiver separation distance in an office environment.
VII. REFERENCES [1] Theodore S. Rappaport, Wireless Communications: Principles & Practice, Prentice Hall, 1996. [2] John G. Proakis, Digital Communications, McGraw Hill, Inc., 1995. [3] J. Foerster, E. Green, S. Somayazulu and D. Leeper, ``Ultra-Wideband Technology for Short- or MediumRange Wireless Communications," Intel Technology Journal, Q2, 2001. [4] M.Z. Win and R.A. Scholtz, ``Ultra-Wide Bandwidth Signals in Multipath Environments," IEEE Communications Letters, Vol. 2, No. 2, February 1998. [5] R.A. Scholtz and M.Z. Win, ``Impulse Radio," IEEE PIMRC ’97. [6] Lawrence Milstein, “Wideband Code Division Multiple Access,” Procedings of RAWCON 2001.
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