Peak Minimization for Reference-Based Ultra-Wideband ... - IEEE Xplore
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 8, AUGUST 2012
Peak Minimization for Reference-Based Ultra-Wideband (UWB) Radio Kyle Morrison, Ça˘gatay Çapar, and Dennis Goeckel
Abstract—We introduce a peak mitigation technique for reference based systems that is similar to the tone reservation scheme employed in orthogonal frequency division multiplexing (OFDM) systems but without the cost in data rate. A comparison of reference-based systems under either peak or average power constraints is presented. Index Terms—Communication system signaling, ultrawideband (UWB), peak-to-average power ratio (PAPR), receivers.
I. I NTRODUCTION HE natural approach to ultra-wideband (UWB) receiver design is to apply a rake receiver similar to that seen in traditional wideband wireless systems. But, because UWB systems spread energy over a large bandwidth, a rake receiver implementation is difficult because of the channel estimation and circuit complexity required to support a large number of taps [1], [2]. An early solution offered was the transmitted reference (TR-UWB) system [3], [4]. But the wideband delay element required is difficult to implement in low power integrated circuits [5]. Therefore an alternative receiver to the traditional rake receiver design and transmitted reference is frequency shifted reference (FSR-UWB) [6], where, instead of offsetting the wideband data signal and wideband reference signal in the time domain, a slight translation is done in the frequency domain which results in a simple (analog) receiver implementation. Motivated by the FSR-UWB scheme, two research teams independently proposed more general square-wave approaches: code-shifted reference UWB (CSR-UWB) [8], [9], and code-multiplexed reference UWB (CM-UWB) [10]. In these schemes, rather than separating the reference and data with a frequency shift, the reference and data signals are modulated with unique code sequences from the rows of a Hadamard matrix and thus are offset by an orthogonal code shift. There are many possible code combinations between reference and data signal that maintain orthogonality but, for the single-user case, choosing the optimal set of codes yields again a system similar to the standardized 802.15.4a version of pulse position modulation (PPM-UWB) [10]. Multi-differential frequency shifted reference (MD-FSR) [11] improves the performance of FSR-UWB under an average
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Paper approved by L. Yang, the Editor for Ultra Wideband of the IEEE Communications Society. Manuscript received June 15, 2011; revised December 6, 2011 and February 17, 2012. The authors are with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003, USA (e-mail: {kmorrison, ccapar, goeckel}@ecs.umass.edu). This paper is based in part upon work supported by the National Science Foundation under Grants ECS-0725616, CNS-0831133 and by the STTR Program of the Army Research Office. Portions of this work were presented at the 2009 IEEE International Conference on Ultra-Wideband (ICUWB) and the 2008 Asilomar Conference on Signals, Systems, and Computers. Digital Object Identifier 10.1109/TCOMM.2012.060112.110128
power constraint but suffers under a peak power constraint. In [11], multiple carriers, each carrying data differentially encoded relative to a single reference (called multi-differential (MD)), are employed. Orthogonality is preserved since the frequency offsets are well below the channel coherence bandwidth. This allows a strong reference with cost amortized over a number of data bits, which improves the performance of each of the data streams. In particular, if K data signals are transmitted over K orthogonal carrier frequencies, MD-FSR has a 5 log10 K dB gain in performance in terms of average signal-to-noise ratio (SNR) over standard single-differential (SD) FSR; in the limit of large K, the reference becomes essentially noiseless, thereby eliminating the dominating “noise cross noise” term of reference-based systems and leading to a system whose performance is only a fixed energy loss worse than the coherent system whose implementation has proven so difficult [11]. Per above, the major limitation of MD-FSR is a high peak-to-average power ratio (PAPR). The multiple sinusoids with random data modulation essentially lead to a similar PAPR problem to that of orthogonal frequency division multiplexing (OFDM) systems. Per above, peak constraints are pertinent, and as described in [6], systems with a higher peak-to-average power ratio (PAPR) will need a higher number of frames and thus will need to be able to tolerate even more significant inter-frame interference (IFI). But the inability to tolerate arbitrary IFI puts a limitation on the number of frames. Therefore, strategies need to be considered to reduce the PAPR of many systems. There are a number of peak-reduction techniques in the literature for standard OFDM systems [12], but the FSR-UWB system has some important nuances that need to be considered. In particular, we have a small number of data carriers and the potential to employ peak-reducing carriers at frequency separations beyond the channel coherence frequency where data cannot be carried. Standard coding approaches to peak reduction would necessarily have all of the coded bits on carriers corresponding to frequency separations less than the coherence frequency, and hence come at a cost in data rate. Selective mapping and partial transmit sequences are widely employed in conventional OFDM systems, but the overhead required for the side information does not make sense with the short data blocks considered here. Hence, we turn to the tone reservation and active constellation expansion (ACE) family of techniques; since we can pre-compute the values for the peak-reducing tones, algorithmic complexity is not an issue and we employ a version of tone-reservation. But we hasten to emphasize that, because of the differences in the operation and constraints for this UWB application versus those for an OFDM system, these additional waveforms come at no cost in data rate. In particular, the constraint on the data rate for a multi-differential reference-based system is that the
c 2012 IEEE 0090-6778/12$31.00
MORRISON et al.: PEAK MINIMIZATION FOR REFERENCE-BASED ULTRA-WIDEBAND (UWB) RADIO
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frequency offset of each data carrier from the reference must be less than the coherence frequency of the channel. But we will demonstrate that this constraint does not apply to the additional tones added for peak reduction. The remainder of this paper is organized as follows. Section II introduces the peak mitigation solution and Section III gives the numerical results for the reference-based systems considered. Finally, Section IV provides the conclusions. II. P EAK R EDUCTION V IA T ONE R ESERVATION In this paper, a baseband UWB system will be assumed. Since low data rate applications are targeted, a symbol interval Ts = Nf Tf consists of Nf 1 frames, each of duration Tf and containing a single UWB pulse p(t) with normalized energy p2 (t)dt = 1/Nf . Thus, for a given symbol interval, a data-carrying signal modulated onto the unit-energy train Nisf −1 of impulses u(t) = j=0 p(t−jTf ). In a general referencebased framework with K data-carriers, the transmitted signal consists of a reference signal and a collection of data signals, modulating the impulse train; hence, over the lth symbol period, the transmitted signal is given by: x(t) = u(t − lTs )xenv (t − lTs )
(1)
where xenv (t) =
K−1 (k) (k) bl Er + (−1) Ed φk (t)
(2)
k=0 (k)
Er and Ed , k = 0, 1, ..., K − 1, represent the energy in (k) the reference and k th data-bearing signal respectively, bl is the k th data bit in the lth symbol period, and {φk (t), k = 0, 1, . . . , K − 1} is an orthogonal set of waveforms with T normalization 0 s φ2k (t)dt = Ts for all k. Although it is not T strictly necessary, we assume 0 s φk (t)dt = 0 for all k, which is true of all of the major systems that have been introduced. For multi-differential (MD-FSR-UWB), [11] where multiple sinusoidal are used in parallel as data carrying signals with a single reference, a critical constraint is that the frequency separation between the reference signal and most distant data carrier must lie below the coherence frequency (Δf )c of the channel so that the reference sounds the proper channel for all of the data carriers. This puts a constraint on the maximum combined data rate of the MD-FSR-UWB system. The transmitted signal on [lTs , (l + 1)Ts ] is given by: xMD−F SR (t) = Er u(t − lTs ) K−1 (k) (k) + (−1)bl 2Ed cos(2πfk t)u(t − lTs ) (3) k=0
where K is the number of sinusoidal carriers, and f0 , f1 , · · · , fK−1 are set by fk = (2k + 1)f0 to ensure that interference among carriers does not fall at one of the frequency offsets of the data carriers from the reference. In the extensions of FSR-UWB and CM-UWB, and, for other systems with K > 1, a high PAPR is encountered. Per above, peak reduction is obtained by adding additional “dummy” carriers whose “data” values are set to minimize
Fig. 1. An example illustrating the difference of interleaving versus placing peak reducing tones at the end for peak reduction
the peak of the signal as shown in Fig. 1. The transmit signal now becomes: x(t) = Er u(t − lTs ) K−1 (k) (k) (−1)bl Ed u(t − lTs )φk (t − lTs ) + +
k=0 P −1
mp u(t − lTs )φK+p (t − lTs ),
(4)
p=0
where P is the number of peak reduction waveforms. Here, the amplitudes of these extra waveforms, mp , p = 0, 1, ..., P − 1, are chosen optimally via a convex optimization so that the peak power of the signal x(t) is minimized. In practice, since we envision the number of data carriers will be on the order of 4 to 8, this optimization would be done off-line for each possible set of data bits and the results stored in a table for efficient on-line usage. In choosing the number of these carriers, P , there are a few issues to consider. First, the average energy of the transmitted signal needs to be carefully observed, since it gets larger by adding these carriers. Although the system must pay a total average power penalty to accommodate the extra waveforms, a system that is IFI-limited will not be able to increase Nf to deal with PAPR problems (see [6]), and thus the reduction of the absolute peak allows more average energy to be put into the data carriers. Secondly, Nf should be large enough to prevent the extra carriers from effectively getting aliased to lower frequency tones. A simple example is given in Fig. 2 for the envelope of the signal x(t) given in (4). For a given P , optimal coefficients, mp ’s, can be found to minimize the peak value of the resulting signal. Clearly, the
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 8, AUGUST 2012
For y ∈ X and x ∈ X ∗ , < y, x > denotes the result of the functional when x acts upon y. The vector that maximizes the value < y, x >, yopt , lies in the subspace M , while the vector mopt lies in the space M ⊥ . M ⊥ is defined as the set of vectors which result in zero when applied on any vector in M . The peak reduction problem in (9) is a problem in the space L∞ . Thus, we find the minimum peak value by solving the corresponding (easier) maximization problem in the space L1 which has its dual L∞ . In other words,
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max{ |xenv(t)| }
7.5 7 6.5 6 5.5 5
0
10
20 30 40 Number of peak reduction carriers
50
min
60
m(t)∈M ⊥ ⊂L∞
= Fig. 2. Peak value of the envelope of an MD-FSR signal as a function of the number of peak reduction carriers. For the example signal, √ K = 4, Er = (k) 4, Ed = 1, b(k) = 1, ∀k. The original peak value is 2 + 4 2. The peak value of the signal decreases with increasing P and approaches a limit.
peak value decreases with increasing P and approaches to a limit. Knowing this limit reveals how close a given system is to the optimum achievable. It is difficult to find this limit directly as it involves optimizing a countably infinite number of coefficients. Hence, we employ a dual space technique. Let K−1 (k) (k) Er + (−1)bl Ed φk (t − lTs ) xenv (t) = k=0 P −1
+
mp φK+p (t − lTs )
(5)
p=0
be the envelope of the MD-FSR signal which modulates the pulse train. Let xref (t), xdata (t) be the part of this signal that carries the reference and the data, respectively, and m(t) = ∞ p=0 mp φK+p (t − lTs ) be the signal added for peak reduction. With these definitions, xenv (t) = xref (t) + xdata (t) + m(t)
(6)
is the signal which is required to have minimal peak value. Note that the peak value of a signal is its infinity norm, denoted by ·∞ and notice that =
xref (t) + xdata (t) + m(t)∞ xdata (t) + m(t)∞ + Er .
(7)
Then, the peak minimization problem can be equivalently posed as minimize
xdata (t) + m(t)∞ .
(8)
The requirement here is that m(t) is a linear combination of functions in the set {φk (t), k = K, K + 1, · · · }, which is a set of orthogonal functions. Thus, the minimum peak value is xdata (t) + m(t)∞ + Er . (9) min m(t)∈span{φk (t),k=K,K+1,··· }
To find this value, we make use of the following theorem: Theorem 1 ( [15, pg. 119] ): Let M be a subspace in a real normed space X. Let X ∗ be the dual space of X. Then, min
m∈M ⊥ ⊂X ∗
x + m =
sup y∈M⊂X, y≤1
< y, x > .
(10)
xdata (t) + m(t)∞
sup y(t)∈M⊂L1 , y(t)1 ≤1
< y(t), xdata (t) > .
(11)
Here, mopt (t) lies in the subspace M ⊥ = span{φk (t), k = K, K + 1, · · · }. In MD-FSR, φk (t) = cos(2π(2k + 1)f0 t), hence M = span {cos(2π(2k + 1)f0 t), k = 0, 1, · · · , K − 1} ∪
{sin(2πkf0 t), k = 1, 2 · · · }
∪
{cos(2π(2k)f0 t), k = 0, 1, 2 · · · }
(12)
Within this subspace M , finding yopt (t) is not easier than finding mopt (t). However, we narrow down the subspace M by the following observations: 1) The projection of yopt (t) onto span{sin(2πkf0 t), k = 1, 2 · · · } is the zero signal, because < sin(2πkf0 t), xdata (t) >= 0, ∀k, and adding a sine function to y(t) only increases the 1-norm. 2) The projection of yopt (t) onto span{cos(2π(2k)f0 t), k = 0, 1, 2 · · · }} is the zero signal, because < cos(2π(2k)f0 t), xdata (t) >= 0, ∀k, and adding an even frequency cosine function to y(t) only increases the 1-norm. Hence, the subspace in which yopt (t) lies is confined to span{cos(2π(2k + 1)f0 t), k = 0, · · · , K − 1}. Thus y(t) = K−1 k=0 yk cos(2π(2k + 1)f0 t), where yk ’s are real scalars to be optimized. Then, < y(t), xdata (t) >=
K−1
(k)
(−1)bl
(k) Ed yk ,
(13)
k=0
which results in the following optimization problem: maximize
K−1
(−1)
k=0
Ts
subject to 0
≤ 1.
(k)
bl
(k) Ed yk
(14)
K−1 y cos(2π(2k + 1)f t) k 0 dt k=0
(15)
This K-parameter maximization problem (e.g., K = 4 in Fig. 2) is clearly easier to solve than the norm minimization problem with countably infinite parameters in the √ dual space. The result of this maximization problem plus Er is equal to the minimum peak value that can be achieved. Numerical results are given in the next section.
MORRISON et al.: PEAK MINIMIZATION FOR REFERENCE-BASED ULTRA-WIDEBAND (UWB) RADIO
0
10
III. N UMERICAL R ESULTS
−1
bit error rate
10
−2
3.9 Mbps, CSR 3.9 Mbps, FSR 3.9 Mbps, MFSR 3.9 Mbps, FSR2 7.8 Mbps, CSR 7.8 Mbps, FSR 7.8 Mbps, MFSR 7.8 Mbps, FSR2
10
−3
10
10
15
20
25
Es/N0 (dB)
Fig. 3. Simulated results for the performance of various binary schemes (Code Shifted Reference (CSR), Frequency Shifted Reference (FSR), a Modified FSR (MFSR) system that uses a sample-and-hold approach across each frame, and FSR-2 that is the same as FSR but with the frequency offset halved) on the IEEE 802.5.14a indoor NLOS model (CM4) with a fixed frame time of Tf = 16 ns, Nf = 8 (dashed curves) and Nf = 16 (solid curves) which correspond to Rb = 7.8 Mbps and 3.9 Mbps, respectively. For each point, 106 data symbols have been simulated and inter-frame interference is considered.
10
10 bit error rate
The parameters of the UWB system are derived, as follows. First, the data rate Rb and the number of data carriers K sets the symbol time Ts = K/Rb . The peak power constraint derived from the UWB transmitter hardware, along with the peak-to-average energy of the system, yields the average energy per UWB pulse. The number of pulses per symbol period Ts can then be increased until the system is limited by interframe interference (IFI) or the FCC limit is reached [6]. The performance (e.g. probability of error) given these parameter settings is then readily derived [6], [11]. For devices to meet the circuit constraints, FCC spectral mask, and maximize system performance from a commercialization perspective under a peak power constraint would suggest large Nf (on the order of approximately 75 pulses, in [14]). This is further exacerbated in systems with significant PAPR, because they must further increase Nf to boost average power [6]. Consider the comparison of various reference-based systems under an average power constraint. The pulse shape is the second derivative Gaussian with a zero-to-zero pulse width of .25 ns. The noise bandwidth, corresponding to that of a front end filter, is 4.0 GHz (one-sided). Simulation results were conducted over multipath channels from the IEEE 802.15.4a standardization effort [7]. An example of the performance comparison is given in Fig. 3 for the IEEE 802.15.4a indoor non-light-of-sight (NLOS) model (CM4). For the high data rate of 7.8 Mbps (Nf = 8) FSR-UWB has poor performance compared to the other systems considered. In the FSR system there is signal degradation contributed to the sampling of sinusoidal pulse over a frame. As an alternative solution, we propose ’modified’ FSR (MFSR) where we sample and hold the sinusoidal pulse across the frame. But the frequency offset for the original FSR system was chosen larger than necessary, we introduce “FSR-2-UWB” where the frequency offset is half. M-FSR-UWB shows improved performance compared to FSR, and FSR-2-UWB, shows slightly better performance compared to CSR-UWB. Overall, the systems considered show similar results under an average power constraint for reasonably large Nf . In Fig. 4 the simulated bit error rate performance of an MDFSR system with and without peak reducing dummy carriers is compared. As expected, for all three channels simulated, bit error rates remain the same under both cases for all SNR values. This supports the claim that adding orthogonal carriers to the transmitted signal does not affect the detection performance. Next, how the number of peak reduction carriers, P , affects the amount of peak reduction is considered. Of course, P can be increased to achieve smaller peaks and this increase will have no effect on the bit error rate performance. However, the increased number of carriers gives diminishing gains and there is a nonzero limit to how much the peak can be reduced, as considered in Section II. The change of peak reduction as a function of the number of extra carriers is shown in Fig. 5, where peak reducing dummy carriers interleaved and at frequency separations beyond those of the data carriers are compared. For the example shown in Fig. 5, as the number of peak reduction carriers increase, interleaving shows a minimal performance gain. So we conclude that peak reduction is not
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10
0
−1
−2
−3
AWGN, no peak reduction CM3, no peak reduction CM4, no peak reduction AWGN, PRCs interleaved CM3, PRCs interleaved CM4, PRCs interleaved AWGN, PRCs at the end CM3, PRCs at the end CM4, PRCs at the end
−4
0
5
10
15 20 Es/N0 (dB)
25
30
35
Fig. 4. BER computed for an MD-FSR system with 5 data carriers for different SNR values, where the signal power is the average power in the data carriers. Three different channels are simulated for cases (1) without peak reduction and (2) with peak reduction by adding 3 extra carriers. As expected, for all the three channels simulated, bit error rates remain the same under both cases for all SNR values. This justifies the fact that adding orthogonal carriers to the transmitted signal do not affect the detection performance. Note that the additional average energy required for the peak reduction carriers is addressed separately in Section II
significantly below optimal when the peak reduction carriers are placed at higher frequency offsets than the data carriers. It might also seem possible to add extra carriers to a CSR/CMUWB signal for peak reduction. However, while a CSR/CMUWB signal has a lower initial peak compared to FSR-UWB, the specific separating waveforms used in CSR/CM-UWB do not allow peak reduction by adding extra carriers. Table I presents the peak values before and after peak reduction for CSR/CM-UWB and FSR-UWB for the case of K = 1, 2, 4 carriers and P = 20 peak reducers.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 8, AUGUST 2012
TABLE I P EAK REDUCTION RESULTS FOR CSR/CM-UWB AND FSR-UWB SCHEMES (Nf = 128, Tf = 31.25 ns, AND P = 20)
FSR, K = 4 CSR/CM, K = 4 FSR, K = 2 CSR/CM, K = 2 FSR, K = 1 CSR/CM, K = 1
initial peak (max|xenv (t)|) 0.8263 0.6475 0.6475 0.5211 0.5211 0.4317
initial and final energy 1, 1.7487 1, 1 1, 1.3185 1, 1 1, 1.0978 1, 1
final peak 0.5899 0.6475 0.5059 0.5211 0.4559 0.4317
UWB. Such peak mitigation techniques can lead to small PAPR gains of frequency-shifted reference UWB versus codemultiplexed and code-shifted reference UWB systems.
4 3.5 3 peak reduction (dB)
peak reduction (dB) 2.9270 0 2.1436 0 1.1605 0
2.5
R EFERENCES
2 1.5 1 peak reduction carriers at the end
0.5 0
peak reduction carriers interleaved
1
3
5 7 9 11 number of peak reduction carriers
13
15
Fig. 5. Peak Reduction of Multi-Differential FSR-UWB with additional tones (K=5), while either interleaved in triangle markers or at the end in circle markers.
IV. C ONCLUSION We have considered the comparison of recently proposed reference-based systems under both peak and average constraints. In reference based systems with a single data carrier, halving the frequency offset between reference and data in the frequency-shifted reference UWB system, as suggested in recent work, leads to slight performance advantages over alternatives under peak and average power constraints. However, since the gains are slight, this suggest code-multiplexed or code-shifted reference UWB systems for implementation, since they avoid the amplitude modulation required in the frequency-shifted reference UWB system. Systems that employ multiple data carriers with a single reference have the potential to greatly improve performance versus their single data carrier counterparts; however, all considered systems then require amplitude modulation, and can suffer from a significant PAPR, which limits performance under peak power constraints. Here, we have introduced peak mitigation alternatives that do not restrict the data rate and are effective for some systems in the class, most notably for the multiple data carrier version of frequency-shifted reference
[1] R. Wilson and R. Scholtz, “Comparison of CDMA and modulation schemes for UWB radio in a multipath environment,” in Proc. 2003 Global Telecommun. Conf. [2] H. Sheng, R. You, and A. Haimovich, “Performance analysis of ultrawideband rake receivers with channel delay estimation errors,” in Proc. 2004 Conf. Inf. Sci. Syst. [3] R. Hoctor and H. Tomlinson, “Delay-hopped transmitted-reference RF communications,” in Proc. 2002 IEEE Conf. Ultra-Wideband Syst. Technol., pp. 265–270. [4] M. Casu and G. Durisi, “Implementation aspects of a transmittedreference UWB receiver,” Wireless Commun. Mobile Comput. vol. 5, pp. 537–549, May 2005. [5] N. van Stralen, A. Dentinger, K. Welles II, R. Gaus Jr., R. Hoctor, and H. Tomlinson, “Delay hopped transmitted reference experimental results,” in Proc. 2002 IEEE Conf. Ultra-Wideband Syst. Technol. [6] D. Goeckel and Q. Zhang, “Slightly frequency-shifted reference ultrawideband (UWB) radio,” IEEE Trans. Commun., vol. 55, 508–518, Mar. 2007. [7] A. F. Molisch, K. Balakrishnan, and C. C. Chong, “IEEE 802.15.4a channel model - final report,” Tech. Rep., Dec. 2004. [8] H. Nie and Z. Chen, “Code-shifted reference ultra-wideband radio,” in Proc. 2008 Commun. Netw. Services Research Conf. [9] H. Nie and Z. Chen, “Code-shifted reference transceiver for impulse radio ultra-wideband systems,” Physical Commun., vol. 2, pp. 274–284, Dec. 2009. [10] A. D’Amico and U. Mengali, “Code-multiplexed UWB transmittedreference radio,” IEEE Trans. Commun., vol. 56, pp. 2125–2132, Dec. 2008. [11] Q. Zhang and D. Goeckel, “Multi-differential slightly frequency-shifted reference ultra-wideband (UWB) radio,” in Proc. 2006 Conf. Inf. Sci. Syst. [12] C. Ciochina, F. Buda, and H. Sari, “An analysis of OFDM peak power reduction techniques for WiMAX systems,” IEEE Trans. Commun., June 2006. [13] J. Tellado-Mourelo, “Peak to average power reduction for multi-carrier modulation,” Ph.D. dissertation, Stanford University, 1999. [14] K. Morrison, C. Capar, Z. Lai, D. Goeckel, and R. Jackson, “A unified framework for reference-based ultra-wideband signaling,” 2009 IEEE International Conf. Ultra-Wideband. [15] D. G. Luenberger, Optimization by Vector Space Methods. Wiley, 1990.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 8, AUGUST 2012
Peak Minimization for Reference-Based Ultra-Wideband (UWB) Radio Kyle Morrison, Ça˘gatay Çapar, and Dennis Goeckel
Abstract—We introduce a peak mitigation technique for reference based systems that is similar to the tone reservation scheme employed in orthogonal frequency division multiplexing (OFDM) systems but without the cost in data rate. A comparison of reference-based systems under either peak or average power constraints is presented. Index Terms—Communication system signaling, ultrawideband (UWB), peak-to-average power ratio (PAPR), receivers.
I. I NTRODUCTION HE natural approach to ultra-wideband (UWB) receiver design is to apply a rake receiver similar to that seen in traditional wideband wireless systems. But, because UWB systems spread energy over a large bandwidth, a rake receiver implementation is difficult because of the channel estimation and circuit complexity required to support a large number of taps [1], [2]. An early solution offered was the transmitted reference (TR-UWB) system [3], [4]. But the wideband delay element required is difficult to implement in low power integrated circuits [5]. Therefore an alternative receiver to the traditional rake receiver design and transmitted reference is frequency shifted reference (FSR-UWB) [6], where, instead of offsetting the wideband data signal and wideband reference signal in the time domain, a slight translation is done in the frequency domain which results in a simple (analog) receiver implementation. Motivated by the FSR-UWB scheme, two research teams independently proposed more general square-wave approaches: code-shifted reference UWB (CSR-UWB) [8], [9], and code-multiplexed reference UWB (CM-UWB) [10]. In these schemes, rather than separating the reference and data with a frequency shift, the reference and data signals are modulated with unique code sequences from the rows of a Hadamard matrix and thus are offset by an orthogonal code shift. There are many possible code combinations between reference and data signal that maintain orthogonality but, for the single-user case, choosing the optimal set of codes yields again a system similar to the standardized 802.15.4a version of pulse position modulation (PPM-UWB) [10]. Multi-differential frequency shifted reference (MD-FSR) [11] improves the performance of FSR-UWB under an average
T
Paper approved by L. Yang, the Editor for Ultra Wideband of the IEEE Communications Society. Manuscript received June 15, 2011; revised December 6, 2011 and February 17, 2012. The authors are with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003, USA (e-mail: {kmorrison, ccapar, goeckel}@ecs.umass.edu). This paper is based in part upon work supported by the National Science Foundation under Grants ECS-0725616, CNS-0831133 and by the STTR Program of the Army Research Office. Portions of this work were presented at the 2009 IEEE International Conference on Ultra-Wideband (ICUWB) and the 2008 Asilomar Conference on Signals, Systems, and Computers. Digital Object Identifier 10.1109/TCOMM.2012.060112.110128
power constraint but suffers under a peak power constraint. In [11], multiple carriers, each carrying data differentially encoded relative to a single reference (called multi-differential (MD)), are employed. Orthogonality is preserved since the frequency offsets are well below the channel coherence bandwidth. This allows a strong reference with cost amortized over a number of data bits, which improves the performance of each of the data streams. In particular, if K data signals are transmitted over K orthogonal carrier frequencies, MD-FSR has a 5 log10 K dB gain in performance in terms of average signal-to-noise ratio (SNR) over standard single-differential (SD) FSR; in the limit of large K, the reference becomes essentially noiseless, thereby eliminating the dominating “noise cross noise” term of reference-based systems and leading to a system whose performance is only a fixed energy loss worse than the coherent system whose implementation has proven so difficult [11]. Per above, the major limitation of MD-FSR is a high peak-to-average power ratio (PAPR). The multiple sinusoids with random data modulation essentially lead to a similar PAPR problem to that of orthogonal frequency division multiplexing (OFDM) systems. Per above, peak constraints are pertinent, and as described in [6], systems with a higher peak-to-average power ratio (PAPR) will need a higher number of frames and thus will need to be able to tolerate even more significant inter-frame interference (IFI). But the inability to tolerate arbitrary IFI puts a limitation on the number of frames. Therefore, strategies need to be considered to reduce the PAPR of many systems. There are a number of peak-reduction techniques in the literature for standard OFDM systems [12], but the FSR-UWB system has some important nuances that need to be considered. In particular, we have a small number of data carriers and the potential to employ peak-reducing carriers at frequency separations beyond the channel coherence frequency where data cannot be carried. Standard coding approaches to peak reduction would necessarily have all of the coded bits on carriers corresponding to frequency separations less than the coherence frequency, and hence come at a cost in data rate. Selective mapping and partial transmit sequences are widely employed in conventional OFDM systems, but the overhead required for the side information does not make sense with the short data blocks considered here. Hence, we turn to the tone reservation and active constellation expansion (ACE) family of techniques; since we can pre-compute the values for the peak-reducing tones, algorithmic complexity is not an issue and we employ a version of tone-reservation. But we hasten to emphasize that, because of the differences in the operation and constraints for this UWB application versus those for an OFDM system, these additional waveforms come at no cost in data rate. In particular, the constraint on the data rate for a multi-differential reference-based system is that the
c 2012 IEEE 0090-6778/12$31.00
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frequency offset of each data carrier from the reference must be less than the coherence frequency of the channel. But we will demonstrate that this constraint does not apply to the additional tones added for peak reduction. The remainder of this paper is organized as follows. Section II introduces the peak mitigation solution and Section III gives the numerical results for the reference-based systems considered. Finally, Section IV provides the conclusions. II. P EAK R EDUCTION V IA T ONE R ESERVATION In this paper, a baseband UWB system will be assumed. Since low data rate applications are targeted, a symbol interval Ts = Nf Tf consists of Nf 1 frames, each of duration Tf and containing a single UWB pulse p(t) with normalized energy p2 (t)dt = 1/Nf . Thus, for a given symbol interval, a data-carrying signal modulated onto the unit-energy train Nisf −1 of impulses u(t) = j=0 p(t−jTf ). In a general referencebased framework with K data-carriers, the transmitted signal consists of a reference signal and a collection of data signals, modulating the impulse train; hence, over the lth symbol period, the transmitted signal is given by: x(t) = u(t − lTs )xenv (t − lTs )
(1)
where xenv (t) =
K−1 (k) (k) bl Er + (−1) Ed φk (t)
(2)
k=0 (k)
Er and Ed , k = 0, 1, ..., K − 1, represent the energy in (k) the reference and k th data-bearing signal respectively, bl is the k th data bit in the lth symbol period, and {φk (t), k = 0, 1, . . . , K − 1} is an orthogonal set of waveforms with T normalization 0 s φ2k (t)dt = Ts for all k. Although it is not T strictly necessary, we assume 0 s φk (t)dt = 0 for all k, which is true of all of the major systems that have been introduced. For multi-differential (MD-FSR-UWB), [11] where multiple sinusoidal are used in parallel as data carrying signals with a single reference, a critical constraint is that the frequency separation between the reference signal and most distant data carrier must lie below the coherence frequency (Δf )c of the channel so that the reference sounds the proper channel for all of the data carriers. This puts a constraint on the maximum combined data rate of the MD-FSR-UWB system. The transmitted signal on [lTs , (l + 1)Ts ] is given by: xMD−F SR (t) = Er u(t − lTs ) K−1 (k) (k) + (−1)bl 2Ed cos(2πfk t)u(t − lTs ) (3) k=0
where K is the number of sinusoidal carriers, and f0 , f1 , · · · , fK−1 are set by fk = (2k + 1)f0 to ensure that interference among carriers does not fall at one of the frequency offsets of the data carriers from the reference. In the extensions of FSR-UWB and CM-UWB, and, for other systems with K > 1, a high PAPR is encountered. Per above, peak reduction is obtained by adding additional “dummy” carriers whose “data” values are set to minimize
Fig. 1. An example illustrating the difference of interleaving versus placing peak reducing tones at the end for peak reduction
the peak of the signal as shown in Fig. 1. The transmit signal now becomes: x(t) = Er u(t − lTs ) K−1 (k) (k) (−1)bl Ed u(t − lTs )φk (t − lTs ) + +
k=0 P −1
mp u(t − lTs )φK+p (t − lTs ),
(4)
p=0
where P is the number of peak reduction waveforms. Here, the amplitudes of these extra waveforms, mp , p = 0, 1, ..., P − 1, are chosen optimally via a convex optimization so that the peak power of the signal x(t) is minimized. In practice, since we envision the number of data carriers will be on the order of 4 to 8, this optimization would be done off-line for each possible set of data bits and the results stored in a table for efficient on-line usage. In choosing the number of these carriers, P , there are a few issues to consider. First, the average energy of the transmitted signal needs to be carefully observed, since it gets larger by adding these carriers. Although the system must pay a total average power penalty to accommodate the extra waveforms, a system that is IFI-limited will not be able to increase Nf to deal with PAPR problems (see [6]), and thus the reduction of the absolute peak allows more average energy to be put into the data carriers. Secondly, Nf should be large enough to prevent the extra carriers from effectively getting aliased to lower frequency tones. A simple example is given in Fig. 2 for the envelope of the signal x(t) given in (4). For a given P , optimal coefficients, mp ’s, can be found to minimize the peak value of the resulting signal. Clearly, the
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For y ∈ X and x ∈ X ∗ , < y, x > denotes the result of the functional when x acts upon y. The vector that maximizes the value < y, x >, yopt , lies in the subspace M , while the vector mopt lies in the space M ⊥ . M ⊥ is defined as the set of vectors which result in zero when applied on any vector in M . The peak reduction problem in (9) is a problem in the space L∞ . Thus, we find the minimum peak value by solving the corresponding (easier) maximization problem in the space L1 which has its dual L∞ . In other words,
8
max{ |xenv(t)| }
7.5 7 6.5 6 5.5 5
0
10
20 30 40 Number of peak reduction carriers
50
min
60
m(t)∈M ⊥ ⊂L∞
= Fig. 2. Peak value of the envelope of an MD-FSR signal as a function of the number of peak reduction carriers. For the example signal, √ K = 4, Er = (k) 4, Ed = 1, b(k) = 1, ∀k. The original peak value is 2 + 4 2. The peak value of the signal decreases with increasing P and approaches a limit.
peak value decreases with increasing P and approaches to a limit. Knowing this limit reveals how close a given system is to the optimum achievable. It is difficult to find this limit directly as it involves optimizing a countably infinite number of coefficients. Hence, we employ a dual space technique. Let K−1 (k) (k) Er + (−1)bl Ed φk (t − lTs ) xenv (t) = k=0 P −1
+
mp φK+p (t − lTs )
(5)
p=0
be the envelope of the MD-FSR signal which modulates the pulse train. Let xref (t), xdata (t) be the part of this signal that carries the reference and the data, respectively, and m(t) = ∞ p=0 mp φK+p (t − lTs ) be the signal added for peak reduction. With these definitions, xenv (t) = xref (t) + xdata (t) + m(t)
(6)
is the signal which is required to have minimal peak value. Note that the peak value of a signal is its infinity norm, denoted by ·∞ and notice that =
xref (t) + xdata (t) + m(t)∞ xdata (t) + m(t)∞ + Er .
(7)
Then, the peak minimization problem can be equivalently posed as minimize
xdata (t) + m(t)∞ .
(8)
The requirement here is that m(t) is a linear combination of functions in the set {φk (t), k = K, K + 1, · · · }, which is a set of orthogonal functions. Thus, the minimum peak value is xdata (t) + m(t)∞ + Er . (9) min m(t)∈span{φk (t),k=K,K+1,··· }
To find this value, we make use of the following theorem: Theorem 1 ( [15, pg. 119] ): Let M be a subspace in a real normed space X. Let X ∗ be the dual space of X. Then, min
m∈M ⊥ ⊂X ∗
x + m =
sup y∈M⊂X, y≤1
< y, x > .
(10)
xdata (t) + m(t)∞
sup y(t)∈M⊂L1 , y(t)1 ≤1
< y(t), xdata (t) > .
(11)
Here, mopt (t) lies in the subspace M ⊥ = span{φk (t), k = K, K + 1, · · · }. In MD-FSR, φk (t) = cos(2π(2k + 1)f0 t), hence M = span {cos(2π(2k + 1)f0 t), k = 0, 1, · · · , K − 1} ∪
{sin(2πkf0 t), k = 1, 2 · · · }
∪
{cos(2π(2k)f0 t), k = 0, 1, 2 · · · }
(12)
Within this subspace M , finding yopt (t) is not easier than finding mopt (t). However, we narrow down the subspace M by the following observations: 1) The projection of yopt (t) onto span{sin(2πkf0 t), k = 1, 2 · · · } is the zero signal, because < sin(2πkf0 t), xdata (t) >= 0, ∀k, and adding a sine function to y(t) only increases the 1-norm. 2) The projection of yopt (t) onto span{cos(2π(2k)f0 t), k = 0, 1, 2 · · · }} is the zero signal, because < cos(2π(2k)f0 t), xdata (t) >= 0, ∀k, and adding an even frequency cosine function to y(t) only increases the 1-norm. Hence, the subspace in which yopt (t) lies is confined to span{cos(2π(2k + 1)f0 t), k = 0, · · · , K − 1}. Thus y(t) = K−1 k=0 yk cos(2π(2k + 1)f0 t), where yk ’s are real scalars to be optimized. Then, < y(t), xdata (t) >=
K−1
(k)
(−1)bl
(k) Ed yk ,
(13)
k=0
which results in the following optimization problem: maximize
K−1
(−1)
k=0
Ts
subject to 0
≤ 1.
(k)
bl
(k) Ed yk
(14)
K−1 y cos(2π(2k + 1)f t) k 0 dt k=0
(15)
This K-parameter maximization problem (e.g., K = 4 in Fig. 2) is clearly easier to solve than the norm minimization problem with countably infinite parameters in the √ dual space. The result of this maximization problem plus Er is equal to the minimum peak value that can be achieved. Numerical results are given in the next section.
MORRISON et al.: PEAK MINIMIZATION FOR REFERENCE-BASED ULTRA-WIDEBAND (UWB) RADIO
0
10
III. N UMERICAL R ESULTS
−1
bit error rate
10
−2
3.9 Mbps, CSR 3.9 Mbps, FSR 3.9 Mbps, MFSR 3.9 Mbps, FSR2 7.8 Mbps, CSR 7.8 Mbps, FSR 7.8 Mbps, MFSR 7.8 Mbps, FSR2
10
−3
10
10
15
20
25
Es/N0 (dB)
Fig. 3. Simulated results for the performance of various binary schemes (Code Shifted Reference (CSR), Frequency Shifted Reference (FSR), a Modified FSR (MFSR) system that uses a sample-and-hold approach across each frame, and FSR-2 that is the same as FSR but with the frequency offset halved) on the IEEE 802.5.14a indoor NLOS model (CM4) with a fixed frame time of Tf = 16 ns, Nf = 8 (dashed curves) and Nf = 16 (solid curves) which correspond to Rb = 7.8 Mbps and 3.9 Mbps, respectively. For each point, 106 data symbols have been simulated and inter-frame interference is considered.
10
10 bit error rate
The parameters of the UWB system are derived, as follows. First, the data rate Rb and the number of data carriers K sets the symbol time Ts = K/Rb . The peak power constraint derived from the UWB transmitter hardware, along with the peak-to-average energy of the system, yields the average energy per UWB pulse. The number of pulses per symbol period Ts can then be increased until the system is limited by interframe interference (IFI) or the FCC limit is reached [6]. The performance (e.g. probability of error) given these parameter settings is then readily derived [6], [11]. For devices to meet the circuit constraints, FCC spectral mask, and maximize system performance from a commercialization perspective under a peak power constraint would suggest large Nf (on the order of approximately 75 pulses, in [14]). This is further exacerbated in systems with significant PAPR, because they must further increase Nf to boost average power [6]. Consider the comparison of various reference-based systems under an average power constraint. The pulse shape is the second derivative Gaussian with a zero-to-zero pulse width of .25 ns. The noise bandwidth, corresponding to that of a front end filter, is 4.0 GHz (one-sided). Simulation results were conducted over multipath channels from the IEEE 802.15.4a standardization effort [7]. An example of the performance comparison is given in Fig. 3 for the IEEE 802.15.4a indoor non-light-of-sight (NLOS) model (CM4). For the high data rate of 7.8 Mbps (Nf = 8) FSR-UWB has poor performance compared to the other systems considered. In the FSR system there is signal degradation contributed to the sampling of sinusoidal pulse over a frame. As an alternative solution, we propose ’modified’ FSR (MFSR) where we sample and hold the sinusoidal pulse across the frame. But the frequency offset for the original FSR system was chosen larger than necessary, we introduce “FSR-2-UWB” where the frequency offset is half. M-FSR-UWB shows improved performance compared to FSR, and FSR-2-UWB, shows slightly better performance compared to CSR-UWB. Overall, the systems considered show similar results under an average power constraint for reasonably large Nf . In Fig. 4 the simulated bit error rate performance of an MDFSR system with and without peak reducing dummy carriers is compared. As expected, for all three channels simulated, bit error rates remain the same under both cases for all SNR values. This supports the claim that adding orthogonal carriers to the transmitted signal does not affect the detection performance. Next, how the number of peak reduction carriers, P , affects the amount of peak reduction is considered. Of course, P can be increased to achieve smaller peaks and this increase will have no effect on the bit error rate performance. However, the increased number of carriers gives diminishing gains and there is a nonzero limit to how much the peak can be reduced, as considered in Section II. The change of peak reduction as a function of the number of extra carriers is shown in Fig. 5, where peak reducing dummy carriers interleaved and at frequency separations beyond those of the data carriers are compared. For the example shown in Fig. 5, as the number of peak reduction carriers increase, interleaving shows a minimal performance gain. So we conclude that peak reduction is not
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−2
−3
AWGN, no peak reduction CM3, no peak reduction CM4, no peak reduction AWGN, PRCs interleaved CM3, PRCs interleaved CM4, PRCs interleaved AWGN, PRCs at the end CM3, PRCs at the end CM4, PRCs at the end
−4
0
5
10
15 20 Es/N0 (dB)
25
30
35
Fig. 4. BER computed for an MD-FSR system with 5 data carriers for different SNR values, where the signal power is the average power in the data carriers. Three different channels are simulated for cases (1) without peak reduction and (2) with peak reduction by adding 3 extra carriers. As expected, for all the three channels simulated, bit error rates remain the same under both cases for all SNR values. This justifies the fact that adding orthogonal carriers to the transmitted signal do not affect the detection performance. Note that the additional average energy required for the peak reduction carriers is addressed separately in Section II
significantly below optimal when the peak reduction carriers are placed at higher frequency offsets than the data carriers. It might also seem possible to add extra carriers to a CSR/CMUWB signal for peak reduction. However, while a CSR/CMUWB signal has a lower initial peak compared to FSR-UWB, the specific separating waveforms used in CSR/CM-UWB do not allow peak reduction by adding extra carriers. Table I presents the peak values before and after peak reduction for CSR/CM-UWB and FSR-UWB for the case of K = 1, 2, 4 carriers and P = 20 peak reducers.
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TABLE I P EAK REDUCTION RESULTS FOR CSR/CM-UWB AND FSR-UWB SCHEMES (Nf = 128, Tf = 31.25 ns, AND P = 20)
FSR, K = 4 CSR/CM, K = 4 FSR, K = 2 CSR/CM, K = 2 FSR, K = 1 CSR/CM, K = 1
initial peak (max|xenv (t)|) 0.8263 0.6475 0.6475 0.5211 0.5211 0.4317
initial and final energy 1, 1.7487 1, 1 1, 1.3185 1, 1 1, 1.0978 1, 1
final peak 0.5899 0.6475 0.5059 0.5211 0.4559 0.4317
UWB. Such peak mitigation techniques can lead to small PAPR gains of frequency-shifted reference UWB versus codemultiplexed and code-shifted reference UWB systems.
4 3.5 3 peak reduction (dB)
peak reduction (dB) 2.9270 0 2.1436 0 1.1605 0
2.5
R EFERENCES
2 1.5 1 peak reduction carriers at the end
0.5 0
peak reduction carriers interleaved
1
3
5 7 9 11 number of peak reduction carriers
13
15
Fig. 5. Peak Reduction of Multi-Differential FSR-UWB with additional tones (K=5), while either interleaved in triangle markers or at the end in circle markers.
IV. C ONCLUSION We have considered the comparison of recently proposed reference-based systems under both peak and average constraints. In reference based systems with a single data carrier, halving the frequency offset between reference and data in the frequency-shifted reference UWB system, as suggested in recent work, leads to slight performance advantages over alternatives under peak and average power constraints. However, since the gains are slight, this suggest code-multiplexed or code-shifted reference UWB systems for implementation, since they avoid the amplitude modulation required in the frequency-shifted reference UWB system. Systems that employ multiple data carriers with a single reference have the potential to greatly improve performance versus their single data carrier counterparts; however, all considered systems then require amplitude modulation, and can suffer from a significant PAPR, which limits performance under peak power constraints. Here, we have introduced peak mitigation alternatives that do not restrict the data rate and are effective for some systems in the class, most notably for the multiple data carrier version of frequency-shifted reference
[1] R. Wilson and R. Scholtz, “Comparison of CDMA and modulation schemes for UWB radio in a multipath environment,” in Proc. 2003 Global Telecommun. Conf. [2] H. Sheng, R. You, and A. Haimovich, “Performance analysis of ultrawideband rake receivers with channel delay estimation errors,” in Proc. 2004 Conf. Inf. Sci. Syst. [3] R. Hoctor and H. Tomlinson, “Delay-hopped transmitted-reference RF communications,” in Proc. 2002 IEEE Conf. Ultra-Wideband Syst. Technol., pp. 265–270. [4] M. Casu and G. Durisi, “Implementation aspects of a transmittedreference UWB receiver,” Wireless Commun. Mobile Comput. vol. 5, pp. 537–549, May 2005. [5] N. van Stralen, A. Dentinger, K. Welles II, R. Gaus Jr., R. Hoctor, and H. Tomlinson, “Delay hopped transmitted reference experimental results,” in Proc. 2002 IEEE Conf. Ultra-Wideband Syst. Technol. [6] D. Goeckel and Q. Zhang, “Slightly frequency-shifted reference ultrawideband (UWB) radio,” IEEE Trans. Commun., vol. 55, 508–518, Mar. 2007. [7] A. F. Molisch, K. Balakrishnan, and C. C. Chong, “IEEE 802.15.4a channel model - final report,” Tech. Rep., Dec. 2004. [8] H. Nie and Z. Chen, “Code-shifted reference ultra-wideband radio,” in Proc. 2008 Commun. Netw. Services Research Conf. [9] H. Nie and Z. Chen, “Code-shifted reference transceiver for impulse radio ultra-wideband systems,” Physical Commun., vol. 2, pp. 274–284, Dec. 2009. [10] A. D’Amico and U. Mengali, “Code-multiplexed UWB transmittedreference radio,” IEEE Trans. Commun., vol. 56, pp. 2125–2132, Dec. 2008. [11] Q. Zhang and D. Goeckel, “Multi-differential slightly frequency-shifted reference ultra-wideband (UWB) radio,” in Proc. 2006 Conf. Inf. Sci. Syst. [12] C. Ciochina, F. Buda, and H. Sari, “An analysis of OFDM peak power reduction techniques for WiMAX systems,” IEEE Trans. Commun., June 2006. [13] J. Tellado-Mourelo, “Peak to average power reduction for multi-carrier modulation,” Ph.D. dissertation, Stanford University, 1999. [14] K. Morrison, C. Capar, Z. Lai, D. Goeckel, and R. Jackson, “A unified framework for reference-based ultra-wideband signaling,” 2009 IEEE International Conf. Ultra-Wideband. [15] D. G. Luenberger, Optimization by Vector Space Methods. Wiley, 1990.
