Unified Capacity Limit of Non-Coherent Wideband Fading Channels
1
Unified Capacity Limit of Non-Coherent Wideband Fading Channels
arXiv:1501.04905v2 [cs.IT] 1 Sep 2015
Felipe G´omez-Cuba∗, Student Member, IEEE, Jinfeng Du∗ , Member, IEEE, Muriel M´edard, Fellow, IEEE, and Elza Erkip, Fellow, IEEE
Abstract—In non-coherent wideband fading channels where energy rather than spectrum is the limiting resource, peaky and non-peaky signaling schemes have long been considered species apart, as the first approaches asymptotically the capacity of a wideband AWGN channel with the same average SNR, whereas the second reaches a peak rate at some finite critical bandwidth and then falls to zero as bandwidth grows to infinity. In this paper it is shown that this distinction is in fact an artifact of the limited attention paid in the past to the product between the bandwidth and the fraction of time it is in use. This fundamental quantity, that is termed bandwidth occupancy, measures average bandwidth usage over time. As it turns out, a peaky signal that transmits in an infinite bandwidth but only for an infinitesimal fraction of the time may only have a small bandwidth occupancy, and so does a non-peaky scheme that limits itself to the critical bandwidth even though more spectrum is available, so as to not degrade rate. The two types of signaling in the literature are harmonized to show that, for all signaling schemes with the same bandwidth occupancy, rates converge to the wideband AWGN capacity with the same asymptotic behavior as the bandwidth occupancy approaches its critical value, and decrease to zero as the occupancy goes to infinity. This unified analysis not only recovers previous results on capacity bounds for (non-)peaky signaling schemes, but also reveals the fundamental tradeoff between precision and accuracy when characterizing the maximal achievable rate.
across all the available bandwidth and time slots, the desired signal would be buried by the channel uncertainty if bandwidth is too large. M´edard and Gallager proved this [6] through an upper bound to rate proportional 4 to the ratio between the fourth moment of the signal (E |x| ) and its bandwidth (B), i.e., R D/Tc , the idle stage serves as the “zero-padding prefix” that justifies our no-ISI approximation. For a signal a(t) drawn from a stochastic sequence process A, its peakiness is measured by its kurtosis EA |a(t)|4 (7) κ(A) = 2. EA [|a(t)|2 ] When a signal x is zero a fraction 1−δ of the time, its kurtosis can be written as a function of the kurtosis of the nonzero elements, κ(x)= κ(x6δ=0) . Therefore determining peakiness using the on/off ratio δ and the kurtosis statistic κ are in accordance with each other. III. BANDWIDTH O CCUPANCY L IMIT Our analysis is a generalization of the the SISO analysis with non-peaky signaling in [17]. The analysis follows four steps, represented in Fig. 2. 1) Find a bell-shaped lower bound RLB (B) ≤ I (X; Y ); 2) Determine the unique maximum of RLB (B), RLB (B ∗ ); 3) Find a bell-shaped upper bound RUB (B) ≥ I (X; Y ); 4) Determine B + and B − such that B − ≤ B ∗ ≤ B + and RUB (B ± ) = RLB (B ∗ ). The result of [17] is that capacity in a non-coherent fading channel only grows with bandwidth below a critical bandwidth Bcrit which falls into the range [B − , B + ]. A system operating with insufficient bandwidth BBcrit , the channel-uncertainty
4
3 4
2
and
4
RLB ((δB)∗ )≥
s
P Nr 1− N0
log(Bc Tc ) (κ−2+Nt +Nr ) log π . Bc T c (10)
Proof: See Appendix B-B.
1 Figure 2. The four-step approach [17] to set the range of critical bandwidth.
induced penalty grows with increasing bandwidth and the achievable rate decreases to zero as B→∞. So, contrary to the wideband AWGN channel where “the deeper into the lowSNR regime, the better”, in the non-coherent fading channel the optimal guideline is “enter only marginally into the lowSNR regime”. The optimal operation point occurs at Bcrit . Our first contribution is a generalization of this argument to arbitrary levels of signal peakiness δ and identification of the fundamental quantity bandwidth occupancy (δB). We obtain bell-shaped lower and upper bounds on the achievable rate, find (δB)∗ that maximizes the lower bound, and then determine the range (δB)± that contains the unknown critical bandwidth occupancy (δB)crit . For any B>Bcrit it is possible to operate with peaky signalling with δ=Bcrit /B to bring the system back into the same optimal operation point. A. Lower bound Lemma 1. The achievable rate in a wideband non-coherent channel with duty cycle δ ∈ (0, 1] is lower bounded by P (κ − 2 + Nt + Nr ) P Nr 1− RLB (δB) = N0 2δBNt N0 (8) BNt Nr P −δ log 1 + Bc T c , Bc T c δBNt N0 where κ = κ(H) is the kurtosis of the channel. Proof: See Appendix B-A. The kurtosis κ for many fading distributions are in the range of [1, 2]. For example, as given in [17], κ = 2 for Rayleigh fading, κ = 2−4k 2 /(1+2k)2 for Rice fading with factor k > 0, and κ = 1+1/m for Nakagami-m fading channels. Remark 1. Even though the capacity may be a twodimensional function of δ and B, the lower bound is only a function of the product δB. This allows optimization with respect to the product in the next lemma. B. Maximum of RLB Lemma 2. RLB (δB) is maximized at RLB ((δB)∗ ) with s Bc T c P ∗ (κ − 2 + Nt + Nr ), (9) (δB) ≃ N0 Nt log(Bc Tc )
Remark 2. We would like to emphasize that the optimal bandwidth occupancy (δB)∗ is very large given the fact that the channel coherence Bc Tc usually ranges from a few hundreds to hundreds of thousands. For example, assuming 2 × 2 MIMO over Rayleigh fading (κ=2) with P/N0 =107 , we have (δB)∗ ≃120 MHz with capacity gap ∆
Unified Capacity Limit of Non-Coherent Wideband Fading Channels
arXiv:1501.04905v2 [cs.IT] 1 Sep 2015
Felipe G´omez-Cuba∗, Student Member, IEEE, Jinfeng Du∗ , Member, IEEE, Muriel M´edard, Fellow, IEEE, and Elza Erkip, Fellow, IEEE
Abstract—In non-coherent wideband fading channels where energy rather than spectrum is the limiting resource, peaky and non-peaky signaling schemes have long been considered species apart, as the first approaches asymptotically the capacity of a wideband AWGN channel with the same average SNR, whereas the second reaches a peak rate at some finite critical bandwidth and then falls to zero as bandwidth grows to infinity. In this paper it is shown that this distinction is in fact an artifact of the limited attention paid in the past to the product between the bandwidth and the fraction of time it is in use. This fundamental quantity, that is termed bandwidth occupancy, measures average bandwidth usage over time. As it turns out, a peaky signal that transmits in an infinite bandwidth but only for an infinitesimal fraction of the time may only have a small bandwidth occupancy, and so does a non-peaky scheme that limits itself to the critical bandwidth even though more spectrum is available, so as to not degrade rate. The two types of signaling in the literature are harmonized to show that, for all signaling schemes with the same bandwidth occupancy, rates converge to the wideband AWGN capacity with the same asymptotic behavior as the bandwidth occupancy approaches its critical value, and decrease to zero as the occupancy goes to infinity. This unified analysis not only recovers previous results on capacity bounds for (non-)peaky signaling schemes, but also reveals the fundamental tradeoff between precision and accuracy when characterizing the maximal achievable rate.
across all the available bandwidth and time slots, the desired signal would be buried by the channel uncertainty if bandwidth is too large. M´edard and Gallager proved this [6] through an upper bound to rate proportional 4 to the ratio between the fourth moment of the signal (E |x| ) and its bandwidth (B), i.e., R D/Tc , the idle stage serves as the “zero-padding prefix” that justifies our no-ISI approximation. For a signal a(t) drawn from a stochastic sequence process A, its peakiness is measured by its kurtosis EA |a(t)|4 (7) κ(A) = 2. EA [|a(t)|2 ] When a signal x is zero a fraction 1−δ of the time, its kurtosis can be written as a function of the kurtosis of the nonzero elements, κ(x)= κ(x6δ=0) . Therefore determining peakiness using the on/off ratio δ and the kurtosis statistic κ are in accordance with each other. III. BANDWIDTH O CCUPANCY L IMIT Our analysis is a generalization of the the SISO analysis with non-peaky signaling in [17]. The analysis follows four steps, represented in Fig. 2. 1) Find a bell-shaped lower bound RLB (B) ≤ I (X; Y ); 2) Determine the unique maximum of RLB (B), RLB (B ∗ ); 3) Find a bell-shaped upper bound RUB (B) ≥ I (X; Y ); 4) Determine B + and B − such that B − ≤ B ∗ ≤ B + and RUB (B ± ) = RLB (B ∗ ). The result of [17] is that capacity in a non-coherent fading channel only grows with bandwidth below a critical bandwidth Bcrit which falls into the range [B − , B + ]. A system operating with insufficient bandwidth BBcrit , the channel-uncertainty
4
3 4
2
and
4
RLB ((δB)∗ )≥
s
P Nr 1− N0
log(Bc Tc ) (κ−2+Nt +Nr ) log π . Bc T c (10)
Proof: See Appendix B-B.
1 Figure 2. The four-step approach [17] to set the range of critical bandwidth.
induced penalty grows with increasing bandwidth and the achievable rate decreases to zero as B→∞. So, contrary to the wideband AWGN channel where “the deeper into the lowSNR regime, the better”, in the non-coherent fading channel the optimal guideline is “enter only marginally into the lowSNR regime”. The optimal operation point occurs at Bcrit . Our first contribution is a generalization of this argument to arbitrary levels of signal peakiness δ and identification of the fundamental quantity bandwidth occupancy (δB). We obtain bell-shaped lower and upper bounds on the achievable rate, find (δB)∗ that maximizes the lower bound, and then determine the range (δB)± that contains the unknown critical bandwidth occupancy (δB)crit . For any B>Bcrit it is possible to operate with peaky signalling with δ=Bcrit /B to bring the system back into the same optimal operation point. A. Lower bound Lemma 1. The achievable rate in a wideband non-coherent channel with duty cycle δ ∈ (0, 1] is lower bounded by P (κ − 2 + Nt + Nr ) P Nr 1− RLB (δB) = N0 2δBNt N0 (8) BNt Nr P −δ log 1 + Bc T c , Bc T c δBNt N0 where κ = κ(H) is the kurtosis of the channel. Proof: See Appendix B-A. The kurtosis κ for many fading distributions are in the range of [1, 2]. For example, as given in [17], κ = 2 for Rayleigh fading, κ = 2−4k 2 /(1+2k)2 for Rice fading with factor k > 0, and κ = 1+1/m for Nakagami-m fading channels. Remark 1. Even though the capacity may be a twodimensional function of δ and B, the lower bound is only a function of the product δB. This allows optimization with respect to the product in the next lemma. B. Maximum of RLB Lemma 2. RLB (δB) is maximized at RLB ((δB)∗ ) with s Bc T c P ∗ (κ − 2 + Nt + Nr ), (9) (δB) ≃ N0 Nt log(Bc Tc )
Remark 2. We would like to emphasize that the optimal bandwidth occupancy (δB)∗ is very large given the fact that the channel coherence Bc Tc usually ranges from a few hundreds to hundreds of thousands. For example, assuming 2 × 2 MIMO over Rayleigh fading (κ=2) with P/N0 =107 , we have (δB)∗ ≃120 MHz with capacity gap ∆
