Capacity Analysis of MMSE Pilot Patterns for Doubly-Selective Channels. Arun Kannu and Phil Schniter T. H. E OHIO STATE UNIVERSITY.

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1 Capacity Analysis of MMSE Pilot Patterns for Doubly-Selective Channels Arun Kannu and Phil Schniter T. H. E OHIO STATE UIVERSITY June 6, 2005 Ohio State Univ 1

2 Pilot Aided Transmission: Assume that the transmitter and receiver both know the channel statistics but not the channel realization. Pilot-aided Transmission (PAT) defined as follows. 1. The transmitter sends an -block including both data and pilots. 2. The receiver estimates the channel once using only the pilots. 3. The receiver attempts coherent data detection using the estimated channel matrix. Key observations about our definition of PAT: 1. Iterative channel/data estimation is prohibited. 2. PAT is actually a form of non-coherent communication. Ohio State Univ 2

3 MMSE-PAT: Many authors have suggested PAT with: 1. Wiener channel estimation at the receiver. 2. The pilot sequence chosen to minimize the MSE of channel estimates (subject to a pilot power constraint), 3. The pilot power chosen via some other criterion. This problem has been investigated for various channel classes (e.g., time-selective, frequency-selective, doubly-selective). However, most investigations have assumed non-superimposed (SI) pilot/data patterns. What about MMSE-PAT with superimposed pilot/data? Ohio State Univ 3

4 Problem Setup: Observation: y = H(p + d) + v p, d C = (P + D)h + v Channel estimate: ĥ = f(y, P) h := h ĥ Detection: y Pĥ }{{} y eff y = Pĥ + P h + Dĥ + D h + v = Dĥ + P h + D h + v }{{} v eff y eff = Ĥd + v eff The structures of H, P,D depend on the modulation scheme (e.g., CP-OFDM, SCCP) and the channel properties (e.g., TS, FS, DS). Ohio State Univ 4

5 Generic Conditions for MMSE-PAT: Say where y = (P + D)h + v h = Uλ U H U = I M, E[λ] = 0, E[λλ H ] = diag(σλ 2 0,..., σλ 2 M 1 ) 0, E[D] = 0, E[v] = 0, E[vv H ] = σvi, 2 uncorrelated {D, λ, v}, and p 2 E p. Can show that E{ h 2 } is minimized if and only if D, (PU) H DU = 0 (1) (PU) H PU = diag(α 0,..., α M 1 ) (2) where the water-filling coefs {α m } depend on {σλ 2 m }, σv, 2 and E p. Ohio State Univ 5

6 Generic Conditions for MMSE-PAT (cont.): Interpretation of (1)-(2): 1. D, (PU) H DU = 0: Pilot/data subspaces remain orthogonal at channel output. 2. (PU) H PU = diag(α 0,..., α M 1 ): Pilot excitation proportional to strength of channel mode. Implication: Pilot/data superposition is tolerated as long as pilot/data can be separated by a linear receiver, a consequence of our not allowing iterative channel estimation. Ohio State Univ 6

7 Application: The Doubly-Dispersive Channel: Consider a SISO, WSSUS, Rayleigh fading channel. Assume t ISI coefficients, i.i.d. with uniform Doppler spectrum over [ f d, f d ) Hz, approximated by a basis expansion model: h(n, l) = 1 ( f 1)/2 k= ( f 1)/2 λ(k, l)e j 2π kn, for 0 n <. where f := 2f d T s + 1. For y = (P + D)h + v with length-( t 1) CP, this implies h = Uλ ( U = I t F :, f 1 : f 1) 2 2 λ C ( ) 0, f t I f t, where F is the unitary -DFT matrix. ote U H U = I f t. Ohio State Univ 7

8 DS-Channel Conditions for MMSE-PAT: With this -block DS model, the necessary and sufficient conditions for MMSE-PAT become: k t, m f, E p δ(k)δ(m) = 1 n=0 p(n)p (n k)e j 2π mn (3) 0 = 1 n=0 d(n)p (n k)e j 2π mn (4) t := { t + 1,..., t 1} f := { f + 1,..., f 1}. To construct such a pilot/data pattern, 1. Find pilot sequence p satisfying (3). 2. Write (4) as W p d = 0 and set d = Bs, where the s columns of B form an O basis for null(w p ). We call this the (p, B) MMSE-PAT pattern. Ohio State Univ 8

9 The Data Dimension s : In MMSE-PAT, the data is represented by the s symbols in s. It is relatively easy to bound the data dimension s as (2 f 1)(2 t 1) s f t. A more careful analysis, however, reveals the strict upper bound s < f t when t > 1 and f > 1 (i.e., the strictly-ds case). Ohio State Univ 9

10 Example MMSE-PAT Pilot Patterns: 1. Time-domain Kronecker Delta (TDKD): s = f (2 t 1) data 2 t 1 f = data data 0 time (This non-superimposed PAT was suggested by Ma/Giannakis/Ohno.) 2. Freq-domain Kronecker Delta (FDKD): s = t (2 f 1) data 2 f 1 t = data data 0 3. Orthogonal Chirps: s = 2 t f + 1. freq p(n) = E p ej 2π f 2 n2 b k (n) = E p p(n)e j 2π (k+ f t )n, 0 k < s Ohio State Univ 10

11 Capacity of (p, B) MMSE-PAT: Say p 2 E p, E[ s 2 ] E s, and define σs 2 := E s, σp 2 := E p. s t f Then where C mmse-pat C mmse-pat C mmse-pat C mmse-pat := 1 E log det( I + ρ l B H H H HB ) C mmse-pat := 1 E log det( I + ρ u B H H H HB ) ρ l := σ2 s σ 2 v ( σ 2 p σ 2 p + σ 2 s + σ 2 v ) and ρ u := σ2 s. σv 2 (For the lower bound, we assumed the worst-case h via independent CWG, and for the upper bound the best-case h via h = 0.) Ohio State Univ 11

12 Power Allocation that Maximizes C mmse-pat : Say α (0, 1) is used to allocate the total power E t = E s + E p : E s = αe t and E p = (1 α)e t. Then C mmse-pat maximized by β β 2 β s t f α = 1 2 s = t f β := 1 + f t ρ 1 f t s ρ := E t σ 2 v. Ohio State Univ 12

13 High-SR Capacity of MMSE-PAT: With the C mmse-pat -maximizing power allocation, C mmse-pat (ρ) = s log(ρ) + O(1), as ρ Recall that s differed among the different MMSE-PAT examples. ote that, when t > f : FDKD-PAT dominates TDKD-PAT and Chirp-PAT. Superimposed PAT has advantages over non-superimposed PAT. Ohio State Univ 13

14 umerical Example: =128, t =16, f =2 Example: 6 Lower TDKD f c = 12 GHz, 5 Lower FDKD Upper TDKD T 1 s = 2 MHz, v = 3 133kmh, T delay = 8µs, Yields: = 128. Capacity Bounds Bits/sec/Hz Upper FDKD Lower oncoherent Coherent t = 16, f = SR in db Ohio State Univ 14

15 High-SR Capacity Summary of Known Results: coherent: log(ρ) + O(1) noncoherent TS: noncoherent FS: noncoherent DS: SI-PAT TS: SI-PAT FS: SI-PAT DS: MMSE-PAT: f log(ρ) + O(1) t log(ρ) + O(1) t f log(ρ) + O(1) f log(ρ) + O(1) t log(ρ) + O(1) f (2 t 1) log(ρ) + O(1) s log(ρ) + O(1) Ohio State Univ 15

16 On the on-optimality of MMSE-PAT: ote that for TS and FS channels, C mmse-pat (ρ) achieves the same slope as C ts (ρ) and C fs (ρ) as ρ. But, for DS channels (i.e., f > 1 and t > 1) as ρ, C ds (ρ) = t f log(ρ) + O(1), C mmse-pat (ρ) = s log(ρ) + O(1) for s < t f, and thus MMSE-PAT is strictly suboptimal. This motivates non-pat schemes, e.g., schemes based on iterative channel/data estimation. Ohio State Univ 16

17 Summary: Derived nec/suff conditions for MMSE-PAT design in LTV channels. Derived nec/suff conditions for MMSE-PAT design in DS channels, yielding novel MMSE-PAT schemes. Established bounds on the capacity of MMSE-PAT over DS chans. Suggested data/pilot power allocation for MMSE-PAT via C mmse-pat maximization. Showed advantages of superimposed over non-superimposed MMSE-PAT when time-spreading dominates frequency-spreading. Established high-sr noncoherent capacity of the DS channel. Showed that MMSE-PAT is strictly suboptimal in DS channels. Ohio State Univ 17