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Computing the Matched Filter in Linear Time

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Computing the Matched Filter in Linear Time Computing the Matched Filter in Linear Time To Solomon Golomb for the occasion of his 80 birthday mazel tov Alexander Fish Shamgar Gurevich Ronny Hadani Department of Mathematics Department of Mathematics Department of Mathematics University of Wisconsin University of Wisconsin University of Texas Madison, WI 53706, USA Madison, WI 53706, USA Austin, TX 78712, USA Email: afi[email protected] Email: [email protected] Email: [email protected] Akbar Sayeed Oded Schwartz Department of Electrical Engineering Department of Computer Science University of Wisconsin University of California Madison, WI 53706, USA Berkeley, CA 94720, USA Email: [email protected] Email: [email protected] Abstract—A fundamental problem in wireless communica- station, !j encodes the radialp velocity of user j with respect tion is the time-frequency shift (TFS) problem: Find the time- to the base station, and i = −1: frequency shift of a signal in a noisy environment. The shift is The base station ”knows” the signals S ’s and R. The the result of time asynchronization of a sender with a receiver, j and of non-zero speed of a sender with respect to a receiver. objective is: A classical solution of a discrete analog of the TFS problem Problem I.1 (Mobile communication problem): Extract the is called the matched filter algorithm. It uses a pseudo-random waveform S(t) of the length p; and its arithemtic complexity bits bj; j = 1; :::; r: 2 is O(p · log(p)), using fast Fourier transform. In these notes A resolution of Problem I.1 will be deduced (see Section we introduce a novel approach of designing new waveforms that allow faster matched filter algorithm. We use techniques from I-F) from our solution to the following problem. group representation theory to design waveforms S(t); which enable us to introduce two fast matched filter (FMF) algorithms, B. The time-frequency shift (TFS) problem called the flag algorithm, and the cross algorithm. These methods We have r signals Sj 2 H, j = 1; :::; r; called the sender solve the TFS problem in O(p · log(p)) operations. We discuss waveforms. Additionally, we are given the receiver waveform applications of the algorithms to mobile communication, GPS, R 2 H, which satisfies and radar. r 2πi X !j ·t I. INTRODUCTION R(t) = e p · Sj(t + τ j) + W(t); t 2 Fp; (I.2) j=1 Denote by H = C(Fp) the vector space of complex valued functions on the finite field Fp = f0; 1; :::; p − 1g; where where W 2 H denotes a random white noise of mean zero, and addition and multiplication is done modulo the odd prime (τ j;!j) 2 Fp ×Fp, j = 1; :::; r: We will call the pairs (τ j;!j) number p: The vector space H is equipped with the standard the time-frequency shifts, and the vector space V = × P Fp Fp inner product hf1; f2i = f1(t)f2(t), for f1; f2 2 H, and the time-frequency plane. arXiv:1112.4883v1 [cs.IT] 20 Dec 2011 t2Fp time-frequency shift problem will be referred to as the Hilbert space of digital signals. The precise formulation of the Let us start with a motivational problem. is the following: Problem I.2 (TFS problem): Given the waveforms S ; j = A. Mobile communication problem j 1; :::; r; and R; extract the time-frequency shifts (τ j;!j) 2 V , We consider the following mathematical model of mobile j = 1; :::; r: communication [10]. There exists a collection of users j = 1; :::; r, each holding a bit bj 2 {±1g; and a private signal C. The matched filter (MF) algorithm Sj 2 H. User j transmits its message bj · Sj to a base station (antenna), and the base station receives the superposition sum A classical solution [3], [4], [5], [7], [10], [11], [12] to Problem I.2, is the matched filter algorithm. For a fixed k 2 r 2πi X !j ·t f1; :::; rg; we define the following matched filter (MF) matrix R(t) = bj · e p · Sj(t + τ j) + W(t); t 2 Fp; (I.1) of the sender Sk, and the receiver R: j=1 D 2πi !·t E where W 2 H denotes a random white noise of mean zero, M[Sk;R](τ; !) = e p · Sk(t + τ);R(t) ; (τ; !) 2 V: τ j encodes the time asynchronization of user j with the base (I.3) 2πi p (τ!j −!τ j ) A direct verification shows that for ζj = e , j = the fastest method which exists in the literature [9]. Note 1; :::; r, we have that computing one entry in M[Sk;R] costs already O(p) operations. This leads to the following fast matched filter M[Sk;R](τ; !) = ζk ·M[Sk;Sk](τ − τ k;! − !k) (I.4) (FMF) problem: X + ζj ·M[Sk;Sj](τ − τ j;! − !j) Problem I.4 (FMF problem): Design waveforms Sj 2 H, j6=k j = 1; :::; r; to solve TFS problem in almost linear time for NSR +O( p ); shift. p D. The flag method where NSR = 1 is the inverse of the signal-to-noise ratio SNR flag method between the waveform Sk and W: For simplicity, we assume We introduce the to propose a solution to FMF that the NSR is not too large, and, for the rest of the paper, problem. We will show how to associate with the p + 1 lines, we will omit the last term in (I.4). through (0; 0) in the time-frequency plane, Lj; j = 1; :::; p+1; a system of almost orthogonal waveforms S 2 H; that we In order to extract the time-frequency shift (τ k;!k); using Lj the matched filter, it is ”standard” (see [3], [4], [5], [7], [10], will call flags. The system satisfies [11], [12]) to use almost-orthogonal pseudo-random signals 8 < 2 + "r;p, if (τ; !) = (τ k;!k); Sj 2 H of norm one. Namely, all the summands in right-hand 0 jM[SLk ;R](τ; !)j = 1 + "r;p, if (τ; !) 2 Lk r (τ k;!k); side of (I.4) are of size O( p1 ), with the exception that for : 0 p "r;p, if (τ; !) 2 V r Lk; (τ; !) = (τ k;!k) we have M[Sk;Sk](τ − τ k;! − !k) = 1: (I.6) pr Hence, where "r;p = O( p );R is the receiver waveform (I.2), defined 0 with respect to any r flags containing SL ; and L is the 1 + "r;p, if (τ; !) = (τ k;!k); k k jM[Sk;R](τ; !)j = shifted line Lk + (τ k;!k). "r;p, if (τ; !) 6= (τ k;!k); (I.5) pr where "r;p = O( p ): Fig. 2. M[SL1 ;R] with two flags SL1 ;SL2 , and (τ 1;!1) = (50; 50) Identity (I.6) suggests the ”flag” algorithmic solution to Fig. 1. jM[S ;R]j with pseudo-random S ;S , and (τ ;! ) = (50; 50) 1 1 2 1 1 FMF problem in the case that the number of waveforms p Identity (I.5) suggests the following ”entry-by-entry” r p; and p is sufficiently large. In the following we algorithmic solution to TFS problem: Compute the assume that R and SLk are as in (I.6). matrix M[Sk;R]; and choose (τ k;!k) for which ? Algorithm I.5 (Flag algorithm): • Choose a line Lk dif- jM[Sk;R](τ k;!k)j ≈ 1: However, this solution of TFS ferent from Lk: problem is very expensive in terms of arithmetic complexity, ? • Compute M[SL ;R] on L . Find (τ; !) such that i.e., the number of arithmetic (multiplication, and addition) k k jM[SL ;R](τ; !)j ≈ 1, i.e., (τ; !) on the shifted line is O(r · p3): One can do better using a ”line-by-line” k Lk + (τ k;!k): computation. This is due to the next observation. • Compute M[SLk ;R] on Lk + (τ k;!k) and find (τ; !) Remark I.3 (FFT): The restriction of the matrix M[Sk;R] such that jM[SLk ;R](τ; !)j ≈ 2: to any line (not necessarily through the origin) in the time- The arithmetic complexity of the flag algorithm is O(r · frequency plane V; is a convolution that can be computed, p log(p)); using the FFT (Remark I.3). using the fast Fourier transform algorithm (FFT), in O(p · log(p)) arithmetic operations. E. The cross method As a consequence of Remark I.3, one can solve TFS Another solution to the TFS problem, and subsequently to problem in O(r · p2 · log(p)) arithmetic operations. To the the mobile communication problem, is the cross method. The best of our knowledge, the ”line-by-line” computation is also idea is similar to the flag method, i.e., first to find a line on of a pseudorandom signal and a structural signal. The first has the MF matrix which is almost delta function at the origin, and the MF matrix of the second is supported on a line. The designs of these waveforms are done using group representation theory. The pseudorandom signals are designed [4], [5], [12] using the Weil representation, and will be called Weil (peak) signals1. The structural signals are designeded [6], [7] using the Heisenberg representation, and will be called Heisenberg (lines) signals. We will call the collection of all flag waveforms, the Heisenberg–Weil flag system. In this section we briefly recall constructions, and properties of Fig. 3. Diagram of the flag algorithm these waveforms. A more comprehensive treatment, including proofs, will appear in [2].
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