Computing the Matched Filter in Linear Time
To Solomon Golomb for the occasion of his 80 birthday
A fundamental problem in wireless communication is the time-frequency shift (TFS) problem: Find the time-frequency shift of a signal in a noisy environment. The shift is the result of time asynchronization of a sender with a receiver, and of non-zero speed of a sender with respect to a receiver. A classical solution of a discrete analog of the TFS problem is called the matched filter algorithm. It uses a pseudo-random waveform of the length and its arithemtic complexity is , using fast Fourier transform. In these notes we introduce a novel approach of designing new waveforms that allow faster matched filter algorithm. We use techniques from group representation theory to design waveforms which enable us to introduce two fast matched filter (FMF) algorithms, called the flag algorithm, and the cross algorithm. These methods solve the TFS problem in operations. We discuss applications of the algorithms to mobile communication, GPS, and radar.
Denote by the vector space of complex valued functions on the finite field where addition and multiplication is done modulo the odd prime number The vector space is equipped with the standard inner product , for , and will be referred to as the Hilbert space of digital signals.
Let us start with a motivational problem.
I-a Mobile communication problem
We consider the following mathematical model of mobile communication . There exists a collection of users , each holding a bit and a private signal . User transmits its message to a base station (antenna), and the base station receives the superposition sum
where denotes a random white noise of mean zero, encodes the time asynchronization of user with the base station, encodes the radial velocity of user with respect to the base station, and
The base station ”knows” the signals ’s and . The objective is:
Problem I.1 (Mobile communication problem)
Extract the bits
I-B The time-frequency shift (TFS) problem
We have signals , called the sender waveforms. Additionally, we are given the receiver waveform , which satisfies
where denotes a random white noise of mean zero, and , We will call the pairs the time-frequency shifts, and the vector space the time-frequency plane.
The precise formulation of the time-frequency shift problem is the following:
Problem I.2 (TFS problem)
Given the waveforms and extract the time-frequency shifts ,
I-C The matched filter (MF) algorithm
A direct verification shows that for , , we have
where is the inverse of the signal-to-noise ratio between the waveform and For simplicity, we assume that the is not too large, and, for the rest of the paper, we will omit the last term in (I-C).
In order to extract the time-frequency shift using the matched filter, it is ”standard” (see [3, 4, 5, 7, 10, 11, 12]) to use almost-orthogonal pseudo-random signals of norm one. Namely, all the summands in right-hand side of (I-C) are of size , with the exception that for we have Hence,
Identity (I.5) suggests the following ”entry-by-entry” algorithmic solution to TFS problem: Compute the matrix and choose for which However, this solution of TFS problem is very expensive in terms of arithmetic complexity, i.e., the number of arithmetic (multiplication, and addition) is One can do better using a ”line-by-line” computation. This is due to the next observation.
Remark I.3 (Fft)
The restriction of the matrix to any line (not necessarily through the origin) in the time-frequency plane is a convolution that can be computed, using the fast Fourier transform algorithm (FFT), in arithmetic operations.
As a consequence of Remark I.3, one can solve TFS problem in arithmetic operations. To the best of our knowledge, the ”line-by-line” computation is also the fastest method which exists in the literature . Note that computing one entry in costs already operations. This leads to the following fast matched filter (FMF) problem:
Problem I.4 (FMF problem)
Design waveforms , to solve TFS problem in almost linear time for shift.
I-D The flag method
We introduce the flag method to propose a solution to FMF problem. We will show how to associate with the lines, through in the time-frequency plane, a system of almost orthogonal waveforms that we will call flags. The system satisfies
where is the receiver waveform (I.2), defined with respect to any flags containing and is the shifted line .
Algorithm I.5 (Flag algorithm)
Choose a line different from
Compute on . Find such that , i.e., on the shifted line
Compute on and find such that
The arithmetic complexity of the flag algorithm is using the FFT (Remark I.3).
I-E The cross method
Another solution to the TFS problem, and subsequently to the mobile communication problem, is the cross method. The idea is similar to the flag method, i.e., first to find a line on which the time-frequency shift is located, and then to search on the line to find the time-frequency shift. We will show how to associate with the distinct pairs of lines a system of almost-orthogonal waveforms that we will call crosses. The system satisfies
where is the receiver waveform (I.2), defined with respect to any different crosses containing and ,
The arithmetic complexity of the cross method is using the FFT (Remark I.3).
I-F Solution to the mobile communication problem
Ii The Heisenberg–Weil flag system
The flag waveforms, that play the main role in the flag algorithm, are of a special form. Each of them is a sum 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) signals111For the purpose of the Flag method, other pseudorandom signals may work.. 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 these waveforms. A more comprehensive treatment, including proofs, will appear in .
Ii-a The Heisenberg (lines) system
Consider the following collection of unitary operators, called Heisenberg operators, that act on the Hilbert space of digital signals:
where is the time-shift operator, is the frequency-shift operator, for every , and denotes composition of operators.
The operators (II.1) do not commute in general, but rather obey the Heisenberg commutation relations The expression vanishes if , belong to the same line. Hence, for a given line we have a commutative collection of unitary operators
We use the theorem from linear algebra about simultaneous diagonalization of commuting unitary operators, and obtain [6, 7] a natural orthonormal basis consisting of common eigenfunctions for all the operators (II.2). The system of all such bases where runs over all lines through the origin in will be called the Heisenberg (lines) system. We will need the following result [6, 7]:
The Heisenberg system satisfies the properties
Line. For every line , and every , we have
Almost-orthogonality. For every two lines , and every , we have
Ii-B The Weil (peaks) system
Consider the following collection of matrices
Note that is in a natural way a group  with respect to the operation of matrix multiplication. It is called the special linear group of order two over Each element acts on the time-frequency plane via the change of coordinates For every , let be a linear operator on which is a solution of the following system of linear equations:
Theorem II.2 (Stone–von Neumann–Schur-Weil)
There exist a unique collection of solutions which are unitary operators, and satisfy the homomorphism condition
Denote by the collection of all unitary operators on the Hilbert space of digital signals . Theorem II.2 establishes the map , which is called the Weil representation . The group is not commutative, but contains a special class of maximal commutative subgroups called tori222There are order of tori in [4, 5]. Each torus acts via the Weil representation operators
This is a commutative collection of diagonalizable operators, and it admits [4, 5] a natural orthonormal basis for , consisting of common eigenfunctions. The system of all such bases where runs over all tori in will be called the Weil (peaks) system. We will need the following result [4, 5]:
The Weil system satisfies the properties
Peak. For every torus , and every , we have
Almost-orthogonality. For every two tori , and every , with we have
Ii-C The Heisenberg–Weil system
We define the Heisenberg–Weil system of waveforms. This is the collection of signals in , which are of the form , where and are Heisenberg and Weil waveforms, respectively. The main technical result of this paper is:
The Heisenberg–Weil system satisfies the properties
Flag. For every line , torus , and every flag with , we have
Almost-orthogonality. For every two lines , tori and every two flags with , , we have
As a consequence of Theorem II.4 we obtain families of almost-orthogonal flag waveforms which can be used for solving the TFS and mobile communication problems in almost linear time.
Iii The Heisenberg cross system
We define the Heisenberg cross system of waveforms. This is the collection of signals in , which are of the form , where are Heisenberg waveforms defined in Section II-A. The following follows immediately from Theorem II.1:
The Heisenberg cross system satisfies the properties
Cross. For every pair of distinct lines , and every cross , with , we have
Almost-orthogonality. For every four distinct lines , and every two crosses we have
As a consequence of Theorem III.1 we obtain families of almost-orthogonal cross waveforms which can be used for solving the TFS and mobile communication problems in almost linear time.
Iv Applications to GPS and radar
In the introduction we described application of flag and cross methods to mobile communication. In this section we demonstrate applications to global positioning system (GPS), and discrete radar.
Iv-a Application to global positioning system (GPS)
The model of GPS works as follows . A client on the earth surface wants to know his geographical location. Satellites send to earth their location. For simplicity, the location of satellite is a bit Satellite transmits to the earth its signal multiplied by its location The client receives the signal
where encodes the radial velocity of satellite with respect to the client, encodes the distance between satellite and the client333From the we can find  the distance between satellite and the client, given that and the clocks of all satellites are synchronized., and is a random white noise of mean zero.
Problem IV.1 (GPS problem)
By using Heisenberg–Weil or Heisenberg cross waveforms we find the pairs in arithmetical operations.
Iv-B Application to discrete radar
The model of discrete radar works as follows . A radar sends a waveform which bounds back by targets. The signal which is received as an echo has the form444In practice there are intensity coefficients such that Assuming that ’s are sufficiently large our methods are applicable verbatim.
where encodes the radial velocity of target with respect to the radar, encodes the distance between target and the radar, and is a random white noise of mean zero.
Problem IV.2 (Discrete radar problem)
By sending Heisenberg–Weil waveform we get555For simplicity we assume that all the shifted lines ’s are distinct. The general case is treated similarly.
This means that by using the flag algorithm we solve the radar problem in arithmetical operations.
Remark IV.3 (Important)
Note that the cross method is not applicable for the discrete radar problem if the number of targets
Acknowledgement. Warm thanks to Joseph Bernstein for his support and encouragement in interdisciplinary research. We are grateful to Anant Sahai, for sharing with us his thoughts, and ideas on many aspects of signal processing and wireless communication. The project described in this paper was initiated by a question of Mark Goresky and Andy Klapper during the conference SETA2008, we thank them very much. We appreciate the support and encouragement of Nigel Boston, Robert Calderbank, Solomon Golomb, Guang Gong, Olga Holtz, Roger Howe, Peter Sarnak, Nir Sochen, and Alan Weinstein.
-  Artin M., Algebra. Prentice Hall, Inc., Englewood Cliffs, NJ (1991).
-  Fish A., Gurevich S., Hadani R., Sayeed A., Schwartz O., Fast matched filter and group representation theory. In preparation (2011).
-  Golomb, S.W. and Gong G., Signal design for good correlation. For wireless communication, cryptography, and radar. Cambridge University Press, Cambridge (2005).
-  Gurevich S., Hadani R., Sochen N., The finite harmonic oscillator and its associated sequences. PNAS, July 22, 2008 vol. 105 no. 29 9869–9873.
-  Gurevich S., Hadani R., Sochen N., The finite harmonic oscillator and its applications to sequences, communication and radar . IEEE Transactions on Information Theory, vol. 54, no. 9, September 2008.
-  Howe R., Nice error bases, mutually unbiased bases, induced representations, the Heisenberg group and finite geometries. Indag. Math. (N.S.) 16 (2005), no. 3–4, 553–583.
-  Howard, S. D., Calderbank, R., and Moran W., The finite Heisenberg–Weyl groups in radar and communications. EURASIP J. Appl. Signal Process (2006).
-  Kaplan E., Understanding GPS Principles and Applications. Artech house, INC (1996).
-  O’Toole J.M., Mesbah M., and Boashash B., Accurate and efficient implementation of the time–frequency matched filter. IET Signal Process., 2010, Vol. 4, Iss. 4, pp. 428–437.
-  Tse D., and Viswanath P., Fundamentals of Wireless Communication. Cambridge University Press (2005).
-  Verdu S., Multiuser Detection, Cambridge University Press (1998).
-  Wang Z., and Gong G., New Sequences Design From Weil Representation With Low Two-Dimensional Correlation in Both Time and Phase Shifts. IEEE Transactions on Information Theory, vol. 57, no. 7, July 2011.
-  Weil A., Sur certains groupes d’operateurs unitaires. Acta Math. 111, 143-211 (1964).