Computing the Matched Filter in Linear Time
To Solomon Golomb for the occasion of his 80 birthday
mazel tov
Abstract
A fundamental problem in wireless communication is the timefrequency shift (TFS) problem: Find the timefrequency shift of a signal in a noisy environment. The shift is the result of time asynchronization of a sender with a receiver, and of nonzero 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 pseudorandom 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.
I Introduction
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.
Ia Mobile communication problem
We consider the following mathematical model of mobile communication [10]. 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
(I.1) 
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
IB The timefrequency shift (TFS) problem
We have signals , called the sender waveforms. Additionally, we are given the receiver waveform , which satisfies
(I.2) 
where denotes a random white noise of mean zero, and , We will call the pairs the timefrequency shifts, and the vector space the timefrequency plane.
The precise formulation of the timefrequency shift problem is the following:
Problem I.2 (TFS problem)
Given the waveforms and extract the timefrequency shifts ,
IC The matched filter (MF) algorithm
A classical solution [3, 4, 5, 7, 10, 11, 12] to Problem I.2, is the matched filter algorithm. For a fixed we define the following matched filter (MF) matrix of the sender , and the receiver :
(I.3) 
A direct verification shows that for , , we have
where is the inverse of the signaltonoise 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 (IC).
In order to extract the timefrequency shift using the matched filter, it is ”standard” (see [3, 4, 5, 7, 10, 11, 12]) to use almostorthogonal pseudorandom signals of norm one. Namely, all the summands in righthand side of (IC) are of size , with the exception that for we have Hence,
(I.5) 
where
Identity (I.5) suggests the following ”entrybyentry” 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 ”linebyline” 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 timefrequency 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 ”linebyline” computation is also the fastest method which exists in the literature [9]. 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.
ID 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 timefrequency plane, a system of almost orthogonal waveforms that we will call flags. The system satisfies
(I.6) 
where is the receiver waveform (I.2), defined with respect to any flags containing and is the shifted line .
Identity (I.6) suggests the ”flag” algorithmic solution to FMF problem in the case that the number of waveforms and is sufficiently large. In the following we assume that and are as in (I.6).
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).
IE 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 timefrequency shift is located, and then to search on the line to find the timefrequency shift. We will show how to associate with the distinct pairs of lines a system of almostorthogonal 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).
IF 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) signals^{1}^{1}1For 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 [2].
Iia The Heisenberg (lines) system
Consider the following collection of unitary operators, called Heisenberg operators, that act on the Hilbert space of digital signals:
(II.1) 
where is the timeshift operator, is the frequencyshift 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
(II.2) 
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]:
Theorem II.1
The Heisenberg system satisfies the properties

Line. For every line , and every , we have

Almostorthogonality. For every two lines , and every , we have
for every
IiB The Weil (peaks) system
Consider the following collection of matrices
Note that is in a natural way a group [1] with respect to the operation of matrix multiplication. It is called the special linear group of order two over Each element acts on the timefrequency 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:
(II.3) 
where is defined by (II.1). Denote by the space of all solutions to System (II.3). The following is a basic result [13]:
Theorem II.2 (Stone–von Neumann–SchurWeil)
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 [13]. The group is not commutative, but contains a special class of maximal commutative subgroups called tori^{2}^{2}2There are order of tori in [4, 5]. Each torus acts via the Weil representation operators
(II.4) 
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]:
Theorem II.3
The Weil system satisfies the properties

Peak. For every torus , and every , we have

Almostorthogonality. For every two tori , and every , with we have
for every
IiC 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:
Theorem II.4
The Heisenberg–Weil system satisfies the properties

Flag. For every line , torus , and every flag with , we have
where and

Almostorthogonality. For every two lines , tori and every two flags with , , we have
for every
Remark II.5
As a consequence of Theorem II.4 we obtain families of almostorthogonal 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 IIA. The following follows immediately from Theorem II.1:
Theorem III.1
The Heisenberg cross system satisfies the properties

Cross. For every pair of distinct lines , and every cross , with , we have
where

Almostorthogonality. For every four distinct lines , and every two crosses we have
for every
Remark III.2
As a consequence of Theorem III.1 we obtain families of almostorthogonal 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.
Iva Application to global positioning system (GPS)
The model of GPS works as follows [8]. 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 client^{3}^{3}3From the we can find [8] 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)
Find
By using Heisenberg–Weil or Heisenberg cross waveforms we find the pairs in arithmetical operations.
IvB Application to discrete radar
The model of discrete radar works as follows [7]. A radar sends a waveform which bounds back by targets. The signal which is received as an echo has the form^{4}^{4}4In 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)
Find
By sending Heisenberg–Weil waveform we get^{5}^{5}5For simplicity we assume that all the shifted lines ’s are distinct. The general case is treated similarly.
where
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.
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