Signal transmission through an unidentified channel Dae Gwan Lee * , G¨ otz E. Pfander * and Volker Pohl † * Mathematisch-Geographische Fakult¨ at, Katholische Universit¨ at Eichst¨ att-Ingolstadt, 85071 Eichst¨ att, Germany Email: daegwans@gmail.com, pfander@ku.de † Lehrstuhl f¨ ur Theoretische Informationstechnik, Technische Universit¨ at M¨ unchen, 80333 M¨ unchen, Germany Email: volker.pohl@tum.de Abstract—We formulate and study the problem of recovering a signal x in X⊂ C L which, after adding with a pilot signal c ∈ C L \{0}, is transmitted through an unknown channel H in H⊂L(C L , C L ). Here, X and H are a priori known and fixed while c is designed by the user. In particular, we consider the case where H is generated by a subset of time-frequency shift operators on C L , which leads to investigation of properties of Gabor matrices. I. I NTRODUCTION To transmit data efficiently over frequency-selective and time-varying channels, a communication channel is generally identified or estimated before its use. For this, communication systems often adapt the following two stage transmission scheme. In the first stage, a known pilot signal is transmitted based on which the channel is identified or estimated [2], [5], [11]. In the second stage, the actual data signal is transmitted through the channel and the receiver uses the channel informa- tion to recover the data signal. However, for rapidly varying channels such a scheme is no longer applicable. In that case it may be of advantage to combine the two stages so as to estimate the channel and the data signal simultaneously. This paper investigates one of such signal recovery schemes. In our model, channels and data signals are assumed to be in some known sets, and the data signal is combined additively with a pilot signal before transmitting it through the channel. As we shall see, the design freedom in the pilot signal is what enables the exact recovery of the data signal even if the data signal cannot be recovered from its corresponding output (see Example 10 below). While deriving some necessary and/or sufficient conditions for the recovery of data signals that passed through an unidentified channel in the general setup, we consider the application relevant case where the channel space is spanned by a subset of time-frequency shift operators on C L . This leads to the investigation of properties of Gabor matrices [10]. The special case where the channel space is spanned only by time shift operators on C L corresponds to circular convolutions; the study of simultaneous channel identification and signal recovery in this case is referred to as blind deconvolution [1], [4], [8], [13]. We also give some examples to illustrate our results. II. BACKGROUND AND PROBLEM FORMULATION A. Channel Identification and Signal Recovery In communications engineering, it is often required to identify a channel before using it to transmit signals. Definition 1. A class of linear operators H⊂L(C L , C L ) is said to be identifiable if there exists a vector c ∈ C L such that the map Φ c : H −→ C L , H → Hc is injective. Such a vector c is called an identifier for H. Once the communication channel is identified, we use it to transmit a signal x from a set X⊂ C L . The receiver observes the channel output y = Hx where the information of H is now known, and therefore x can be successfully recovered from y provided that H is injective on X . B. Problem Formulation We aim to recover signals that are transmitted through an unidentified channel. To achieve this, we combine the process of channel identification and signal recovery by modeling the input signal to be of the form x + c, where x ∈X is the data signal to be sent and c ∈ C L \{0} is a pilot signal which is designed by the user. We formulate our problem precisely as follows. Main Problem. What conditions on H⊂L(C L , C L ) and X⊂ C L are necessary and/or sufficient so that there exists a vector c ∈ C L \{0} with the property that x ∈X can be recovered uniquely from y = H(x+c) with H ∈H unknown? Let us mention that in applications it is often possible to design the data space X⊂ C L as well as c, while the channel space H⊂L(C L , C L ) is a priori fixed. III. NECESSARY AND SUFFICIENT CONDITIONS Certainly, a naive approach to our problem is to first identify the channel H ∈H and then to use the channel information to recover x ∈X . However, in principle it is not necessary to find the exact channel H ∈H in order to recover x ∈X . In fact, we have the following necessary and sufficient condition for the recovery of x. Proposition 2. Let ∅ = H⊂L(C L , C L ), ∅ = X⊂ C L , and c ∈ C L \{0}. Then every x ∈X is uniquely recoverable from y = H(x + c) with H ∈ H\{0} unknown, if and only if (•) H(x + c)= H ′ (x ′ + c) for some H, H ′ ∈ H\{0} and x, x ′ ∈X implies x = x ′ . Proposition 2 gives a general solution to our problem. Unfortunately, the necessary and sufficient condition (•) for the recovery of x is not practical and usually hard to verify. For