Totally blind channel identification by exploiting guard intervals☆
Introduction
Estimating the impulse response of the channel over which a communication system operates is complicated by the source symbols being a-priori unknown to the receiver. Blind channel identification techniques [1], [3], [4], [6], [7], [17], [21], [24], [25] have been applied extensively to this problem. Blind techniques must exploit a known property of the transmitted signal, such as a statistical or finite alphabet property or a bandwidth constraint if the technique requires the received signal to be over-sampled. This paper presents a different approach; it makes no assumptions on the source symbols themselves but rather relies on the communication system introducing a guard interval between each transmitted block. It is proved that the channel is identifiable (up to an inherent scaling factor) from just two received blocks. This complements the recent result that one block suffices if finite alphabet properties of the source symbols in addition to guard intervals are exploited [9].
The main result of this paper is captured in the following unrealistically small but otherwise representative example of a communication system using guard intervals. Example 1 Assume the source symbols 1,2,3,4 are broken into two blocks and guard intervals inserted, forming the transmitted signal 0,1,2,0,3,4,0. Let with h0=1 and h1=−1 denote the impulse response of the channel. The receiver sees the transmitted signal convolved with the channel impulse response, namely 1,1,−2,3,1,−4. More generally, if s1,s2,s3,s4 denote the source symbols then the received symbols are and y6=h1s4. The proposed idea is to invert this set of polynomial equations, thus identifying the channel. If y1=y2=1 and y3=−2 then the first three equations reduce to 2h02+h0h1−h12=0. Assume h0≠0 and set λ=h1/h0. The equation becomes h02(2+λ−λ2)=0 with solutions λ=−1,2. Note that λ=−1 is indeed the ratio h1/h0 of the impulse response used to generate the output 1,1,−2. Repeating the process with the second block, namely and y6=−4, yields the two solutions . In particular, the only solution in common is λ=−1. The ratio h1/h0 of the channel impulse response has been recovered without knowledge of the source symbols.
Three observations are made. The channel is recoverable only up to a scaling factor because the equations in Example 1 are bilinear; if the source symbols are doubled and the impulse response halved, the output remains the same. A single received block does not provide enough information to identify the channel but it narrows the possibilities down to a finite number. Two blocks suffice to recover the channel. The main result of this paper is that these three observations generalise to arbitrarily sized communication systems and even remain true if additive noise is present.
Since the proposed identification technique relies on guard intervals, a brief discussion on guard intervals ensues. Guard intervals, also known as null guards or zero padding, are sequences of L−1 consecutive zeros inserted between blocks to prevent inter-block interference [2], [16], [19], [26]. Here, L is an upper bound on the length of the channel over which the system operates and inter-block interference refers to the problem of the kth output block depending not only on the kth transmitted block but also on the (k−1)th transmitted block due to the memory of the channel. A sequence of L−1 zeros empties the memory of the channel and thus prevents inter-block interference. These guard intervals are often used in time division multiple access (TDMA) systems, such as the current GSM standard for mobile telephones. They can also be used in orthogonal frequency division multiplexing (OFDM) systems [18], [19], [20]. Although originally there to prevent inter-block interference, it is now known that guard intervals have additional advantages, such as always allowing the source symbols to be recovered if the channel is known (and non-zero) and ensuring that channel deconvolution is always a stable operation [8], [12]; see also [11]. The present paper presents yet another advantage of guard intervals; they enable the receiver to identify the channel without any knowledge of the source symbols whatsoever. Note that the parameter L must be known a-priori to both the transmitter and the receiver.
The remainder of this paper is organised as follows. Section 2 defines two types of channel identifiability, one related to the noise free case and the other to when the received signal is corrupted by additive noise. It also states the result from [15] that not only are both definitions equivalent, but that channel identification is possible if and only if a related polynomial map is rationally invertible. Determining if a polynomial map is rationally invertible is difficult in general. However, Section 3 shows this task simplifies if a certain repeated structure is present. Section 4 then applies this result to prove that if guard intervals are inserted between transmitted blocks, the receiver can identify the channel using just two received blocks. Section 4 remarks that the main ideas in this paper extend to proving identifiability of a wider class of linearly and affinely precoded communication systems. Section 5 concludes the paper.
Section snippets
Problem statement and known results
In a communication system operating over a channel of length at most L and with guard intervals of length L−1 inserted between blocks, the kth block of p source symbols is received aswhere is the lower triangular Toeplitz matrix having [h0,…,hL−1,0,…,0]T, the impulse response of the channel padded with p zeros, as its first column. Since the impulse response is unknown, it is appropriate to write (1) in the form of a polynomial equation,
Partially coupled polynomial equations
Proving invertibility of a polynomial map is difficult in general [15], [22], [23]. This section derives a necessary and sufficient condition for an augmented map of the form found in Theorem 4 to be rationally invertible.
Throughout this section, denotes a full rank polynomial map; see Appendix A for the meaning of full rank. As in Section 2, the notation G[2] denotes the augmented map defined by . Theorem 8, the main result of this
Totally blind channel identification
This section uses Theorem 4 and Corollary 9 to prove the system (2) is identifiable using just two blocks. Lemma 10 For any p⩾1 and L⩾1, let be lower triangular Toeplitz matrices whose first column ends in p−1 zeros. The range space of H is contained in the range space of if and only if for some . Proof A straightforward proof by induction is possible. Alternatively, a z-domain argument can be used, as in [9, Proof of Theorem 23]. □ Theorem 11 For any p⩾1 and L⩾2, the system (2) is identifiable,
Conclusion
Guard intervals are often inserted between transmitted blocks in a communication system to prevent inter-block interference. This paper proved that the repeated structure introduced by the guard intervals enables the receiver to identify the channel after receiving just two blocks by solving a system of polynomial equations. Moreover, it was remarked that the same method of proving identifiability extends to more general linearly and affinely precoded communication systems.
Acknowledgements
The first author would like to thank Prof. Philippe Loubaton for informative discussions on guard intervals in communication systems. The second author acknowledges support of the NSF.
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2014, Pattern RecognitionCitation Excerpt :It is worth noting that the received signal of each block of a communication system containing guard intervals satisfies (1) in signal format [32]. It has been proved [32] that to recover the source signal r(n), the received signal needs to accumulate a minimum of two blocks. In other words, if the number of available samples in the received signal is insufficient, then the source symbols cannot be identified.
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This work was undertaken in part while the first author was visiting ENST/TSI in Paris, France.