An OFDM interpretation of zero padded block transmissions☆
Introduction
A wireless communications channel is often modelled by a finite impulse response channel whose impulse response varies with time. For high-speed transmissions, it may be assumed that n consecutive symbols can be transmitted without the impulse response changing significantly, where n⪢1 (for example, n=64 or higher). This paper shows that two popular and apparently distinct block based transmission systems, both of which transmit data in blocks of n symbols, are in fact more closely related than initially suspected. The usefulness of this finding is discussed at the end of this section.
The two transmission systems considered in this paper are orthogonal frequency division multiplex (OFDM) systems [12] and the more recently proposed zero padded systems [10], both of which are special cases of linearly precoded block transmission systems. These terms are defined as follows.
A linearly precoded block transmission system first breaks the source symbols {…,s−1,s0,s1,…} into blocks of length p. A linear precoder matrix is then used to encode each block prior to transmission. Assuming the system operates over a finite impulse response (FIR) channel of length at most L, the ith received block iswhere is the m×m lower triangular Toeplitz channel matrix whose first column is [h0,…,hL−1,0,…,0]T, is the m×m upper triangular Toeplitz matrix with first row equal to [0,…,0,hL−1,…,h1] and is additive white Gaussian noise (AWGN). Here, models the inter-block interference (IBI) caused by the memory of the channel while models the effect of the channel on the current block.
This paper is concerned with the following special cases of (1). An OFDM system uses the precoder P=CDH where D is used throughout to denote the normalized (so that DHD=I) discrete Fourier transform (DFT) matrix whose dimensions can be determined from its context and C is the cyclic prefix matrixwhich adds a cyclic prefix of length L−1. Due to the cyclic prefix, an OFDM system transmits L−1 extra symbols per block, that is, m=p+L−1. A channel coded OFDM system uses a precoder of the form for some precoder where m⩾p+L−1. The interpretation of P̃, if it has more rows than columns, is that it spreads the source symbols out over the frequency domain in each block (see Section 3). A TZ-OFDM system [10] uses P=ZDH where Z is the zero padded matrixwhich adds L−1 trailing zeros. (As in an OFDM system, m=p+L−1.) More generally, a zero padded system is used here to refer to any system (1) which uses a precoder P of the form , where P̃ is an arbitrary precoder having equal or more rows than columns (that is, m⩾p+L−1). Remark Note that in both channel coded OFDM systems and zero padded systems, P̃ can cancel the IDFT operation DH. For example, P=Z is a zero padded system since if .
Trailing zero OFDM (TZ-OFDM) systems are so named in the literature because they replace the cyclic prefix in an OFDM system by a sequence of trailing zeros [10]. Ironically, this paper (see also [2]) shows that TZ-OFDM systems are more closely related to OFDM systems than originally suspected. Indeed, in Section 3 it is shown that the DFT of the received symbols in a zero padded system is related to the source symbols in exactly the same way as in a channel coded OFDM system. In fact, this is a consequence of the more general result, proved in Section 2, that for any zero padded system, there is an equivalent channel coded OFDM system with the same statistical performance.
These results imply that the statistical performance of any zero padded system can be understood by investigating the statistical performance of the equivalent channel coded OFDM system. Three examples in which this proves beneficial are given in 4 Spectrally balanced nature of TZ-QFDM systems, 5 OFDM versus TZ-OFDM, 6 Pilot tones in zero padded systems. Section 4 proves that TZ-OFDM systems spread the source symbols uniformly in the frequency domain; this is a desirable property if the channel is unknown to the transmitter. Section 5 explains intuitively why TZ-OFDM systems do not always perform better than OFDM systems. (Note though that this is no reason not to use a TZ-OFDM system; for most channels, a TZ-OFDM system performs better than an OFDM system.) Section 6 demonstrates how TZ-OFDM systems can use pilot tones to identify the channel, just as in OFDM systems. Concluding remarks are made in Section 7.
Related work: In [8], it was shown that it is possible to make a TZ-OFDM system resemble an OFDM system by using an appropriate reduced complexity equaliser. However, such an equaliser is statistically sub-optimal. The connection made in the present paper is stronger because it is based on statistically optimal equalisers being used for both systems. Furthermore, it is mentioned that although zero padded systems are explained in the literature as OFDM systems using a different precoder matrix [8], [11], this does not necessarily mean zero padded systems are related to OFDM systems in any way. Indeed, without the presence of a cyclic prefix, a system cannot be called a (channel coded or linearly precoded) OFDM system.
Relevance: The design and understanding of linearly precoded systems is currently an important area of research. Prior to this work, it was not clear what the mathematical difference was of using a zero padded system instead of a cyclic prefixed system. This paper proves that the only difference is that the zero padded system inherently incorporates a precoding operation which uniformly spreads the spectrum of the source symbols. Therefore, it suffices to understand cyclic prefixed systems in order to understand zero padded systems. This is advantageous because cyclic prefixed systems have been studied extensively in the past whereas zero padded systems are relatively new.
Section snippets
Zero forcing equalisers and sufficient statistics
To understand the performance of various linearly precoded transmission systems, it is insufficient to study only the precoding operation in (1). This is because the receiver must cope with the problem of IBI (that is, the term in (1)), and in particular, the best way of dealing with IBI depends on the actual precoder used. Therefore, it is necessary to study the statistical information about the source vector present in the received vector. Indeed, two systems will be defined to be
Frequency domain interpretation of zero padded systems
It is well-known that OFDM systems have a particularly simple frequency domain interpretation. This section first reviews this interpretation and then applies the results of the previous section to derive a novel frequency domain interpretation of zero padded systems.
The received symbols of a channel coded OFDM system in the frequency domain are given by taking the DFT of (4), namely:It is a standard result that is a diagonal matrix, a consequence of being circulant.
Spectrally balanced nature of TZ-QFDM systems
As (13) shows, a zero padded system uses the matrix to spread the source symbols over the frequency domain. This section draws attention to the fact that D1ZD2H can be interpreted as spreading the symbols “uniformly”, and moreover, this is a desirable property if the channel is unknown to the transmitter.
Consider the channel coded OFDM system where spreads the spectrum of the source symbols . That is, as (10) shows, the ith element of is transmitted on the ith
OFDM versus TZ-OFDM
It is proposed in [3], [4] to measure the intrinsic performance of linearly precoded systems by the mean-square error of the source symbol estimates if a minimum variance unbiased estimator is used; see Section 2. Although a TZ-OFDM system outperforms an OFDM system over most channels , an example of a channel over which a TZ-OFDM system performs worse than an OFDM system appears in [5]. This section uses the frequency domain interpretation of TZ-OFDM systems derived in Section 3
Pilot tones in zero padded systems
One way of allowing the receiver to estimate the channel is to transmit sinusoids at various frequencies and with known amplitudes. These sinusoids are called pilot tones. It is straightforward to generate pilot tones in an OFDM system; simply set various elements of the source symbols to known values [6]. This section shows that pilot tones can also be generated in zero padded systems. Remark The use of pilot tones in a TZ-OFDM system has already been proposed in [8]. This section shows that pilot
Conclusion
OFDM systems are best understood in the frequency domain. This paper showed that zero padded systems can also be understood more easily in the frequency domain than in the time domain. Indeed, it was proved that associated with every zero padded system is a channel coded OFDM system such that both systems receive exactly the same symbols after taking into account that OFDM receivers discard the guard interval whereas zero padded systems do not. The advantages of this interpretation were
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Cited by (12)
A frequency domain subspace blind channel estimation method for trailing zero OFDM systems
2011, Journal of Network and Computer ApplicationsCitation Excerpt :In this paper, we propose a frequency domain subspace approach to blind channel estimation for TZ-OFDM systems. Although the frequency domain is particularly suitable for analyzing TZ-OFDM systems (Manton, 2002), due to the fact that there is no trivial link in terms of problem characterization between the time and frequency domains, to the best of our knowledge, there has been no frequency domain blind TZ-OFDM channel estimation method previously available. The significance of our proposed frequency domain method is threefold.
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2005, Systems and Control LettersCitation Excerpt :The motivation for this work is that the use of linear precoders in wireless transmission systems is quite common; see [8,11–13,15,20] and the references therein.
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2003, Systems and Control LettersTotally blind channel identification by exploiting guard intervals
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2018, International Journal on Electrical Engineering and Informatics
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This work was supported by the Australian Research Council.