An OFDM interpretation of zero padded block transmissions

Abstract

Orthogonal frequency division multiplex (OFDM) systems are particularly straightforward to understand in the frequency domain. The purpose of this paper is to draw attention to the fact that zero padded block transmissions also have a straightforward frequency domain interpretation. The advantages of this interpretation are illustrated by three additional results which further the understanding of zero padded block transmission systems; it is shown that the zero padding spreads the spectrum of the source symbols uniformly, it is explained why zero padded block transmission systems do not always outperform OFDM systems, and it is shown how pilot tones can be incorporated into 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 $\text{s}{}^{\mathrm{(i)}}\text{=[s}{}_{\mathrm{ip}}\text{,s}{}_{\mathrm{ip+1}}\text{,…,s}{}_{\mathrm{ip+p-1}}\text{]}{}^{\mathrm{<mtext>T</mtext>}}\text{∈}\text{C}{}^{\mathrm{p}}$ of length p. A linear precoder matrix $\text{P∈}\text{C}{}^{\mathrm{m\times p}}$ is then used to encode each block prior to transmission. Assuming the system operates over a finite impulse response (FIR) channel $\text{h}\text{=[h}{}_0\text{,…,h}{}_{\mathrm{L-1}}\text{]}{}^{\mathrm{<mtext>T</mtext>}}\text{∈}\text{C}{}^{\mathrm{L}}$ of length at most L, the ith received block is

$$\text{y}{}^{\mathrm{(i)}}\text{=}\text{H}{}^{\mathrm{<mtext>IBI</mtext>}}\text{P}\text{s}{}^{\mathrm{(i-1)}}\text{+}\text{H}{}^{\mathrm{<mtext>CB</mtext>}}\text{P}\text{s}{}^{\mathrm{(i)}}\text{+}\text{n}{}^{\mathrm{(i)}}\text{∈}\text{C}{}^{\mathrm{m}}\text{,}$$

where $\text{H}{}^{\mathrm{<mtext>CB</mtext>}}$ is the m×m lower triangular Toeplitz channel matrix whose first column is [h₀,…,h_L−1,0,…,0]^T, $\text{H}{}^{\mathrm{<mtext>IBI</mtext>}}$ is the m×m upper triangular Toeplitz matrix with first row equal to [0,…,0,h_L−1,…,h₁] and $\text{n}{}^{\mathrm{(i)}}\text{∈}\text{C}{}^{\mathrm{m}}$ is additive white Gaussian noise (AWGN). Here, $\text{H}{}^{\mathrm{<mtext>IBI</mtext>}}$ models the inter-block interference (IBI) caused by the memory of the channel while $\text{H}{}^{\mathrm{<mtext>CB</mtext>}}$ 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=CD^H where D is used throughout to denote the normalized (so that D^HD=I) discrete Fourier transform (DFT) matrix whose dimensions can be determined from its context and C is the cyclic prefix matrix

$$\text{C=}\text{0}{}_{\mathrm{(L-1)\times (m-2L+2)}}\text{I}{}_{\mathrm{L-1}}\text{I}{}_{\mathrm{m-L+1}}\text{,}$$

which 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 $\text{P=CD}{}^{\mathrm{<mtext>H</mtext>}}\text{P}\text{̃}$ for some precoder $\text{P}\text{̃}\text{∈}\text{C}{}^{\mathrm{(m-L+1)\times p}}$ 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=ZD^H where Z is the zero padded matrix

$$\text{Z=}\text{I}{}_{\mathrm{(m-L+1)}}\text{0}{}_{\mathrm{(L-1)\times (m-L+1)}}\text{,}$$

which 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 $\text{P=ZD}{}^{\mathrm{<mtext>H</mtext>}}\text{P}\text{̃}$, 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 D^H. For example, P=Z is a zero padded system since $\text{P=ZD}{}^{\mathrm{<mtext>H</mtext>}}\text{P}\text{̃}$ if $\text{P}\text{̃}\text{=D}$.

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.