Single carrier frequency division multiple access

Note that the high PAPR of OFDM signals is mostly problematic for transmitters, because of the linearity requirements and the consequent limited efficiency of power amplifiers. At the receiving side it is less dramatic because a low noise amplifier (LNA) has a lower power consumption, even if the linearity requirements are strict. It is therefore possible to combine individual SC transmissions (each with low PAPR) into one OFDM signal. This is exactly what happens in single carrier frequency division multiple access (SC-FDMA), which has been selected as the encoding for the uplink in long term evolution(LTE), aka 4G [159], [58].

Conceptually, all users simultaneously transmit an OFDM signal in which only a few exclusive tones are active for that user and the others are set to zero. Because of the smart choice of the assigned tones, this individual OFDM signal is equivalent to a SC transmission, which has the advantageous low PAPR and which explains the name of the encoding schema. In practice the assignment of the tones to each user varies over time to level out the good and bad tones. The uplink receiver at the base station can process the sum of all individual SC signals simultaneously as one large OFDM block.

Receiver windowing

This chapter is based on the article Combined per tone equalization and receiver windowing in DSL receivers: WiPTEQ[42], as published in Elsevier Signal Processing. Only the layout and the numbering of the references, equations and figures have been changed to accomodate for the different page size and to improve consistency. The research in this chapter was done in collaboration with Alcatel and was patented [38], [35].

List of symbols

Lower case bold-faced letters are used to denote vectors and upper case bold-faced letters to denote matrices. In addition, the following notation is used throughout the text:

AT _{the transpose of A}

A(k, :) row k of matrix A Ir×c unity matrix of size r × c

Or×c zero matrix of size r × c

J cost function min

x J the minimization of J over x

E{·} expected value Number of pages: 42 (including figures and tables) Number of figures: 12

Number of tables: 5

Keywords: DSL, equalization, window functions, interference, PTEQ

Abstract

A novel technique is described for the combination of per-tone equalization (PTEQ) with receiver windowing. The PTEQ is an equalization technique for discrete multitone modulation (DMT) based modems, such as asymmetric digital subscriber line (ADSL) modems and very high bitrate digital subscriber line (VDSL) modems, optimizing the SNR (and thus capacity) of each carrier separately. Windowing functions are very useful in multitone communications systems, to prevent a narrow band noise source from causing wide band interference. Combining both techniques in a windowed PTEQ (referred to as WiPTEQ) leads to a robust communication system. The described technique is especially useful in case of a trapezoidal or raised cosine window, and when the window taper length is large compared to the number of equalizer filter taps.

3.1

Introduction and motivation

Asymmetric Digital Subscriber Loop (ADSL) makes use of discrete multitone modulation(DMT): the spectrum is divided into a large number of bands. Carriers (tones) in these bands are (de)modulated in the digital domain, through a (inverse) Discrete Fourier Transform(DFT), in practice carried out using the Fast Fourier Transform(FFT) algorithm [31]. Equalization is facilitated by a cyclic prefix (CP) preceding each symbol [145]. As long as the channel impulse response length does not exceed the CP length the equalization of each tone can be done easily through a one-tap frequency domain equalizer (FEQ), consisting of a multiplication and phase shift for each tone individually.

Since the prefix does not carry any useful information, it is kept as short as possible, implying that often the channel impulse response length exceeds the cyclic prefix, such that the aforementioned condition for easy equalization does not hold, and inter carrier interference (ICI) results ([149], [165], [87], [21]). Classical DMT receiver schemes (e.g. for ADSL) make use of a time domain equalizer (TEQ) to shorten the channel such that the cascade of the channel and TEQ is shorter than the CP, shown in Fig.3.1. This TEQ is mostly implemented as a finite impulse response (FIR) filter of length

T taps. Many algorithms have been developed to initialize the TEQ using training

sequences. The final result depends on the optimization criterion used, but in general the resulting capacity is suboptimal (although there are exceptions, e.g. [5], [201]). In [196] a new receiver structure, based on so-called per tone equalization (PTEQ), has been developed as an alternative to the classical TEQ. For each tone seperately, a

T -taps FEQ is constructed that maximizes the SNR on that tone. In [196], it is shown

that the PTEQ of length T offers a performance upper bound for any TEQ design of the same length.

S/P ,φ ,φ ... To demapper at sample rate

T−taps FIR filter

FEQ=

N 1−tap complex filter 2 at symbol rate A A_{N N} 2 2 , A_iφ_i DFT REMOVE CP front−end From analog TEQ

Figure 3.1: Classical receiver block scheme with a time domain equalizer, serial-to- paraller converter, removal of the cyclic prefix DFT and one-tap complex frequency domain equalizer

The spectrum occupied by high speed digital subscriber line (DSL) modems overlaps with the bands used for radio communications. AM broadcast stations and amateur radio transmitters introduce radio frequency interference (RFI) impairing the DSL- receiver. For discrete multitone (DMT) based Very high-bitrate Digital Subscriber Loop(VDSL) modems, receiver windowing is used to reduce the effect of spectral leakage of such narrow band RFI. However, also for ADSL, the use of windowing functions is important in order to sustain high rates under RFI conditions, as will be shown later. In [154], a procedure is given to calculate an optimal window, given an existing TEQ.

In [197] it was shown that applying a time domain window is equivalent to applying a fixed per-tone equalizer, be it with different equalizer coefficients for different tones. However, the straightforward implementation of the windowing operation as a per tone equalization is computationally demanding.

In the presence of RFI, the PTEQ effectively behaves as a windowing function for each tone separately, but obviously only if the RFI is also present during equalizer training. In case RFI emerges only after the training phase, neither the classical TEQ nor the PTEQ handle this interference satisfactory. We describe a novel method for combining windowing with per tone equalization. The windowing operation stays in place and is done before the FFT. The PTEQ is modified to take windowing into account, hence the name window incorporating PTEQ (WiPTEQ). More specifically, a new scheme is developed, based on the efficient implementation of a sliding windowed FFT (as opposed to the simple sliding FFT formula used by the PTEQ when no windowing is applied). We will consider two specific choices for the window functions to be applied, namely the trapezoidal window [44] and the raised cosine window [41], as these correspond to practical choices in view of e.g. implementation complexity. In Fig.3.2, the side lobe behaviour of these windows is compared to the classical rectangular (Dirichlet)window.

analysis followed by performance simulation results in section 3.4. Finally in section 3.5 conclusions are drawn.

0 50 100 150 200 250 300 −150 −100 −50 0 tones

sidelobe envelope rolloff [dB]

Sidelobe envelope rolloff

rectangular trapezoidal (µ=16) raised cosine (µ=16)

Figure 3.2: Side lobe decay of the rectangular (Dirichlet), raised cosine and trapezoidal window. The window taper length µ equals 16.