Spline Pulse Shaping with Isi-Free Matched Filter Receiver

SPLINE PULSE SHAPING WITH ISI-FREE MATCHED FILTER RECEIVER JesúsIbáñez, Carlos Pantaleón, JoséDiez, Ignacio Santamar´ıa DICOM, ETSII y Telecomunicación,Universidad de Cantabria. Avda Los Castros s.n., 39005 Santander, Spain. Tel: +34 942 201388; fax: +34 942 201488 e-mail:[email protected] ∗ ABSTRACT idea, in this paper we propose to interpolate the discrete se- Although the raised cosine pulse shaping filter is a well- quence of symbols using splines. In this way, we construct an established standard in digital communications, in many ISI-free signal when sampled at the symbol rate. Although practical systems simpler approaches are useful. In this any reconstruction technique from the communications sym- paper a new family of pulse shaping filters with zero in- bols that exactly goes through the symbols is a zero ISI tersymbol interference after matched filtering is proposed. communications signal, splines have a number of advantages The method, based on the spline interpolation of the symbol that make them an interesting choice when spectral con- data, exploits two properties of splines: that the B-spline tent as well as complexity are of concern. The main advan- coefficients can be efficiently obtained via digital filtering, tage is that any spline interpolation model of odd-order may and that the B-spline kernels of order n can be constructed be computed by filtering the sequence of symbols with two from the convolution of n + 1 rectangular pulses. These matched filters, so it is quite simple to design an optimal re- two facts suggest how to decompose the filtering operations ceiver. Besides this advantage, spline interpolation achieves needed to obtain any spline interpolation of odd-order into better spectral characteristics than rectangular pulses with two matched filters. The proposed architecture allows an a moderate increase in complexity. efficient implementation, produces optimal performance and The rest of the paper is organized as follows. In Section 2 shows better spectral characteristics than other suboptimal we review some basic notions of spline interpolation. In Sec- approaches. tion 3 we show how to use splines as ISI-free shaping filters. Section 4 is concerned with implementation aspects of the 1 INTRODUCTION proposed spline shaping technique. Section 5 shows some simulation results to illustrate the advantages of spline con- In digital communications it is important to select signal formation. The paper ends with the conclusions presented shapes that produce zero intersymbol interference (ISI) after in Section 6. the matched filtering stage when the channel is bandlimited. In his seminal paper, Nyquist set down the necessary condi- 2 SPLINE INTERPOLATION tions to achieve ISI-free transmission [1]. Among them, the In this section we review the fundamental properties of spline first Nyquist criterion, which states that the equivalent im- interpolation; a more detailed review is carried out in [6, 7, 8]. pulse response of the transmitting and receiving filters should A B-spline of order n, βn(t), is a symmetrical, bell-shaped have zero crossings at multiples of the symbol period T , is function that can be constructed from the convolution of the most used in practice. Also due to Nyquist, the vestigial n + 1 identical rectangular pulses symmetry theorem allows the design of realizable filters that satisfy the first Nyquist criterion, including the raised cosine ( 0 1, |t| < 1/2; filter, that is used in many practical systems when ISI is a rel- β (t) = 0, |t| ≥ 1/2. evant problem. Despite the success of the raised cosine pulse shape, still some research is conducted in the design of filters Spline interpolation of a given input data signal s[k] con- that satisfy the Nyquist-I constraint, searching for different sists in determining a set of coefficients c[k] so that the con- objectives such as maximizing the energy in a certain time tinuous time signal interval [2], or having ISI-free properties with or without X n t matched filtering [3, 4]. In some cases, however, due to the s(t) = c[k]β − k (1) T computational cost associated to this optimal raised cosine k∈ filtering, practical applications rely on much simpler shaping Z filters with worse spectral characteristics, mainly rectangular goes exactly through the data points; i.e., waves or cosine functions. s[k] = s(kT ), k ∈ Z. (2) Although not commonly stated, the first Nyquist criterion is equivalent to interpolating the communication symbols us- One of the advantages of the B-spline expansion (1) is that ing a basis function with the interpolatory property [5], i.e., the coefficients c[k] can be easily obtained by digital filtering with zero crossings at the sampling points. Following this techniques. To clarify this idea, it is convenient to intro- duce the discrete B-spline kernel bn[k], which is obtained by ∗ This work has been partially supported by the European sampling the B-spline of degree n at integer instants Commission and the Spanish Government under grants 1FD97- 1066-C02-01 and TIC2001-0751-C04-03. bn[k] = βn(k) = βn(t)| . t=k∈Z 1 3 CUBIC SPLINE FILTER n=3 In this section we discuss how to split the zero ISI cardinal 1, n=5 spline (3) between the transmitter and the receiver in order n=9 to match the transmitting and receiving filters and achieve a ∞ 0,8 n= practical implementation. Even if we restrict our exposition to cubic spline interpolation, all the results may be extended 0,6 to any odd-order spline. The direct B-spline filter for the cubic spline is 0,4 3 1 − z1 |k| d [k] = z1 , 0,2 1 + z1 Impulse response √ 0 where z1 = −2 + 3 is the filter pole [7]. It is easy to show that the direct cubic B-spline filter can be factored into two 3+ 3− −0,2 filters, one causal (d [k]) and one anticausal (d [k]), given by −4 −2 0 2 4 3± ±k t/T d [k] = (1 − z1)z1 u[±k]. (4) Since d3+[k] = d3−[−k], their Fourier transforms are con- jugated and hence both filters are matched. As a pair of n Figure 1: Impulse response of the pulse shaping filter ϕ (t) matched filters we propose a transmission filter hT (t), com- for n = 3, 5, 9 and ∞. posed by the cascade connection of d3+[k] and β1(t/T ), X 3+ 1 t h (t) = d [k]β − k , T T k∈ Using the discrete B-spline kernel, (2) may be expressed as Z the convolution and a receiving filter hR(t), composed by the cascade of β1(t/T ) and d3−[k] X n n s[k] = c[m]b [k − m] = c[k] ∗ b [k]. m∈ X 3− 1 t Z h (t) = h (−t) = d [k]β − + k . R T T According to this equation, s[k] is obtained by filtering c[k] k∈Z with bn[k]. Hence, to obtain c[k] from s[k] it is enough to ap- We will denote hR(t) and hT (t) as square root cubic spline ply the filter dn[k] that verifies bn[k]∗dn[k] = δ[k]. The filter n (SQRCS) filters. d [k], known as the direct filter, is all-pole and has symme- Since the frequency response of the SQRCS transmitting try properties that permit to implement it as a cascade of n+ n− filter is a causal, d [k], and an anticausal, d [k], recursive filters. sinc2(ωT/2π) As we show in the next section, this property is crucial to HT (ω) = (1 − z1) −jωT , 1 − z1e obtain a ISI-free matched filter using splines. if we assume that the symbol sequence s[k] is white with The overall interpolation process may be expressed as a variance σ2 , then, the power spectral density (PSD) of the convolution using the discrete sequence s[k] —the communi- S transmitted signal s (t) will be cation symbols— as coefficients T 2 2 4 σS 2 3σS sinc (ωT/2π) X n t ST (ω) = |HT (ω)| = . s(t) = s[k]η − k , T T 2 + cos(ωT ) T k∈ Z 4 IMPLEMENTATION ASPECTS n where η (t) is the cardinal spline of degree n, which is ob- In this section we briefly discuss some implementation as- tained as the convolution of the direct filter and the B-spline pects of the proposed spline pulse shaping technique. We kernel n X n n focus on the cubic spline conformation, but the results are η (t) = d [k]β (t − k). easily extended to any odd-order spline. k∈Z In principle, the major drawback of the proposed approach If, for notational convenience, we define our zero ISI pulse- is the implementation of the IIR (infinite impulse response) shaping filter as digital filters d3±[k]. The causal filter d3+[k] may be imple- t mented using just one multiplication and one addition for ϕn (t) = ηn , (3) T T each input symbol the communications signal can be rewritten as sd+ [k] = s[k] + z1sd+ [k − 1]. X n s(t) = s[k]ϕT (t − kT ). The anticausal filter, however, must be truncated. More- over, to keep a matched transmission system, the causal filter k∈Z should also be truncated. In this way, both filters d3+[k] and Fig. 1 shows the impulse response of the pulse shaping d3−[k] are implemented as finite impulse response (FIR) fil- n filters ϕT (t) of degree 3, 5, 9 and ∞. In particular, for ters. The truncation procedure introduces some degradation; n 3+ 3− n = ∞, ϕT (t) is the ideal Nyquist filter. It is clear from the however, d [k] and d [k] are both fast falling filters, and zero crossings at the multiples of T that the proposed spline we have found by simulation that with just three coefficients interpolation technique produces ISI-free signals.

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