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An Analysis of Out-Of-Band Emission and In-Band Interference for Precoded and Classical OFDM Systems

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An Analysis of Out-Of-Band Emission and In-Band Interference for Precoded and Classical OFDM Systems An analysis of out-of-band emission and in-band interference for precoded and classical OFDM systems Medhat Mohamad∗, Rickard Nilsson∗ and Jaap van de Beek∗† ∗Department of Computer Science, Electrical and Space Engineering Lulea˚ University of Technology, SE–97187, Lulea,˚ Sweden. {medhat.mohamad, rickard.o.nilsson, jaap.vandeBeek}@ltu.se †Huawei Technologies, Sweden {jaap.vandebeek}@huawei.com Abstract—In this paper we present analytical expressions for OFDM symbol to zero. In [5], the main goal is to introduce the out-of-band (OOB) emission and in-band interference of nulls at well chosen frequencies in the OFDM spectrum. In two different OOB suppressed OFDM-systems; classical low-pass [7] the constraints on the precoding matrix suggested in [5] filtered and precoded. Then, we analytically compare their per- are relaxed. In [8], the precoder is designed to limit the OOB formance in terms of OOB emission suppression and introduced emission suppressed below a particular power mask. level of in-band interference. We analyse the fact that the in- band interference introduced by the filter depends on the length A comparison between the different schemes regarding the of the cyclic prefix as well as the behavior of the channel while OOB emission suppression is carried out [4]. Yet, to our that of the precoded OFDM does not. The analysis confirms that knowledge, no comparison at the level of the in-band inter- edged subcarriers suffer higher in-band interference than central ference introduced by the different schemes has been carried subcarriers. Moreover, frequency precoders, for a specific choice out. of notching frequencies, can outperform time precoders. In this paper, we focus on the two precoding approaches presented in [6] and [7] respectively and compare them with I. INTRODUCTION the classical OFDM (we refer to the low pass filtered OFDM One of the few drawbacks with traditional OFDM is its as classical OFDM). We present analytical expressions for relative high levels of out-of-band (OOB) emission which the introduced in-band interference at different subcarriers in results in strong interference into neighbouring frequency both cases. Then, we make a comparative analysis of the bands. This plays a crucial role in the legitimacy of OFDM in OOB emission suppression performance and the amount of the coming communication systems and standards especially in-band interference. We investigate the effect of the channel with the cognitive radio where many different radio-systems behaviour as well as the length of the cyclic prefix on the must be able to co-reside politely in densely packed spectrum in-band interference level. We also discuss the fact that in bands. both cases edge subcarriers are more affected by interference One track is to replace the whole OFDM system with lower than centrally located subcarriers. and finally, we support our OOB emission systems such as the filter bank multicarrier analytical results with simulations of the bit error rate (BER). (FBMC) or the universal filtered multicarrier (UFMC) [1]–[3]. In Section II, we describe the two different OOB suppression Although those techniques show impressive results at the level approaches: classical and precoded OFDM. In section III, we of supression of OOB emission, yet their drawbacks appear in derive analytical expressions of the in-band interference for the implementation complexity issues as well as their degraded two approaches. We check the effect of cyclic prefix length on performance in MIMO channels. the level of in-band interference introduced. In section IV, On the other track, suppression of the OOB emission can we analyse and compare the OOB suppression performance be done by processing the plain OFDM signal. This is done and in-band interference effect for both precoded and classical by either treating the OFDM signal after modulation (low OFDM under different channels behaviour. Finally, we sum- pass filtering and windowing) or before modulation (carrier marize our work in Section V. cancellation and precoding) [4]. While any of these approaches can be efficient in suppress- II. OUT-OF-BAND EMISSION SUPPRESSION ing the OOB emission, all of them inevitably comes with We will investigate the baseband OFDM transmitted signal, a certain price. Carrier cancellation is spectrally inefficient given by due to sacrificing of subcarriers. Filtering and windowing ∞ introduce in-band interference by smearing the OFDM signal s(t)= si(t − i(Ts + Tg)), (1) to suppress the OOB emission. Precoding introduces ICI by i=−∞ linearly combining the independent data symbols. where Ts is the symbol duration, Tg is the duration of the Many precoding schemes are presented in the literature. th cyclic prefix and si(t) is the i OFDM transmitted symbol In [5], OOB emission are suppressed by modulating the K−1 −j2π k t subcarriers with well-chosen precoded data symbols. In [6] represented by si(t)= dk,ie Ts , −Tg ≤ t ≤ Ts. the main goal is to push the OFDM signal continuous in k=0 time, as well as its N-derivatives by forcing the edges of the Here, dk,i represents an information symbol taken from some s (t) f di i s (t) IFFT Filter i Th d d¯ sp(t) i Precoder i IFFT i Tg Tg Fig. 1. OFDM system model with a filter (top) or a precoder (bottom) used s (t) s (t) for the suppression of out-of-band emission. i−1 i Fig. 2. ISI of symbol si−1(t) on symbol si(t). The cyclic prefix of length Tg T symbol constellation C where k ∈ [0, 1, ..., K − 1] represents mitigates part of the filter’s impulse response of length h but the remaining part will interfere with the following OFDM symbol. the subcarrier index. The power spectrum of the OFDM signal (1) is [7] th 1 The i precoded OFDM symbol is ( )= ||a( )||2 S f f 2, (2) K−1 T p ¯ −j2π k t s (t)= dk,ie Ts , −Tg ≤ t ≤ Ts, (7) a(f) i where is a vector collecting the modulated subcarriers in k=0 the frequency domain such that ¯ ¯ ¯ ¯ T where di =[d1,i, d2,i,...,dK−1,i] is the linearly precoded T a(f)=[a0(f),a1(f) ...aK−1(f)] , (3) data symbol vector such that th d¯ = Gd and ak(f) is the k modulated subcarrier given by i i, (8) d =[ ]T G −jπ(T −T )(f− k ) k where i d1,i,d2,i,...,dK−1,i and is the precoding ( )= s g Ts ( ( − )) ak f Te sinc πT f , (4) matrix given by [6], [7] Ts −1 H H where T = Ts +Tg. As is well known, the spectrum (2) decays G I − B (BB ) B. (9) with 1/f 2. The precoding matrix G, then, is the projection onto the In what follows, we study the suppression of OOB emission nullspace N (B) of B. The choice of B depends on the of signal (1) by a classical approach and a precoding approach. precoding technique followed. We recall two choices from the literature here. A. Classical approach In the time precoding approach proposed in [6], the OOB The first approach we study applies a low pass filter to (1) emission are treated by managing the discontinuity property to suppress the OOB emission as shown in Figure 1 (top). of the OFDM signal. In [6], the OOB emission are suppressed h(t) Assuming that the filter has impulse response with length by rendering the OFDM signal represented by (1) as well as T [0,T ] s (t) h and support on h , then after introducing symbol i its first N derivatives continuous by pushing the beginning and ith into the filter, the classical filtered output symbol is the end of the OFDM symbol to the origin, i.e. f ( )= ( ) ∗ ( ) n n si t si t h t , (5) d p d p s (t) = s (t) =0. (10) dtn i dtn i and the power spectrum of the classical signal becomes t=−Tg t=Ts f 2 This is accomplished by choosing B in (9) as S (f)=|H(f)| S(f), (6) AΦ B where H(f) is the frequency response of the filter (the Fourier A , (11) transform of the impulse response of the filter). The spectrum of a classical OFDM system is reperesented in Figure 3. where A is an (N +1)× K matrix with entries [A]ij = f i The classical baseband transmitted signal s (t) will be longer kj , i =0...N , j =0...K − 1 and Φ = 2 2 2 than the baseband transmitted signal s(t) since the filtering diag(ej πk0 ,ej πk1 ...ej πkK−1 ) [6]. process extends the length of the output signal. If the length In the frequency precoding approach proposed in [7], the core of the cyclic prefix is not long enough to mitigate the joint idea falls in nulling the spectrum of s(t) , S(f), at certain set of dispersion caused by the filter and the channel impulse re- frequencies, M = {f0 ...fm ...fM−1}, such that S(fm)=0. sponse, the following OFDM symbol is affected by ISI. Figure Here, the frequency precoder, B is an M ×K matrix collecting 2 illustrates this situation. It shows the tail of the impulse the notching vectors at different notching frequencies. response from symbol s −1 that is not absorbed by the cyclic i B =[a( ) a( ) a( )]T prefix, thus causing interference with the following OFDM f0 ... fm ... fM−1 , (12) symbol si. In section III we show how much ISI will affect where a(fm) is the subcarriers vector at frequency fm as the OFDM symbols subcarriers. defined in (4). For both the time precoder and the frequency precoder, the B.
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