1 Active Interference Cancellation for OFDM Spectrum Sculpting: Linear Processing is Optimal Jorge F. Schmidt, Member, IEEE, Daniel Romero, Student Member, IEEE, and Roberto L´ opez-Valcarce, Member, IEEE Abstract— Active interference cancellation (AIC) is a multi- carrier spectrum sculpting technique which reduces the power of undesired out-of-band emissions by adequately modulating a subset of reserved cancellation subcarriers. In most schemes online complexity is a concern, and thus cancellation subcarriers have traditionally been constrained to linear combinations of the data subcarriers. Recent AIC designs truly minimizing out-of- band emission shift complexity to the offline design stage, moti- vating the consideration of more general mappings to improve performance. We show that there is no loss in optimality incurred by constraining these mappings to the set of linear functions. Index Terms— Active Interference Cancellation, Out-of-band radiation, Spectrum Sculpting, Cognitive OFDM. I. I NTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) has become the modulation format of choice in modern high- speed wireless and wireline systems, due to its many well- known qualities. Nevertheless, one shortcoming of OFDM resides in the large sidelobes of the Inverse Discrete Fourier Transform (IDFT), which result in substantial leakage across subcarriers with the ensuing adjacent channel interference. This issue is often dealt with by deactivating a number of guard subcarriers at the edges of the signal spectrum, with the consequent penalty in data rate. In order for OFDM to be adopted by future high-performance systems, e.g., 5G, a number of enhancements will become necessary to overcome this and other drawbacks [1], [2]. The leakage problem is also of concern in wideband OFDM-based cognitive systems in which deep notches must be sculpted in the spectrum in order to avoid interfering to narrowband licensed users [3]. An appealing approach to IDFT leakage reduction is active interference cancellation (AIC), first proposed in [4]: undesired emission is reduced by judiciously modulating a number of cancellation subcarriers (CSs), while using the remaining data subcarriers (DSs) for transmission as usual. Thus, operation is transparent to the receiver, which just discards the CSs after demodulation. The advantage of AIC resides in that the number of CSs required to achieve a given level of undesired emission is typically much smaller than the number of guard subcarriers to be turned off in the traditional approach. Several AIC designs have been subsequently proposed [5]- [9]. These works minimize w.r.t. the CS values a cost function The authors are with the Department of Signal Theory and Communi- cations, University of Vigo, 36310 Vigo, Spain {jschmidt, dromero, val- carce}@gts.uvigo.es. Work supported by the Spanish Government, ERDF funds (TEC2013- 47020-C2-1-R COMPASS, CONSOLIDER-INGENIO 2010 CSD2008-00010 COMONSENS), FPU Grant AP2010-0149, and the Galician Regional Gov- ernment (CN 2012/260 AtlantTIC). given by the magnitude of the instantaneous signal spectrum for a given OFDM symbol at a number of frequencies within the protected band, subject to different constraints to con- trol the power allocated to CSs. These designs suffer from one main drawback. Finding the optimal CS values requires solving a constrained optimization problem for each OFDM symbol, as the solution is dependent on the specific DS values; this results in significant online complexity. In this context, AIC has been recast in terms of minimization of the average undesired emission power under a total power constraint, assuming a linear map from DS to CS values [10]. This formulation drastically reduces the online computational cost of AIC, which is a main concern in practice. Specifically, the resulting matrix defining the optimal mapping is indepen- dent of the instantaneous data, so no optimization problem has to be solved on the fly: online complexity remains low, without sacrificing performance, and with tight control on the transmit power. In view of this, as a next step it is reasonable to ask whether performance could be further improved by allowing more general (nonlinear) relations between DS and CS values under the framework of [10]. Were the answer affirmative, then it would make sense to approach the design of such nonlinear mappings in an optimal way. Our contribution is to answer this question: we prove that performance cannot improve by incorporating nonlinear de- pendencies. Hence, there is no loss of optimality by restricting the mapping from DS to CS values to the class of linear functions, which have the advantage of simple implementation. We note that the fact that certain earlier AIC designs such as [6] and [9] directly result in a linear relation for the optimum CS values does not readily imply that this should also be the case for the setting in [10] because, as stated above, the use of instantaneous values (as opposed to statistical averages) in the cost function and constraints results in structurally different problems. As it turns out, the corresponding proof is not trivial due to a number of constraints inherent to the problem. The letter is organized as follows. Sec. II describes the psd- based AIC design from [10]. Optimality of linear processing is established in Sec. III, and conclusions are drawn in Sec. IV. II. PSD- BASED ACTIVE I NTERFERENCE CANCELLATION Consider the transmission of an OFDM signal with N subcarriers. The power radiated in some band B, covered by N P contiguous subcarriers within the transmission band- width, is to be minimized 1 . The AIC scheme reserves N A = 1 If B is outside the transmission bandwidth, as would be the case for out-of- band radiation minimization, then N P =0, as no system subcarriers overlap with the target band.