Signal Processing 93 (2013) 1259–1267 Contents lists available at SciVerse ScienceDirect Signal Processing journal homepage: www.elsevier.com/locate/sigpro A generalized Fourier domain: Signal processing framework and applications Jorge Martinez n, Richard Heusdens, Richard C. Hendriks Signal and Information Processing (SIP) Lab. Delft University of Technology, Mekelweg 4, 2628CD Delft, Netherlands article info abstract Article history: In this paper, a signal processing framework in a generalized Fourier domain (GFD) is Received 9 August 2011 introduced. In this newly proposed domain, a parametric form of control on the periodic Received in revised form repetitions that occur due to sampling in the reciprocal domain is possible, without the 18 October 2012 need to increase the sampling rate. This characteristic and the connections of the Accepted 24 October 2012 generalized Fourier transform to analyticity and to the z-transform are investigated. Core Available online 2 November 2012 properties of the generalized discrete Fourier transform (GDFT) such as a weighted Keywords: circular correlation property and Parseval’s relation are derived. We show the benefits of Generalized Fourier domain using the novel framework in a spatial-audio application, specifically the simulation of Generalized discrete Fourier transform room impulse responses for auralization purposes in e.g. virtual reality systems. Weighted circular convolution & 2012 Elsevier B.V. All rights reserved. 1. Introduction are derived such as the weighted circular correlation property and Parseval’s energy relation for the GFD that, In this paper we introduce a framework for signal together with the previously introduced weighted circular processing in a generalized Fourier domain (GFD). In this convolution theorem for the GDFT, are fundamental to domain a special form of control on the periodic repeti- build a general-purpose GFD-based signal processing tions that occur due to sampling in the reciprocal domain framework. To finalize our discussion in Section 4 we is possible, without the need to increase the sampling show how the novel framework can be used in spatial- rate. First in Section 2 we review the definition of the audio applications such as the simulation of multichannel generalized discrete Fourier transform (GDFT) and its room impulse responses for auralization purposes in e.g. associated generalized Poisson summation formula virtual reality and telegaming systems. (GPSF), both previously introduced in [1]. Analogous to the periodic extension of a finite-length signal that occurs in standard Fourier theory [2,3], here we introduce the 2. A generalized Fourier domain concept of ‘‘weighted periodic signal extension’’ that naturally occurs when working in the GFD. Next we study Let us define the generalized discrete Fourier trans- the connections of the presented theory to spectral form for finite-length signals x(n), n ¼f0, ...,NÀ1g, with \ sampling, analyticity and the z-transform. This analysis parameter a 2 C f0g as, also serves as a discussion of the generalized Fourier NXÀ1 bn Àjð2p=NÞkn transform and its relationship to the standard Fourier F afxðnÞg9XaðkÞ¼ xðnÞe e , ð1Þ transform. In Section 3 important properties of the GDFT n ¼ 0 for k ¼f0, ...,NÀ1g, where b ¼ logðaÞ=N. The inverse GDFT is given by [1], n Corresponding author. Tel.: þ31 15 27 82 188; fax: þ31 15 27 81 843. eÀbn NXÀ1 F À1fX ðkÞg9xðnÞ¼ X ðkÞejð2p=NÞkn: ð2Þ E-mail address: [email protected] (J. Martinez). a a N a URL: http://siplab.tudelft.nl (J. Martinez). k ¼ 0 0165-1684/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2012.10.015 1260 J. Martinez et al. / Signal Processing 93 (2013) 1259–1267 relationships are used in a practical application of the theory in Section 4. Let us begin with the connection to analyticity. Define the standard spectrum of a discrete-time signal by SðoÞ, where o 2 R represents angular frequency and let SðoÞ2L2½p,p. Since the original signal s(n), n 2 Z,is defined for discrete-time it is clear that SðoÞ is a periodic function of o with period equal to 2p. The signal s(n) could represent the samples of a continuous-time signal, but without the sampling interval that information is lost and no particular analog representation is to be inferred. Let us assume for a moment that SðoÞ can be analytically continued into the complex angular–frequency plane. This is SðoÞ-SðozÞ, where oz 2 C is the complex-valued Fig. 1. Geometrically weighted extension of a finite length signal when angular frequency. Let or and oi denote the real and evaluated outside its original domain, for a ¼ 0:5 and N¼10. imaginary parts respectively of oz. Then from the defini- tion of the discrete-time Fourier transform we have, The GDFT (1) is equivalent to the ordinary discrete Fourier X1 bn Àjozn transform of the modulated signal xðnÞe . The finite-length SðozÞ¼ sðnÞe , signal ebn for n ¼f0, ...,NÀ1g,isoffiniteenergyforall n ¼1 X1 a 2 C\f0g, therefore for x(n) a signal of finite energy, the Àjðor þ joiÞn Sðor þjoiÞ¼ sðnÞe , GDFT can be properly defined [2,1]. Note that when a ¼ 1, n ¼1 the transform pair correspond to the standard DFT pair. X1 oin Àjor n Sðor þjo Þ¼ sðnÞe e : ð4Þ Let us denote the periodic extension of XaðkÞ by X~ aðkÞ for i ~ n ¼1 k 2 Z. Clearly we have that X aðkÞ¼XaððkÞÞN,where 9 From here we see that if s(n) is a causal sequence and the XaððkÞÞN Xaðk mod NÞ, i.e. the circular shift of the sequence is represented as the index modulo N. On the other hand (1) analytic continuation of SðoÞ is done on the lower half of and (2) imply a geometrically weighted periodic extension of the complex plane then oi o0, the signal x(n) when evaluated outside f0, ...,NÀ1g.Thisis X1 oin Àjor n stated by the generalized Poisson summation formula Sðor þjoiÞ¼ sðnÞe e , ¼ (GPSF) associated with the transform [1], n 0 and the extra factor eoin can only improve the conver- X eÀbn NXÀ1 x~ ðnÞ9 apxððnÞÞN ¼ X ðkÞejð2p=NÞkn, ð3Þ gence rate of the series. Now, a N a p2Z k ¼ 0 X1 Àjor n þ oin lim Sðor þjoiÞ¼ lim sðnÞe , o -À0 o -À0 i i n ¼ 0 where n 2 Z, p ¼bn=Nc and ððnÞÞN ¼ nþpN. We can regard X1 x~ ðnÞ as a superposition of infinitely many translated and a ¼ lim sðnÞeÀjor n þ oin, o -À0 geometrically weighted ‘‘replicas’’ of x(n). The replicas out- n ¼ 0 i p side the support of x(n)areweightedbya and x~aðnÞ¼xðnÞ ¼ SðoÞ: for n ¼f0, ...,NÀ1g. This is illustrated in Fig. 1,wherea On the other hand we have that finite signal and (a part of) its geometrically weighted Z p extension are depicted for a ¼ 0:5andN¼10. Therefore to 2 9Sðo þjo Þ9 do work in the generalized Fourier domain implies a manip- r i r Àp Z p ulation of the signals involved via their geometrically n ¼ Sðor þjoiÞS ðor þjoiÞ dor, weighted extensions. This is an important fact as we will Àp Z ! seethroughtherestofthepaper. p X1 X1 Àjor n þ oin n jor m þ oim Signals of the form x~ aðnÞ although infinitely long and not ¼ sðnÞe s ðmÞe dor, Àp n ¼ 0 m ¼ 0 being of finite energy can be decomposed into its generalized Z ! X1 X1 p Fourier transform components by means of (1), evaluating n ¼ sðnÞeoin s ðmÞeoim ejor ðmÀnÞ do , thetransformoverasignalinterval(‘‘period’’)oflengthN. r n ¼ 0 m ¼ 0 Àp This fact follows directly from the generalized Poisson X1 oin n oin summation formula (3)Pwhich shows, that the inverse trans- ¼ 2p sðnÞe s ðnÞe , NÀ1 n ¼ 0 form ðexpðbnÞ=NÞ k ¼ 0 XaðkÞexpðjð2p=NÞknÞ is of the X1 X1 form x~ ðnÞ when evaluated over n ¼ Z. 2 2 a ¼ 2p 9sðnÞ9 e2oin o2p 9sðnÞ9 , n ¼ 0 n ¼ 0 2.1. Connection to sampling, analyticity and the z-transform where n denotes complex conjugation. Recalling Parse- val’s relation and noting that SðoÞ2L2½p,p implies The summation formula (3) has an important relation- sðnÞ2l2ðZÞ [3,2], then for a positive constant C, ship to spectral sampling. The connection of the (discrete- Z p time) generalized Fourier transform to analyticity and to 2 9Sðor þjoiÞ9 dor oC: the z-transform follows as part of the analysis. These Àp J. Martinez et al. / Signal Processing 93 (2013) 1259–1267 1261 The function Sðor þjoiÞ is the analytic continuation from subinterval points. Let 2p=N be the sampling interval, then the real line into the lower half of the complex plane of Z 2p 2p NXÀ1 2p the spectrum of the causal signal s(n). By the same S ðoÞejon do S k ejð2p=NÞnk : a N a N arguments if sðnÞ¼0 for n40 (i.e. is an anticausal signal), 0 k ¼ 0 then its spectrum admits analytic continuation into the An approximation of s(n), call it s~aðnÞ,isthusobtainedas upper half of the complex angular–frequency plane. Define now a discrete-time generalized Fourier trans- Àbn NXÀ1 e 2p jð2p=NÞnk form as, s~aðnÞ¼ Sa k e : N N k ¼ 0 X1 bn Àjon SaðoÞ¼ sðnÞe e , ð5Þ n ¼1 Substituting (5) into this last expression reveals the connec- tion of s~aðnÞ to the original signal s(n), with inverse transformation, ! Z Àbn NXÀ1 X1 Àbn p e bm Àjð2p=NÞkm jð2p=NÞkn e s~aðnÞ¼ sðmÞe e e sðnÞ¼ ¼ S ðoÞejon do, ð6Þ N a k ¼ 0 m ¼1 2p Àp ! eÀbn X1 NXÀ1 where, as in (1), b ¼ logðaÞ=N and a 2 C=f0g.