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, ...,N 1g, with \ sampling, analyticity and the z-transform. This analysis parameter a 2 C f0g as, also serves as a discussion of the generalized Fourier NX 1 bn jð2p=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, ...,N 1g, 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. e bn NX 1 F 1fX 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 SðoÞ2L2½ p,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 jozn transform of the modulated signal xðnÞe . The finite-length SðozÞ¼ sðnÞe , signal ebn for n ¼f0, ...,N 1g,isoffiniteenergyforall n ¼ 1 X1 a 2 C\f0g, therefore for x(n) a signal of finite energy, the jðor þ joiÞn Sðor þjoiÞ¼ sðnÞe , GDFT can be properly defined [2,1]. Note that when a ¼ 1, n ¼ 1 the transform pair correspond to the standard DFT pair. X1 oin jor n Sðor þjo Þ¼ sðnÞe e : ð4Þ Let us denote the periodic extension of XaðkÞ by X~ aðkÞ for i ~ n ¼ 1 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, ...,N 1g.Thisis X1 oin jor 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 e bn NX 1 x~ ðnÞ9 apxððnÞÞN ¼ X ðkÞejð2p=NÞkn, ð3Þ gence rate of the series. Now, a N a p2Z k ¼ 0 X1 jor n þ oin lim Sðor þjoiÞ¼ lim sðnÞe , o - 0 o - 0 i i n ¼ 0 where n 2 Z, p ¼ bn=Nc and ððnÞÞN ¼ nþpN. We can regard X1 x~ ðnÞ as a superposition of infinitely many translated and a ¼ lim sðnÞe jor n þ oin, o - 0 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, ...,N 1g. 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 p 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 p Z ! seethroughtherestofthepaper. p X1 X1 jor n þ oin n jor m þ oim Signals of the form x~ aðnÞ although infinitely long and not ¼ sðnÞe s ðmÞe dor, p 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 ðm nÞ do , thetransformoverasignalinterval(‘‘period’’)oflengthN. r n ¼ 0 m ¼ 0 p 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 , N 1 n ¼ 0 form ðexpð bnÞ=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 SðoÞ2L2½ p,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 p 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 NX 1 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- bn NX 1 e 2p jð2p=NÞnk form as, s~aðnÞ¼ Sa k e : N N k ¼ 0 X1 bn jon SaðoÞ¼ sðnÞe e , ð5Þ n ¼ 1 Substituting (5) into this last expression reveals the connec- tion of s~aðnÞ to the original signal s(n), with inverse transformation, ! Z bn NX 1 X1 bn p e bm jð2p=NÞkm jð2p=NÞkn e s~aðnÞ¼ sðmÞe e e sðnÞ¼ ¼ S ðoÞejon do, ð6Þ N a k ¼ 0 m ¼ 1 2p p ! e bn X1 NX 1 where, as in (1), b ¼ logðaÞ=N and a 2 C=f0g. By our ¼ sðmÞebm ejð2p=NÞkðn mÞ : N previous discussion we can write m ¼ 1 k ¼ 0
SaðoÞ¼SðoþjbÞ¼Sðo bi þjbrÞ, For p 2 Z, p ¼ bn=Nc we have [3], ( where b and b are the real and imaginary parts respec- NX 1 N, m ¼ nþpN, r i ejð2p=NÞkðn mÞ ¼ tively of parameter b. Clearly if b o0 (or equivalently 0, otherwise: r k ¼ 0 9a9o1) the transform is well defined for causal signals. In 9 9 the same way if br 40 (i.e. a 41) the transform is well Then defined for anticausal signals. In both cases the general- X1 bn bðn þ pNÞ ized spectrum can be obtained via analytic continuation s~aðnÞ¼e sðnþpNÞe (into the proper half of the complex plane) of the standard p ¼ 1 Fourier spectrum. Also note that b implies a frequency X1 X1 i ¼ sðnþpNÞebpN ¼ apsðnþpNÞ: shift. If the principal value of logðaÞ is to be taken this shift p ¼ 1 p ¼ 1 is limited from p=N to p=N. Recall now the definition of the z-transform, So that X1 1 N 1 X e bn X 2p SðzÞ¼ sðnÞz n: s~ ðnÞ¼ apsðnþpNÞ¼ S k ejð2p=NÞnk : a N a N n ¼ 1 p ¼ 1 k ¼ 0
Since parameter a is constant, we find the transform (5) We have arrived to the generalized Poisson summation to be a particular case of the z-transform, with formula (3), which states that uniform spectral sampling of 1=N br jðbi oÞ 9 9 9 9 z ¼ e . This implies z ¼ a , with the real the generalized Fourier spectrum SaðoÞ implies a geometri- positive Nth-root being the only root satisfying the cally weighted periodic summation of the original discrete- equation. The discrete-time GFT can be viewed as the time signal s(n). This property is used in a spatial-audio z-transform of the signal evaluated on a circle of radius application in Section 4, but first we analyze some properties 1=N 9a9 . When 9a9 ¼ 1 the evaluation is done on the unit of the GDFT. circle and the GFT is equivalent to the standard Fourier transform shifted in frequency, this is, S ðoÞ¼Sðo y=NÞ, a 3. Properties of the GDFT with a ¼ ejy. We can now extend the definition of the GFT to signals other than single-sided (causal or anticausal). If In this section we present some important properties the z-transform of the signal has a region of convergence 1=N of the GDFT, these are a fundamental part in any GFD- that includes the circle of radius 9a9 , then the GFT based signal processing framework. In the following let exists. Note that finite-length signals have as region of x(n) and y(n) for n ¼f0, ...,N 1g be two finite-duration convergence the whole z-plane with exception of the and in general complex signals of length N, and X ðkÞ and points z ¼ 0 and/or z ¼1. Since a 2 C=f0g the GFD always a Y ðkÞ for k ¼f0, ...,N 1g their respective GDFTs. exists for finite-length signals of finite energy. a The relationship of (3) to spectral sampling is now Property 1. Weighted circular convolution. Point-wise multi- explored. Since S ðoÞ is a periodic function of o the a plication of XaðkÞ and YaðkÞ in the GFD corresponds to the integral in (6) can be taken over any interval of length 2p. weighted circular convolution of x(n) and y(n) in the time To make the derivation simpler let domain, i.e. Z bn 2p e jon bn NX 1 sðnÞ¼ ¼ SaðoÞe do: 1 e jð2p=NÞnk 2 F fXaðkÞYaðkÞg ¼ XaðkÞYaðkÞe p 0 a N k ¼ 0 The integral can be approximated using a rectangular quad- NX 1 rature rule, dividing the integration interval uniformly into N ¼ xðmÞy~ aðn mÞ, subintervals and using the samples of the integrand at the m ¼ 0 1262 J. Martinez et al. / Signal Processing 93 (2013) 1259–1267
NX 1 respect to a. Notice that if the extension is time reversed 1 p F a fXaðkÞYaðkÞg ¼ xðmÞa yðn mþpNÞ the geometrically weighted ‘‘replicas’’ outside the support m ¼ 0 of xð nÞ no longer correspond to a weight ap but to a p. N 1 X 1 p Therefore to obtain Property 4 a GDFT with parameter a ¼ xðmÞa yððn mÞÞN, ð7Þ m ¼ 0 has to be applied to the time reversed extension, x~að nÞ. The proof is given in Appendix C. or
F a ðxny~ aÞðnÞ2XaðkÞYaðkÞ, Property 5. Time domain complex-conjugate.
n 2F an ~ n x ðnÞ Xað kÞ: where n is the linear convolution operator. The operation can thus be seen as the linear convolution of one of the signals e.g. x(n), with the respective signal extension of n ~ n To take the inverse GDFT with parameter a of X að kÞ the other, y~ aðnÞ. Note that for n ¼f0, ...,N 1g we have is equivalent to take the complex conjugate of the time that ðn mÞ2f N þ1, ...,N 1g, and thus p 2f0,1g, so that domain signal xnðnÞ. The proof is given in Appendix D. (7) can be rewritten as Consider that the real part of a complex signal is given n Xn NX 1 by RfxðnÞg ¼ ð1=2ÞðxðnÞþx ðnÞÞ, and its imaginary part is xðmÞyðn mÞþa xðmÞyðN þn mÞ, ð8Þ given by IfxðnÞg ¼ ð1=2jÞðxðnÞ xnðnÞÞ. The last property n m ¼ 0 m ¼ n þ 1 (making a-a ) can then be used to derive the following results, where the left hand summation represents the contribu- tion of N linear convolution terms, and the right hand 2F a 1 ~ n RfxðnÞg ðXa þXan ð kÞÞ: summation the contribution of N circular convolution 2 terms (which are in fact the last terms of the linear The real part of a complex signal x(n) can be obtained by convolution). The factor a effectively weights the amount taking the inverse GDFT with parameter a of a linear of circular convolution that is obtained. The proof is combination of the GDFT of x(n) with parameter a, and given in [1]. Property 1 can be exploited to compute the GDFT of x(n) with parameter an conjugated and linear convolutions in the GFD without the need of zero- reversed (modulo N). Correspondingly we also have that, padding, using GDFTs with e.g. parameter a ¼ 7j or a51 2 R, [1]. n F a 1 ~ IfxðnÞg2 ðX X n ð kÞÞ: Next we present the shifting properties for the GDFT, 2j a a the first accounts for a shift in the time-domain the second for a shift in the GFD. Property 6. GFD complex-conjugate. Property 2. Time-domain shift. F n For n0 2 Z, n 2a n x~ að nÞ XaðkÞ, F a bn0 jð2p=NÞkn0 x~ aðn n0Þ2e e XaðkÞ: where a n ¼ðanÞ 1. To take the complex conjugate of the n spectrum, XaðkÞ is equivalent to take the GDFT with The GDFT with parameter a of the shifted signal n parameter a n of x~ ð nÞ. The proof is given in Appendix E. x~ ðn n Þ is equal to the modulated generalized spectrum a a 0 Before we proceed with the next property let us define of the original signal x(n). The proof is given in Appendix the weighted circular correlation of two in general com- A. plex length N signals, x(n) and y(n) by the linear (deter-
~ n Property 3. GFD shift. For k0 2 Z, ministic) correlation function of x(n) and ya ðnÞ, i.e., n F a ~ ð Þ¼ ð Þn ~ n ð Þ jð2p=NÞk0n2 ~ ra,xy n x n ya n xðnÞe X aðk k0Þ¼Xaððk k0ÞÞN: NX 1 n To circularly shift the generalized spectrum XaðkÞ of a ¼ xðmÞy~ a n ðm nÞ signal is equivalent in the time-domain to modulate the m ¼ 0 signal with the function ejð2p=NÞk0n. The proof is given in NX 1 Appendix B. ¼ xðmÞða pynðm nþpNÞÞ m ¼ 0 Property 4. Time reversal. F NX1 2a 1 ~ p n x~ að nÞ X að kÞ: ¼ xðmÞða y ððm nÞÞNÞ, ð9Þ m ¼ 0 The GDFT with parameter a 1 of the time-reversed extension of x with parameter a, x~ að nÞ, is thus equivalent where we have used the fact that linear correlation can be to reversing (modulo N) the GDFT of x(n) with parameter expressed in terms of the linear convolution of the signals a. This result is a direct consequence of the reciprocal- with one of them time-reversed and conjugated. Note that symmetric structure of the signal extension x~aðnÞ with for n ¼f0, ...,N 1g we have, J. Martinez et al. / Signal Processing 93 (2013) 1259–1267 1263 8 > NX 1 (a monopole) and produces a perfect delta pulse at a > n > xðmÞy ðmÞ, n ¼ 0 > certain time, then the resulting sound field measured at a > m ¼ 0 > single point in space is called the room impulse response < Xn 1 1 n (RIR) [4,5]. Current approaches to model the sound field in r~a,xyðnÞ¼ a xðmÞy ðN þm nÞ ð10Þ > > m ¼ 0 a room although accurate are computationally complex > > NX 1 [4,6–9]. In the context of spatial-audio applications like > n :> þ xðmÞy ðm nÞ, otherwise: virtual reality systems, real-time or interactive simulation m ¼ n of RIRs at all positions in a room becomes a challenging Let us now state the following property. problem. Consider now a room with fully reflective walls. In this Property 7. Weighted circular correlation. case, the sound field inside the room is given by a periodic
F a n summation of the sound field of the source [10]. Intui- r~ ðnÞ2R ðkÞ¼X ðkÞY n ðkÞ, a,xy a,xy a a tively this summation represents the effect of reverbera- tion, since the reflections of the sound field produced by the source(s) on the walls can be modeled by spatial where by definition Ra,xyðkÞ is the GDFT of r~a,xyðnÞ¼xðnÞ n copies of the sources outside the room. If a room could n ~ n ya ð nÞ. The proof of this property follows directly have fully reflective walls, these copies would be perfect from the complex-conjugate property (Property 6) with and the summation would be perfectly periodic. A key - n a a , the weighted circular convolution property observation to derive a fast algorithm to model the sound (Property 1), and Eq. (9). field in a room is then the following, sampling of a function In analogy to the weighted circular convolution prop- results in a periodic summation of its Fourier transform. This erty a weighted circular correlation can be obtained by relation is given by the Poisson summation formula [2,1] point-wise multiplication of the spectra in the GFD. and it is a well known property in digital signal processing However in this case one of the two spectra corresponds (see, e.g. [3]). If we carefully sample the spatio-temporal to the complex conjugate of the GDFT of the signal with spectrum of the sound field produced by the source and n 9 9 parameter a . Notice that when a ¼ 1 we have, apply an inverse Fourier transform on this sampled 2F a n spectrum we can obtain the required periodic summation r~a,xyðnÞ Ra,xyðkÞ¼XaðkÞYaðkÞ, that constitutes the sound field in the whole room [10]. this includes the standard DFT correlation theorem (a ¼ 1). Using this method we dramatically reduce the complexity The following property is a direct consequence of the needed to compute individual impulse responses from weighted circular correlation theorem. 3 OðNt Þ per receiver position (with Nt proportional to the desired reverberation time T ) of approaches related or Property 8. Parseval’s energy relation. 60 based on the mirror image source method [7],to NX 1 NX 1 n 1 n OðNologðNoÞÞ (with No proportional to the maximum xðnÞy ðnÞ¼ XaðkÞY n ðkÞ: ð11Þ N a desired temporal bandwidth say, ) taking advantage n ¼ 0 k ¼ 0 ob of the FFT. On the other hand in virtually all real-life cases the walls in a room are at least partially absorptive, the This equality represents Parseval’s theorem for the GDFT. summation defining the sound field in a room is therefore 1 never perfectly periodic. Using standard Fourier theory is It follows by evaluating F a fRa,xyg¼r~a,xyðnÞ at n¼0. For the case yðnÞ¼xðnÞ we further have, however impossible to obtain something different than a periodicity-sampling relation in reciprocal domains. And NX 1 NX 1 2 1 n therefore although of theoretical importance, the method 9xðnÞ9 ¼ X ðkÞX n ðkÞ: ð12Þ N a a n ¼ 0 k ¼ 0 in [10] has no direct practical use. If the walls are no longer fully reflective in [11] is The energy in the finite duration signal x(n)is shown that the sound field in the room can be modeled by expressed in terms of the frequency components a weighted periodic summation, of the form given by the n N 1 fX ðkÞX n ðkÞg . From here we see that if 9 9 ¼ 1 the a a k ¼ 0 a generalized Poisson summation formula [1] (in this paper ð Þ energy in x(n) equals 1=N times the energy in Xa k i.e., a discrete version of the formula is given by (3)). Every NX 1 1 NX 1 time the sound field is reflected on a wall, part of its 9xðnÞ92 ¼ 9X ðkÞ92 for all a : 9a9 ¼ 1: ð13Þ N a energy is absorbed and its frequency components might n ¼ 0 k ¼ 0 experience a phase change. Higher order reflections can then be seen as geometrically weighted copies of the sound field of the source. A GFD based method for the 4. A spatial-audio signal processing application. simulation of RIRs is then derived as follows. For every source in the room first and second-order reflections on In this section we consider the simulation of room orthogonal walls [10,11] are considered first. To model impulse responses as an application for the GFD- these, another seven virtual sources are added at positions framework just presented. Let us start with the sound outside the room. A total of eight mother sources are then field inside a box-shaped room which always contains considered. The sound field of each mother source is then reverberation (at least in the vast majority of real-life factored into waves traveling only in the direction of each cases). If the source of sound is perfectly omnidirectional of the eight space octants. This gives rise to a total of 2ð3 2Þ 1264 J. Martinez et al. / Signal Processing 93 (2013) 1259–1267 spatio-temporal functions to be considered. Let p ðx,tÞ T ¼ 2p= O is the interval of temporal periodicity. Then lq s s 0 1 represent these functions where l ¼f0, ...,7g is the X X Y B ðiÞn mother source index and q ¼f0, ...,7g is an enumeration 9 @ q i A p~ alqðx,tÞ R plqðxþKn,t þTsnÞ: ð17Þ T i of the octants, x ¼½x,y,z is the space variable vector n2Z3 n2Z i2fx,y,zg where the superscript T denotes matrix transposition, and a t 2 R denotes time. The reverberated sound field in the The summation over n 2 Z, n 0, is the temporal aliasing. 5 room is then modeled by [11], We can neglect it making Os 1, (Ts becomes large), this 0 1 increases computational complexity since a smaller sam- X7 X7 X Y pling interval implies more samples needed in the recon- @ 7 A pðx,tÞ¼ Rlq Ri ni plqðxþKn,tÞ, ð14Þ struction. Taking advantage of the GFD framework, l ¼ 0 q ¼ 0 n2Z3 i2fx,y,zg temporal aliasing can be further reduced using a temporal component different than 1 in the a parameter of (16), where K is the generator matrix of the periodicity lattice without the need to change the sampling rate. We derive L (i.e. the multidimensional signal ‘‘period’’), Rlq are this below, after the current discussion. constants required, n ¼½nx,ny,ny is a triplet of integers, The sound field is therefore approximated,
Ri, for i 2fx,y,zg are the reflection factors of the walls, e.g. T X7 X7 bq x bt X e jðkT WT ½xT ,t T Þ rx ¼ rx0rx1 where Rx0 is the reflection factor of the wall pðx,tÞ R P ðWkÞe : ð18Þ 9D9 lq alq perpendicular to the x direction at the origin of coordi- l ¼ 0 q ¼ 0 k2Z4 nates and Rx1 the reflection factor of the opposite wall. The sign of the exponent in the product over i 2fx,y,zg Further, the infinite summation over k in (18), must be depends on the particular octant in the definition of the limited to a finite number of elements. Extending peri- odically this finite set of spectral coefficients, we impose a function Rlq. The reader is referred to [11] for the details of this derivation. The important result behind (14) is that discretization of the space-time function, so that a the infinite summation over n is a weighted periodic sampled (in both space and time) sound field is approxi- summation, of the form given by the generalized Poisson mated (which can then be handled on a computer). The summation formula, which for multidimensional signals spectral set must be big enough to cover the support of takes the form, the spectrum if corruption due to aliasing is to be avoided. D ! Let S C, be the spectral periodicity lattice, and G the X nY 1 bT x X spatio-temporal sampling lattice, assume D . Then e T T D G ani pðxþKnÞ¼ P ðUkÞejðk U xÞ, ð15Þ T i 9K9 a C ¼ 2pR , so that, n2Zn i ¼ 0 k2Zn bT Cn X e T T p~ ðCnÞ¼ 9C9 1P ðWkÞejðk W CnÞ, ð19Þ where n 2 Z is the dimension of the space, 9K9 is the alq NðD=GÞ alq k2VRð0Þ absolute value of the determinant of K, U ¼ 2pK T is the generator matrix of the spectral sampling lattice F (the where VSð0Þ is the (central) Voronoi region around the (scaled) reciprocal lattice of the periodicity lattice L), origin of lattice S, NðD=GÞ is the number of lattice points T n b ¼ K logðaÞ, a 2 C : aia0 8i ¼ 0, ...,n 1, is the para- of G that lie in VDð0Þ (the central Voronoi region of lattice D). meter of the multidimensional generalized Fourier trans- Making VRð0Þ larger implies a finer sampling of p~ alq.Further T form Pa, and logðaÞ9½logða0Þ, ...,logðan 1Þ . we have that, The main result in [11] relates the sound field in a 9D9 ð2pÞ49W T 9 9R9 room with a sampling condition on the generalized Four- NðD=GÞ¼ ¼ ¼ ¼ NðS=CÞ: 9C9 ð2pÞ49R T 9 9W9 ier spectrum. This is, if K denotes the generator matrix of the lattice specifying the spatial periodic packing of the The sampled sound field is thus obtained by, sound fields plqðx,tÞ, and U denotes the generator matrix X7 X7 of the lattice specifying the sampling points of the spatial- pðCnÞ Rlqp~ alqðCnÞ for Cn 2 VDð0Þ: ð20Þ T generalized spectra, then making U ¼ 2pK , the func- l ¼ 0 q ¼ 0 tions P ðUk,oÞ, k 2 Z3 are the generalized Fourier coef- alq Considering that the spectrum energy is concentrated in ficients of J/Jr9o=c9 [12] we can evaluate up to a given o .Notethat 0 1 b X Y (19) has the form of a generalized Poisson summation B ðiÞn @ q i A formula (the multidimensional extension of (3)), the right Ri plqðxþKn,tÞ, ð16Þ n2Z3 i2fx,y,zg hand term is thus a multidimensional GDFT and the inner summation over k corresponds to a DFT (this comes from BqðxÞ BqðyÞ BqðzÞ T with a ¼½Rx ,Ry ,Rz ,1 , where o is the temporal the fact that the GDFT is equivalent to the DFT of the 7 frequency variable and Bqð Þ ¼ 1, depending on the modulated input signal). Using the FFT, the operation will 4 coordinates defining the qth octant of the space. take only OðNo log NoÞ operations for computing NðD=GÞ To apply the method on a computer, all frequency spatio-temporal positions, with No proportional to ob. variables (not only the spatial-frequency variable) must Since NðD=GÞ¼NðS=CÞ, the method is of complexity be sampled. Sampling the temporal-frequency variable o OðNo log NoÞ per receiver position. Again the reader is introduces temporal aliasing. Let W be the matrix of the referred to [11] for detailed experimental results. spectral sampling lattice W ¼ diagðU,OsÞ, Os is the Returning to Eq. (17), we see that temporal aliasing temporal-frequency sampling interval. Define D ¼ diag is introduced due to spectral sampling of o. Clearly, by T T ðK,TsÞ, so that W ¼ 2pD ¼ 2pdiagðK,TsÞ , where making the spectral sampling period Os smaller, the J. Martinez et al. / Signal Processing 93 (2013) 1259–1267 1265
In principle by making e.g. at 51 we can further reduce the temporal aliasing without the need to increase the spectral sampling period Os. In practice however, the accuracy of the computations is limited by the arithmetic precision used, moreover the causality of the RIR in time is only strictly valid in non-bandlimited scenarios. This is, by limiting the summation over k in (18) to a finite number of elements, we are effectively multiplying the discrete spatio-temporal spectrum by a multidimensional rectangular window. In space-time this has the effect of a convolution with a multidimensional sinc function, mak- ing the resulting band-limited RIR non-causal. In this case (17) for na0 defines the aliasing terms, but still the terms for no0 have less corruptive influence. Despite these practical issues it is still possible to reduce the temporal
aliasing using a temporal component e.g. at 51inthe generalized Fourier synthesis (19).InFig. 3(a)–(d) results of the generalized Fourier synthesis are given setting 2 3 4 at ¼ 0:5, at ¼ 10 , at ¼ 10 and at ¼ 10 respectively. Indeed aliasing corruption decreases for the first two cases, 3 4 but for at ¼ 10 and at ¼ 10 the repeated terms to the right of the temporal support of the RIR become too large, Fig. 2. Simulated room impulse responses. having a negative impact in the reconstruction. Clearly the corruption is more pronounced to the right of the support of 2 the RIR. In this case setting the parameter at ¼ 10 gives a aliasing components in the time dimension appear further good reconstruction, especially when compared with the apart from each other, reducing the error. In time the RIR result obtained setting at ¼ 1depictedinFig. 2.Workingin does not have compact support, but it is a causal (single the GFD on the temporal dimension allows a non-negligible sided) function, having a starting point in time and an gain in accuracy. exponential decay afterwards (see e.g. [5] for a parametric The method for multichannel simulation of RIRs has an characterization of this decay). A simulated example RIR important application in immersive virtual gaming (using using the Mirror Image Source Method (MISM) [7],is for example stereo headphones). In this case many RIRs depicted in Fig. 2, together with a RIR simulated using the for different (virtual) room conditions need to be com- GFD approach described above and in [11], the error puted and later fast convolved with a given audio signal power between both signals is plotted in dB. The band- (i.e. auralization) to give the users an audio experience width frequency is ob ¼ 2pð2 kHz, so that the temporal such that they have the impression of being in the game sampling frequency is f s ¼ 4 kHz. The length of the RIR is field. For example, at one moment the users could be at an Th ¼ 1:02 s or 4096 taps. The GFD method RIR shown in open location such as a park, and at another moment they Fig. 2 is obtained according to (19) and (20) using a could be inside a room. To create a satisfactory experi- BqðxÞ BqðyÞ BqðzÞ T parameter a ¼½Rx ,Ry ,Rz ,1 , so that the spatial part ence, the system have to reproduce the acoustic charac- of the a parameter applied in the generalized Fourier teristics of different scenarios for moving sources/ synthesis gives the required spatial weighted periodicity, receivers. The computation of all the necessary RIRs can and in time a standard Fourier synthesis is applied. The be done with low-complexity using the GFD method RIR is thus one spatial sample of the set pðCnÞ. The presented in [11]. A GDFT in the temporal dimension spectral sampling period is set to Os ¼ 2p=ð2ThÞ, so that can be applied to reduce aliasing corruption as explained the interval of temporal periodicity (and thus temporal above. The novel GFD framework presented in this paper aliasing) Ts is 2 times the reverberation time. can then be used to perform fast convolution for auraliza- Using a temporal component ata1 in the a parameter, tion or other signal processing tasks in the GFD. we can further reduce the temporal aliasing. Since the RIR is a causal function of time, the repeated terms to the 5. Concluding remarks right of the temporal support of the RIR (for no0), do not contribute to the time-domain aliasing. Therefore we can In this work, a generalized Fourier domain (GFD) signal rewrite (17) as, processing framework is introduced. The proposed frame- 0 1 work allows a special form of control on the periodic X Y @ BqðiÞni A repetitions that occur due to sampling in the reciprocal p~ alqðx,tÞ¼ R plqðxþKn,tÞ i domain. We show that this property can be expressed in n2Z3 i2fx,y,zg 0 1 terms of a weighted periodic extension of a signal. We X1 X Y demonstrate that the (discrete-time) generalized Fourier n @ BqðiÞni A þ ðat Þ Ri plqðxþKn,tþTsnÞ: transform can be seen as a special case of the z-transform, 3 n ¼ 1 n2Z i2fx,y,zg and relate the analytic continuation of the standard ð21Þ Fourier spectrum to the generalized Fourier spectrum. 1266 J. Martinez et al. / Signal Processing 93 (2013) 1259–1267
Fig. 3. Comparison of RIRs simulated with the GFD method setting parameter at to different values, and the same RIR simulated with the MISM depicted 2 3 4 in Fig. 2. (a) at ¼ 0:5, (b) at ¼ 10 , (c) at ¼ 10 , (d) at ¼ 10 .
Core properties of the generalized discrete Fourier trans- Appendix A. Proof of Property 2 form (GDFT) are given. These allow to concisely work in the GFD. The close relationship of the GDFT to the DFT allows a generalized fast Fourier transform (GFFT) to be Proof. Time-domain shift property. directly obtained via the FFT. For n 2 , we have that The novel framework opens possibilities for signal 0 Z processing applications where working on the GFD results p p x~ aðn n0Þ¼a xððn n0ÞÞ ¼ a xðn n0 þpNÞ, in a computational or analytical advantage. As an exam- N ple, we review a method for low-complexity simulation of room impulse responses (RIRs) [10,11] based on the GFD. where p ¼ bðn n0Þ=Nc. Then,
The framework presented in this paper can then be used NX 1 bn jð2p=NÞkn to perform e.g. auralization, adaptive filtering or other F afx~aðn n0Þg ¼ x~ aðn n0Þe e acoustic signal processing operations in the GFD. n ¼ 0 s MATLAB code to implement the GDFT is available on- NX 1 p bn jð2p=NÞkn line for educational and non-profit purposes at the web- ¼ a xðn n0 þpNÞe e , page of TuDelft SIPLAB (http://siplab.tudelft.nl). n ¼ 0 J. Martinez et al. / Signal Processing 93 (2013) 1259–1267 1267
!n make m ¼ n n0 þpN, then NX 1 ¼ xðnÞebnejð2p=NÞkn NX 1 p bðm þ n0 pNÞ jð2p=NÞkðm þ n0 pNÞ n ¼ 0 F afx~ aðn n0Þg ¼ a xðmÞe e !n m ¼ 0 NX 1 bn jð2p=NÞðN kÞn NX 1 ¼ xðnÞe e bn jð2p=NÞkn bm jð2p=NÞkm ¼ e 0 e 0 xðmÞe e n ¼ 0 n m ¼ 0 ~ & bn jð2p=NÞkn ¼ X að kÞ: ¼ e 0 e 0 XaðkÞ: &
Appendix B. Proof of Property 3 Appendix E. Proof of Property 6
Proof. GFD shift property. Proof. GFD complex-conjugate.
For k0 2 Z, n b n NX 1 1 n e n jð2p=NÞkn NX 1 fX ðkÞg ¼ X ðkÞe F ðanÞ 1 a a jð2p=NÞk0n jð2p=NÞk0n bn jðÞð2p=NÞkn N F afxðnÞe g¼ xðnÞe e e k ¼ 0 ! n ¼ 0 n n b n NX 1 NX 1 e jð2p=NÞkn ¼ XaðkÞe ¼ xðnÞebne jðÞð2p=NÞnðk k0Þ N k ¼ 0 n ¼ 0 ( n NX 1 x ð0Þ, n ¼ 0 bn jðÞð2p=NÞnððk k0ÞÞ ¼ n n ¼ xðnÞe e N eb nxnðN nÞeb ðN nÞ, otherwise n ¼ 0 ( n ¼ X ððk k ÞÞ ¼ X~ ðk k Þ: & x ð0Þ, n ¼ 0 a 0 N a 0 ¼ ðaxðN nÞÞn, otherwise
n ¼ x~ að nÞ, Appendix C. Proof of Property 4 since b ¼ logðaÞ=N. &
Proof. Time reversal. References For ( xðnÞ for n ¼ 0 [1] J. Martinez, R. Heusdens, R.C. Hendriks, A generalized Poisson x~að nÞ¼ summation formula and its application to fast linear convolution, axðN nÞ for n ¼f1, ...,N 1g, IEEE Signal Processing Letters 18 (9) (2011) 501–504. [2] R. Bracewell, The Fourier Transform & Its Applications, 3rd edition, we have that, McGraw-Hill, New York, 1999. [3] J.G. Proakis, D.G. Manolakis, Digital Signal Processing, Principles, NX 1 bn jð2p=NÞkn Algorithms and Applications, 4th edition, Prentice-Hall, New York, 1 fx~ ð nÞg ¼ xð0Þþ xðN nÞe e , F a a a 2007. n ¼ 1 [4] H. Kuttruff, Room Acoustics, Taylor and Francis, London, U.K., 2000. set m ¼ N n, then, [5] Y. Huang, J. Benesty, J. Chen, Acoustic MIMO Signal Processing (Signals and Communication Technology), Springer-Verlag, Secau- NX 1 cus, NJ, 2006. bðN mÞ jð2p=NÞkðN mÞ F a 1 fx~að nÞg ¼ xð0Þþ axðmÞe e [6] B. Xu, S.D. Sommerfeldt, A hybrid modal analysis for enclosed m ¼ 1 sound fields, Journal of the Acoustical Society of America 128 (5) NX 1 (2010) 2857–2867. ¼ xð0Þþ xðmÞebme jð2p=NÞðN kÞm [7] J. Allen, D. Berkley, Image method for efficiently simulating small room acoustics, Journal of the Acoustical Society of America m ¼ 1 65 (1979) 943–950. NX 1 bm jð2p=NÞðN kÞm [8] A. Berkhout, D. de Vries, J. Baan, B. van den Oetelaar, A wave field ¼ xðmÞe e extrapolation approach to acoustical modeling in enclosed spaces, m ¼ 0 Journal of the Acoustical Society of America 105 (1999) 1725–1733. [9] S.R. Bistafa, J.W. Morrissey, Numerical solutions of the acoustic ¼ XaðN kÞ¼X~ að kÞ, eigenvalue equation in the rectangular room with arbitrary (uniform) since b ¼ logðaÞ=N. & wall impedances, Journal of Sound and Vibration 263 (1) (2003) 205–218. [10] J. Martinez, R. Heusdens, On low-complexity simulation of multi- Appendix D. Proof of Property 5 channel room impulse responses, IEEE Signal Processing Letters 17 (7) (2010) 667–670. [11] J. Martinez, R. Heusdens, R.C. Hendriks, A spatio-temporal general- Proof. Time domain complex-conjugate. ized fourier domain framework to acoustical modeling in enclosed spaces, in: Proceedings of the ICASSP, 2012, pp. 529–532. NX 1 n [12] T. Ajdler, L. Sbaiz, M. Vetterli, The plenacoustic function and its n n b n jð2p=NÞkn F an fx ðnÞg ¼ x ðnÞe e sampling, IEEE Transactions on Signal Processing 54 (10) (2006) n ¼ 0 3790–3804.