Generalized Sampling Theorem for Bandpass Signals

Transcription

1 Hindawi Publishing Corporation EURASIP Journal on Applied Signal Processing Volue 26, Article ID 59587, Pages 6 DOI 55/ASP/26/59587 Generalized Sapling Theore for Bandpass Signals Ales Prokes Departent of Radio Electronics, Brno University of Technology, Purkynova 8, 62 Brno, Czech Republic Received 29 Septeber 25; Revised 9 January 26; Accepted 26 February 26 Recoended for Publication by Yuan-Pei Lin The reconstruction of an unknown continuously defined function f t fro the saples of the responses of linear tieinvariant LTI systes sapled by the /th Nyquist rate is the ai of the generalized sapling Papoulis 977 provided an elegant solution for the case where f t is a band-liited function with finite energy and the sapling rate is equal to 2/ ties cutoff frequency In this paper, the scope of the Papoulis theory is extended to the case of bandpass signals In the first part, a generalized sapling theore GST for bandpass signals is presented The second part deals with utilizing this theore for signal recovery fro nonunifor saples, and an efficient way of coputing iages of reconstructing functions for signal recovery is discussed Copyright 26 Hindawi Publishing Corporation All rights reserved INTRODUCTION A ultichannel sapling involves passing the signal through distinct transforations before sapling Typical cases of these transforations treated in any works are delays [2 6] or differentiations of various orders [7] A generalization of both these cases on the assuption that the signal is represented by a band-liited tie-continuous real function f t with finite energy was introduced in [] and developed in [8] Under certain restrictions entioned below, a siilar generalization can be fored for bandpass signals represented by a function f t whose spectru Fω isassued to be zero outside the bands ω U, ω L andω L, ω U asdepicted in Figure a while its other properties are identical as in the case of bandpass signals If a signal is undersapled ie, a higher sapling order is used, then its original spectru coponents and their replicas overlap and the frequency intervals ω L, ω U and ω U, ω L are divided into several subbands, whose nuber depends for given frequencies ω U and ω L on the sapling frequency ω S,[4, 9] For the introduction of GST, the nuber of overlapped spectru replicas has to agree with the sapling order, with the nuber of subbands inside the frequency ranges ω L, ω U and ω U, ω L, and with the nuber of linear systes As presented in [9], to eet the above deands, ust be an even nuber and the sapling frequency ω S = 2π/T S,whereT S is the sapling period, and bandwidth ω B = ω U ω L has to eet the following conditions: ω L ω U ω L = ω L ω B =, where is any positive integer nuber, and ω S ω B = 2 2 An exaple of a fourth-order sapled signal spectru in the vicinity of positive and negative original spectral coponents, if conditions and2 are fulfilled, is shown in Figure b and Figure c Figure 2 shows a graphical interpretation of the sapling orders = 2 to 6 in the plane ω S /ω B versus ω C /ω B,where ω C = ω U + ω L /2, if the coon case of bandpass signal sapling is assued ie, frequency ω S for given ω B and ω C is chosen arbitrary [4, 9] Odd orders correspond to the grey areas, whereas even orders correspond to the white ones The solutions of and2 for =,, 2, when = 2and = 4 are arked by black points 2 GENERALIZED SAMPLING THEOREM FOR BANDPASS SIGNALS This expansion deals with the configuration shown in Figure 3 Function f t is led into LTI prefilters channels with

2 2 EURASIP Journal on Applied Signal Processing Fω syste functions H ω, H 2 ω,, H ω 3 ω U ω C ω L ω L ω C ω U ω a G s i ω ω L ω C + +ω S ω U ω G s i ω b ω U ω C +ω S ω L ω c Figure : a Spectru of bandpass signal, spectru of sapled responses g i t at the output of LTI prefilters in the vicinity of b positive and c negative original spectru coponents The output functions of all prefilters g i t = ωl FωH i ωe jωt dω 2π ω U + 2π ωu ω L FωH i ωe jωt dω are then sapled at /th Nyquist rate related to the cutoff frequency ω B If utual independence of the prefilters is assued and if no noise is present in the syste, function f t can be exactly reconstructed fro saples g i nt S, where T S = π/ω B For this purpose, the following syste of equations has to be fored: 4 HY = R, 5 where atrix H and vectors Y and R are of the following for: H ω, H 2 ω, H ω H ω + ωs, H2 ω + ωs, H ω + ωs H ω + ω 2 S, H 2 ω + ω 2 S, H ω + 2 ω S H = H ω ω S, H 2 ω ω S, H ω k, 6 ω S H ω ω S, H 2 ω ω S, H ω ω S H ω + k + ω S, H2 ω + k + ω S, H ω + k + ω S Y ω, t Y 2 ω, t Y =, R = Y ω, t exp Y ω, t exp j exp jω S t 2 ω S t exp j 2 + ω S t j ω S t exp j + ω S t 7

3 Ales Prokes 3 ωs/ωb = = = 2 = 2 9 = = 4 = = 2 = 4 = = ω C /ω B f t H ω H 2 ω H ω g t g 2 t g t g s t g s 2t g s t n δ t nt s Y ω, t Y 2 ω, t Y ω, t f t f 2 t f t Figure 3: Multichannel sapling configuration + f t Figure 2: Graphical interpretation of the sapling orders In the above forulae t is any nuber and ω ω U, ω U + ω S This syste defines functions Y i ω, t, Y 2 ω, t,, Y ω, t 8 of ω and t because the coefficients in atrix H depend on ω and the right-hand side depends on t Functions H i ω are on the one hand general, but on the other hand they cannot be entirely arbitrary: they ust eet the condition that the deterinant of the atrix of coefficients differs fro zero for every ω ω U, ω U + ω S Since the sapled responses g s i t are of the for g s i t = g i t n= δ t nt s, i =, 2,,, 9 the function f t at the output of the ultichannel sapling configuration can be described by the following forula: f t = gi s t y i t = where n= g i nts yi t nts, y i t = T ωu +ω S s Y i ω, te jωt dω, i =,, 2π ω U The GST 5,, and can be proven in a siilar way as published in [] If we assue that = and put it into, then we obtain ω L = It eans that the bandpass function turns into the band-liited function with cutoff frequency ω B and the above sapling theore turns into a generalized sapling expansion [] We can say that [] isaspecialcaseof the above GST 3 FUNCTION RECOVERY FROM NONUNIFORM SPACED SAMPLES As one of the typical applications of the GST, the reconstruction of a signal f t fro periodically repeated groups of nonunifor spaced saples can be considered [2 5] It can be obtained if the following forulae hold: H i ω = e jαiω, H i ω + qωs = Hi ωe jαiqωs, 2 where α i denotes tie delay in ith branch, that is, the distance between ith saple and the centre of the group Substituting 2 into 6, a set of linear equations is obtained, which can be solved by Craer s rule [] For this purpose, + deterinants of the following types have to be solved: s, s 2, s s 2, s 2 2, s 2 D = H i ω s /2, s /2 2, s /2, s /2+k, s /2+k 2, s /2+k s /2+k+, s /2+k+ 2, s /2+k+ s /2+k, s /2+k 2, s /2+k 3 where s i = e jαiωs, i =, 2,, Then and consequently are applied to the resulting set of functions Yt, ω The calculation of 3 using soe of the classical ethods eg, the Laplace expansion [] can be difficult for large values of One of the ways leading under several conditions to a siplification is based on the fact that 3 is of a siilar for to Vanderonde s one [] However, there is adifference between the, which is hidden in the fact that a jup change in the power for the value of appears in the

4 4 EURASIP Journal on Applied Signal Processing lower part of deterinant Finding systeatic results, which can be explored for reconstruction, is probleatic A possibly way for >2and consists in transforation of 3 into Vanderonde s for by appending the auxiliary ters X, X 2, X 3,andS 2 : where X, S D V = H i ω X 2, S 2 = H i ω, 4 X 3, S 3,,, x, x k, S, S x /2, x /2, S /2, S /2 x /2, x /2, S /2, S /2 = x /2+k, x /2+k, S /2+k, S /2+k x /2+k, x /2+k, S /2+k, S /2+k x /2+k+, x /2+k+, S /2+k+, S /2+k+ x +k, x +k, S +k, S +k 5 and in using of Vanderonde s rule D V = H i ω j= j=i+ k sj x i sj s i j=i+ xj x i 6 The interediate ter can be rewritten in the following for: s x s2 x s x s x 2 s2 x 2 s x 2 sj x i = j= s x k s2 x k s x k = [ x σ x + σ 2 x 2 + ] σ [x 2 σ x2 + σ 2 x2 2 + ] σ [x k σ xk + σ 2 xk 2 + ] σ, 7 where σ k are syetric polynoials consisting of products of all the perutations of k =, 2,, ters s, s 2,, s That is, σ = s + s s, σ 2 = s s 2 + s s s s + s 2 s s 2 s + + s s, σ = s s 2 s 8 Finally, we assue that deterinant Δ in 4 is expanded according to the interediate band of X 2, S 2 using the Laplace expansion Because the desired deterinant 3 ters S and S 3 is an algebraic copleent of ter X 2,it can be revealed as a factor in every ultiplication of all the perutations of ters of block X 2 in the result of 7 Therefore, only one product corresponding to the ain diagonal or an adjacent one of the subdeterinant X 2 is sufficient For the final expression of result, special cases of syetric polynoials have to be defined They are σ = and σ k = fork<andk> In this way, deterinant 3 is obtained in the for D = H i ω j=i+ sj s i σ /2 k+, σ /2 k+2, σ /2 kk /2 σ /2 k+2, σ /2 k+3, σ /2+ σ /2, σ /2+, σ /2+k 9 In a siilar way, the deterinants D i, i =, 2,,, which can be fored by replacing the ith colun vector with the right-hand side vector R, can be coputed Finally, the desired functions Y i ω, t can be obtained fro the ratio D i /D The deterinant in 9 is called the per-syetric deterinant In the case, it contains nonzero ters only near the secondary diagonal The efficiency of the described ethod depends on the values and It is very high in the case <,because the order of the deterinant that has to be coputed is lower than the order of deterinant 3, and the resulting expression of 9 contains a large aount of products s j s i, soe of which vanish due to divisions D i /D Functions Y i ω, t are then obtained in a very siple for If increases, the deterinant order in 9 also increases while the efficiency decreases In the case >>, the order approaches the value 2 Although the deterinant contains the ajority of zero ters, the result is ore coplicated copared to the result of the classical ethods of coputing the deterinants

5 Ales Prokes 5 4 EXAMPLE OF GST APPLICATION Although a band-liited function with finite energy is assued in paragraph, in reality ost the signals can be regarded as tie-unliited A siple exaple = 2 of a signal with infinite-energy recovery is shown below By choosing H ω = andh 2 ω = e jαω and substituting the into, we obtain [ ] [ ] [, e jαω Y ω, t, e jα[ω+k+ωs] = Y 2 ω, t e jαk+ωst ] t, s y t y 2 t α = 25 s Figure 4: Reconstructing functions y tandy 2 t The iages of reconstructing functions Y ω, tandy 2 ω, t can be found in the for Y ω, t = e jk+/2ωst sin [ + /2 ω s t α ] sin [ + ω s α/2 ], Y 2 ω, t = e jk+/2ωstt α e jωα sin [ + ω s t/2 ] sin [ + ω s α/2 ] 2 By evaluating under the condition that ω S = ω B, the reconstructing functions can be expressed as [ πt sin k + π t α ] /T s y t = sin c sin [ + ], πα/t s T s [ ] [ πt α sin k + ] πt/t s y 2 t = sin c sin [ + ], πα/t s T s 22 where sincx = sinx/x The final reconstruction fro a liited nuber of saple groups n can be rewritten in the for f r t = [ f nts y t nts + f nts + α y 2 t nts ] 23 Inaccordancewith and2, the bandwidth, sapling frequency, carrier frequency, and coefficient are chosen as follows: ω B = π/2rad/s,ω S = π/2rad/s,ω C = 2π rad/s, and = 7 Let function f t be given by the forula f t = [ + 5sin ω t +sinω 2 t] cos 2πt The spectru of f t is then coposed of five Dirac pulses at the frequencies 2π±ω,2π± ω 2,and2π To deonstrate the reconstruction of a function whose spectru is inside or partially outside the frequency interval ω L, ω U, the odulation frequencies of f t were chosen as follows: ω = 2rad/s,ω 2 = 6rad/s, and ω 2 = 85 rad/s Reconstructing functions y t andy 2 t are shown in Figure 4 The relation between spectru Fω and the spectru of sapled coon responses G s iω is shown in Figure 5 G s i ω Fω Fω ω S = 578 rad/s, ω C = rad/s, ω B = 578 rad/s ω, rad/s a ω S = 64 rad/s, ω = 3 rad/s, ω 2 = 6rad/s ω, rad/s b ω S = 64 rad/s, ω = 3 rad/s, ω 2 = 85 rad/s ω, rad/s c Figure 5: a Spectru of sapled responses in the vicinity of positive original spectru coponent for = 2 Spectru of f tfor both cases of odulation frequencies ω and ω 2 b, c Function f t and its reconstruction f r t froseven groups of saples n [ 3, 3] are plotted in Figure 6 Itis obvious that in the case of an aliasing occurrence Figure 6b, the reconstruction exhibits a easurable error 5 CONCLUSION A generalized sapling theore for tie-continuous bandpass signal and the application of this theore to signal recovery fro nonunifor saples have been presented An efficient ethod of coputing the Fourier iages of reconstructing functions for signal recovery fro periodically repeated groups of nonunifor spaced saples has then been discussed As entioned above, the ethod presented is suitable for lower values of wideband applications Finding a siplification siilar to 9 in the case >> narrowband applications is very difficult and the classical ethods of coputing the deterinant sees to be the best approach

6 6 EURASIP Journal on Applied Signal Processing ω S = 64 rad/s, ω = 3 rad/s, ω 2 = 6 rad/s, t, s f t f r t Saples a ω S = 64 rad/s, ω = 3 rad/s, ω 2 = 85 rad/s, t, s f t f r t b Figure 6: Function f t and its reconstruction f r t in cases that spectru of f t is a inside or b outside frequency interval ω L, ω U T S = 4s,n [ 3, 3], α = 25 s [5] Y-P Lin and P P Vaidyanathan, Periodically nonunifor sapling of bandpass signals, IEEE Transactions on Circuits and Systes II: Analog and Digital Signal Processing, vol 45, no 3, pp 34 35, 998 [6] Y C Eldar and A V Oppenhei, Filterbank reconstruction of bandliited signals fro nonunifor and generalized saples, IEEE Transactions on Signal Processing, vol 48, no, pp , 2 [7] D A Linden and N M Abrason, A generalization of the sapling theore, Inforation and Control,vol3,no,pp 26 3, 96 [8] J Brown Jr, Multi-channel sapling of low-pass signals, IEEE Transactions on Circuits and Systes, vol28,no2,pp 6, 98 [9] A Prokeš, Paraeters deterining character of signal spectru by higher order sapling, in Proceedings of the 8th International Czech-Slovak Scientific Conference Radioelektronika 98, vol 2, pp , Brno, Czech Republic, June 998 [] C D Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, Pa, USA, 2 Ales Prokes was born in Znojo, Czech Republic, in 963 He received the MS and PhD degrees in electrical engineering fro the Brno University of Technology, Czech Republic, in 988 and 2, respectively He is currently an Assistant Professor at the Brno University of Technology, Departent of Radio Electronics His research interests include nonunifor sapling, velocity easureent based on spatial filtering and free-space optical counication with ephasis on optical receiver perforance analysis and optiization Note that in the case of frequency-liited signal recovery =, the deterinant of syetrical polynoials is equal to one and the solution is very siple ACKNOWLEDGMENT This work has been supported by the Grant GACR Czech Science Foundation no 2/4/557 Developent of the Digital Wireless Counication Resources, and by the Research Prograe MSM26353 Advanced Electronic Counication Systes and Technologies REFERENCES [] A Papoulis, Generalized sapling expansion, IEEE Transactions on Circuits and Systes, vol 24, no, pp , 977 [2] A Kohlenberg, Exact interpolation of band-liited functions, Journal of Applied Physics, vol 24, no 2, pp , 953 [3] D A Linden, A discussion of sapling theores, Proceedings of the IRE, vol 47, pp , 959 [4] A J Coulson, A generalization of nonunifor bandpass sapling, IEEE Transactions on Signal Processing, vol 43, no 3, pp , 995