Nonuniform Sampling for Random Signals Bandlimited in the Linear Canonical Transform Domain

Non uniform Sampling for Random Signals Bandlim ited in the Linear Canonical T ransform Domai n ∗ Haiy e Huo a , W enc hang Sun b a Departmen t of Mathematics, Sc ho ol of Science, Nanc hang Univ ersit y , Nanc hang 33 0031, Jiangxi, China b Sc ho ol of Mathematical Sciences and LPMC, Nank ai Unive r sit y , Tianjin 300071, China Emails: hyh uo@ncu.edu.cn; sun w ch@nank ai.edu.cn Abstr act . In this p ap er, w e mainly inv estigate the non u n iform sampling for rand om signals whic h are bandlimited in the linear canonical transf orm (LCT) domain. W e sho w that the non u niform sampling for a random signal bandlimited in the LCT domain is equal to the u niform sampling in the sense of second order s tatistic c h aracters after a pre- filter in the LC T d omain. Moreo v er, we pr op ose an appr o ximate reco ve ry ap p roac h for non u niform sampling of r an d om signals bandlimited in the LC T domain. F urther m ore, w e study the mean square error of th e nonuniform sampling. Finally , w e do some s imulations to verify the correctness of our theoretical results. Keywor ds. Non un iform sampling; Linear canonical transform; Random signals; Ap- pro ximate reconstru ction; Sin c inte r p olation 1 In tro d uction Sampling is very fundamental in signal pro cessing, as it pro vides an effectiv e w ay to con- nect the analogue signals and d igital signals. Since S hannon [18] in tro duced the concept of sampling theorem in 1949 , the sampling theorem has b een widely studied in v arious academic fields. In particular, u niform sampling theorems f or deterministic signals or random signals, which are bandlimited in the F ourier domain, fractional F ourier trans- form domain, or linear canonical tr ansform (LCT) d omain, hav e b een int en sely stud ied in literatures [4, 5, 10, 18, 19, 20, 21, 22, 24, 27, 28, 29, 30, 34]. In practice, w e might only obtain nonuniform samples, for instance, in the areas of geoph ysics, biomedical imaging, or comm u nication theory [8, 12, 17]. T herefore, n onuni- form sampling has aroused m u ch more atten tion on the theoretical and practical sides in the literature. There are man y kinds of approac hes for reco vering the original signals from their non u niform samples. F or example, Y ao and Thomas [32] derive d th e reconstruction form u la for band limited signals from their nonuniform samples by u sing th e Lagrange ∗ This w ork was partially supp orted by the N ational N atural Science F oundation of China (11525104, 1153101 3 and 11371200). 1 in terp olatio n functions. Ho wev er, since the Lagrange in terp olation functions often ha ve distinct formats at differen t sampling times, it is very complicated to reco ver the bandlim- ited signals b y u tilizing Lagrange in terp olation functions. Ma ny other appr oac hes h a ve b een presented to solv e this problem. Time-w arping tec h nique w as u sed in [15] for r eco v er- ing b andlimited signals from their j ittered samples. In [16], the author made some revision on tr ad itional Lagrange in terp olation functions, in order to impro ve the accuracy of reco v- ering b an d limited signals f r om their non uniform samples. Due to the p erfect reco ve r y of bandlimited signals from their nonuniform samples w ith sinc in terp olati on, Ma ymon and Opp enheim [14] prop osed a class of appro ximate reco very approac hes for bandlimited sig- nals f rom their non u niform samples b y utilizing sin c interp olation functions. F urthermore, Xu, Zhang and T ao [31] generalized the results mentio n ed in [14] from traditional F ourier domain to fractional F ou r ier trans f orm domain. In ord er to learn more information on non u niform samplin g, we refer the r eaders to [1, 3, 6, 9 , 13, 23, 25, 33]. F or random signals which are bandlimited in the L CT domain, there exist few r esu lts on sampling theorems. In [10], based on the framewo r k of L CT auto-correlation fu nction and p ow er sp ectral density , we in vestig ated the un iform sampling theorem and m ultic h an- nel samp lin g theorem for random signals which are bandlimited in L C T domain. I n this pap er, we derive the relationship b et we en the LCT auto-p o wer sp ectral densities of th e inputs and outp u ts. In addition, we study the n on u niform sampling for random s ignals bandlimited in the LCT d omain and giv e an approxima te r eco v ery approac h with sinc in- terp olation functions. Moreo v er, w e in ve stigate the err or estimate of non u niform samp ling for r andom signals b andlimited in the LC T domain in the mean square sense. Finally , some s imulations are carried out to illustrate the effectiv eness of our metho d s. The rest of the pap er is presented as follo ws. In S ection 2, we first in tro duce the concepts of the LCT, the LCT correlation f u nction, and the LC T p o wer sp ectral densit y . Then, we sho w the connection b et we en the LCT auto-p ow er sp ectral d ensit y of the inputs and outputs. In Section 3, w e stud y th e non u niform samp ling, its approximate reco very metho d, the corresp onding reconstruction error for rand om signals bandlimited in the LCT domain in the mean square sense. Moreo ver, we analyze the p erformances of our theoretical results by sim ulation. I n Section 4, we conclude the p ap er. 2 Preliminaries 2.1 The Linear Canonical T ransform Definition 2.1. The LCT of a signal f ( t ) ∈ L 2 ( R ) is denote d by [10] L A { f ( t ) } ( u ) =    R + ∞ −∞ f ( t ) q 1 j 2 πb e j a 2 b t 2 − j 1 b ut + j d 2 b u 2 d t, b 6 = 0 , √ de j cd 2 u 2 f ( du ) , b = 0 , (1) wher e A =  a b c d  , and p ar ameters a, b, c, d ∈ R satisfy ad − bc = 1 . Since the LC T is a C hirp m ultiplication op erator when b = 0, w e assume without loss of the generalit y that b > 0 in the rest of the p ap er. F rom (1), we can easily d eriv e the 2 connection b et ween the LCT an d the F ourier transform as follo ws: L A { f ( t ) } ( u ) = r 1 j 2 π b e j d 2 b u 2 F ( f ( t ) e j a 2 b t 2 )  u b  , (2) where th e F ourier transform of f ( t ) is defined b y F ( f )( u ) = Z + ∞ −∞ f ( t ) e − j ut d t. (3) 2.2 The LCT Po w er Sp ect ral Density In this pap er, we consider a sp ecia l class of random signals that is wide sense stationary . Giv en a probabilit y sp ace (Ω , F , P ), a sto c hastic pr o cess x ( t ) is called stationary in a wide sense, if its mean is zero, its s econd moment is finite, and its auto-correlati on fu nction R xx ( t + τ , t ) = E [ x ( t + τ ) x ∗ ( t )] (4) is ind ep endent of t ∈ R , where E denotes m athematical exp ectation, and x ∗ stands for the complex conjugate of x . Two sto c hastic pro cesses x ( t ) and y ( t ) are said to b e join tly stationary in a wide sens e, if x ( t ) and y ( t ) are b oth wide sense stationary , and their cross-correlatio n function R xy ( t + τ , t ) = E [ x ( t + τ ) y ∗ ( t )] (5) is indep end en t of t ∈ R . Next, w e introdu ce the LCT auto-co r relation function, the LCT cross-correlatio n fun c- tion, the LCT auto-p ow er sp ectral density and the LCT cross-p o we r sp ectral densit y as follo ws. Definition 2.2. Given r andom sig nals x ( t ) and y ( t ) , the LCT auto-c orr elation function of x ( t ) is define d by R A xx ( t 1 , t 2 ) = E [ x ( t 1 ) x ∗ ( t 2 ) e j a b t 2 ( t 1 − t 2 ) ] = R xx ( t 1 , t 2 ) e j a b t 2 ( t 1 − t 2 ) , (6) and the LCT cr oss-c orr elation function of y ( t ) and x ( t ) i s define d by R A y x ( t 1 , t 2 ) = E [ y ( t 1 ) x ∗ ( t 2 ) e j a b t 2 ( t 1 − t 2 ) ] = R y x ( t 1 , t 2 ) e j a b t 2 ( t 1 − t 2 ) . (7) One can see that if ˜ x ( t ) = x ( t ) e j a 2 b t 2 is stationary , that is, R ˜ x ˜ x ( t 1 , t 2 ) = R ˜ x ˜ x ( τ ) , (8) where τ = t 1 − t 2 , then the function R A xx ( t 1 , t 2 ) also dep ends only on τ . In fact, R A xx ( t 1 , t 2 ) = E [ x ( t 1 ) x ∗ ( t 2 ) e j a b t 2 ( t 1 − t 2 ) ] = E [ x ( t 1 ) e j a 2 b t 2 1 x ∗ ( t 2 ) e − j a 2 b t 2 2 ] e − j a 2 b ( t 1 − t 2 ) 2 = R ˜ x ˜ x ( t 1 , t 2 ) e − j a 2 b ( t 1 − t 2 ) 2 = R ˜ x ˜ x ( τ ) e − j a 2 b τ 2 . (9) 3 Definition 2.3. Given r andom signals x ( t ) and y ( t ) , and two p ar ameters A =  a b c d  , A ′ =  a − b − c d  . The LCT auto-p ower sp e ctr al density of x ( t ) is define d by P A xx ( u ) = r 1 − j 2 π b e − j d 2 b u 2 L A { R A xx ( τ ) } ( u ) = r 1 j 2 π b e j d 2 b u 2 L A ′ { R A, 2 xx ( τ ) } ( − u ) , (10) and the LCT cr oss-p ower sp e ctr al density of y ( t ) and x ( t ) is define d by P A y x ( u ) = r 1 − j 2 π b e − j d 2 b u 2 L A { R A y x ( τ ) } ( u ) = r 1 j 2 π b e j d 2 b u 2 L A ′ { R A, 2 y x ( τ ) } ( − u ) , (11) wher e R A, 2 xx ( t 1 , t 2 ) = E [ x ( t 1 ) x ∗ ( t 2 ) e j a b t 1 ( t 1 − t 2 ) ] = R xx ( t 1 , t 2 ) e j a b t 1 ( t 1 − t 2 ) (12) and R A, 2 y x ( t 1 , t 2 ) = E [ y ( t 1 ) x ∗ ( t 2 ) e j a b t 1 ( t 1 − t 2 ) ] = R y x ( t 1 , t 2 ) e j a b t 1 ( t 1 − t 2 ) . (13) It follo ws from (1 ) and (10) that R A xx ( τ ) = Z + ∞ −∞ P A xx ( u ) e − j a 2 b τ 2 + j 1 b uτ d u. (14) In [10, 7 ], a mo del of LCT multiplicat ive filter has b een introd uced as in Fig. 1, where X ( u ) = L A { x ( t ) } ( u ), Y ( u ) = X ( u ) H ( u ), and the output fu nction y ( t ) is giv en by y ( t ) = L A − 1 { Y ( u ) } ( t ) = L A − 1 { X ( u ) H ( u ) } ( t ) . (15) With th e L CT multiplicativ e filter describ ed in Fig. 1, w e can obtain the relationship b et ween the LCT auto-p o wer sp ectral densit y P A xx ( u ) and cross-p ow er s p ectral d ensit y P A y x ( u ). Prop osition 2.4. [10, The or em 2.3] L et r andom signals x ( t ) and y ( t ) b e the input and output of the LCT multiplic ative filter, and the tr ansfer function H ( u ) satisfy h ( t ) = 1 √ 2 π Z + ∞ −∞ H ( u ) e j ut/b d u. (16) Then, P A y x ( u ) = H ( u ) P A xx ( u ) . (17) Similarly , we can get the connection b et ween P A y y ( u ) and P A xx ( u ) as follo ws. 4 Theorem 2.5. L et r andom signals x ( t ) and y ( t ) b e the input and output of the LCT multiplic ative filter, and the tr ansfer fu nction H ( u ) satisfy h ( t ) = 1 √ 2 π Z + ∞ −∞ H ( u ) e j ut/b d u. (18) Then, P A y y ( u ) = | H ( u ) | 2 P A xx ( u ) . (19) Pr o of. By (15), w e hav e y ( t ) = L A − 1 {L A { x ( t ) } ( u ) H ( u ) } ( t ) . (20) Using the con volutio n theorem ([7 , Theorem 1]), we can rewrite (20) as y ( t ) = 1 √ 2 π b Z + ∞ −∞ x ( t − τ ) e j a 2 b ( τ 2 − 2 tτ ) h ( τ )d τ . (21) Th us, R y y ( t 1 , t 2 ) = E [ y ( t 1 ) y ∗ ( t 2 )] = 1 √ 2 π b Z + ∞ −∞ h ∗ ( u ) e − j a 2 b ( u 2 − 2 t 2 u ) R y x ( t 1 , t 2 − u )d u = 1 √ 2 π b Z + ∞ −∞ h ∗ ( u ) e j a b t 1 u [ R y x ( t 1 , t 1 − ( τ + u )) e − j a 2 b ( u 2 +2 τ u ) ]d u, (2 2) where τ = t 1 − t 2 . Combining (13) and (22), we get R A, 2 y y ( τ ) = R y y ( t 1 , t 1 − τ ) e j a b t 1 τ = 1 √ 2 π b Z + ∞ −∞ h ∗ ( u ) e j a b t 1 ( τ + u ) R y x ( t 1 , t 1 − ( τ + u )) e − j a 2 b ( u 2 +2 τ u ) d u = 1 √ 2 π b Z + ∞ −∞ h ∗ ( u ) R A, 2 y x ( τ + u ) e − j a 2 b ( u 2 +2 τ u ) d u = 1 √ 2 π b Z + ∞ −∞ h ∗ ( − u ) R A, 2 y x ( τ − u ) e j a 2( − b ) ( u 2 − 2 τ u ) d u. (23) Therefore, w e obtain L A ′ { R A, 2 y y ( τ ) } ( u ) = H ∗ ( − u ) L A ′ { R A, 2 y x ( τ ) } ( u ) . (24) Substituting (10) and (11) in to (24), w e ha ve P A y y ( u ) = H ∗ ( u ) P A y x ( u ) . (25) It follo ws from (17) that P A y y ( u ) = | H ( u ) | 2 P A xx ( u ) . This completes th e pro of. 5 3 Non uniform Sampling and Error Estimate for Random Signals Bandlimited in the LCT Domain In this section, we stud y the nonuniform samp ling and error estimate for random signals whic h are bandlimited in the LCT d omain. First, we in tro duce the definition. Definition 3.1. [10] We c al l a r andom signal x ( t ) b and limite d in the LCT domain, if its LCT p ower sp e ctr al density P A xx ( u ) satisfies P A xx ( u ) = 0 , | u | > u r , (26) wher e u r is the b andwidth. Before stating our main results, we pr esent a lemma that is u s eful in the follo wing. Lemma 3.2. Assume that a r andom signal x ( t ) is b and limite d in the LCT domain with b andwidth u r , and ˜ x ( t ) = x ( t ) e j a 2 b t 2 is a wide sense stationary pr o c ess. Then, ˜ x ( t ) is b and limite d in the F ourier domain with b andwidth u r /b. Pr o of. Since ˜ x ( t ) is stationary in a wide sense, it follo ws fr om (9) and (10 ) that P A xx ( u ) = r 1 − j 2 π b e − j d 2 b u 2 L A { R A xx ( τ ) } ( u ) = r 1 − j 2 π b e − j d 2 b u 2 L A { R ˜ x ˜ x ( τ ) e − j a 2 b τ 2 } ( u ) = r 1 − j 2 π b r 1 j 2 π b h Z + ∞ −∞ R ˜ x ˜ x ( τ ) e − j 1 b uτ d τ i = 1 2 π b P ˜ x ˜ x ( u b ) . (27) Note th at P A xx ( u ) = 0 , | u | > u r . (28) W e ha v e P ˜ x ˜ x ( u ) = 0 , | u | > u r b . This completes th e pro of. 3.1 Non uniform Sampl ing First, we restate a n on un iform sampling theorem for deterministic s ignals which are ban- dlimited in the LCT domain, as men tioned in [23]. Prop osition 3.3. [23, The or em 4] Supp ose that a deterministic signal f ( t ) is b and limite d in the LCT domain with b andwidth u r . If | t n − n bπ u r | ≤ D < bπ 4 u r , (29) 6 then the function f ( t ) c an b e p erfe ctly r e c over e d by its samples f ( t n ) with the fol lowing formula, f ( t ) = e − j a 2 b t 2 + ∞ X −∞ f ( t n ) e j a 2 b t 2 n G ( t ) G ′ ( t n )( t − t n ) , (30) wher e G ( t ) = e αt ( t − t 0 ) Y n 6 =0  1 − t t n  e t/t n , α = X n 6 =0 1 t n , D ∈ R , and G ′ ( t ) is the derivative of G ( t ) . It is k n o wn that Lagrange in terp olatio n functions often ha ve distinct formats at dif- feren t sampling times. Hence it is v ery complicated to reco ve r signals b y using Pr op o- sition 3.3. In this p ap er, w e giv e another reco very app roac h instead. W e b egin with a non u niform sampling mo del [31] as in Fig. 2, where { x ( t n ) } is the sampling sequence of a random signal x ( t ), and { t n } is the sequence of sampling p oints. W e assume that t n = nT + ξ n , where T ≤ π b u r is the a ve r age samp ling in terv al, and { ξ n } is a sequence of indep en d en t identic ally distributed (i.i.d.) random v ariables with zero mean in the in terv al ( − T / 2 , T / 2). Th is non uniform sampling mod el is also called jitter sampling. F or jitter sampling, th e sampling noise is int ro duced to the exp ected sampling time, i.e., t n = nT + τ n with τ n ∈ ( − T / 2 , T / 2) and E [ τ n ] = 0. F or example, the time b ase jitter of a 50 GHz sampling oscilloscope is identified to ha ve standard deviation 0 . 965 ps, that is, the actual measurement time is corrupted by zero-mean Gaussian noise [26]. The jittered samples often o ccur in biomedical d evices and A/D con verters du e to the in tern al clo ck imp erfections [2, 11]. Next we show that in the sense of second order statistic c h aracters, n on u n iform s am- pling is identica l to un iform sampling after a pre-filter. Theorem 3.4. Assume that a r andom signal x ( t ) i s b and limite d i n the LCT domain with b andwidth u r , and ˜ x ( t ) = x ( t ) e j a 2 b t 2 is a wide sense stationary pr o c ess. Then, in the sense of se c ond or der statistic c har acters, the nonuniform sampling of x ( t ) is identic al to the uniform sampling after a LCT filter h 1 ( t ) establishe d i n Fig. 3, i.e. , h 1 ( t ) = 1 √ 2 π Z + ∞ −∞ H 1 ( u ) e j ut/b d u, (31) wher e T is the aver age sampling interval, t n = n T + ξ n is the sampling p oint, v ( t ) is an additive noise with zer o me an and i s indep endent of x ( t ) , and the LCT auto-p ower sp e ctr al density P A vv ( u ) of v ( t ) is P A xx ( u )(1 − | H 1 ( u ) | 2 ) . Her e, H 1 ( u ) = φ ξ ( u b ) , wher e φ ξ ( u ) denotes the char acteristic function of ξ n . Pr o of. Since ˜ x ( t ) = x ( t ) e j a 2 b t 2 is a wide sense stationary pr o cess, the non u niform sampling can b e describ ed as in Fig 4. By the d esign of LCT filter in Fig 3, we ha ve P A y y ( u ) = | H 1 ( u ) | 2 P A xx ( u ) . (32) 7 Since v ( t ) is an add itiv e n oise with zero mean and v ( t ) is indep enden t of x ( t ), the LCT auto-correlat ion fu n ction R A z z ( τ ) of z ( t n ) is identica l to R A y y ( τ ). Thus, com bining (14 ) and (32), w e ha ve R A z z ( nT , ( n − k ) T ) = R A y y ( nT , ( n − k ) T ) = Z u r − u r P A y y ( u ) e − j a 2 b ( kT ) 2 + j 1 b uk T d u = Z u r − u r | H 1 ( u ) | 2 P A xx ( u ) e − j a 2 b ( kT ) 2 + j 1 b uk T d u. (33) Hence R ˜ z ˜ z ( nT , ( n − k ) T ) = E [ z ( nT ) z ∗ ( nT − k T ) e j a 2 b (( nT ) 2 − ( nT − k T ) 2 ) ] = e j a 2 b ( kT ) 2 R A z z ( nT , ( n − k ) T ) = Z u r − u r | H 1 ( u ) | 2 P A xx ( u ) e j 1 b uk T d u. (34) Since x ( t n ) an d ξ n are tw o random v ariables, by (14), we obtain R A xx ( t n , t n − k ) = E [ R A xx ( k T + ξ n − ξ n − k )] = Z u r − u r P A xx ( u ) E [ e − j a 2 b ( kT + ξ n − ξ n − k ) 2 + j 1 b u ( kT + ξ n − ξ n − k ) ]d u. (35) Com bin ing (9) and (14), we hav e E [ R ˜ x ˜ x ( k T + ξ n − ξ n − k )] = E [ e j a 2 b ( kT + ξ n − ξ n − k ) 2 R A xx ( k T + ξ n − ξ n − k )] = Z u r − u r P A xx ( u ) e j 1 b uk T E [ e j 1 b u ( ξ n − ξ n − k ) ]d u. (36) Let Z = ξ n − ξ n − k and f Z ( η ) b e the p robabilit y density function of Z . Note that ξ n and ξ n − k are indep endent and hav e identica l distrib utions. Let f ξ ( η ) b e their common probabilit y den s it y function. Th en we hav e f Z ( η ) = f ξ ( η ) ⋆ f ξ ( − η ) , (37) where ⋆ denote as the con volution op erator. Hence E [ e j 1 b u ( ξ n − ξ n − k ) ] = Z + ∞ −∞ f Z ( η ) e j 1 b uη d η = Z + ∞ −∞ [ f ξ ( η ) ⋆ f ξ ( − η )] e j 1 b uη d η = Z + ∞ −∞ f ξ ( η ) e j 1 b uη d η × Z + ∞ −∞ f ξ ( − η ) e j 1 b uη d η = | φ ξ ( u b ) | 2 , (38) 8 where φ ξ ( u ) = Z + ∞ −∞ f ξ ( η ) e j uη d η . Substituting (38) into (36), w e get E [ R ˜ x ˜ x ( k T + ξ n − ξ n − k )] = Z u r − u r P A xx ( u ) e j 1 b uk T E [ e j 1 b u ( ξ n − ξ n − k ) ]d u = Z u r − u r P A xx ( u ) | φ ξ ( u b ) | 2 e j 1 b uk T d u. (39) By s etting H 1 ( u ) = φ ξ ( u b ) in (34), w e get (39). Hence, the auto- corr elation fu nction of ˜ x ( t n ) in Fig. 4 is equal to that of the output in Fig. 3. Therefore, the nonuniform sampling is iden tical to the uniform sampling in Fig. 4, in the sense of second order statistic c haracters. This completes the pr o of. 3.2 Appro ximate reco very approac h As we m en tion ab o ve, it is m u c h easier to deal with un iform sampling than non u niform sampling. S ince sinc in terp olation leads to exact r eco v ery for u niform sampling, Maymon and O pp enh eim [14] in tro duced a new approximat e r eco v ery form u la of nonuniform sam- pling for a random signal x ( t ) b andlimited in F ourier domain by utilizing sinc in terp olation function. The app ro ximate reco v ery formula can b e represented as f ollo ws: ˆ x ( t ) = T T N + ∞ X n = −∞ x ( t n ) s ( t − ˜ t n ) , (40) where s ( t ) = sin c  π t T N  , π T N is the ban d width of x ( t ), ˜ t n = nT + ζ n . Here, ˜ t n is not required to b e iden tical to the original samp ling p oints t n . But, if the original random signal x ( t ) is not bandlimited in th e F ourier domain, th e approxi mate reco v ery appr oac h migh t n ot w ork. M otiv ated by [14], Xu, Zh ang, and T ao [31] considered the case when the rand om signal is band limited in the fractional F ourier domain. Since LCT is a more general transf orm, whic h includes F ourier tran s form and fractional F our ier transform as its sp ecial cases, it is p ossible that a s ignal which is non-band limited in the F ou r ier d omain or the fr actional F our ier domain, is bandlimited in the LCT domain. So, it is necessary to in vestig ate the corresp onding app ro ximate reco very result for random signal bandlimited in the LCT domain. Theorem 3.5. Assume that r andom a signal x ( t ) is b and limite d in the LCT domain with b andwidth u r , and ˜ x ( t ) = x ( t ) e j a 2 b t 2 is a wide se nse stationa ry pr o c ess. Then x ( t ) c an b e appr oximate d fr om its nonuniform samples by utilizing the sinc interp olation function, ˆ x ( t ) = T T N e − j a 2 b t 2 + ∞ X n = −∞ x ( t n ) e j a 2 b t 2 n h 2 ( t − ˜ t n ) , (41) wher e h 2 ( t ) = sin c  u r t b  , T is the uniform sampling interval, T N is the Nyq u ist sampling interval, t n = nT + ξ n , ˜ t n = nT + ζ n , and ξ n is not ne c essarily e qual to ζ n . 9 Pr o of. F r om Lemma 3.2, w e kno w that ˜ x ( t ) is band limited in the F ourier domain with bandwidth u r b . By (40), w e h a ve ¯ x ( t ) = + ∞ X n = −∞ T T N ˜ x ( t n )sinc π ( t − ˜ t n ) T N . (42) Substituting ˜ x ( t ) = x ( t ) e j a 2 b t 2 and ¯ x ( t ) = ˆ x ( t ) e j a 2 b t 2 in to (42), w e obtain (41), whic h completes the p ro of. F rom Theorem 3.5 we kno w that the appro x im ate reco v ery approac h for rand om signals whic h are bandlimited in the LCT d omain, can b e expressed as the mo del presen ted in Fig. 5. 3.3 Error estimate of reconstruction for random signals In th is su b section, w e s tudy the reconstruction err or in the mean square sense by consid- ering the sampling and r econstruction pro cedu r es as the s ystem whose frequency resp onse is d ep endent on the p r obabilit y densit y fu nction of the p erturb ations. It f ollo ws from Theorem 3.4 th at, if the a v erage sampling inte rv al T is greater than the Nyqu ist sampling in terv al T N , then the mo del presented in Fig. 6 is identica l to the pr o cedure, whic h includes the nonuniform samp lin g men tioned in su bsection 3.1, and the appro ximate reconstruction approac h b y us in g sinc in terp olation fun ction d escrib ed in Fig. 5, in the sense of second order s tatistic characte r s. Theorem 3.6. Assume that a r andom signal x ( t ) i s b and limite d i n the LCT domain with b andwidth u r , ˆ x ( t ) is the appr oximation of x ( t ) obtaine d in Fig. 6, and ˜ x ( t ) = x ( t ) e j a 2 b t 2 is a wide se nse stationa ry pr o c ess. L et the fr e qu ency r esp onse of the filter h 3 ( t ) b e the joint char acteristic function of the r andom variables ξ n and ζ n , i.e ., φ ξ ζ ( u, − u ) . And let v ( t ) b e an additive noise with zer o me an, which is unc orr elate d with x ( t ) and has the p ower sp e ctr al density P vv ( u ) = T Z u r − u r P A xx ( u 1 )[1 − φ ξ ζ ( u 1 b , − u ) | 2 ]d u 1 , | u | < u r . (43) Then the mo del describ e d in Fig. 6 is identic al to the pr o c e dur e, which includes the nonuni- form sampling mentione d in subse ction 3.1 and the appr oximate r e c onstruction appr o ach by utilizing sinc interp olation function r epr esente d in Fig. 5, in the sense of se c ond or der statistic c har acters. F urthermor e, we have E [ | ˆ x ( t ) − x ( t ) | 2 ] = Z u r − u r P A xx ( u ) | 1 − φ ξ ζ ( u b , − u b ) | 2 d u + T 2 π b Z u r − u r P A xx ( u ) Z u r − u r 1 − | φ ξ ζ ( u b , − u 1 b ) | 2 d u 1 d u. Pr o of. By Theorem 3.5, we hav e ¯ x ( t ) = ˆ x ( t ) e − j a 2 b t 2 = T T N + ∞ X n = −∞ x ( t n ) e j a 2 b t 2 n h 2 ( t − ˜ t n ) , (44) 10 where h 2 ( t ) = sin c  u r b t  . Th u s, we can get the auto-correlat ion function of ¯ x ( t ) as follo ws: R ¯ x ¯ x ( t, t − τ ) = E [ ¯ x ( t ) ¯ x ( t − τ )] = E h T T N + ∞ X n = −∞ x ( t n ) e j a 2 b t 2 n h 2 ( t − ˜ t n ) × T T N + ∞ X k = −∞ x ∗ ( t k ) e − j a 2 b t 2 k h ∗ 2 ( t − τ − ˜ t k ) i =  T T N  2 E h + ∞ X n = −∞ x ( nT + ξ n ) e j a 2 b ( nT + ξ n ) 2 h 2 ( t − nT − ζ n ) × + ∞ X k = −∞ x ∗ ( k T + ξ k ) e − j a 2 b ( kT + ξ k ) 2 h ∗ 2 ( t − τ − k T − ζ k ) i =  T T N  2 + ∞ X n = −∞ + ∞ X k = −∞ E h R ˜ x ˜ x ( nT − k T + ξ n − ξ k ) × h 2 ( t − nT − ζ n ) h ∗ 2 ( t − τ − kT − ζ k ) i =  T T N  2 R ˜ x ˜ x (0) + ∞ X n = −∞ E h h 2 ( t − nT − ζ n ) h ∗ 2 ( t − τ − nT − ζ n ) i +  T T N  2 X n 6 = k E h R ˜ x ˜ x ( nT − k T + ξ n − ξ k ) h 2 ( t − nT − ζ n ) × h ∗ 2 ( t − τ − k T − ζ k ) i , I + I I . (45) Next, w e use t wo steps to compute R ¯ x ¯ x ( t, t − τ ). Note that X n e j ( u 2 − u 1 ) nT = 2 π X k δ (( u 2 − u 1 ) T − 2 π k ) (46) and h 2 ( t ) = 1 √ 2 π Z + ∞ −∞ H 2 ( u ) e j u b t d u = b √ 2 π Z + ∞ −∞ H 2 ( ub ) e j ut d u. (47) W e obtain I =  T T N  2 R ˜ x ˜ x (0) + ∞ X n = −∞ E h h 2 ( t − nT − ζ n ) h ∗ 2 ( t − τ − nT − ζ n ) i = b 2 2 π  T T N  2 R ˜ x ˜ x (0) Z + ∞ −∞ Z + ∞ −∞ H 2 ( bu 1 ) H ∗ 2 ( bu 2 ) e j ( u 1 − u 2 ) t e j u 2 τ 11 × + ∞ X n = −∞ e j ( u 2 − u 1 ) nT E [ e j ( u 2 − u 1 ) ζ n ]d u 1 d u 2 =  T b T N  2 R ˜ x ˜ x (0) Z u r b − u r b 1 T | H 2 ( bu ) | 2 e j uτ d u = T 4 π 2 Z u r b − u r b e j uτ h Z u r b − u r b P ˜ x ˜ x ( u 1 )d u 1 i d u, (48) where we use the fact that H 2 ( bu ) = T N b √ 2 π χ [ − π T N , π T N ] ( u ) in the last step. Similarly , we hav e I I =  T T N  2 X n 6 = k E h R ˜ x ˜ x ( nT − k T + ξ n − ξ k ) h 2 ( t − nT − ζ n ) × h ∗ 2 ( t − τ − k T − ζ k ) i = b 2 (2 π ) 2  T T N  2 X n 6 = k E h Z u r b − u r b P ˜ x ˜ x ( u ) e j u ( nT − k T + ξ n − ξ k ) d u × Z + ∞ −∞ H 2 ( bu 1 ) e j u 1 ( t − nT − ζ n ) d u 1 Z + ∞ −∞ H ∗ 2 ( bu 2 ) e − j u 2 ( t − τ − k T − ζ k ) d u 2 i = b 2 (2 π ) 2  T T N  2 X n 6 = k Z u r b − u r b Z + ∞ −∞ Z + ∞ −∞ P ˜ x ˜ x ( u ) H 2 ( bu 1 ) H ∗ 2 ( bu 2 ) e j u 2 τ × e j ( u 1 − u 2 ) t e j ( u − u 1 ) nT e − j ( u − u 2 ) kT E h e j uξ n e − j uξ k × e − j u 1 ζ n e j u 2 ζ k i d u 1 d u 2 d u = b 2 (2 π ) 2  T T N  2 Z u r b − u r b Z + ∞ −∞ Z + ∞ −∞ P ˜ x ˜ x ( u ) H 2 ( bu 1 ) H ∗ 2 ( bu 2 ) φ ξ ζ ( u, − u 1 ) × φ ∗ ξ ζ ( u, − u 2 ) e j u 2 τ e j ( u 1 − u 2 ) t X n e j ( u − u 1 ) nT × X k e − j ( u − u 2 ) kT d u 1 d u 2 d u − b 2 (2 π ) 2  T T N  2 Z u r b − u r b Z + ∞ −∞ Z + ∞ −∞ × P ˜ x ˜ x ( u ) H 2 ( bu 1 ) H ∗ 2 ( bu 2 ) φ ξ ζ ( u, − u 1 ) φ ∗ ξ ζ ( u, − u 2 ) e j u 2 τ e j ( u 1 − u 2 ) t × X n e j ( u 2 − u 1 ) nT d u 1 d u 2 d u =  b T N  2 Z u r b − u r b P ˜ x ˜ x ( u ) | H 2 ( bu ) | 2 | φ ξ ζ ( u, − u ) | 2 e j uτ d u − T 2 π  b T N  2 Z u r b − u r b Z u r b − u r b P ˜ x ˜ x ( u 1 ) | H 2 ( bu ) | 2 | φ ξ ζ ( u 1 , − u ) | 2 e j uτ d u 1 d u =  b T N  2 Z u r b − u r b | H 2 ( bu ) | 2 e j uτ h P ˜ x ˜ x ( u ) | φ ξ ζ ( u, − u ) | 2 − T 2 π Z u r b − u r b P ˜ x ˜ x ( u 1 ) × | φ ξ ζ ( u 1 , − u ) | 2 d u 1 i d u 12 = 1 2 π Z u r b − u r b e j uτ h P ˜ x ˜ x ( u ) | φ ξ ζ ( u, − u ) | 2 − T 2 π Z u r b − u r b P ˜ x ˜ x ( u 1 ) × | φ ξ ζ ( u 1 , − u ) | 2 d u 1 i d u. (49) Com bin ing (48 ) and (49), we obtain R ¯ x ¯ x ( t, t − τ ) = T 4 π 2 Z u r b − u r b  Z u r b − u r b P ˜ x ˜ x ( u 1 )[1 − | φ ξ ζ ( u 1 , − u ) | 2 ]d u 1  e j uτ d u + 1 2 π Z u r b − u r b e j uτ P ˜ x ˜ x ( u ) | φ ξ ζ ( u, − u ) | 2 d u. (50) Similarly , we can obtain the cross-correlation function of ¯ x ( t ) and ˜ x ( t ) as follo ws, R ¯ x ˜ x ( t, t − τ ) = 1 2 π Z u r b − u r b e j uτ P ˜ x ˜ x ( u ) φ ξ ζ ( u, − u )d u. (51) Therefore, w e hav e P ¯ x ¯ x ( u ) = P ˜ x ˜ x ( u ) | φ ξ ζ ( u, − u ) | 2 + T 2 π  Z u r b − u r b P ˜ x ˜ x ( u 1 )[1 − | φ ξ ζ ( u 1 , − u ) | 2 ]d u 1  (52) and P ¯ x ˜ x ( u ) = P ˜ x ˜ x ( u ) φ ξ ζ ( u, − u ) . (53) Hence, the first term P ˜ x ˜ x ( u ) | φ ξ ζ ( u, − u ) | 2 in (52) is the p o w er sp ectral d ensit y of ˜ y ( t ) in Fig. 6. Sub stituting (27) in to (43), w e ha ve P vv ( u ) = T Z u r − u r P A xx ( u 1 )[1 − φ ξ ζ ( u 1 b , − u ) | 2 ]d u 1 = T Z u r − u r 1 2 π b P ˜ x ˜ x ( u 1 b )[1 − φ ξ ζ ( u 1 b , − u ) | 2 ]d u 1 = T 2 π Z u r b − u r b P ˜ x ˜ x ( u 1 )[1 − φ ξ ζ ( u 1 , − u ) | 2 ]d u 1 . Th us, the second term T 2 π  R u r b − u r b P ˜ x ˜ x ( u 1 )[1 − | φ ξ ζ ( u 1 , − u ) | 2 ]d u 1  in (52) is identica l to the p o w er sp ectral densit y of v ( t ). Consequently , the m o del describ ed in Fig. 6 is equal to the pro cedur e, whic h includes the nonuniform sampling mentioned in sub section 3.1 and th e appro xim ate reconstruction approac h by utilizing sin c interpolation function r epresen ted in Fig. 5, in th e sense of second order statistic characte r s. Next, w e estimate th e error E [ | ˆ x ( t ) − x ( t ) | 2 ]. Let ǫ ( t ) = ˆ x ( t ) − x ( t ). C om binin g (27) and (52), we get P A ˆ x ˆ x ( u ) = 1 2 π b P ¯ x ¯ x ( u b ) 13 = 1 2 π b P ˜ x ˜ x ( u b ) | φ ξ ζ ( u b , − u b ) | 2 + T 4 π 2 b  Z u r b − u r b P ˜ x ˜ x ( u 1 )[1 − | φ ξ ζ ( u 1 , − u b ) | 2 ]d u 1  = P A xx ( u ) | φ ξ ζ ( u b , − u b ) | 2 + T 2 π b  Z u r − u r P A xx ( u 1 )[1 − | φ ξ ζ ( u 1 b , − u b ) | 2 ]d u 1  . (54) Similarly , we obtain P A ˆ xx ( u ) = 1 2 π b P ¯ x ˜ x ( u b ) = 1 2 π b P ˜ x ˜ x ( u b ) φ ξ ζ ( u b , − u b )) = P A xx ( u ) φ ξ ζ ( u b , − u b ) . (55) Hence, the LCT auto-p o wer sp ectral density of the r econstru ction err or ǫ ( t ) in Fig. 6 is P A ǫǫ ( u ) = P A ˆ x ˆ x ( u ) − P A ˆ xx ( u ) − P A x ˆ x ( u ) + P A xx ( u ) = P A xx ( u ) | φ ξ ζ ( u b , − u b ) | 2 + T 2 π b  Z u r − u r P A xx ( u 1 ) × [1 − | φ ξ ζ ( u 1 b , − u b ) | 2 ]d u 1  − P A xx ( u ) φ ξ ζ ( u b , − u b ) − [ P A xx ( u ) φ ξ ζ ( u b , − u b )] ∗ + P A xx ( u ) = P A xx ( u ) | 1 − φ ξ ζ ( u b , − u b ) | 2 + T 2 π b  Z u r − u r P A xx ( u 1 ) × [1 − | φ ξ ζ ( u 1 b , − u b ) | 2 ]d u 1  . (56) Therefore, it f ollo w s from (9 ) and (14) that E [ | ǫ ( t ) | 2 ] = R ˜ ǫ ˜ ǫ (0) = R A ǫǫ (0) = Z u r − u r P A ǫǫ ( u )d u = Z u r − u r P A xx ( u ) | 1 − φ ξ ζ ( u b , − u b ) | 2 d u + T 2 π b Z u r − u r P A xx ( u ) Z u r − u r 1 − | φ ξ ζ ( u b , − u 1 b ) | 2 d u 1 d u. This completes th e pro of. Note that the reconstruction error E [ | ˆ x ( t ) − x ( t ) | 2 ] is related to the LCT auto-correlat ion p o w er sp ectral density P A xx ( u ) of the random signal x ( t ) and the join t c haracteristic func- tion φ ξ ζ ( u, − u ) of t wo random v ariables ξ n and ζ n . In particular, when ξ n and ζ n are constan ts and b oth are equal to zeros, i.e., the nonuniform sampling studied in th is pap er 14 reduces to uniform sampling, w e hav e E [ | ˆ x ( t ) − x ( t ) | 2 ] = 0 from Th eorem 3.6 . Therefore the result of uniform sampling prop osed in [10, Theorem 3.4] is a sp ecial case of Theorems 3.5 and 3.6 in this pap er. On the other hand, the LCT in clud es man y wid ely used linear transf orm s as sp ecial cases. F or example, b y setting A =  cos θ sin θ − sin θ cos θ  with θ ∈ [ − π , π ) in (1), the LCT of f ( t ) b ecomes the fractional F ourier transform of f ( t ) with angle θ . In this case, our results coincide with those in [31], and in particular, w hen ξ n and ζ n are constan ts and equal to zeros, our resu lts coincide with the uniform sampling results in [24]. F u r thermore, by setting A =  0 1 − 1 0  , the LCT of f ( t ) b ecomes the F ourier transform of f ( t ) multiplied b y a constan t q 1 j 2 π . In this case, our results coincide with those in [14]. 3.4 Sim ulations In this sub section, we consider a random signal x ( t ) = e j 5 πt + j ψ − j 3 2 π t 2 , − 4 ≤ t ≤ 4, wh ere ψ follo ws th e standard Gaussian distribution. It is easy to v erify that ˜ x ( t ) = x ( t ) e j a 2 b t 2 is wide sense stationary , and x ( t ) is appro ximately band limited with bandwidth 10 Hz in the LC T domain with parameter A =  a b c d  =  3 1 /π π 2 / 3  . Let T = T N = 0 . 1. First, the appr o ximate signal reco very results based on (41) for one realization of x ( t ) are resp ectiv ely sh o wn in Fig. 7 in terms of t wo different nonuniform samp lin g mo dels. Let ξ n follo w the uniform distribu tion in the int erv al [ − 0 . 01 − 0 . 002 ∗ k , 0 . 01 + 0 . 002 ∗ k ] and in teger k tak e a v alue from 0 to 15. Then, for eac h k , 0 ≤ k ≤ 15, we implement 5000 realizatio n s of x ( t ) and estimate th e mean square error of the r econstruction in terms of four differen t non un iform sampling mo dels as sh o wn in Fig. 8. One can see from Fig. 8 that the r econstru ction from the nonuniform sampling with ζ n = 0 is preferable. In fact, the reconstruction is an appr o ximate solution, and w e cannot claim which sampling mo del is the b est in general. Ho wev er, according to (44) in Theorem 3.6, a lo we r m ean square error of the reconstruction migh t b e obtained by c h o osing a prop er join t charac teristic function of ξ n and ζ n . 4 Conclusion In this pap er, we mainly discus s the non u niform samp ling for random signals, whic h are bandlimited in the LCT domain. A t the b eginning, based on the concepts of the LC T correlation fu n ction and p o we r sp ectral densit y , w e get the connection b etw een the LCT auto-p o wer sp ectral densit y of the inputs and outputs. Moreo ver, w e sho w th at non uniform sampling for random signals band limited in the LCT d omain can b e identic al to u niform sampling after a pre-filter in th e sense of second ord er statistic charac ters. F u rthermore, w e derive an approximat e reconstruction form u la for rand om signals bandlimited in LCT domain f r om their nonuniform samples, by utilizing the sinc interp olation fu nctions. 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uncertain ty principles asso ciated with the linear canonical tr an s form’, IET Signal Pr o c ess , 2016, 10 , (7), pp. 791-7 97 18 x ( t ) LCT O H ( u ) In verse LCT y ( t ) X ( u ) Y ( u ) Figure 1: The LC T multiplica tive filter. t 0 t 1 t n − 1 t n T ξ 1 ξ 2 ξ n x ( t ) O P n δ ( t − t n ) x ( t n ) Figure 2: The non u niform sampling representa tion. x ( t ) O e j a 2 b t 2 h 1 ( t ) O e − j a 2 b t 2 Sampling M v ( nT ) O e j a 2 b ( nT ) 2 ˜ z ( nT ) ˜ x ( t ) ˜ y ( t ) y ( t ) y ( nT ) z ( n T ) Figure 3: T he equiv alen t system of the non u niform sampling, w here v ( t ) is an add itiv e noise with zero mean, v ( t ) is ind ep endent of x ( t ), and the LCT auto-pow er sp ectral dens it y of v ( t ) is P A xx ( u )(1 − | H 1 ( u ) | 2 ). 19 x ( t ) Sampling t n = nT + ξ n O e j a 2 b t 2 n ˜ x ( t n ) x ( t n ) Figure 4: Another v ers ion of nonuniform sampling. x ( t n ) O e j a 2 b t 2 n Synt hesis O e − j a 2 b t 2 ˆ x ( t ) ˜ x ( t n ) ¯ x ( t ) Figure 5: The appro ximate reconstruction with sinc in terp olation function, where ¯ x ( t ) = P n T T N ˜ x ( t n )sinc π ( t − ˜ t n ) T N . x ( t ) O e j a 2 b t 2 h 3 ( t ) M v ( t ) O e − j a 2 b t 2 ˆ x ( t ) ˜ x ( t ) ˜ y ( t ) ¯ x ( t ) Figure 6: The non uniform sampling and r econstruction system, where v ( t ) is an additiv e noise w ith zero mean, which is uncorrelated with x ( t ) and has the p o w er sp ectral d ensit y defined by (43). 20 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2 −1 0 1 2 Time Real part of the signal Original signal Reconstructed signal −5 −4 −3 −2 −1 0 1 2 3 4 5 −2 −1 0 1 2 Time Imaginary part of the signal Original signal Reconstructed signal (a) −5 −4 −3 −2 −1 0 1 2 3 4 5 −2 −1 0 1 2 Time Real part of the signal Original signal Reconstructed signal −5 −4 −3 −2 −1 0 1 2 3 4 5 −2 −1 0 1 2 Time Imaginary part of the signal Original signal Reconstructed signal (b) Figure 7: The app ro ximate signal reconstruction: (a) when ξ n is un iformly distributed in the interv al [ − 0 . 01 , 0 . 01] and ζ n = 0; (b) when ξ n and ζ n are i.i.d. w ith uniform distribution in the inte rv al [ − 0 . 01 , 0 . 01]. 21 0 5 10 15 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 k Mean square error ζ n =0 ζ n is independent of ξ n and uniformly distributed in [−0.01,0.01] ζ n = ξ n ζ n and ξ n are i.i.d. Figure 8: Mean square er r or of the reconstru ction when ξ n is u niformly d istributed in the in terv al [ − 0 . 01 − 0 . 002 ∗ k , 0 . 01 + 0 . 002 ∗ k ], where k = 0 , 1 , · · · , 15. 22