Difference between Reconstruction from Uniform and Non-Uniform Samples using Sinc Interpolation

IJRRES INERNAIONAL JOURNAL OF RESEARCH REVIEW IN ENGINEERING SCIENCE & ECHNOLOGY Difference between Reconstruction from Uniform and Non-Uniform Samples using Sinc Interpolation *Apurva H. Ghate, **Dr. K. B. Khanchandani Abstract Discrete time signal is nothing but samples continuous time signal and this process is called as sampling. If inter-sampling distance is constant it is called as uniform sampling and if distance is unequal, it is non-uniform sampling. Irrespective of type of sampling, reconstruction is possible if sampling rate do not exceed Nyquist rate. Sinc interpolation is one of the methods which results in reconstruction of original signal. In this paper we will see how to reconstruct input from non-uniform and uniform samples using sinc interpolation and why it is said that perfect reconstruction from non-uniform samples is more difficult than uniform samples. 1 INRODUCION he most common form of sampling used in the context of discrete-time processing of continuous time signals is uniform sampling corresponding to samples of continuous-time signals obtained at equally spaced time intervals. Under certain conditions, specified by the Nyquist- Shannon sampling theorem, the original signal can be reconstructed from this set of equally-spaced samples. he reconstruction is possible by variety of approaches discussed and proposed before. he reconstruction is done through sinc interpolation corresponding to the impulse response of a linear time-invariant ideal low pass filter. Various extensions of the uniform sampling theorem are well Known, as given by Papoulis [23]. In [11] some special non-uniform sampling processes are examined in detail and generalized sampling theorems are obtained. Yao and homas [20] discuss extensions of the uniform sampling expansion and establish that a band-limited signal can be uniquely determined from non-uniform samples, provided that the average sampling rate exceeds the Nyquist rate. However, in contrast to uniform sampling, reconstruction of the continuous-time signal from non-uniform samples using direct interpolation is computationally difficult. One practical approach to recovering a signal from its nonuniform samples has been the use of non-uniform splines [5]. Iterative reconstruction methods for non-uniform sampling which are computationally demanding and have potential issues of convergence have also been previously proposed ([25], [28], [13], [14],[15],[9],[24]. In a different approach, time-warping methods were applied by Papoulis in [10] to reconstruct band limited signals from jittered samples. In [1] and [19], time-warping was used for reconstruction from samples of signals with time-varying frequency content. A method of designing FIR Apurva Ghate is currently pursuing masters degree program in Digital Electronics Engineering in Amravati University, India E-mail: Apurva.ghate@gmail.com Dr. K. B. Khnachandani is currently Head of Department (Electronics and elecommunication) at Shri Sant Gajanan Maharaj College Of Engineering, Shegaon in Amravati University, India IJRRES, ijrrest.org filters in such a way that the effect of input clock jitter is diminished is discussed in [26]. In ([29], [22]) several approaches are suggested and analyzed for approximate reconstruction from jittered samples. Mean-square comparison of various interpolators is done in [12] for the case of uniform sampling, uniform sampling with skips, and Poisson sampling. A modification of the conventional Lagrange interpolator is proposed in [30] which allows approximating a band limited signal from its Non-uniform samples with high accuracy. A comprehensive review of literature concerning other techniques in non-uniform sampling can be found in [3] and [6]. Uniform sampling is the most common form of sampling in discrete time processing. We can also say that uniform sampling is a form of ideal sampling. While dealing with practical sampling, we can observe that in a variety of contexts, non-uniform sampling naturally arises or is preferable to uniform sampling. Non uniform sampling arises in practical applications like Biomedical devises, self timed circuits which tend to introduce non-uniformity in sampling clock, RADARS, in time interleaved ADCs etc. In many cases non-uniform sampling is deliberate and advantageous. But reconstruction from non-uniform samples is really difficult and erroneous as compared to uniform samples. In this paper we will discuss reconstruction using sinc interpolation for both uniform and non-uniform samples and perform error comparison between both based on Mean squared error and Mean absolute deviation. 2 SINC INERPOLAION As we have already discussed, non-uniform sampling is the way complicated than the uniform sampling as it needs the time variant system to deal with. Hence to reconstruct signals from non-uniform samples, a time dependent signal representation has been developed. his framework is used to review some of the sampling theorems presented in [7] and [16]. In a classic paper on non-uniform sampling of band limited signals [18], Yen introduced several reconstruction theorems to address the cases of a finite 17 P a g e

IJRRES INERNAIONAL JOURNAL OF RESEARCH REVIEW IN ENGINEERING SCIENCE & ECHNOLOGY number of non-uniform samples on an otherwise uniform grid, a single gap in uniform sampling and recurrent nonuniform sampling. here are different approaches that can be used for reconstruction as proposed in( [17],[4],[27],[8],[2]) which follows the recurrent non-uniform sampling approach,but here in this paper we will use approach contributed by Yao and homas [21]. Yao and homas the Lagrange interpolation functions were applied to the reconstruction of band limited signals from nonuniform samples. It is shown there that a finite-energy signal x ( band limited N can be reconstructed from its nonuniform samples x ( t n ) using Lagrange interpolation when the sampling instants t n do not deviate by more than N 4 from a uniform grid with spacing of N. Specifically, If tn nn d N / 4, nz (1) hen x ( ln(, (2a) n where G( ln( ( G'( ( t (2b) t G ( t t0 1 (2c) k, tk k0 e over, given that the N N condition of eq. (1) is satisfied, i.e., jt Ln ln( e 0, (3) N and 1 N jt L k n( ) e d ln( tk ) n k (4) 2 N Equation (4) utilizes the interpolation condition of the Lagrange kernel which ensures that the property of consistent resampling is upheld, i.e., that sampling the reconstructed signal on the non-uniform grid t n yields the dg( and G'(. Interpolation using equations (2a, 2b, 2c ) are referred to as Lagrange s interpolation. his theorem is based on a theorem proved by Levinson [8], which states that the functions L n defined as the Fourier transforms of l n t are band limited and form a sequence bi- j t orthogonal to n original samples x t n Note that expressing L n in (4) as the Fourier transform of l n t results in bi-orthogonality of the sequences l n t and sin c t. i.e., l sinc ( t t n N t n ( n k (5) N k N and perfect reconstruction of band-limited signals from nonuniform samples uses the Lagrange interpolation formula subject to constraints imposed on the sample instants. In this section, the series expansion of a band-limited function based on its non-uniform samples will be discussed, and the properties related to Lagrange reconstruction will be elaborated upon. he non-uniform sampling theorem in [14] states that a signal x (, belonging to the class of functions band-limited to 0 rad/sec can be represented as a series expansion. We denote by x [n] a sequence of non-uniform samples of x ( i.e., [ n ] x ( t n) tn represents a nonuniform grid which can be given as t n, (6) x, where n n where n are the deviations from the non-uniform grid. We can define sin tn Gn(. (7) he derivative of above equation is G' n ( cos tn which can be evaluated at n G '( t n ) ( 1) t n, has the value. Let the scaled Hilbert transformer remain as h( 1 t. his gives approximation as Gn( (8) n ( t G' n ( sin t tn t n t tn sinc t t n. (9) n sin t tn he sinc kernel ln( does not satisfy t tn the interpolation property sinc l n( t n) 1 while l n ( t k ) 0 for n k. Hence for non-uniform sample instants, this approximation method causes inter-symbol interference and does not result in consistent sampling. Fig.1 gives block diagram for reconstruction using sinc interpolation of non-uniform samples.. he reconstruction is done through sinc interpolation corresponding to the impulse response of a linear time-invariant ideal low pass filter. If n are deviations from uniform grid which can be given as in equation (6). If we put equation (6) in (9) we get: IJRRES, ijrrest.org 18 P a g e

IJRRES INERNAIONAL JOURNAL OF RESEARCH REVIEW IN ENGINEERING SCIENCE & ECHNOLOGY sin t n n). n t n (10) Equation (10) gives reconstructed signal x ( t ) using sampled signal x ( n). We can observed that x ( n) is non-uniform in nature. he simulated results for the same are given in Fig.2 For special case if deviations from uniform grid are equal and constant such that n then equation (10) becomes, sin t n. (11) n t n Equation (11) gives reconstructed signal x ( t ) from its sampled version x (n), where this sampling is assumed to be equal to the uniform sampling. Fig.3 depicts simulation result for equation (11). Fig.3: Reconstruction from uniform samples using sinc interpolation 3 ERROR COMPARISON OF RECONSRUCION USING SINC INERPOLAION OF UNIFORM SAMPLING AND NON- UNIFORM SAMPLING Here we can compare the reconstruction results of Figure 2 and Figure 3. From these simulation results we cannot analyze which one is more accurate and consistent. So for this we will compare performance of both methods by numerical experiments. And for that, we will calculate various errors like Mean Squared error and Mean absolute deviation. Fig.1: Block diagram of sinc interpolation Normalized mean squared error (MSE) of the output is given by, 1 2 1 MSE, (15) 1 2 1 Mean absolute deviation is a measure of dispersion, a measure of by how much the values in the data set are likely to differ from their mean. he absolute value is used to avoid deviations with opposite signs cancelling each other out. he mean absolute deviation can be given by formula 1 n MAD x i x, (16) n i1 where n is the number of observed values x is the mean of the observed values and x 1 are the individual values. Fig.2: Reconstruction from non-uniform samples using sinc interpolation Figure 4 (ii) shows Mean Squared error (MSE) for reconstruction using sinc interpolation for uniform and nonuniform samples. he error has been calculated for 100 instances and we can observe in the Figure 4 (ii) that MSE_uniform is constant and greater in magnitude than MSE_non-uniform. IJRRES, ijrrest.org 19 P a g e

IJRRES INERNAIONAL JOURNAL OF RESEARCH REVIEW IN ENGINEERING SCIENCE & ECHNOLOGY Numerically, MSE_uniform is steady with value of 0.06. While MSE_non-uniform is fluctuating and is in the range of 0.02-0..While in Figure 4 (i), we can observe the behavior of Mean Absolute Deviation (MAD) also. It is also calculated for 100 events and we can observe results similar MSE, MAD_uniform is having constant deviation than MAD uniform. MAD_uniform is obsedrved to be 0.13 and MAD_Nonuniform is in range 0.06 to 0.1. reconstruction from NU samples using sinc interpolation deviation comparison (ii) Mean squared error comparison of reconstruction from NU samples using sinc interpolation deviation comparison 4 CONCLUSION From all above discussion we can conclude that if we observe both reconstruction results, we can say that reconstruction using sinc interpolation method from uniform samples more constant than that of non-uniform. Hence the performance of approximation methods can be measured in the form of errors in the reconstruction with respect to input. As the eroor in the case of uniform is constant, we can say that, the error detection and minimization is easier because error is predictable in each and every case. But in case of non-uniform samples as errors are unpredictable, perfect reconstruction is really inaccurate and difficult. Fig. 4 (i) Mean absolute deviation comparison of reconstruction from NU samples using sinc interpolation deviation comparison (ii) Mean squared error comparison of reconstruction from NU samples using sinc interpolation deviation comparison Figure 5 shows comparison of average MSE which is calculated for 100 instances. We iterated 25 times to observe consistency of results. From which we conclude that the method of approximation using Sinc for uniform gives constant deviation as compared to that of with non-uniform. Fig.6 shows Mean absolute deviations for 25 iterations which are again calculated by averaging 100 instances. And here also we can say that deviation is constant and consistent for non-uniform samples Fig. 5 MAD and MSE comparison for 25 iterations Figure 5 (i) Mean absolute deviation comparison of REFERENCES IJRRES, ijrrest.org [1] Balakrishnan, A. V., On the problem of time jitter in sampling, IRE rans. Inform. heory, vol. 8, pp. 226 236, 1962. [2] Boor, C. D., On calculating with b-splines, J. Approximat. heory, vol. 6, pp. 50 62, 1970. [3] Clark, J. J., Palmer, M. R. and Lawrence, P. D. A transformation method for the reconstruction of functions from non-uniformly spaced samples, IEEE rans. Acoust., Speech, Signal Processing, vol. ASSP-33, no. 4, pp. 1151 1165, October 1985 [4] Eldar, Y. C. and Oppenheim, A. V., 2000, IEEE rans. Signal Processing, 48, (10), 2864 2875. [5] Feichtinger, H. G. and Grochenig, K. Irregular sampling theorems and series expansions of bandlimited functions, J. Math. Anal. and Appl., vol. 167, pp. 530 556, 1992. [6] Feichtinger, H. G., Grochenig, K. and Strohmer,., Efficient numerical methods in non-uniform sampling theory, Numerische Mathematik, vol. 69, no. 4, pp. 423 440, July 1995 [7] Grochenig, K. Reconstruction algorithms in irregular sampling, Math. Comp., vol. 59, no. 199, pp. 181 194, July 1992. [8] Higgins, J. R., 1976, " IEEE rans.information heory, 22 (5), 621-622. [9] Higgins, J. R., Sampling heory in Fourier and SignalAnalysis: Foundations. Oxford: Oxford Science Publications, Clarendon Press, 1996. [10] Jerri, A. J., Proceedings of the IEEE, 65 (11), 1565 1596. [11] Levinson, N., 1940, Gap and Density heorems, New York: A.M.S. [12] Leneman, O. A. Z. and Lewis, J. B., Random sampling of random processes: Mean-square comparison of various interpolators, IEEE 20 P a g e

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