Implementation of Digital Unbiased FIR Filters with Polynomial Impulse Responses

Transcription

1 Circuits Syst Signal Process (2012) 31: DOI /s Implementation of Digital Unbiased FIR Filters with Polynomial Impulse Responses Paula Castro-Tinttori Oscar Ibarra-Manzano Yuriy S. Shmaliy Received: 16 July 2010 / Revised: 7 June 2011 / Published online: 9 July 2011 Springer Science+Business Media, LLC 2011 Abstract This paper addresses the design and implementation of digital unbiased finite impulse response (FIR) filters with polynomial impulse response functions. The transfer function, its fundamental properties, and a general block-diagram are discussed for the impulse response represented with the l-degree Taylor series expansion. As a particular results, we show a fundamental identity uniquely featured to such filters in the transform domain. For low-degree impulse responses, the transfer functions are found in simple closed forms and represented in compact block-diagrams. The magnitude and phase responses are also analyzed along with the group delays. A comparison with predictive FIR filters is given. As examples of applications, filtering of time errors of local clocks is discussed along with the low-pass filter design employing a cascade of the unbiased FIR filters. Keywords Unbiased FIR filter Transfer function Unbiasedness condition Polynomial impulse response Block-diagram 1 Introduction The finite impulse response (FIR) filters, predictors, and smoothers are commonly used whenever a linear phase response is required. Among known solutions, there is a special class of devices [2, 5, 14] intended for unbiased 1 FIR filtering of signals in tracking, control, measurement, positioning, timekeeping, etc. If such filters match 1 The unbiased filter is known to be optimal in the sense of the minimal produced bias between the average input and average output [14]. P. Castro-Tinttori O. Ibarra-Manzano Y.S. Shmaliy ( ) Department of Electronics, Guanajuato University, Salamanca 36855, Mexico shmaliy@ugto.mx O. Ibarra-Manzano irarrao@ugto.mx

2 612 Circuits Syst Signal Process (2012) 31: the signal model in the order, the group delay reaches a minimum. Otherwise, it grows, however, at a lower rate than that in the case of infinite impulse response (IIR) filters. A well-known consequence of this is a higher order of the FIR filters against the IIR ones. Simple implementation of near optimal FIR structures ignoring noise and initial conditions has become available in discrete time n after Heinonen and Neuvo developed the approach, originally discussed by Johnson in [13], and designed predictive FIR filters with the polynomial impulse response functions in [10]. Immediately, these filters have been studied and used by many authors [1, 6, 8, 16, 18]. Later, in [19], Shmaliy showed that Heinonen Neuvo s solution is unbiased. Thereafter, the theory of unbiased FIR filters, predictors, and smothers with polynomial impulse responses has been developed in [19, 21, 22, 24]. A generalization made in [19] has become available by employing the well-known fact from the Kalman filter theory that the order of the optimal filter is the same as that of the system. This means that for signals represented on an interval of N past points, from n N + 1ton, with the finite l-degree Taylor series, the FIR filter impulse response must be represented with the same-degree polynomial [19], in order for the output to be unbiased. Because the white noise variance reduces in averaging filters as a reciprocal of N [26], the unbiased one becomes virtually optimal in the sense of the minimal mean square error (MSE) when N is large [20]. That makes it almost an ideal engineering solution for near optimal filtering. To introduce this technique, let us suppose that there is a measurement y n = x n + v n of a signal x n in the presence of an additive zero mean noise v n. The FIR filtering estimate ˆx n of x n can be found via the discrete convolution as N 1 ˆx n = h li (N)y n i, (1) i=0 where h ln (N) is the l-degree impulse response, to be formally defined below, existing from n N + 1ton. It is known [14] that such an estimate will be unbiased if E{ˆx n }=E{x n }, (2) where E{x} is the expected value of x. Now expand x n to the finite l-degree 2 Taylor series (the higher degree terms are supposed to be identically zero) from n N + 1ton. It was shown in [19, 22] that the unbiasedness between ˆx n given as (1) and x n can be achieved, by (2), if one represents h ln (N) with the l-degree polynomial h ln (N) = l a ml (N)n m (3) m=0 2 In the state space, a signal represented with K states has been expended in [19, 22] to the K-degree Taylor series with l = K 1.

3 Circuits Syst Signal Process (2012) 31: Fig. 1 Low-degree polynomial impulse responses of unbiased FIR filters, after [19] and specifies the coefficient in (3) as a ml (N) = ( 1) m M (m+1)1(n), (4) D(N) where D(N) is the determinant and M (m+1)1 (N) is the minor of the (l + 1) (l + 1) quadratic matrix D(N), d 0 (N) d 1 (N)... d l (N) d 1 (N) d 2 (N)... d l+1 (N) D(N) =......, d l (N) d l+1 (N)... d 2l (N) which generic component d v (N) = N 1 i=0 iv, v [0, 2l], can be found via the Bernoulli polynomials (see Appendix A in [19]). The impulse response (3) has the following fundamental properties: it exists from zero to N 1; the sum of its coefficients is unity, N 1 n=0 h ln(n) = 1, for N 2; and its moments are zero, N 1 h ln (N)n u = 0, 1 u l. (5) n=0 For low-degree polynomial signals, the unique uniform (l = 0), ramp (l = 1), quadratic (l = 2), and cubic (l = 3) impulse responses (Fig. 1) were originally found in [19, 21] to be, respectively, h 0n (N) = 1 N, (6) h 1n (N) = 2(2N 1) 6n, (7) N(N + 1)

4 614 Circuits Syst Signal Process (2012) 31: h 2n (N) = 3(3N 2 3N + 2) 18(2N 1)n + 30n 2, (8) N(N + 1)(N + 2) h 3n (N) = 8(2N 3 3N 2 + 7N 3) 20(6N 2 6N + 5)n + 120(2N 1)n 2 140n 3. N(N + 1)(N + 2)(N + 3) (9) It has to be remarked now that, unlike the predictive FIR filters studied in [1, 6, 8, 16], the responses (6) (9) still have not been represented in closed compact z-transform forms with low-complexity necessary for efficient practical implementations. Their properties in the transform domain still remain unclear as well. In this paper, we consider the design and implementation of the unbiased FIR filters, by finding compact transfer functions and simple block-diagrams for lowdegree polynomial impulse responses. The rest of the paper is organized as follows. In Sect. 2, we discuss the transfer function of the unbiased FIR filter and observe its fundamental properties. The case of low-degree ramp, quadratic, and cubic impulse responses is considered in Sect. 3. Examples of applications for filtering of clock errors and design of low-pass (LP) filters employing a cascade of the unbiased FIR filters, l = 1, are given in Sect. 4. Finally, concluding remarks are in Sect Unbiasedness of Digital FIR Filters in the z-domain The transfer function of the l-degree unbiased FIR filter is specialized with the z- transform applied to the impulse response (3) as N 1 H l (z, N) = h ln (N)z n = n=0 l m=0 N 1 a lm (N) n m z n, n=0 (10a) (10b) where z = e jωt, ω is the angular frequency, T is the sampling time, and a lm (N) is given by (4). The following properties of H l (z, N) can be listed in addition to the inherent ones of 2π-periodicity, symmetry of H l (z, N), and asymmetry of arg H l (z, N). Transfer function at k = 0. By ω = 0, we have z = 1 and, referring to (10b) and (5), obtain H l (z, N) = 1 (11) for all l and N 2, meaning that the unbiased FIR filter is essentially a LP filter.

5 Circuits Syst Signal Process (2012) 31: Impulse response at n = 0. By the inverse z-transform, the value of h ln (N) at n = 0 is defined as h l0 (N) = 1 H l (z, N) dz (12a) 2πj C 1 z 2π = 1 ( H l e jωt,n ) d(ωt). (12b) 2π 0 Because h l0 (N) is always positive-valued, h l0 (N) > 0, for all l and N 2 [19], the counterclockwise circular integration in (12a) always produces a positive imaginary value that gives us H l (z, N) C 1 jz dz = 2πh l0 (N) > 0. (13) Transfer function at ωt = π. transform (10b) to With ωt = π, wehavez n = e jπn = ( 1) n and H l (z = e jπ,n)= l m=0 a lm (N) 2 [ ( 1) N E m (N) E m (0) ], (14) where E m (x) is the Euler polynomial. For low-degree impulse responses, 0 l 3, the Euler polynomials are E 0 (x) = 1, E 1 (x) = x 1 2, E 2(x) = x 2 x, and E 3 (x) = x x Energy. If H l (z, N) is the z-transform of h ln (N), then, by the Parceval theorem and (5), one has 1 2π 2π 0 N 1 ( H l e jωt,n ) 2 d(ωt) = h 2 ln (N) = n=0 l j=0 N 1 a jl (N) h li (N)i j i=0 (15a) (15b) = a 0l (N) = h l0 (N), (15c) and notes that the impulse response energy in the transform domain is equal to the value of h ln (N) at n = 0. The unbiasedness condition in the transform domain. The following theorem establishes the unbiasedness condition fundamentally and uniquely featured to digital FIR filters in the transform domain. Theorem 1 Given a digital FIR filter with the l-degree transfer function H l (e jωt,n) (10a) and input signal x n represented on an interval of N past points from n N + 1

6 616 Circuits Syst Signal Process (2012) 31: to n with the l-degree Taylor series. Then the filter output will be unbiased if the following unbiasedness condition is obeyed in the transform domain: 2π 0 H l ( e jωt,n ) d(ωt) = 2π 0 ( Hl e jωt,n ) 2 d(ωt). (16) Proof Substitute z in (13) with e jωt, compare the result to (15c), and arrive at (16). The proof is complete. Noise power gain. Similar to the noise figure used in wireless communications, a concept of the noise power gain (NPG) was introduced for signal processing by Blum in [4]. The NPG g l (N) can be defined by the energy of h ln (N) [4] to characterize noise amount in the output of a digital filter [9, 11, 15, 17, 23]. By Parceval s theorem, (10b) and (16), NPG acquires the following forms of g l (N) = 1 2π = 1 2π 2π 0 2π 0 ( Hl e jωt,n ) 2 d(ωt) H l ( e jωt,n ) d(ωt) (17a) (17b) = h l0 (N) = a 0l (N). (17c) Note that an analysis of g l (N), 0 l 3, in the time domain is given in [19]. 3 Transfer Function of a Digital Unbiased FIR Filter Although the inner sum in (10b) has no closed form for arbitrary m, the most general form of H l (z) can be performed as By assigning H l (z, N) = li=0 β i z i + z N l i=0 γ i z i 1 + l+1 i=1 α iz i. (18) ( l ) / l+1 H βl (z) = β i z (1 i + α i z ), i (19) i=0 i=1 ( l ) / l+1 H γl (z) = γ i z (1 i + α i z ), i (20) i=0 we go to the generalized block-diagram of the l-degree unbiased FIR filter shown in Fig. 2. For 0 l 3, the coefficients to (18) are listed in Table 1, where a 0l (N) can be defined by letting n = 0 in (6) (9). That allows us to find the low-complexity block-diagrams for practical design of such filters. i=1

7 Circuits Syst Signal Process (2012) 31: Fig. 2 A generalized block-diagram of the l-degree unbiased FIR filter Table 1 Transfer function coefficients of the low-degree unbiased FIR filters l β 0 1 N a 01 (N) a 02 (N) a 03 (N) β 1 0 N 4 18(N 1) N(N+1) 48(N2 2N+2) N(N+1)(N+2) β (2N 3) N N(N+1) β N γ 0 N 1 2 N N 3 γ 1 0 2(N 2) 6(N 3) N(N+1) N(N+1) γ (N 2)(N 3) N(N+1)(N+2) 12(N 4) N(N+1) 12(N 3)(N 4) N(N+1)(N+2) γ (N 2)(N 3)(N 4) N(N+1)(N+2)(N+3) 4 N α α α α Ramp Impulse Response The transfer function associated with the ramp impulse response, l = 1, can be found in the following compact form, provided the transformations of (18) using Table 1, H 1 (z, N) = 2 N a 01 N 2 2z 1 + z N (1 N 2 N+1 z 1 ) (1 z 1 ) 2. (21) A simple analysis reveals that the region of convergence in (21) is for all z and that the filter is both stable and causal. Figure 3 sketches the relevant block-diagram, which structure is N-invariant, although some blocks must be tuned to N. Thisdiagram utilizes 6 multipliers, 4 adders, and 3 time-delays. Note that the optimized recursive form found in [6] for the predictive FIR filter requires 3 multipliers, 5 adders, and 3 time-delays. The magnitude and phase responses of the structure shown in Fig. 3 are illustrated in Figs. 4 and 5, respectively, case l = 1. Figure 4(a) assures us that unbiasedness is achieved by shifting and elevating the side lobes of the uniform FIR filter, l = 0. Accordingly, the l-degree filter passes frequencies close to zero without alteration, magnifies components falling within the first lobe, and then attenuates powers of higher frequency components with 10 db per decade (Fig. 4(b)). The phase response

8 618 Circuits Syst Signal Process (2012) 31: Fig. 3 Block-diagram of the unbiased FIR filter with the ramp impulse response of this filter is linear on average (Fig. 5(a)). However, its function oscillates, similar to the predictive FIR filters [1, 6, 16], making the group delay also oscillating around a small constant value (Fig. 5(b)). 3.2 Quadratic Impulse Response After routine transformations, the transfer function for l = 2 becomes H 2 (z, N) = 3 N a 02 N 3 6(N 1) N+1 z 1 + 3z 2 z N [1 2(N 3) N+1 z 1 + (N 2)(N 3) (N+1)(N+2) z 2 ] (1 z 1 ) 3 (22) requiring 9 multipliers, 5 adders, and 4 time-delays for the block-diagram (Fig. 6). Note that the optimized recursive solution for the relevant predictive filter requires 5 multipliers, 12 adders, and 6 time-delays [6]. Observing Figs. 4 and 5, one infers that there is a certain rule in forming the transfer function by increasing l. In fact, for l = 2, all of the lobes are shifted and elevated with respect to the ramp case of l = 1 (Fig. 4(a)), although the fundamental slope remains the same (Fig. 4(b)). Oscillations in the phase response (Fig. 5(a)) acquire smaller amplitudes that leads to a similar effect in the group delay (Fig. 5(b)). 3.3 Cubic Impulse Response The unbiased FIR filter with l = 3 can be represented with the transfer function H 3 (z, N) = a 03 N 4 12(N 2 2N+2) (N+1)(N+2) z 1 + 6(2N 3) N+1 z 2 4z 3 N(1 z 1 ) 4 /4 + z N (1 b 1 z 1 + b 2 z 2 b 3 z 3 ) N(1 z 1 ) 4, (23) /4 in which the coefficients are given by b 1 = b 2 = 3(N 4) N + 1, (24) 3(N 3)(N 4) (N + 1)(N + 2), (25)

9 Circuits Syst Signal Process (2012) 31: b 3 = (N 2)(N 3)(N 4) (N + 1)(N + 2)(N + 3). (26) The block-diagram corresponding to (23) and sketched in Fig. 7 requires 12 multipliers, 5 adders, and 4 time-shifters. The relevant magnitude and phase characteristics are shown in Figs. 4 and 5, respectively, case l = 3. Observing these figures, one can easily trace an evolution in the filter transfer function, by increasing the filter degree. Moreover, the block-diagrams shown in Figs. 3, 6, and 7 can logically be Fig. 4 Magnitude response of the low-degree unbiased FIR filters: a H l (e jωt ) for N = 20 and b Bode plot of H l (e jωt ) 2 in db for N = 500

10 620 Circuits Syst Signal Process (2012) 31: Fig. 5 Phase characteristics of the low-degree unbiased FIR filters for N = 20: a phase response argh l (e jωt ) and b group delay d [argh l (e jωt )]/d(ωt ) extended to any degree l. Note that such an extension does not follow in a straightforward manner from the optimized recursive predictive FIR filters via diagrams given in [6]. 4 Applications To illustrate the applicability of the solutions proposed, we consider below examples of filtering of clock errors and design of an LP filter. 4.1 Filtering of Clock Errors An experimental test of the unbiased FIR filters has been provided employing the Global Positioning System (GPS)-based measurement of the time interval error (TIE)

11 Circuits Syst Signal Process (2012) 31: Fig. 6 Block-diagram of the unbiased FIR filter with the quadratic impulse response Fig. 7 Block-diagram of the unbiased FIR filter with the cubic impulse response x n of a precision oven-controlled crystal oscillator (OCXO)-based clock. The one pulse per second (1PPS) signals of a GPS timing receiver have been used. As suggested by the IEEE Standard [12], the clock time interval error (TIE) x n was described using the second-degree Taylor polynomial x n = μ 0 + λ 0 τn+ ς 0 2 τ 2 n 2 + w n, (27) where τ = t n t n 1 is a time step, μ 0 is the initial TIE at zero, λ 0 is the fractional frequency offset at zero, and ς 0 is the linear frequency drift rate at zero. Here w n represents the clock noise components having different natures and colors [12]. For this model, the FIR filter implemented as shown in Fig. 3 should be used if the contribution of z 0 is negligible, and one must employ Fig. 6 otherwise. The TIE measurement has been provided using the SynPaQ III GPS Timing Sensor and Stanford Frequency Counter SR620 in the presence of the sawtooth noise

12 622 Circuits Syst Signal Process (2012) 31: induced in the receiver owing to the principle of the 1PPS signal formation. To obtain the reference trend, the TIE has been simultaneously measured for the Symmetricom Frequency Cesium Standard CsIII employing another SR620. At the early state, the clock was identified to have two states, and the ramp FIR filter (Fig. 3) was thus assigned to be a near optimal estimator. For illustrative purposes, different measurements were chosen to have the near linear, near quadratic, and complex TIE trends Near Linear TIE Trend In the first experiment, the near linear TIE function (Fig. 8(a)) was processed. The optimum N was experimentally found to be N opt = 2060 and then the FIR filter, l = 1, (Fig. 3) run to produce the estimate shown in Fig. 8(a). Analysis has shown that this filter tracks the clock TIE with the root mean square error (RMSE) of about 1.6 ns that is substantially lower than that in the sawtooth corrected measurement Near Quadratic TIE Trend The second experiment was organized by applying the filter to the near quadratic TIE function as shown in Fig. 8(b). In this case, N opt was found to be about 920, and we notice that this value is consistent to that recommended for this clock in [25]. Numerical calculation gives us a larger RMSE of about 8.0 ns that is caused by the GPS time temporary uncertainties neatly seen in Fig. 8 ( GPS measurement ). Observe that there are no such uncertainties in the reference measurement Complex TIE Function We finally exploit a part of the measurement with complex behavior of the TIE function. Traditionally, the GPS-based and reference measurements are shown in Fig. 8(c) along with the unbiased FIR estimate. For this case, the optimum N was found to be N opt = 1150 and RMSE calculated as 7.83 ns. One can deduce that the unbiased FIR filter still traces well the mean measurement and behaves close to the reference one. 4.2 LP Filter Design In its direct applications, the l-degree FIR filter produces an unbiased estimate if applied to signals approximated on a time interval from n N + 1ton with the same l-degree polynomial. Moreover, it becomes near optimal in the MSE sense when N 1 [19]. Such filters are also often employed in hybrid structures [3, 7, 10, 16] with different purposes. As an example of design, we consider below a cascade connection of the unbiased FIR filters, l = 1, represented with the transfer function H LP (z) = H m 1 (z, N 1)H m 1 (z, N 2), (28) where m is an integer and H 1 (z, N) is implemented with the structure given in Fig. 3. Figure 9 shows what goes on with the cascade transfer function if we allow for dif-

13 Circuits Syst Signal Process (2012) 31: Fig. 8 GPS-based unbiased FIR filtering of the crystal clock TIE functions: a near linear,b near quadratic, and c complex ferent windows, N 1 = 60 and N 2 = 78, in order to compensate the side lobes. As can be seen, it is highly effective. If fact, by letting m = 7, we obtain an effective suppression of the side lobes, although with an excursion on about 40 db within the bandwidth. This solution can hence be developed to filter signals with low-frequency components.

14 624 Circuits Syst Signal Process (2012) 31: Fig. 8 (Continued) Fig. 9 Bode plot of the magnitude response of the cascade (24) of the ramp unbiased FIR filters (Fig. 2) with N 1 = 60 and N 2 = 78 5 Concluding Remarks In this paper, we have discussed a generalized block-diagram of the digital l-degree polynomial unbiased FIR filter, its transfer function, and implementation. Fundamental properties of the transfer function were also observed. Filters with the low-degree impulse responses (ramp, quadratic, and cubic) were considered in detail along with

15 Circuits Syst Signal Process (2012) 31: their transfer functions and compact block-diagrams. The magnitude and phase responses of the low-degree filters have been analyzed and compared to those of the predictive ones. As examples of applications, we first discussed filtering of the OCXObased clock errors via the GPS-based measurement of the TIE. It has been demonstrated experimentally that the filter produces no visible time delay with respect to the reference measurement. It was also shown that a cascade connection of unbiased FIR filters having different N opens new horizons in the design of LP filters. An importance of this study resides in the fact that, in line with the low-complexity block-diagrams essential for practical implementation, we have found a fundamental identity uniquely featured to unbiased FIR filters in the transform domain. This identity can serve as an indicator of biasedness in design of such filters. References 1. J. Astola, P. Heinonen, Y. Neuvo, Linear median hybrid filters. IEEE Trans. Circuits Syst. 36(11), (1988) 2. L. Arceo-Miquel, Y.S. Shmaliy, O. Ibarra-Manzano, Optimal synchronization of local clocks by GPS 1PPS signals using predictive FIR filters. IEEE Trans. Instrum. Meas. 58(6), (2009) 3. T.C. Ausal, K.E. Barner, Hybrid polynomial filters for Gaussian and non Gaussian noise environment. IEEE Trans. Signal Process. 54(12), (2006) 4. M. Blum, On the mean square noise power of an optimum linear discrete filter operating on polynomial plus white noise input. IEEE Trans. Inf. Theory 3(4), (1957) 5. E. Brookner, Tracking and Kalman Filtering Made Easy (Wiley, New York, 1998) 6. T.G. Campbell, Y. Neuvo, Predictive FIR filters with low computational complexity. IEEE Trans. Circuits Syst. 38(9), (1991) 7. D. Dang, W. Luo, Color image noise removal algorithm utilizing hybrid vector filtering. AEÜ, Int. J. Electron. Commun. 62(1), (2008) 8. X.Z. Gao, S.J. Ovaska, X. Wang, A fuzzy filter for sinusoidal signals with time-varying frequencies. Int. J. Signal Process. 1(2), (2004) 9. P. Händel, P. Tichavský, Asymptotic noise gain of polynomial predictors. Signal Process. 62(2), (1997) 10. P. Heinonen, Y. Neuvo, FIR-median hybrid filters with predictive FIR structures. IEEE Trans. Acoust. Speech Signal Process. 36(6), (1988) 11. S.Y. Hwang, Minimum uncorrelated unit noise in state-space digital filtering. IEEE Trans. Acoust. Speech Signal Process. ASSP-25(4), (1977) 12. IEEE Standard , Definitions of Physical Quantities for Fundamental Frequency and Time Metrology Random Instabilities (IEEE, Piscataway, 1999) 13. K.R. Johnson, Optimum, linear, discrete filtering of signals containing a nonrandom component. IRE Trans. Inf. Theory 2(2), (1956) 14. W.H. Kwon, S. Han, Receding Horizon Control: Model Predictive Control for State Models (Springer, Berlin, 2005) 15. S.K. Mitra, J.F. Kaiser (eds.), Handbook of Digital Signal Processing (Wiley, Berlin, 1993) 16. S.J. Ovaska, O. Vainio, T.I. Laakso, Design of predictive IIR filters via feedback extension of FIR forward predictors. IEEE Trans. Instrum. Meas. 46(5), (1997) 17. J. Ritzerfeld, Noise gain expressions for low noise second-order digital filter structures. IEEE Trans. Circuits Syst. II, Express Briefs 52(4), (2009) 18. S. Samadi, A. Nishihara, Explicit formula for predictive FIR filters and differentiators using Hahn orthogonal polynomials. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E90-A(8), (2007) 19. Y.S. Shmaliy, An unbiased FIR filter for TIE model of a local clock in applications to GPS-based timekeeping. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53(5), (2006) 20. Y.S. Shmaliy, On real-time optimal FIR estimation of linear TIE models of local clocks. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(11), (2007)

16 626 Circuits Syst Signal Process (2012) 31: Y.S. Shmaliy, An unbiased p-step predictive FIR filter for a class of noise-free discrete-time models with independently observed states. Signal Image Video Process. 3(2), (2009) 22. Y.S. Shmaliy, Unbiased FIR filtering of discrete-time polynomial state-space models. IEEE Trans. Signal Process. 57(4), (2009) 23. Y.S. Shmaliy, Linear optimal FIR estimation of discrete time-invariant state-space models. IEEE Trans. Signal Process. 58(6), (2010) 24. Y.S. Shmaliy, L. Morales-Mendoza, FIR smoothing of discrete-time polynomial signals state-space. IEEE Trans. Signal Process. 58(5), (2010) 25. Y.S. Shmaliy, J. Munoz-Diaz, L. Arceo-Miquel, Optimal horizons for a one-parameter family of unbiased FIR filters. Digit. Signal Process. 18(5), (2008) 26. S.W. Smith, The Scientist and Engineer s Guide to Digital Signal Processing (California Technical Publishing, Poway, 1997)