Circuits Syst Signal Process (2012) 31:2153–2166 DOI 10.1007/s00034-012-9433-y
Computationally Efficient FIR Filtering of Polynomial Signals in DFT Domain Paula Castro-Tinttori · Oscar Ibarra-Manzano · Yuriy S. Shmaliy
Received: 9 August 2011 / Revised: 9 May 2012 / Published online: 26 May 2012 © Springer Science+Business Media, LLC 2012
Abstract Fast unbiased finite impulse response (UFIR) filtering of polynomial signals can be provided in the discrete Fourier transform (DFT) domain employing fast Fourier transform (FFT). We show that the computation time can further be reduced by utilizing properties of UFIR filters in the DFT domain. The transforms have been found and investigated in detail for low-degree FIRs most widely used in practice. As a special result, we address an explicit unbiasedness condition uniquely featured to UFIR filters in DFT domain. The noise power gain and estimation error bound have also been discussed. An application is given for state estimation in a crystal clock employing the Global Positioning System based measurement of time errors provided each second. Based upon it, it is shown that filtering in the time domain takes about 1 second, which is unacceptable for real-time applications. The Kalman-like algorithm reduces the computation time by the factor of about 8, the FFT-based algorithm by about 18, and FFT with the UFIR filter DFT properties by about 20. Keywords Unbiased FIR filter · DFT domain · Polynomial signal · Fast algorithm
1 Introduction Approximation with finite-degree polynomials is useful for many applications related to signal, image, voice, and speech processing [20]. It is especially efficient if signals are oversampled or highly oversampled. In such cases, finite impulse response (FIR) P. Castro-Tinttori · O. Ibarra-Manzano · Y.S. Shmaliy () Department of Electronics, University of Guanajuato, Salamanca 36855, Mexico e-mail:
[email protected] O. Ibarra-Manzano e-mail:
[email protected]
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filtering often serves better than infinite impulse response (IIR) one owing to the imbedded bounded input/bounded output (BIBO) stability [16] and better robustness against model uncertainties [18]. For signals processed on an interval of some N neighboring points, a number of optimal and suboptimal FIR structures have been proposed for decades. Among them, there is a class of unbiased estimators whose algorithms can be designed ignoring noise and initial errors [17, 25]. We meet applications of unbiased filtering in positioning [19], tracking [5], timekeeping and clock synchronization [25], speech and image processing [11], etc. Unbiasedness is imbedded to the maximum likelihood estimators. It is strongly desirable for nonlinear filtering [1, 14], channel estimation in wireless systems [21], estimation of systems with unknown inputs [12] and periodic time-varying structures [3], channel identification and equalization [10], image denoising [30], etc. An important peculiarity of UFIR estimators is that they become virtually optimal when N 1 [26]. The effect is due to averaging reducing the white noise variance by the factor of N . On the other hand, FIR estimators are convolution-based, which imposes a computational problem when N is large. Different methods exist of fast convolution computation [22]. Most often, designers employ the circular convolution theorem in the discrete Fourier transform (DFT) domain [24] using the fast Fourier transform (FFT) algorithms. Although the approach suggests an appreciable reduction of the computation time, further reduction can be achieved if to utilize properties of the UFIR filter response in DFT domain, thereby avoiding direct FFT. Since the UFIR filter has a unique l-degree response, its transform and properties in DFT domain are also unique for each signal model. Studies of such properties in the z-domain have been provided for different UFIR structures in Refs. [6, 7, 9, 23]. To the best of out knowledge, there are no results reported in the DFT domain. We notice that fast UFIR filtering is also available using the recently proposed iterative Kalman-like structures [26, 27]. In this paper, we find compact DFT forms for the low-degree unbiased FIR filter responses, investigate their properties in DFT domain in detail, and employ them for fast FFT-based filtering of polynomial signals. The rest of the paper is organized as follows. Section 2 discusses the polynomial signal model and formulates the problem. DFT properties of the unbiased FIRs along with the unbiasedness condition, noise power gain, and error bound are studied in Sect. 3. Section 4 suggests compact transforms for the low-degree FIRs. Applications to clock state estimation are given in Sect. 5 and concluding remarks are drawn in Sect. 6.
2 Polynomial Signal Model and Problem Formulation Consider a band-limited deterministic signal xn in discrete time n. A signal can be represented on an interval of N past-neighboring points from m = n − N + 1 to n with the finite (K − 1)-degree Taylor series expansion as xn =
K−1 p=0
x(p+1)m
τ p np p!
(1)
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= x1m + x2m τ n +
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x3m 2 2 xKm τ n + ··· + (τ n)K−1 , 2 (K − 1)!
(2)
where xqm , q ∈ [1, K], is the signal qth state at m and τ = tn − tn−1 is the sampling time. In is tacitly assumed that the terms of order more than K are negligible. Let xn be measured in the presence of zero mean, E{vn } = 0, additive noise vn having the covariance, Rij = E{vi vj }, as yn = xn + v n .
(3)
Provided yn , the UFIR filter output1 xˆn|n can be found by the discrete convolution employing data from m to n as xˆn|n =
N −1
hli (N )yn−i ,
(4)
i=0
where hln (N) is the UFIR filter polynomial impulse response of l-degree, to be formally defined below. In such filters, hln (N ) obeys the unbiasedness condition E{xˆn|n } = xn ,
(5)
meaning that an average of the estimate is required to be equal to its origin. We notice that different forms of (5) can be used [2, 8, 13, 31]. When N is large, N 1, and the computational problem arises, fast computation of (4) is commonly provided in DFT domain using FFT. The problem now formulates as follows. Given a polynomial signal (1) and its measurement (3), we would like to minimize the computation time featured to (4) by utilizing properties of hln (N ) in DFT domain. We also wish to consider a practical example implying N 1 and estimate the computation time based on different algorithms available.
3 DFT Properties of UFIR Filters The DFT of hln (N ) can be found as Hlk (N ) =
N −1
hln (N )WNnk ,
(6)
n=0 2π
where WN = e−j N . In addition to the inherent N -periodicity, symmetry of |Hlk (N )|, and asymmetry of argHlk (N ), the following properties can be listed. 3.1 Unbiasedness Condition in the DFT Domain The following theorem establishes the unbiasedness condition fundamentally featured to UFIR filters in the DFT domain. 1 Here and in the following, xˆ n|v means the estimate of xn at n via measurement from the past to v.
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Theorem 1 Given a signal xn represented with the (K − 1)-degree Taylor series (1) and measurement yn (3), the estimate xˆn|n (4) of xn will be unbiased by (5) if the filter impulse response hln (N ) obeys the following unbiasedness condition in the DFT domain: N −1
N −1 H(K−1)k (N )2 = H(K−1)k (N ).
k=0
(7)
k=0
Proof To prove (7), invoke the fundamental observation following from the Kalman filter theory: the order of the optimal (and also unbiased [11, 25]) filter is the same as that of the system. Then let l = K − 1 and represent hln (N ) with a polynomial h(K−1)n (N ) =
K−1
am(K−1) (N )nm .
(8)
m=0
The coefficient am(K−1) (N ) was defined for (8) in [25] as am(K−1) (N ) = (−1)m
M(m+1)1 (N ) , |D(N )|
(9)
where |D(N )| is the determinant and M(m+1)1 (N ) is minor of the K × K quadratic matrix ⎡ ⎤ d0 d1 . . . dK−1 ⎢ ⎥ d2 . . . dK ⎥ ⎢ d1 ⎢ ⎥ (10) D(N ) = ⎢ . ⎥, .. .. .. ⎢ . ⎥ . . . ⎣ . ⎦ dK−1
dK
...
d2(K−1)
whose generic component dv (N ) =
N −1
iv ,
v ∈ 0, 2(K − 1)
(11)
i=0
can be determined by the Bernoulli polynomials (see Appendix A in Ref. [25]). The response (8) has the following fundamental properties [4]: the sum of its coefficients is unity and the moments are zeros, respectively: 1=
N −1
(12)
h(K−1)n (N ),
n=0
0=
N −1
h(K−1)n (N )nu ,
1 u K − 1.
(13)
n=0
Now employ the Parceval relation [24], use (8), (12), and (13), and go to N −1 −1 2 N 1 H(K−1)k (N ) = h2(K−1)n (N ) N k=0
n=0
(14)
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=
K−1
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aj (K−1) (N )
j =0
N −1
h(K−1)i (N )i j
(15)
i=0
(16) = a0(K−1) (N ) = h(K−1)0 (N ). −1 By the inverse DFT (IDFT) at n = 0 we have h(K−1)0 (N ) = N1 N k=0 H(K−1)k (N ) that leads to (7). The proof is complete. 3.2 DFT Value at k = 0 n0 = 1 and then, referring to (12), DFT (6) becomes By k = 0, we have W2N
Hl0 (N ) = 1
(17)
for all l. The UFIR filter is thus essentially a low-pass (LP) filter. 3.3 Impulse Response at n = 0 By IDFT, the value of hln (N ) at n = 0 is N −1 1 hl0 (N ) = Hlk (N ). N
(18)
k=0
Since hl0 (N ) > 0 holds for all l [25], the sum of the DFT coefficients is real and positive; that is, N −1
Hlk (N ) = N hl0 (N ) > 0.
(19)
k=0
3.4 DFT Value at k = N/2 n0 = (−1)n . Accordingly, (6) and (8) produce At k = N/2 with N even, we have W2N the boundary value for the DFT
Hl N (N ) = − 2
l alm (N )
Em (N ) − Em (0) , 2
(20)
m=0
where Em (x) is the Euler polynomial. For low-degree gains, 0 l 3, the Euler polynomials are: E0 (x) = 1, E1 (x) = x − 12 , E2 (x) = x 2 − x, and E3 (x) = x 3 − 3 2 1 2x + 4. 3.5 Noise Power Gain A measure of the noise amount in the unbiased FIR filter output is the noise power gain (NPG) gl (N ) defined for white Gaussian noise by the energy of hln (N ) [4].
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Utilizing (7), (14), and (18), the NPG can be evaluated in the following forms: N −1 2 1 Hlk (N ) N
(21)
N −1 1 Hlk (N ) N
(22)
= hl0 (N ) = a0l (N ).
(23)
gl (N ) =
k=0
=
k=0
3.6 Estimation Error Bound The concept of NPG was employed in [28] to find the estimation error bound (EB) in the three-sigma sense. For gl (N ) defined with (21), this bound can be written as
(24) β(N) = 3σv gl (N ), where σv is the standard deviation of the measurement white noise vn . The above-discussed properties of UFIR filters hold true for any degree l. However, the low-degree, l 3, digital UFIR filters have gained most currency. Below, we give exact compact forms for their DFT.
4 DFT of the Low-Degree Impulse Responses By (6) and (8), the DFT of the l-degree UFIR filter can be rewritten as Hlk (N ) =
l m=0
alm (N )
N −1
nm WNnk ,
(25)
n=0
where alm (N ) is specified by (9). For arbitrary m, the inner sum in (25) has no closed form. Instead, solutions can be found for low-degree responses found in [25] and listed in the Appendix. 4.1 Ramp Impulse Response The ramp response (35), l = 1, is associated with signals changing linearly from m −1) 6 to n. It is specialized with the coefficients a01 (N ) = 2(2N N (N +1) and a11 (N ) = − N (N +1) and can be shown to have the DFT j 6 k (26) 1− W H1k (N ) = − N +1 2 sin(πk/N ) 2N =
π 2k 3 ej 2 ( N −1) (N + 1) sin(πk/N )
for 0 < k < N . If k = 0 and k = N , then H1k (N ) is unity, by (17).
(27)
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Fig. 1 DFT of the low-degree UFIR filters: (a) |Hlk (N )| for N = 20, (b) Bode plot of |Hlk (N )|2 for N = 200, and (c) argHlk (N ) for N = 20
The magnitude response |H1k (N )| of (26) decreases monotonously from unity at k = 0 to 3/(N + 1) at k = N/2, tending towards zero at k = N/2 with N 1. Note that N must be even for DFT to exist at k = N/2. Figure 1(a) illustrates |H1k (N )| for N = 20. One can observe that the Bode plot of |H1k (N )|2 shown in Fig. 1(b) has a fundamental slope of k −2 that evolves to k 0 at k = N/2. The phase characteristic of (27) is linear with a positive slope as sketched in Fig. 1(c).
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4.2 Quadratic Impulse Response If a signal changes quadratically, from m to n, then unbiasedness (5) is achieved with the quadratic impulse response h2n (N ) represented by (36) with the coefficients 3(3N 2 − 3N + 2) −18(2N − 1) , a12 (N ) = , N (N + 1)(N + 2) N (N + 1)(N + 2) 30 . a22 (N) = N (N + 1)(N + 2) The relevant DFT exists in the region of 0 < k < N as N − 13 − (N − 3)WNk 6(N − 3) H2k (N ) = 1− (N + 1)(N + 2) 4(N − 3) sin2 (πk/N ) a02 (N) =
(28)
and it becomes unity if k = 0 and k = N , by (17). Although not monotonously, (28) decreases from unity at k = 0 to 12j/(N + 1) at k = N/2 and tends towards zero at this point when N 1, provided N is even. One finds the main difference between |H2k (N )| and |H1k (N )| at k = 1, where the former has an excursion. Namely this “deformation” makes H2k (N ) such that it fits better quadratic signals rather than linear ones. It also forces |H2k (N )|2 starting with the slope k −3 at k = 1 and then evolve through k −2 to k 0 at k = N/2 (Fig. 1(b)). As well as in the ramp response, the phase response associated with (11) is fundamentally linear (Fig. 1(c)). However, it looks like rather a smoothed version of the ramp response phase. 4.3 Cubic Impulse Response The more sophisticated response (37) fits cubic signals with the coefficients a03 (N ) =
8(2N − 1)(N 2 − N + 3) , N (N + 1)(N + 2)(N + 3)
a13 (N ) =
−20(6N 2 − 6N + 5) , N (N + 1)(N + 2)(N + 3)
120(2N − 1) −140 , a33 (N ) = . N (N + 1)(N + 2)(N + 3) N (N + 1)(N + 2)(N + 3) After the routine transformations, its DFT can also be found in a compact form to exist in the region of 0 < k < N as a23 (N ) =
H3k (N ) =
15(N − 1) 4(N + 1)(N + 2) sin4 (πk/N ) 3(N − 2)(N + 3) sin(2πk/N ) − (N 2 + 5) sin(4πk/N ) × 1 − 2j 3(N − 1)(N + 3) 2(N + 2)(2N − 1)WNk − (N 2 + 5)WN2k (29) − 3(N − 1)(N + 3)
and take unity when k = 0 and k = N , by (17). As can be seen (Fig. 1(a)), |H3k (N )| also has an excursion at k = 1. It then reduces to 5(2N − 1)/(N + 1)(N + 3) at k = N/2 and becomes 10/N here when
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Table 1 Characteristics of the most general unbiased filters having κ = 0 Signal
Filter Type
Causality
Impulse response
DFT
Any
ideal LP
no
IIR
(31), A = 1
Constant
simple average
yes
FIR (32)
(33)
Linear
ramp
yes
FIR (35)
(27)
Quadratic
quadratic
yes
FIR (36)
(28)
Cubic
cubic
yes
FIR (37)
(29)
(1), K < ∞
polynomial
yes
FIR (8)
(6)
(1), K = ∞
polynomial
no
IIR (8)
(6)
N 1, provided N is even. Setting aside a part of this function with small k (Fig. 1(c)), one concludes that the phase response of (29) is also fundamentally linear (Fig. 1(c)). 4.4 Filters with Imbedded Unbiasedness UFIR filters with l > 3 are rarely met in applications. Although (25), (27), and (29) cover an overwhelming majority of needs, it is worth to connect them with wellknown filters. As a measure of biasedness, we use below a biasedness criterion κ=
N −1 1 |Hk |2 − Hk , N
(30)
k=0
following directly from Theorem 1. The most common unbiased filters fulfilling κ = 0 are listed in Table 1. An ideal LP filter has the rectangular magnitude response and zero phase. Its DFT is A, |k| B, (31) Hk = 0, |k| > B, where A = const and B is associated with the filter bandwidth. Testing (31) with A = 1 by (30) gives us κ = 0 and we infer that an ideal LP filter is unbiased for all signals whose spectral contents fall within the filter bandwidth. However, any magnification with A = 1 makes this filter biased. In fact, for arbitrary A and B + 1 = N/2 we have κ = A(A − 1) B+1 N = A(A − 1)/2. Although an ideal LP filter is the most universal unbiased one, it is IIR and thus noncausal. Any truncation leads to the Gibbs’s phenomenon [24] and the filter loses an ability of filtering signals with zero bias, because its magnitude is no longer uniform and unit within the bandwidth. We thus infer that there exists no universal causal and feasible LP filter providing unbiased filtering of all signals. A counterpart of the ideal LP filter is the one specified with the uniform impulse response associated with simple averaging, 1/N, 0 n N − 1, h0n = (32) 0, otherwise.
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DFT of (32) calculates
Hk =
0, 0 < k < N, 1, k = 0, k = N,
(33)
and we deduce that this filter is also unbiased, because of κ = 0 for all N 2. Unlike (31) with A = 1 suitable for all signals, (33) fits only the constant ones and its region of applications is thus strongly limited. Again, we notice that any magnification in (33) makes this filter biased. Table 1 also gives us a relevant generalization for K < ∞ and K = ∞. Observing Table 1 and referring to the above-given analysis, we come up with an important inference: every particular signal can be filtered without bias only with a unique filter. 5 Applications to the Clock State As an example of applications, below we provide unbiased FIR filtering of the current state of one the most mystical systems called clock. Its peculiarity is in nonstationary Gaussian noise having flicker and random walk components with not fully known origins. Close to zero Fourier frequency, the clock oscillator phase’s power spectral density (PSD) rises dramatically and is still not specified here by the standard documents [15]. Applications of Kalman filtering thus face problems. Simulation of such systems commonly cannot be provided adequately and the best way is to test the estimator for reference measurement. Optimal estimation of clock state is required for the prediction of future time errors, which is of high importance for digital networks operating in common time. Another peculiarity is that errors in precision and master clocks are in nanoseconds, therefore measurement is expressed in long numbers and its estimation implies large computation time. In our experiment, the time interval error (TIE) of an oven controlled crystal oscillator (OCXO)-based clock imbedded in the Stanford Frequency Counter SR620 was measured for the Global Positioning System (GPS) SynPaQ III Timing Sensor with another SR620. To provide the true TIE behavior, measurement was also conducted simultaneously for the Symmetricom cesium standard of frequency CsIII. Measurements of yn , true TIE xn , and noise vn composed of sawtooth induced by the timing sensor and GPS time temporary uncertainty, are shown in Fig. 2(a). Regular errors were removed at the early stage. In turn, Fig. 2(b) sketches the noise structure v n = yn − xn . Although the IEEE Standard [15] suggests that the clock has a polynomial model (1) with K = 3 over all time, particular clocks may have other number of states. We therefore do not restrict ourselves with l = 3, and apply the l-degree unbiased FIR filter with l ∈ [0, 3] to yn , find the estimate xˆn as a function of N , and ascertain Nlopt for each l, by minimizing the mean square error (MSE) in the estimate [29]. Figure 3 gives the root MSE (RMSE), RMSE(N ) = n2 (N ) , (34) as a function of N in the wide range of N on a baseline of four days, where un stands for an average of un over all n starting with n = N − 1. As can be seen from
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Fig. 2 GPS-based and reference measurements of the crystal clock TIE during 96 hours: (a) measurement yn , and (b) measurement noise vn = yn − xn
the figure, RMSE decreases by N in the region where the filter degree fits the model. It then reaches a minimum at Nlopt and thereafter increases. Table 2 summarizes the results and we notice that the values of Nlopt range close to those found following the recommendations given in Ref. [29]. Filtering of clock state was provided with different Matlab-based algorithms implemented in a Laptop HP Pavilion dv5-1131, 800 MHz. The algorithms were run with l ∈ [1, 3] and Nlopt . The first estimate called “Batch” was obtained by (4), the second one by the Kalman-like algorithm recently addressed in [27], and the third one by FFT in DFT domain. Finally, the fourth estimate called “Improved” is the third one in which FFT of hln (N ) was avoided and substituted utilizing properties of (27)–(29). The computation time for all these algorithms was ascertained experimentally as shown in Table 3. One observes that the “Batch” algorithm takes about one second, which is unacceptable for real-time applications. The Kalman-like algorithm reduces the computation time by the factor of about 8, the FFT-based one by about 18, and the “Improved” algorithm by about 20.
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Fig. 3 RMSE in the first state estimates of the OCXO-based clock Table 2 Minimum RMSE and Nlopt for the l-degree unbiased FIR filter applied to yn (Fig. 2(a))
Table 3 Computation time, in s, for different unbiased FIR algorithms
Degree l l=0
l=1
l=2
l=3
RMSEmin , ns
9.437
7.319
7.477
7.652
Nlopt
50
2000
3400
4700
l
Nlopt
Computation time, in s, over Nlopt Batch
Kalman-like
FFT/IFFT
Improved
1
2000
0.974254
0.111782
0.029236
0.026856
2
3400
0.980974
0.122622
0.054407
0.050886
3
4700
0.989804
0.133725
0.088162
0.084541
6 Concluding Remarks In this paper, we have studied the DFT properties of UFIR filters and employed them in FFT-based filtering of polynomial signals. An importance of this study resides in the fact that we have found a fundamental identity, uniquely featured to UFIR filters in DFT domain. Designers can also appreciate the compact transforms found for low-degree impulse responses. As an example of applications, different UFIR filtering algorithms have been tested for the crystal clock polynomial model employing GPS-based one-pulse-per-second measurement of clock time errors. It has been demonstrated experimentally that, for N = 3400, the improved algorithm is faster than the FFT-based and computationally it is about 20 times more efficient than the batch estimator.
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Appendix: Low-degree unbiased FIR filter impulse responses The following impulse responses [25] are unique for unbiased FIR filtering of linear, quadratic, and cubic signals, respectively, on an interval of N past-points from n − N + 1 to n: h1n (N) =
2(2N − 1) − 6n , N (N + 1)
(35)
h2n (N) =
3(3N 2 − 3N + 2) − 18(2N − 1)n + 30n2 , N (N + 1)(N + 2)
(36)
h3n (N) =
8(2N 3 − 3N 2 + 7N − 3) − 20(6N 2 − 6N + 5)n + 120(2N − 1)n2 − 140n3 . N (N + 1)(N + 2)(N + 3) (37)
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