zyxw I096 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40. NO. 5. MAY 1992 zyxwvutsrqp zyxwvu Noise Reduction in Recursive Digital Filters Using High-Order Error Feedback Timo I. Laakso, Student Member, IEEE, and Iiro 0 . Hartimo, Senior Member, IEEE Abstract-The problem of solving the optimal (minimumnoise) error feedback coefficients for recursive digital filters is addressed in the general high-order case. It is shown that when minimum noise variance at the filter output is required, the optimization problem leads to a set of familiar Wiener-Hopf or Yule-Walker equations, demonstrating that the optimal error feedback can be interpreted as a special case of Wiener filtering. As an alternative to the optimal solution, the formulas for suboptimal error feedback with symmetric or antisymmetric coefficients are derived. In addition, the design of error feedback using power-of-two coefficients is discussed. These schemes are often more suitable for practical implementation than the optimal error feedback and in many cases almost as good roundoff noise performance is achieved. The efficiency of high-order error feedback is examined by test implementations of the set of standard filters. It is concluded that the error feedback is a very powerful and versatile method to cut down the quantization noise in any classical IIR filter implemented as a cascade of second-order direct form sections. Second-order error feedback is sufficient for most cascade implementations, whereas the new high-order schemes are attractive for use with high-order direct form sections. I. INTRODUCTION RROR feedback (EF) (or error spectrum shaping, noise shaping, or residue feedback) is a general method that can be used to reduce errors inherent in any quantization operation. To our knowledge, error feedback techniques are widely used in applications like predictive speech coding 1291, predictive image coding [21], and sigma-delta analog-to-digital conversion 1221. Error feedback can also be used to reduce quantization errors generated in finite-wordlength implementations of recursive digital filters. Especially with fixed-point implementations of narrow-band low-pass filters its effect is striking and usually superior to any other low-noise structure [63], 1581, 1241, 1351. The error feedback is implemented by extracting the quantization error after the product quantization and feeding the error signal back through a simple, usually FIR-type filter (Fig. 1). As is well known, the level of the output quantization noise of a recursive filter tends to be high, especially when the poles are located close to the unit circle [27]. By choosing the EF parameters appropri- E zyx i/m 111 A Lrl x'( U Fig. 1, A quantizer with Nth-order error feedback ately, zeros can be placed in the error spectrum to reduce the noise very efficiently 1251. It should be emphasized that the EF affects only the transfer function of the quantization error signal, while the transfer function of the filter itself remains unchanged. Thus, the EF cannot have any effect on the coefficient sensitivity properties of the filter structure, neither can it enhance the overflow properties of the filter implementation. Originally used in a PCM rounding quantizer [60], error feedback has since been applied to direct form secondorder sections [62], [63], [7], [511, [24], to cascaded direct form structures 1251, [35], to high-order direct form (lump) structure [61], to a cascade of fourth-order directform sections [36], to normal and state-space structures VI, [4l, 1581, [591, 1641, [671, [711, to orthogonal polynomial (Gray-Markel) structures [69], [70], and, quite recently, to a certain class of wave-digital-related VGIC structures 1151. Special low-sensitivity second-order structures that are amenable to the EF have been introduced [14]. The ability of the EF to reduce the amplitude to limit cycles as well or, in some cases, to eliminate them completely has been demonstrated [ 11-[3], [7], [51], [58], [67], [41], [40]. Feeding back the error generated in the overflow situation can be used to reduce the amplitude of overflow transients [ 121 or to eliminate zero-input overflow oscillations completely [39]. zyxwvutsrqp zyxwvutsrqp zyxwvutsrqpo zyxwvuts Manuscript received December 14, 1988; revised February 11, 1991. The authors are with the Laboratory of Signal Processing and Computer Technology, Helsinki, University of Technology, SF-02150 Espoo, Finland. IEEE Log Number 9106580. 1053-587X/92$03.00 0 1992 IEEE I097 LAAKSO AND HARTIMO: NOISE REDUCTION IN RECURSIVE DIGITAL FILTERS Efficient hardware implementations of the EF have been advanced [63], [8], [51], [59], [67], [681, [611, [651. It has been shown that the EF can also be efficiently implemented with a signal processor [35], [36], [13]. With tailor made hardware one can save several bits in the internal signal wordlength by investing in error feedback. With general-purpose hardware (signal processors, etc .) it is possible to increase the effective wordlength when limit cycles or quantization noise grows too high. Some basically different ways to formulate and apply the EF can be distinguished in the existing literature. We divide the proposed EF schemes into three categories according to how the parameters of the EF quantizer, the structure, the coefficients, and the order, are determined: 1) Constant Error Feedback: These schemes allow placing one or two zeros in the error transfer function at the points z = 1 or z = - 1. Hence, they are best suited for narrow-band low-pass and high-pass filters, respectively. The properties of these EF schemes and their hardware implementation were studied in the early works [62], [63], [2], [4], [511. The analyzed filters were mostly simple second-order direct form sections. 2) Built-in Error Feedback: With this term we refer to those EF schemes where the filter structure determines the type of error feedback, that is, the order and the structure of the EF quantizer and the values of the EF coefficients. The structure of the EF quantizer is not necessarily a FIR-type filter, but a replica of the recursive structure of the filter (sometimes the nonrecursive parts are included, which is called error feedforward [24]). The coefficients are more or less directly obtained by simply quantizing the recursive filter coefficients into integer values [8], [58l, 1671, [681, 1611, [701, [711. These EF schemes can be viewed as approximations of extended precision arithmetic [24], [50]. When applied to the state-space structure, elegant formulation and analysis of the noise properties of the filter can be obtained [9], [58], [67], [71]. Some schemes also allow several equivalent implementation strategies, as pointed out in [lo]. However, due to their complexity, these structures are often more interesting from the theoretical point of view than real alternatives for practical implementation. 3) Optimal Error Feedback: In this case the EF quantizer is completely general and isolated from the filter structure. Any quantization point in the filter structure where the wordlength is restored after multiplications can be provided with this kind of error feedback. We call this modified quantizer an error feedback quantizer. From the implementation point of view the optimal EF is most attractive when there are only few quantization points in the structure. The EF coefficients are usually optimized by minimizing the quantization noise power at the filter output, that is, by using an least-mean-square (LMS) type criterion 1241, P51, [141, 1151, [361-[381. To our knowledge, only first- and second-order optimal EF have been addressed in the literature so far. However, in high-order systems high-order EF would naturally offer better noise reduction. Also with the direct form (lump) structure, which is still the most practical structure for certain adaptive and time-varying filtering and control applications, as pointed out in [61], the high-order EF can be of much help in enhancing the otherwise rather poor finite-wordlength properties of this structure. Higgins and Munson have derived formulas to calculate the optimal EF coefficients for a second-order EF quantizer to minimize the power of the output quantization noise [25]. In Section I1 of this paper we expand on the work of Higgins and Munson and derive the general formulas for the optimal coefficients of an EF quantizer of arbitrary order. Additionally, instead of numerical integration that Higgins and Munson utilized, an algorithm for calculating these coefficients directly applying the total square integral in the z-domain [31] is derived. From the implementation point of view, the Nth-order optimal error feedback is often too costly due to the N explicit multiplications required. The costs can be reduced, e.g., by constraining the EF polynomial to have symmetric or antisymmetric coefficients, which cuts the number of distinct coefficients in half. In Section 111, formulas for suboptimal error feedback with symmetric or antisymmetric coefficients are derived. In many cases this approach results in minimal losses in noise reduction as compared to the optimal solution. It is thus well suited for implementations where the coefficient symmetry can be utilized. Another strategy to reduce implementation costs is to use EF coefficients with power-of-two values. This allows simple implementation by using mere additions (or subtractions) and shifting. In Section IV we present a simple discrete optimization algorithm to find near-optimal power-of-two EF coefficients that minimize the output noise in the set of available coefficient values. In Section V, second-, third-, and fourth-order EF is applied to some direct form I cascade implementations of some standard test filters. The cascades of both secondand fourth-order DF I sections are studied. In Section VI, the implementation issues are briefly discussed. Two important implementation strategies, signal processors, and custom VLSI techniques are considered from the standpoint of error feedback. zyxwvuts zyxwvutsrq zyxwvutsrqpon 11. NTH-ORDER OPTIMAL ERRORFEEDBACK The error feedback is implemented by modifying the quantizer in the filter structure. In a fixed-point implementation, the quantization is usually performed by discarding the lower bits of the double-precision accumulator (two’s complement truncation), and thus the quantization error equals this residue left in the lower part. The error is fed back through a simple FIR filter, as shown in Fig. 1. (Also a secondary error is introduced due to the finite wordlength of the EF quantizer [24], but assuming that the EF wordlength is sufficient, this error is seldom of any importance and will be neglected in our discussion.) zyxwvu ~ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1098 zyxwvutsrqpon zyxwvutsrq zyxwvutsrqponmlkjihgfe IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40. NO. 5, MAY 1992 As usual, each quantizer is modeled as an independent additive white noise source with variance o 2 = 2 - 2 b / 1 2 where (1 b) is the wordlength (1 b for sign) [27]. With this assumption, the optimal coefficients for the EF parameters that minimize the power of the quantization noise can be determined independently for each quantizer in the filter structure [25]. It can be shown that, with an Nth-order purely recursive direct form section, the optimal coefficients for N th-order EF are directly the recursive coefficients (for the second-order case, see [24], [50]). Whenever the error transfer function from the quantizer to the filter output is not purely recursive (e.g., with direct form cascade structures) or when the order of the EF quantizer is lower than that of the filter section, this is not the case. Let us consider the Nth-order error feedback quantizer shown in Fig. 1 . The EF network affects with the following z-domain transfer function: + B(z) = 1 + The expressions in square brackets are identified as Fourier coefficients of the normalized power spectrum, that is, as autocorrelation coefficients of the output error signal. Denoting these as qk = T jT COS O kwQ(w) dw and observing that the autocorrelation sequence is symmetric ( q - k = qk),the integral can be expressed as zyxwvuts N I = k=l N N c 1=I PkPlqlk-ll c P k q k + 90. + 2 k= + p2F2 + & - I ..* + (1) = B(z)G ( z ) . (2) The normalized noise variance (noise gain) at the filter output is obtained with the following integral [31]: This can be expressed as a quadratic form I = wTRw + 2pTw + qo +'j' T IB(eJw) I I G(ej")l2 dw. O (4) Now it is desired to find the coefficients for B(z) so that this integral is minimized. Clearly, this is an LMS-type minimization problem that can be approached analytically. For convenience, let us denote Qb)= lG(eJW)l2. (5) This quantity is the normalized power spectrum of the error at the filter output. Proceeding as in [25] the integral (4) can be elaborated into the following form: N r. PkP! k=l /=I (9) where superscript T denotes transposition and the matrices and vectors are w = (PI P2 * . PNlT (10a) The matrix R is recognized as the N X N autocorrelation matrix of the output error, which is known to have a symmetric Toeplitz structure [23]. The vector p is the crosscorrelation vector between the input and output error. The optimal solution is found by setting the derivatives with respect to the EF coefficients to zero, yielding the optimal coefficient vector w * as w* = - R - ' p . or, equivalently [27] N (8) zyxwvu E(z) = 1 zyxwvutsrqp Let the transfer function from the quantization point to the filter output be G(z),excluding the effect of error feedback. In general, G(z) is a recursive transfer function of a linear, time-invariant, causal, and stable system of order usually higher than N . Hence, G(z) is of the form N ( z ) / D ( z ) where N ( z ) and D(z) are polynomials of z - I . Here it is assumed that both G(z) and B(z) have real coefficients. The overall error transfer function is then expressed as I (74 T 3 PII 1 COS (k - l ) w Q ( w ) dw O (6) (1 1) This normal equation can be interpreted as a Wiener-Hopf equation [66], [23], thus demonstrating that optimal error feedback can be interpreted as an application of Wiener filtering theory. More precisely, (1 1) is exactly of the form of the Yule-Walker equation (see, e.g., [28]), which means that B(z) can be viewed as an optimal inverse or whitening filter for a given G(z) in the LMS sense. In other words, the inverse of the optimal B(z) can be interpreted as an optimal AR model for the error power spectrum. The corresponding equations have also been derived in the context of inverse filtering of speech [44] and considering the use of noise shaping with predictive coding of speech [43]. Since the matrix R is guaranteed to be positive definite (see, e.g., [28], [ 2 3 ] ) ,the optimal solution is unique and always exists. Once the autocorrelation coefficients are known, the linear systems of equations ( 1 1) is most efficiently solved using the Levinson-Durbin recursion (see, e.g., [ 2 3 ] )which takes advantage of the symmetric ToeDlitz structure of the autocorrelation matrix. In the present problem, the autocorrelation coefficients zyxw zyxwvutsrqponmlkjihgf zyxwvutsr 1099 LAAKSO AND HARTIMO: NOISE REDUCTION IN RECURSIVE DIGITAL FILTERS zyxwvutsrq qk depend only on the given rational transfer function G(z) and can thus be determined exactly. However, the task is not trivial but causes the main computational burden in determining the optimal EF solution. Note that in order to solve the optimal coefficients of an Nth-order EF quantizer, the calculation of the (N + 1) terms of the autocorrelation function is required. In [25], the autocorrelation coefficients were determined by approximating (7a) by numerical integration, which is computationally rather intensive. Another approach is to use the z-domain version of (7a) which gives the autocorrelation coefficients via the inverse z transform as Several exact algorithms for this integral that operate directly in the z-domain have been proposed in the literature [26], [ 191, [ 181, and [33]. We have found yet another algorithm which is believed to be new. Due to the symmetry of the autocorrelation sequence ( q C k= q k ) ,the integral (7b) can as well be given in the form ments constrain the zeroes of the filter to be (in most cases) exactly on the unit circle, as is well known from the exact linear-phase design of FIR filters [52]. The approach is well motivated, since in practice the error transfer function G(z) often has poles not far from the unit circle, resulting in the optimal (unconstrained) inverse filter B(z) with zeros very close to the unit circle. In [24], the solution for second-order symmetric EF was derived. In [38], some low-order solutions were presented. Essentially the same problem was addressed in [20], where the ideal solution for an arbitrary odd-order (even-length!) linear-phase adaptive filter with symmetric coefficients was derived. Let us denote the odd filter order n = 2M 1 . In our notation, the optimal solution can be expressed as + zyxwvu zyxw where the optimal coefficients of the symmetric odd-order solution are given by the M-length vector ( M = (N 1)/2) wso = (PI P2 ..* OM)' (144 and the involved M x M-dimensional matrices and M-length vectors are defined as BY executing spectral factorization of (zk + zk)/2, i.e., expressing it as a product of the form 40 41 ... 91 40 ... 7zyxw ... . 4M-I the integral can be expressed as a total square integral [31]. The details are presented in the Appendix. Many algorithms have been proposed for the total square integral in the literature [31], [72], 1321, 1461, [54]. @cording to our experiments [34], the algorithm due to Astrom e! al. [72] is the most efficient. The relationship of Astrom's algorithm to the Levinson recursion has been discussed in [ 111. It might also be interesting to optimize the EF coefficients with respect to other criterions, e.g., to make the output noise spectrum as flat as possible in the minimax sense. This is equal to determining B(z) so that IB(e'") G(e'")I2 is minimized in the Chebyshev sense. The solution for B(z) can be obtained by using the standard Parks-McClellan algorithm [53] with slight modifications, as pointed out by Diniz [ 161. However, in this work we concentrate on the minimization in the LMS sense only. 4M- 2 ... 40 4 2 M -2 R, = 4 2 M -3 zyxwvutsrq WITH SYMMETRIC OR 111. ERRORFEEDBACK ANTISYMMETRIC COEFFICIENTS In practice, the hardware or softward implementation of Nth-order optimal error feedback is often too costly due to the N explicit multiplications required. One way to to reduce the number of multiplications is to constrain B(z) to be symmetric or antisymmetric, which halves the number of required multiplications. The symmetry require- qM- 1 The matrix R, has a Hankel structure, i.e., the elements on cross diagonals are equal [45]. Hence, the matrix [R, I?,] to be inverted no longer has a Toeplitz structure. However, it is still guaranteed to be positive definite, since it can be interpreted as the autocorrelation matrix of a joint nonstationary process [20]. Efficient algorithms for the solution of systems of equations with this kind of Toeplitz-plus-Hankel structure have been proposed in [49]. It is easily found that, using the same notation, the corresponding antisymmetric odd-order solution is obtained as + w:, = - [ R , - R,]-'[p, - Po]. (15) Note that in both cases (13) and (15) there are painvise two coefficients of the same value. When the filter order is even, say, N = 2L, (i.e., the filter length is odd) there 1100 zyxwvu zyxwvutsrqponmlkjihg zyxwvutsrqpon zyxwvu zyxwvuts zyxwvutsrqp IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40, NO. 5. MAY 1992 TABLE I SUBOPTIMAL SYMMETRIC A N D ANTISYMMETRIC EF COEFFICIENTS FOR FILTER ORDERS 1 TO 4 is an odd middle value. This slightly complicates the solution which we have found to be w,*+ = - w+1- IP+ (16) where the optimal coefficients of the symmetric or antisymmetric even-order solution are given by the L-length vector Suboptimal Solution N Symmetric B(z) Antisymmetric B(z) zyxwvutsrqpon 1 1 1 2 1 -z-I + z - I + plz-' + z - 2 p --2q, 1 - z - 2 I - 40 where L = N / 2 and 4 I + plz-' + P PI = ~ z -+~P I ~ - 3+ z - ~ 1 + 24142 - %(41 + 4 3 ) %(40 + - 41(% + 43) with 2 - %(% + 42) - - biz-' PI 24: =- - Plz-3 - z - ~ -(41 - 43) 40 - 42 242(40 + 42) 47) - 24: choose a fourth-order purely recursive transfer function which contains the poles of a fourth-order elliptic lowpass filter. It is of the form 1/ D ( z ) , where D(z) Re = 9 2 N -- 2 92L - 3 92L - 3 92L - 4 9L- 1 ... ... ... ... = + 0.801564~-~) (1 - 1.833400~-' + 0.927062C2) (1 - 1.7731522-' * = 1 - 3.6065522-I - 3.1134092 -3 92 where the positive sign gives the symmetric solution and the negative sign gives the antisymmetric solution. Note that with the antisymmetric even-order solution the middle coefficient always vanishes, i.e., pL = 0, so that the ( L - 1)-length solution vector is obtained directly from the reduced equation + 4.979522~-~ + 0.7430992 -4. (20) Assuming signal quantization after the accumulation of products, the noise gain (3) of this filter in direct form implementation without error feedback is 43.51 dB. The noise gain figures when first- to fourth-order optimal, symmetric, and antisymmetric EF's applied, as collected in Table 11. The corresponding EF coefficients are also given. From the data of optimal error feedback, it is observed that increasing the order of the EF polynomial reduces the noise gain to the order 4 when the solution B(z) = D(z) is achieved, as expected, in Section 11. In this case, complete noise cancellation occurs (noise gain equals unity or 0 dB). Naturally, this solution is also obtained when trying to solve a higher order EF from (1 1). This solution can also be interpreted as a double-precision implementation of the filter [24], [50]. However, it is to be noted that when the noise transfer function G(z) is not purely recursive (i.e., with cascaded direct form structures), complete cancellation is no longer possible but the solution only asymptotically approaches the 0 dB level when the EF order is increased. The corresponding error spectra are shown in Fig. 2, which clearly illustrates the zeros induced by the optimal error feedback. Unlike the unconstrained solutions, the symmetric and antisymmetric EF polynomials do not offer monotonic zyxwvut Symmetric and antisymmetric solutions of the order 1 to 4 are collected in Table I. It is observed that the first-order solutions have no free parameters but they set a fixed real zero at z = k 1, thus being suitable for narrow-band, lowpass, or high-pass filters when only moderate noise reduction is required. The second- through fourth-order solutions contain at most 2 free parameters which control the locations of the complex-conjugate zeros, thus offering more efficient noise reduction capabilities. Let us illustrate the use of the proposed optimal and suboptimal symmetric/antisymmetric EF schemes. We zyxwvu zyx I I01 LAAKSO AND HARTIMO: NOISE REDUCTION IN RECURSIVE DIGITAL FILTERS TABLE I1 THENOISEGAINSA N D THE CORRESPONDING OPTIMAL A N D SUBOPTIMAL SYMMETRIC AND ANTISYMMETRIC EF COEFFICIENTS APPLIED TO THE SAMPLE FILTEROF (20) Optimal 1-EF Noise (dB) 30.25 -0.9761 PI zyxwvutsrqpon zyxwvutsrq zyxwvuts zyxw zyxwvuts 2-EF Noise (dB) 15.47 PI - 1.9358 P2 0.9832 3-EF Noise (dB) 3.49 PI - 2.8874 Pz 2.8567 -0.9678 P3 4-EF Noise 0.00 -3.6066 4.9795 -3.1134 0.7431 PI Pz P3 04 Symmetric Antisymmetric 49.48 1 30.30 -1 15.51 36.23 - 1.9522 I 0 -1 21.42 -0.9526 -0.9526 3.56 -2.9 190 2.9190 -1 1 0.60 - 3.8552 '"I !I.y I,> 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Normalired Frequcncy Fig. 3. Error power spectra of the direct form implementation of the sample filter of (20) with zero- to fourth-order suboptimal symmetric error feedback. Solid line: even-order EF. Dotted line: odd-order EF. 8.91 -1.9196 0 1.9196 -1 5.7134 -3.8552 1 0.2 I 60, ! zyxwvutsrqponm zyxwvutsr zyxwv Normalized Frcquency -"0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I Fig. 4. Error power spectra of the direct form implementation of the sample filter of (20) with zero- to fourth-order suboptimal antisymmetric error feedback. Solid line: even-order EF. Dotted line: odd-order EF. Norrnalued Frequency Fig. 2. Error power spectra of the direct form implementation of the sample filter of (20) with zero- to fourth-order optimal error feedback. Solid line: even-order EF. Dotted line: odd-order EF. noise reduction when the order is increased. This is due to the fixed zeros inherent in the solutions: as is known from the linear-phase FIR design, odd-order symmetric/ antisymmetric polynomials always have a zero at z = f 1. Therefore a careful choice has to be made between symmetric and antisymmetric solutions on one hand and between odd and even order on the other hand. This is also illustrated in the corresponding spectra which are shown in Figs. 3 and 4. Otherwise the example shows that, when the best one of the symmetric or antisymmetric solutions is chosen, the noise reduction is at most 0.60 dB worse than with the optimal EF of the same order. The corresponding coefficients are also very close to the same. The zeros of the optimal EF polynomial are seen to be very close to the unit circle even though the pole radii of the filter are not particularly critical (in this case rl = 0.90 and r2 = 0.96). IV. ERRORFEEDBACK WITH POWER-OF-TWO COEFFICIENTS The implementation of error feedback is often the most efficient if explicit multiplications are not needed at all. For example, if the coefficients are quantized to powers of 2 (or to a sum of powers of 2 with only few terms), only additions or subtractions with shift are needed for implementation. This is usually advantageous, e.g., in signal processor applications [35]. In the following we discuss some methods to find these coefficients so that the output error power is minimized. A . Direct Rounding of Optimal Coe#cients First- and second-order E F has been shown to retain most of its power when the coefficients are simply rounded I102 zyxwvutsrqponmlkjihg IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40. NO. 5. MAY 1992 off to the nearest powers of 2 [25]. However, with higher order EF this is not the case. Later it is shown that rounding the optimal coefficients of third- or fourth-order EF directly to the nearest powers of 2 can make it practically useless. Hence, either more bits should be allocated to the coefficients or, preferably, more sophisticated quantization schemes should be considered. TABLE I11 THENOISEGAINSAND THE CORRESPONDING OPTIMAL A N D POWER-OF-TWO COEFFICIENTS OBTAINED BY DIRECTROUNDING O N SEARCH ALGORITHM. SAMPLEFILTEROF (20) zyxwvutsrq zyxwvutsrqponm B. Discrete Optimization Obviously, by direct rounding of the optimal coefficients one cannot even guarantee a satisfactory solution, not to speak about the global minimum-noise solution in the desired power-of-two coefficient grid. A standard tool for this kind of discrete optimization problem is dynamic programming [ 6 ] ,which typically involves constructing a search algorithm where the parameter space is constrained to keep the number of combinations to be tested in reasonable limits. Often the global optimum cannot be guaranteed, but at least a suboptimal solution is obtained in finite time. Design of FIR and IIR filters with finite-wordlength coefficients has been addressed extensively in the last years (see, e.g., [42], [5], [30]). Instead of constructing sophisticated and fast discrete optimization algorithms, we tried some simple schemes that gave satisfactory results within reasonable time. Rather than how the discrete (sub)optimal solution is found, we want to focus on what kind of performance can be achieved with error feedback having power-of-two coefficients. Let us denote with N-b power-of-two coefficient the case where the coefficient is represented as a sum of N terms each assumes a power-of-two value with sign. When this kind of coding uses the minimum number of terms for the representation of a conventional two’s complement number, it is called canonic signed-digit (CSD) code. According to [57], CSD code is guaranteed when there are no successive powers of 2 in the number presentation. Here we are interested in EF filters of the order 1 to 4. We used simple search algorithms that check some values close to the continuous optimum, one for 1-b quantization and the other for 2-b quantization. The algorithm for 1-b quantization is as follows: the quantized coefficient was allowed to take the following values: @a = Lp*J . A . 2b’ A = -1, 0, 1; 2-EF Noise (dB) PI P* 3-EF Noise (dB) PI Pz P3 Pz P3 04 15.47 -1.9358 0.9832 l b 19.38 -2 1 2b 15.47 -1.9375 0.9844 3b 15.47 -1.9375 0.9844 l b 19.38 -2 I 2b 16.37 -1.875 0.9375 3.49 -2.8874 2.8567 -0.9678 l b 29.58 -2 2 -I 2b 15.92 -3 3 -0.9688 3b 13.40 -2.8750 2.8750 -0.9688 Ib 19.38 -2 1 0 2b 9.68 -3 3 -1 0.00 -3.6066 4.9795 -3.1134 0.7431 l b 51.32 -4 4 -4 0.5 2b 31.17 -3.5 5 -3 0.75 3b 9.48 -3.6250 4.9844 -3.1250 0.7500 l b 15.44 -2 0 2 -1 (I*) 2b 3.48 -4 6 -4 zyxwvutsrqp 4-EF Noise PI Search Algorithm Direct Rounding Optimal 1 (2*) the next bit. By always quantizing the residue, a canonic code is guaranteed. The algorithm for 2-b quantization is as follows: pp = Lp*J . 2” . (1 b2 -- -2 * * * - 4; +A * A -1, 0, 1. = 2”) b, = 0, 1; (22) Similar algorithms are easily constructed for quantization to 3 or more bits. With a large number of bits it may be sufficient to use a search routine only for some lower digits. The algorithms were used to design power-of-two EF of the order 2 to 4 for the sample filter (20). The results are collected in Table 111. It is observed that direct rounding works quite well with second-order EF, but no longer with third- or fourth-order EF. When the parameters of fourth-order EF are rounded off to the nearest powers of 2, the resulting noise gain is seen to be even higher than without any error feedback! With second-order EF the results of the search algorithm are essentially the same as with direct rounding (the slightly worse performance is due to the finite range of powers-of-two used), whereas with third- and fourth-order EF the search algorithms gave dramatically improved results. Although the best third-order 1-b EF is actually of the second-order, the search algorithm was able to find it. Most impressive are the results with fourth-order EF, where the optima found with the 1- and 2-b search algorithms were 35 and 28 dB less noisy than those obtained by direct rounding, respectively. Interestingly, the polynomials (1*) and (2*) found by the search algorithms are (1 z - I ) ( 1 - z and (1 z respectively.With 1-b rounding there seems to be quite a limited number of possible polynomials. This implies that instead of searching for a solution based on the continuous optimum (1 l ) , it may be profitable to use EF zyxwvutsrqpo b , = -1, 0, 1, 2 zyxwvuts (21) where the floor operation denotes magnitude truncation down to the nearest power of 2. Thus, 4 values in the power-of-two grid around an optimal coefficient p* (obtained from (1 1)) are checked. Additionally, the integer factor A is introduced to include the value 0 in the set and also the values with the opposite sign. The algorithm is used separately for each coefficient to be optimized and all the possible combinations are examined. If it is desired to use two or more bits for all or some of the coefficients, one can apply a similar algorithm for + zyx zyxwvutsrqp zyxwvuts zyxwvutsrqp zyxwvuts zyxwvu zyxw zyxwvutsrqp I103 LAAKSO AND HARTIMO: NOISE REDUCTION IN RECURSIVE DIGITAL FILTERS TABLE IV THETESTSPECIFICATIONS polynomials with coefficients constrained to integer values, like cyclotomic polynomials. This approach has been investigated in [ 171. V. DESIGNEXAMPLES For the sake of comparison, we consider the implementation of the following recursive digital filters: LP10. A tenth-order narrow-band Chebyshev lowpass filter 1271. LP7. A seventh-order broad-band elliptic low-pass filter 1361. BP6. A sixth-order multiband bandpass filter 1351, 1361. BS6. A sixth-order Butterworth bandstop filter [ 151. The filters’ specifications are given in Table IV. The use of fourth-order direct form sections has been studied in 1361, where it was found that second-order optimal EF is very efficient with this structure, too. Third and fourth-order EF are naturally likely to be even better. The fourth-order sections were also found to be very efficient to implement with the current generation of signal processors. Unfortunately, the direct form sections of orders higher than two have significantly worse overflow behavior than second-order sections. Mitra has shown that, even with saturation arithmetics, higher order direct form sections can sustain zero-input overflow oscillations 1471. However, it is possible to determine if a given section implemented with saturation is free from overflow oscillations by using Mitra’s criterion 1481. This criterion restricts the possible pole locations rather severely: only one of our test filters, the broad-band low-pass filter, passed Mitra’s test. The filters were implemented according to the following rules: 1) Direct form I (DF1) second- or fourth-order sections are used [52]. They are here referred to as 2-cascades and 4-cascades, respectively. 2) L , norm scaling procedure is applied [27]. 3) Signal quantization is performed by rounding after additions. 4) The quantization noise sources are assumed to be uncorrelated so that the standard noise model can be applied [27]. 5 ) Second-order roundoff effects are ignored 1251. 6) All possible section orderings and pole-zero pairings are tested and the ordering with the lowest noise gain (3) is chosen. However, the quantization of the EF coefficients is based on the minimum-noise ordering found with unquantized coefficients. 7) The implementation with no EF, with optimal EF (1 l), and EF with 1-b and 2-b power-of-two coefficients was considered. The power-of-two coefficients were found using the described algorithms (21) and (22). We decided to examine only optimal EF for reference and power-of-two EF for implementation, since accord- Passband Specifications Filter TY Pe A , (dB) U,, (T rad) LPlO LP7 BP6 0.2 I .o 0.8 ca. 0.2 0.5 0.02-0.2 Stopband Specifications A , (dB) 2.0 0.0-0.46 0.54-1 .O rad) 70 60 60 40 BS6 U, (T 60 33.9 0.554 0.0-0.002 0.4-0.8 0.8-1.0 0.49-0.5 1 ing to our experience it is the most suitable way to implement the EF, e.g., with a signal processor [35]. In the current-generation signal processors the coefficient symmetry is difficult to utilize and thus the proposed symmetric and antisymmetric solutions do not offer any real advantage over the optimal solution. With the implementations of the low-pass filter (LP7) Mitra’s test [48] was used to form overflow-stable fourthorder sections. Those 4-cascade implementations are thus guaranteed to be free from zero-input overflow oscillations when saturation arithmetic is used, whereas the other 4-cascade implementations are not. The noise figures for DF1 2-cascade implementations of the test filters are shown in Table V. It is observed that with 2-cascades, increasing the EF order above 2 can no longer reduce the noise much. This comes from the fact that, with a good section-ordering and pole-zero pairing, the noise transfer functions from each quantizer to the filter output are usually very close to second-order all-pole transfer functions that can be compensated for with a second-order all-zero EF. Instead of increasing the EF order it seems to be more rewarding to increase the wordlength of the EF coefficients. With the 4-cascades (Table VI) the high-order schemes are more useful. When two bits are used for the EF coefficients, the 4-cascade implementations with 4-EF are all less noisy than the 2-cascades with 2-EF. With 4-cascades there is only half the number of quantizers of the 2-cascade and thus a more efficient implementation can be obtained (for the implementation with signal processors, see 1361). However, since the 4-cascade implementations are not guaranteed to be free from overflow oscillations (except LP7), a more conservative scaling policy may have to be applied which reduces the dynamic range of the filter. It was noticed that very often the optimal third-order EF with 1-b coefficients was actually of the second-order. Due to the symmetry of the bandstop filter (BS6), the oddorder optimal EF coefficients were zero and hence the second- and third-order EF coefficients were equal. It was also observed that error feedback cannot compensate for bad section-ordering and pole-zero pairing. The implementation with the fourth-order optimal EF and nonoptimal ordering can well be much noisier than the optimal ordering with no EF. Hence, it is strongly recommended that all the orderings be checked if possible. zyxwvutsrqpon zyxwvutsrqponmlkjihgfedcbaZYXWVUT zyxwvutsrqponmlkji zyxw IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40, NO. 5 , MAY 1992 I IC4 TABLE V THENOISEGAINSFOR THE TESTFILTER IMPLEMENTATIONS WITH SECOND-, A N D FOURTH-ORDER EF. CASCADE OF DIRECT FORMI SECONDTHIRD-, ORDER SECTIONS TABLE VI THENOISEGAINSFOR THE TESTFILTER IMPLEMENTATIONS WITH SECOND-, THIRD-,A N D FOURTH-ORDER EF. CASCADE OF DIRECT FORMI FOURTH-ORDER SECTIONS LPlO LPl0 EF Version No EF 2-EF 3-EF 4-EF EF Version Noise Gain (dB) 22.09 I -bit 11.64 9.40 8.66 Optimal 6.58 1.52 0.32 Noise Gain (dB) No EF 2-bit 6.68 2.30 0.65 2-EF 3-EF 4-EF 2-EF 3-EF 4-EF EF Version Noise Gain (dB) No EF 13.15 1-bit 6.85 4.91 4.78 Optimal 6.30 4.29 3.08 Noise Gain (dB) No EF 2-bit 6.33 4.36 3.21 2-EF 3-EF 4-EF 2-EF 3-EF 4-EF 34.21 I-bit 4.73 4.73 3.79 Optimal 2.39 0.50 0.13 EF Version Noise Gain (dB) No EF 2-bit 2.53 0.79 0.29 2-EF 3-EF 4-EF 2-EF 3-EF 4-EF Optimal 6.44 6.44 6.40 2-b 8.30 3.68 0.36 BS6 EF Version Noise Gain (dB) No EF 34.59 1-b 12.60 12.60 10.04 Optimal 8.26 1.95 0.02 BS6 EF Version 2-b 5.31 3.96 2.21 BP6 Noise Gain (dB) No EF 12.83 I -b 5.53 4.58 3.93 Optimal 5.31 3.92 2.10 BP6 EF Version 2-b 13.79 7.49 2.47 LP7 LP7 EF Version 29.30 1-b 20.08 16.57 15.15 Optimal 13.22 5.04 0.76 11.00 1-b 6.85 6.85 6.59 Noise Gain (dB) No EF 2-b 6.45 6.45 6.41 With high-order filters heuristic search algorithms, like the ones proposed in [55] or [25], can be used. However, when a reasonable ordering and a sophisticated quantization scheme is utilized, even a simple second-order EF with 1-b coefficients typically results in large reduction in roundoff noise. Our opinion is that this works with all basic filter types, even with bandstop filters provided that they are noisy enough without the EF. In [25] it was found that the EF was not able to reduce the noise of a bandstop filter. Our feeling is that this was partly because the authors did not check all the possible orderings and partly because they used direct form I1 sections where the recursive part (the poles) of the section is before the nonrecursive part (the zeros). According to [27], the DF1 sections are better than DF2 sections, especially for the implementation of bandstop filters. This was also verified in [36]. In general, it can be stated that the noisier the original implementation, the more the use of the error feedback helps. With only moderately noisy filters, as with BS6 of our examples, the noise reduction obtained by error feedback may be too little to compensate for the implementation costs. 2-EF 3-EF 4-EF Optimal 5.67 5.67 2.92 14.36 I-b 5.78 5.78 3.57 2-b 5.72 5.72 3.05 VI. IMPLEMENTATION ISSUES The DSP application sets the constraints to be met as to the roundoff noise and limit cycle behavior. On the other hand, the application also dictates the implementation resources, so that there is usually a tradeoff between the finite-wordlength performance and implementation costs. When a signal proc&sor is used for implementation, most parameters like data and coefficient wordlengths are fixed and must be taken as such. The most critical parameter is usually the maximum achievable sampling frequency which depends on the length of the program code. Due to its simplicity, the cascade of second-order direct form sections is very efficient to be used in a signal processor in terms of the code length [35]. With narrow-band filters, whose poles are close to the unit circle, the structure tends to be rather noisy. A very cost-effective solution is to apply error feedback to only one or two of the noisiest sections, which may easily reduce the noise power by 20-30 dB at the cost of a few additional instructions in the filter code [35]. The amplitude of possible limit cycles typically goes down with the roundoff noise, too 171, W l . LAAKSO AND HARTIMO: NOISE REDUCTION IN RECURSIVE DIGITAL FILTERS In the current signal processors the EF is not supported but it must be implemented with additional instructions. It would be, though, quite a simple enhancement of the ALU of the signal processor to give the possibility of parallel computation of EF, e.g., with 2-b power-of-two coefficients. In that case no time penalty would be caused by the use of EF. In the case of VLSI implementations, there are far better possibilities to optimize the data wordlength of the different blocks. Then the crucial question is: which is more cost effective, to increase the internal wordlength or to use a shorter wordlength supported by error feedback? Another free parameter is the wordlength of the error feedback network. In a recent work the VLSI implementation of recursive filters with distributed arithmetic and error feedback using the VHDL hardware description language was studied [56] .The preliminary results indicate that the error feedback cannot save any silicon when the wordlengths chosen are such that equal roundoff noise performance is achieved with both implementations. However, since the error feedback introduces parallelism in the computations, it was found to result in a slightly shorter delay than the pure distributed arithmetic implementation. It is believed that this is typical so serial arithmetic implementations where the increase of internal wordlength is cheap and simple. Similar results were also obtained in an earlier implementation study using bit-serial LSI techniques 1591. VII. CONCLUSIONS In this work, general formulas for optimal Nth-order error feedback (error-spectrum shaping) were derived and the relations to Wiener filter theory and AR modeling were discussed. A fast and accurate algorithm to obtain the optimal coefficients using the total square integral formula was presented. As alternatives for efficient implementation, suboptimal schemes with symmetric or antisymmetric coefficients were presented and the design of the error feedback quantizer with power-of-two coefficients was considered, The efficiency of the EF schemes was examined by test implementations of some standard filters. It was found that the error feedback is a very powerful and versatile method to cut down the quantization noise in any recursive filter implemented as a cascade of second-order direct form I sections. Second-order error feedback seems to be sufficient for standard cascade implementations, whereas the new high-order schemes are attractive for use with highorder direct form sections. APPENDIX CALCULATION OF AUTOCORRELATION COEFFICIENTS USINGTHE TOTALSQUAREINTEGRAL The total square integral is defined as [31]: 1 I=-$ F(z)F(z -')z dz 2 .Irj zyxwv zyxwvutsrqp zyxw zyxwvut zyxwv zyx zy I IO5 where the real-valued coefficients of F(z) are given as arguments, and F(z) being the z transform of a stable linear shift-invariant system of the form N biz-' i=O F(z) = 7 C ajz-' (A2) i=O where some of the coefficients may take zero values, except ao.Now it is desired to evaluate the integral (7b) using some algorithm for the total square integral. In order to be able to do this, the spectral factorization of the term ( z - ~ z k ) / 2 is required, i.e., it must be expressed in the form C(z)C(z-') so that the argument for the algorithm is F(z) = C(z) G(z).With some algebra, it is easily derived + ('44) Hence, the desired C(z) has complex coefficients and can be chosen as C(Z>= Cre(z) + jCirn(z) (-45) where 1 (1 + z Cirn(Z) = ;(1 - z - k > . Cre(z)= (A64 -k) (A6b) Now the factor C(z) C(z - I ) can be expressed as C(Z>C(Z- I > = CJz> Cre(z - I > - Cirn(z -') ~jm(z> (~7) so that the integral (A3) can be split into a difference of two integrals as follows: with real-coefficient argument functions Fre(z) = 1(1 + z - k ) G ( z ) (A94 Fi,(Z) = ;(1 - z-%O). (A9b) When G(z) is given, the two integrals are readily calculated by using some of the well-known algorithms, e.g., those presented in 1311 .or [72]. In a recent comparison [34], the algorithm by Astrom er al., 1721 was found to be the fastest among several methods for the evaluation of the total square integral. Several other algorithms for the direct evaluation of autocorrelation sequences in the z-domain have been proposed before, e.g., in 1261, 1191, [18], and 1331. In I331 zyxwvutsrqp zyxwvutsrqpo I106 zyxwv zyxwvutsrqponmlkji IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40. NO. 5, MAY 1992 a method that is similar to that of Astrom’s algorithm was proposed. It is believed that our method compares favorably with the other ones, but a more detailed study would be needed to verify that. intra-/interframe DPCM coding of color television signals,” IEEE Trans. Commun., vol. 36, no. 3 , pp. 332-346, Mar. 1988. [22] R. M. Gray, “Oversampled sigma-delta modulation,” IEEE Trans. Commun.,vol. COM-35, pp. 481-489, May 1987. [23] S . Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: PrenticeHall, 1987. [24] W. E. Higgins and D. C. 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Jing and A. T. Fam, “New scheme for designing IIR filters with finite-wordlength coefficients, IEEE Trans. Acoust., Speech, Signal Processing, vol. A S P - 3 4 , pp, 1335-1336, Oct. 1986. (311 E. I. Jury, Theory and Application of the z-Transform Method. New York: Wiley, 1964. (321 E. I. Jury and S . Gutman, “The inner formulation for the total square integral (SUM),’’ Proc. IEEE, vol. 61, pp. 395-397, Mar. 1973. (331 S . Kay, “Generation of the autocorrelation sequence of an ARMA process,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 733-734, June 1985. (341 J. Koski and T. Laakso, “Efficient numerical evaluation of the total square integral: A survey and a comparison,” in Proc. IASTED In?. Conf. Signal Processing Digital Filtering (Lugano, Switzerland), June 18-21, 1990, pp. 260-262. 1351 T. Laakso, 0. Hyvarinen. and M. Renfors, “Signal processor implementation of some recursive digital filter structures,” in Proc. Inr. Con5 Digital Signal Processing (Florence, Italy), Sept. 1987, pp. 220-224. [36] T. Laakso and I. Hartimo, “Direct form revisited: Recursive filter implementation using higher-order direct form sections,” in Proc. Int. Symp. Circuits Syst. ISCAS’88 (Helsinki. Finland), June 7-9, 1988, pp. 791-795. [37] T. Laakso and I. Hartimo, “Determining the optimal coefficients of high-order error feedback,” in Proc. Int. Symp. Circuits Syst. (ISCAS’89) (Portland, OR), May 9-11, 1989, pp. 728-731. (381 T. Laakso and 1. Hartimo, “Efficient implementation of high-order error feedback,” in Proc. In?. Con$ Circuits Syst. (ICCAS’89) (Nanjing, China), July 6-8, 1989, pp. 375-378. [39] T . Laakso, “Suppression of overflow oscillations in recursive digital filters using error feedback,” in Proc. IASTED In?. Conf: Signal Processing Digiral Filtering (Lugano, Switzerland), June 18-21, 1990, pp. 76-80. [40] T. Laakso, “Elimination of limit cycles in recursive digital filters using, error feedback,” in Proc. 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Laakso (S’87) was born in Vantaa, Finland, on February 1, 1961 He received the Diploma Engineer, Licentiate of Technology and Doctor of Technology degrees from the Helsinki University of Technology in 1987, 1990, and 1991. respectively From 1987 to 1988 he was with the Laboratory of Computer and Information Sciences at the Helsinki University of Technology as a Research Assistant. During 1988-1989, he worked in the Laboratory of Communications at the University of Erlangen-Nuremberg, in Erlangen, Germany, as a Visiting Research Scientist Since 1989 he has been with the Laboratory of Signal Processing and Computer Technology at the Helsinki University of Technology, where he is currently working as a Research Scientist. His current research interests include design and implementation methods of DSP algorithms and modeling of musical instruments Iiro 0. Hartimo (S’68-M’72-SM’83) was born in Helsinki, Finland, on October 22, 1943 He received the Diploma Engineer, Licentiate of Technology, and Doctor of Technology degrees from the Helsinki University of Technology, Helsinki, Finland, in 1969, 1975, and 1986, respectively In 1969 he joined the Helsinki University of Technology From 1969 to 1978 he was with the Department of Electrical Engineering, from 1979 to 1988 he was an Associate Professor of Computer and Information Sciences in the Department of Technical Physics, and since 1988 he has been a Professor of Signal Processing and Computer Science leading the DSP research group His research interests are in the field of implementation methods of digital signal processing