Filters. Massimiliano Laddomada and Marina Mondin. Abstract

Transcription

1 Decimation Schemes for Σ A/D Converters 1 based on Kaiser and Hamming Sharpened Filters Massimiliano Laddomada and Marina Mondin Abstract Cascaded-Integrator-Comb (CIC) filters are efficient anti-aliasing rate-conversion filters widely used for Σ A/D converters. High-order structures, attempting to increase the noise rejection within the folding bands, have the drawback of inserting multiple zeroes in the same positions and to increase the edge-band attenuation. In this paper, a combination of sharpened and CIC filters is proposed, with the goal of increasing the rejection of the Σ quantization noise around the folding bands and reducing the pass-band drop of the designed decimation filters with respect to classic CIC structures. Design criteria, leading to optimized structures, and comparisons are given with respect to both classical and modified CIC filters. Keywords CIC-filters, Decimation filters, Σ A/D Converter. I. INTRODUCTION Σ data converters have become widely used as a valid alternative to conventional data converters operating at the Nyquist frequency, because of their simplicity and robustness against circuit imperfections and component mismatch, This work was partially supported by EuroConcepts., S.r.l. ( by Research Funds of CERCOM (Center for Multimedia Communications), and by MURST (Ministero dell Universitá per la Ricerca Scientifica Tecnologica). The authors are with the Dipartimento di Elettronica, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy. E- mails:laddomada@polito.it, marina.mondin@polito.it

2 2 especially for high-speed applications. Their main advantages with respect to other data converters consists in the fact that they can guarantee a high-resolution of the reconstructed data and can be easily implemented in a singlechip high-speed Very Large Scale Integration (VLSI) solution, a key concept for improving the performance of a communications systems, and for enabling third generation air interfaces. Differently from conventional Analog-to-Digital (A/D) data converters, oversampling Σ converters sample the analog input signal with a frequency f s much larger than the signal Nyquist frequency, they employ integration and feedback in iterative loops to obtain high-resolution conversion and to shape the quantization noise out of the useful signal bandwidth. Details about Σ converters can be found in [1], [2], [3] and the references therein. Widely used in practical applications, a second-order Σ data converter, with a 1-bit quantizer inside the inner loop, shows good performance in terms of quantization noise attenuation, and guarantees a good protection against both circuit instability and imperfections. Throughout the paper, the use of a second-order Σ A/D converter will be considered as reference example in order to demonstrate the performance of the proposed decimation architecture. Generalization to higher Σ modulator orders is straightforward since the behavior of the proposed filter is practically independent from the order of the Σ modulator. It is known that, since practical decimation ratios can assume large values, most decimation filter architectures are designed by cascading more than one decimating stage, each one achieving a smaller decimation factor. Good selectivity is generally accomplished by using a very selective low-pass FIR filter at the end of the decimation chain. A filter with these characteristics is expensive to build at the elevated sampling rates required by typical Σ modulators [3], and in practice the overall decimation factor is split between more than one decimating stage in order to relax the specifications of the FIR filter at the end of the chain [4]. A 2-stage example of this architecture is shown in Figure 1. In this paper, the analysis is focused only on the first decimation stage, but the generalization to more stages is straightforward. In fact, since the proposed decimation filter allows for reduction of the quantization noise, the second decimation stage can be designed with relaxed specifications, by using any classical filter design method available in the literature, or by replicating the proposed architecture. Many papers have addressed the problem of realizing efficient decimation filter architectures (see, for example,

3 3 [5], [6]). In [7], the authors present a power efficient multirate multistage Comb decimation filter for mono-bit and multi-bit Σ A/D converters. Polyphase decomposition in all stages, with high decimation factor in the first stage, is used to reduce the sampling frequency of a non-recursive Comb filter. In [8], a low power fifth-order Comb decimation filter with programmable decimation factors and sampling rates suited to GSM and DECT applications is presented. The proposed architecture is based on a non-recursive Comb filter. Other solutions proposed in the literature try to improve the classical CIC [9] performance by cascading digital FIR filters, also said digital pre-filters, which do not require multiplications or have powers-of-two multipliers. Further excellent descriptions of the subject are contained in [10],[11],[12],[13],[14]. Since the overall architecture proposed in this paper is based on a recursive decimation filter, comparisons are given with respect to classical CIC filters [9] and the recently proposed Modified-Sinc architecture [6]. The paper is organized as follows. Section II discusses the mathematical formulation adopted throughout the paper and introduces the proposed decimation filter. Section III is devoted to the optimization of the proposed structure, with particular attention to low-order Σ modulators. Section IV discusses the implementation problem, suggesting a sample realization of the proposed decimation filter. Comparisons and simulations are shown in Section V, also containing a brief summary of results obtained for high-order modulators. The conclusions are drawn in Section VI. II. DECIMATION FILTER DESIGN In an attempt to design a class of filters with reduced pass-band drop and high stopband attenuation, Kaiser and Hamming [15] proposed an efficient method to sharpen the response of a digital filter by using multiple realizations of a low-complexity (or sample ) filter. The family of sharpened filters H nm (f d ) can be expressed as [15]: H nm (f d ) = H n1 p (f d ) m k=0 (n k)! [1 H p (f d )] k (1) n! k! where H p (f d ) is the sample filter (which will be denoted basic filter). The non-negative integers n and m represent the number of non-zero derivatives of H nm (f d ) at the points where H nm (f d ) = 0 and H nm (f d ) = 1, respectively. Notice that we use for simplicity the notation H(f d ) = H(e j2πf d ) = H(z) z=e j2πf d where f d is the digital (normalized) frequency.

4 4 In this paper, the use of the family of filters defined by Equ. 1 is extended to the case of decimation schemes for Σ A/D converters. The mathematical formulation in this paper is referred to the simplest case n = m = 1, which leads to the filter H 11 (f d ) = Hp(f 2 d ) (3 2 H p (f d )) (2) Before proceeding any further, let us define the mathematical terminology adopted throughout the paper. We assume that an analog input signal with maximum frequency f x is fed to a Σ modulator. The Σ A/D converter samples the input signal with rate f s and oversampling ratio ρ, so that f x fs = 1 2ρ. Assuming as reference the 2-stage architecture of Figure 1, we have ρ = D K (D and K are the decimation factors of the first filter and the second filter, respectively). Given this setup, we define = fx f s as the normalized maximum frequency contained in the input signal. In other words, the digital signal at the input of the first decimation filter has frequency components belonging to the range f d [, ]. The main function of a decimation filter is to accomplish an anti-aliasing function, while reducing the sampling rate of the incoming data stream down to a specified rate, satisfying the desired requirements on both the passband drop and the amount of out-of-band quantization noise that will inevitably fold down into the useful signal bandwidth because of the decimation process. These constraints are extremely important when dealing with the quantization noise produced by a Σ A/D converter because of the noise-shaping induced by the Σ modulator. The performance measures generally considered in the design of a decimation filter are the pass-band drop, that indicates the maximum attenuation of the actual designed filter within the useful signal bandwidth with respect to an ideal low-pass filter, the aliasing Error, which characterizes the amount of quantization noise folding into the useful signal bandwidth because of the decimation process, and the worst-case imaging error, which indicates the minimum attenuation applied by the decimation filter to the quantization noise at the lower edge of the first folding band (i.e. for f d = 1 D ). In order to reduce the implementation complexity, in the proposed design the considered basic filter will be a CIC structure. We recall that the overall filter transfer function of a Nth-order CIC can be expressed as [9]: H CN (f d ) = 1 D N ( ) N sin(πdfd ) e j2πf dn D 1 2 (3) sin(πf d )

5 5 where D is the desired decimation factor, f d = ft is the digital frequency, T = 1/f s is the sampling interval and f is the analog frequency. The transfer function of Equ. 3 is equal to zero at integer multiples of the frequency f k = k 1 D, with k = 1,..., D 2 if D is even and k = 1,..., D 1 2 for D odd. Since a first-order CIC filter introduces a delay of D 1 2 samples, an even-order CIC-filter is required to guarantee a group delay equal to an integer number of samples, and a second-order CIC structure will therefore be used as basic filter, setting H p (f d ) = H CN (f d ) N=2 = H C2 (f d ). The paper is aimed at reducing, as much as possible, the effect of the Σ quantization noise aliasing occurring during the decimation of the digital stream output from the Σ modulator. In order to explain the design procedure, let us analyze the transfer functions of Equ. 2. The term (3 2H p (f d )) has the task of imposing m = 1 null derivatives on H 11 (f d ) where H p (f d ) = 1 (i.e where H 11 (f d ) = 1). In other words, this term is used to flatten the pass-band region of the filter, i.e, to reduce the pass-band drop of the transfer function H 11 (f d ). The term H 2 p(f d ), on the other hand, imposes n = 1 null derivatives on H 11 (f d ) where H p (f d ) = 0 (i.e. where H 11 (f d ) = 0), or, in other words, it is used to flatten the zeroes of H 11 (f d ) and to shape the out-of-band filtering mask. In order to further attenuate the quantization noise, a modified CIC structure H q (f d ) proposed in [6] is employed, whose transfer function is equal to: H q (f d ) = 1 D 2 sin[(πf d α/2)d] sin(πf d α/2) sin[(πf d α/2)d] sin(πf d α/2) e j2πf d(d 1) (4) The zeroes of H q (e j2πf d ) are placed at the frequencies k D ± α 2π, where k is an integer whose maximum value is related to the decimation factor D as explained above, and α can be defined as = q2π with q [0, 1] as it will be shown in Section III. The key concept in modifying the filter architecture of Equ. 2 is to substitute a second-order CIC filter cell H p (f d ), which has zeroes at the frequencies k D, with a modified CIC cell H q(f d ), whose zeroes are better distributed in the folding bands. A modified decimation filter H 11 (f d ) will now be described by focusing the attention on Equ. 2, obtained for

6 6 n = m = 1 1, in the hypothesis of using a second-order CIC structure as basic filter H p (f d ) (as defined in Equ. 3 with N = 2). If the rotated-cic cell H q (f d ) of Equ. 4 is substituted to one of the basic filters H p (f d ) in Equ. 2, the proposed decimation filter can be expressed as: H 11 (f d ) = H p (f d ) H q (f d ) H r (f d ) = (5) = H p (f d ) H q (f d ) (3e 2jπ(D 1)f d 2H p (f d )) Comparing now the two expressions (2) and (5) we can notice that: The proposed filter H 11 (f d ) has the same order as the one shown in Equ. 2. The term (3 2H p (f d )) in Equ. 2 is substituted by the term H r (f d ) in Equ. 5, which is responsible for reducing the pass-band drop at the edge of the useful signal bandwidth. The delay taken into account by the term e j2πf d(d 1) guarantees an equal group delay between the two terms 3e j2πf d(d 1) and 2 H p (f d ), since H p (f d ) has a group delay of (D 1) samples. H p (f d ) introduces second-order zeroes exactly at the frequency points k 1 D, where k is a positive integer assuming the values k = 1,..., D 2 D 1 if D is even and k = 1,..., 2 for D odd. H q (f d ), as shown in Eq. 4, introduces two first-order zeros placed on the left and on the right with respect to the frequency f k = k 1 D, for any k = 1,..., D 2 D 1 if D is even or k = 1,..., 2 for D odd. Note that the proposed decimation filter H 11 (f d ) from Equ. 5 has linear phase since it is obtained by cascading linear phase filters. Its frequency behaviour is depicted in Figure 2, and is compared with a fourth-order CIC filter transfer function, which is a CIC structure imposing the same stop-band attenuation order 2. Note that the frequency response of the proposed filter (Equ. 5) shown in figure 2 (dotted line) has been evaluated for q = 0.85, which will be shown in Section III to be an optimal value, since it minimizes the quantization noise around the folding bands f d k/d for k = 1, 2,... Analyzing the frequency behaviour of the filter H 11 (f d ) some considerations can be drawn. The frequencies of the zeros of H q (f d ) are fundamental for achieving good performance in terms of the Σ 1 The generalization to higher order filters, obtainable by increasing the orders n and m in Equ. 1, is straightforward and will be briefly developed in Section V.C. 2 In the stop-band the filter H 11 (f d ) behaves as 3e 2jπ(D 1)f dh p (f d ) H q (f d ), therefore it has a fourth-order behaviour.

7 7 quantization noise rejection around the folding bands. Furthermore, H 11 (f d ) shows good stop-band attenuation behaviour compared to a classic fourth-order CIC filter within the folding bands (i.e. for f d k/d for k = 1, 2,...), while its attenuation is somewhat smaller than a classic fourth-order CIC filter within the so called don t care bands [3], i.e outside the folding bands. However, it is well known that the behaviour of a decimation filter outside the base-band and the folding bands is not very critical [3]. The pass-band drop of the filter H 11 (f d ) is definitely smaller than in a classic CIC filter with an equal stopband order, as it can be observed from the lower rightmost figure. The pass-band drop values of H 11 (f d ) as a function of the decimation factor D and the maximum frequency contained in the input signal bandwidth are shown in Figures 3-4 for the optimal choice of the zeroes rotation q = 0.85 (see Section III). Typical values of of interest for Σ modulators have been considered. From Figure 4 it is possible to note that for practical values of the decimation factor D and for , the pass-band drop assumes small or very small values. III. OPTIMIZATION OF THE PROPOSED FILTER In this section a procedure to optimally select the parameter α in the filters H q (f d ) used to build the proposed decimation filter H 11 (f d ) are presented. By defining α = q2π, it is possible to say that the rotated-cic filter H q (f d ) of Equ. 4 has zeroes located at frequencies k D ± q. The value of the parameter q determines the position of the zeroes of H q (f d ), and therefore the filter performance in terms of noise rejection within the folding bands [k 1 D ; k 1 D ]. We intend to determine the optimal choice of q given the constraint 0 q 1, which makes the zeroes of H q (f d ) fall inside the normalized maximum frequency contained in the input signal. It will be shown that the optimal choice of q is also independent from. It is possible to optimize the parameter q by evaluating the attenuation undergone by the quantization noise around the folding bands as a function of q and then minimizing the quantization noise power P nq inside the folding bands [k 1 D ; k 1 D ] (having a width equal to 2 around the frequencies f k = k D ) as a function of q. The quantization noise power is equal to [2], [3]: P nq = k k 1 D k 1 D H 11 (f d ) 2 S N (f d )df d (6)

8 8 where k = 1,..., D 2 if D is even and k = 1,..., D 1 2 for D odd, and S N(f d ) is the power spectral density of the quantization noise. In the case of a 2-nd order Σ modulator, S N (f d ) can be expressed as S N (f d ) = E(f d ) [2 sin(πf d )] 2 (7) where E(f d ) = f s is the spectral density of the sampled noise under the hypothesis of representing the quantization noise as a white noise [3], is the quantization level of the quantizer contained in the Σ modulator [2], [3], and f s is the Σ converter sampling rate. Results of this optimization are shown in Figure 5, and they suggest the following considerations: The best denoising effect around the folding bands is achieved when q = This value can be simply deduced by figure 5 together with the gain achieved for various normalized signal frequencies. The optimal values of q do not depend on the decimation factor D and on the normalized maximum frequency of the input signal. This reduces the number of parameters to be chosen during the filter design. The maximum achievable gain is related to the normalized digital frequency. In particular, it assumes higher values as increases. This behaviour can be observed from Figure 6 where the achieved gains are plotted as a function of the digital frequency and the decimation factor D. Even though the results shown in Figure 5 have been evaluated for a second-order Σ modulator, we have verified that they still are valid for higher-order Σ modulators. In fact, the power spectral density of the quantization noise S N (f d ) can be considered constant inside the folding bands [k 1 D ; k 1 D ] on which the minimization of the power noise (see Equ. 6) is performed. This consideration suggests that the proposed filter can also be used for decimating digital signals coming from classic (not Σ ) A/D converters in the case in which the input signal is slightly oversampled with respect to the Nyquist frequency. IV. PRACTICAL IMPLEMENTATION OF THE PROPOSED FILTER In this section, we deal with the practical implementation of the filter H 11 (f d ). First of all, we deduce the z- transform of filter cell H q (f d ) in order to justify its practical implementation as a recursive filter employing shift registers for realizing the two multipliers which arise because of the rotated zeroes. Then, we address the practical

9 9 implementation of the filter H 11 (f d ) by highlighting the similarities between the proposed filter and the classical CIC architecture. The scheme of the proposed decimation architecture can be deduced from Equ. 5. Let us consider the z-domain transfer function of a first-order CIC filter, that is H C1 (z) = 1 D (1 z D ) (1 z 1 ). It is straightforward to demonstrate that a clockwise rotation by an angle α = 2qπ applied to any zeros of the previous equation leads to the equation: H q (z) = 1 D 1 z D e jαd 1 z 1 e jα (8) A counterclockwise rotation by the same angle α to any zeros of the classical first order CIC filter leads to the equation: H q (z) = 1 D 1 z D e jαd 1 z 1 e jα (9) By multiplying the latter two equations, we obtain the following filter: H q (z) = H q (z) H q (z) = 1 D cos(αd)z D z 2D 1 2 cos(α)z 1 z 2 (10) which is a linear phase, second order decimation filter with real coefficients. The frequency response of H q (z) is reported in Equ. 4. An efficient, recursive implementation of H q (z) is shown in Fig. 7. It is straightforward to see that the practical implementation of this filter directly corresponds to the Z-transform in Equation 10, in which the numerator 1 2 cos(αd)z D z 2D corresponds to the comb sections, and cos(α)z 1 z 2 corresponds to the integrators inserted on the left side of the decimator by a factor D. This architecture has the advantage that can be embedded in a classical CIC filter, as proposed in [9], by inserting all the integrator and comb sections, respectively, on the left and on the right side of the decimator by D placed in the middle of the overall filter. The drawback of this filter is the presence of the two multipliers b = 2 cos(α) and c = 2 cos(αd). This is the price to be paid to increase the quantization noise rejection around the folding bands with respect to a classical CIC filter. In order to reduce the complexity of this filter, it is important to express both multipliers as two shift registers which do not require real multiplications. An effective procedure in this respect consists in approximating b and c with the expressions b = 2 cos(α) 2 2 L 1 and c = 2 cos(αd) 2 2 L 2, where L 1 and L 2 are two suitable integers.

10 10 By inverting both relations, we obtain: L 1 = 1 log 2 (1 cos(q2π )) (11) and L 2 = 1 log 2 (1 cos(q2π D)), c (1, 2]. (12) where indicates the ceiling of the number. Equ. 12 is valid for any c (1, 2] because only in this range the multiplier c can be expressed in the form c 2 2 L 2. For c (0, 1], a suitable approximation of this multiplier is c 1 2 L2, while for c [ 1, 0] a suitable approximation is c 1 2 L2. The set of values that c can take on, depends on the specific combinations of the parameters q, and D. A design chart which gives the number of extra bits required to implement both multipliers in the form of shift registers is depicted in Fig. 9. Let us consider now the filter cell H r (f d ) = (3z (D 1) 2H p (z)) in Equ. 5, which is responsible for reducing the pass-band drop as already stated in Section II. We can rewrite H r (f d ) as follows: H r (f d ) = (3z (D 1) 2D (1 z D ) 2 ) 2 (1 z 1 ) 2 = 1D (3D 2 2 z D 2z 1 (1 z D ) 2 ) (1 z 1 ) 2. (13) A suitable realization of this filter cell is depicted in Fig. 8 in which the decimator by D of the overall filter is inserted in the middle of any branch. Note that the delay z D in the lower branch becomes z 1 when the decimator by D is inserted in the middle. The overall decimation filter H 11 (f d ) is designed by separating the comb-sections from the integrator-sections of the constituent filters H p (f d ) (a second-order classical CIC filter) and H q (f d ) (a cascade of two first-order rotate-zeros blocks), similarly to how the CIC filter is implemented in practice. A delay of one sample is introduced in the upper branch of the middle section in order to guarantee an equal group delay between the two branches. The overall decimation filter H 11 (f d ) is depicted in Figure 10. The length of the words in the filter H 11 (f d ) depends on the maximum decimation factor D max, the word size M at the input of the filter, and by the number of bits L 1 and L 2 required to implement the two multipliers b and c, and should be finally determined by simulation. It is however possible to give some general guidelines on the basis of the filter structure. The integrator sections of the filter H 11 (f d ) can be sized according to [9].

11 11 In practice, an additional log 2 D max bits should be allocated at each integrator stage to prevent overflow during the time interval between two low rate samples (i.e. two samples at the output of the decimator by D). As a consequence, data words should be sized so that the truncation noise is below the required noise floor. Because of these considerations, the data at the output of the fourth integrator in Fig 10, just before the decimator, need to be represented on M N log 2 (D max ) L 1 bits, where N = 4 is the order of H 11 (f d ) (in fact, there are four integrators), M is the word size of the digital data coming from Σ modulator (M = 1 in the case of 1-bit quantizer within the modulator loop), and L 1 is the extra number of bits required to represent the multiplier by the coefficient b in the form of a shift-register. The data width decreases at each successive stage, so that the least significant bits are lost due to truncation. Truncations between stages is accomplished as 2 s complement truncation with the exception of the output from the final comb filter section which is rounded and scaled on the left in order to produce the final digital word expressed on R bits, as indicated in Fig 10. The value of R can be obtained as a function of both the order N and the word size M of the quantizer of the Σ modulator. Following the mathematical derivations shown in [3], it is possible to obtain: 2π ρ = 2N 10 6R 2N1 10 3(2 M 1) 2 (2N 1) (14) where ρ is the oversampling factor of the Σ A/D converter. A plot of this expression is shown in Fig.11 for N = 1,..., 5 and M = 1. For details on finite-precision implementation olassic CIC filters we refer the interested reader to the reference [18]. V. COMPARISONS AND SIMULATION RESULTS A. Comparisons with Classical and Modified CIC filters In order to compare the proposed decimation filter with the classical CIC architecture, we have evaluated the drop saving obtained by using the decimation filter H 11 (f d ) and a fourth-order CIC filter whose transfer function is represented in Equ. 3 with N = 4. This choice has been motivated by the fact that in the stop-band H 11 (f d ) behaves as an equivalent fourth-order CIC filter.

12 12 Analytically, the pass-band drop of a N-th order CIC filter can be obtained from Equ. 3: d( ) = H C N ( ) H CN (0) 1 sin( αd 2q = 20 log ) N 10 db D N sin( α. (15) 2q ) N where α = 2πq. Applying the same formula to the proposed filter H 11 (f d ), it is possible to obtain the drop savings between the two filters for any combination of the parameter design: d scic ( ) = H 11( ) H 11 (0) H C 4 ( ) db H C4 (0) = (16) db sin 2 ( α 2 ) αd(1 q) sin( 2q ) sin( αd(1q) ( 2q ) sin 2 ( αd 2 ) 3 2 sin 2 ( αd 2q ) )) sin( α(1 q) 2q ) sin( α(1q) 2q ) D 2 sin 2 ( α 2q ) = 20log 10 ( D 2 sin 2 ( α 2q ) sin 2 ( αd 2q ) q=0.85 The values of the drop saving d scic ( ) are shown in Figure 12 for decimation factors D [4 : 32] and [0.001 : 0.011]. It is possible to observe that the drop saving increases as and D increase. We have also compared the decimation filter H 11 (f d ) with the modified CIC structure proposed in [6], and measured the drop saving d sm ( ). In particular, since H 11 (f d ) has an equivalent fourth order behaviour, as seen before, H 11 (f d ) has been compared with a fourth order modified CIC filter composed by the cascade of a rotated zeroes cell H q (f d ) q=0.85 with a second order classical CIC filter H p (f d ) 3. If the same value of q is used for both H 11 (f d ) and the fourth order modified CIC filter H m (f d ) [6], the drop saving induced by H 11 (f d ) can be expressed as: d sm ( ) = H 11( ) H 11 (0) H ( m( ) db H m (0) = 20log sin 2 ( αd 2q ) ) db D 2 sin 2 ( α 2q ). (17) q=0.85 The values of the drop saving d sm ( ) are shown in Figure 13 for decimation factors D [4 : 32] and [0.001 : 0.011]. Notice that the values of d sm ( ) tend to be slightly larger than d scic ( ) for the same values of D and. B. Simulation results Several simulations have been conducted in order to verify the behaviour of the proposed filter H 11 (f d ). In this section we show the results of a sample simulation performed using the Matlab package. 3 The best zeroes rotations for H m(f d ) = H p(f d )H q(f d ) have been found with an optimization procedure and correspond to the value q = 0.85.

13 13 We have simulated a sinusoidal input signal with frequency f x = 100 khz sampled at frequency f s = 2ρf x with a third-order Σ A/D converter whose model has been proposed in [17]. In our setup, the oversampling factor is ρ = 128. The normalized signal frequency is = f x fs Simulation results are shown in Fig. 14. The upper leftmost part of Fig. 14 shows the Power Spectral Density (PSD) at the output of the Σ A/D converter. The oversampled signal has been decimated by a factor D = 16 with the proposed filter H 11 (z) having the optimal zeroes positions previously determined (i.e. q = 0.85), and, for comparison purposes, with a classical fourth-order CIC filter whose frequency response is indicated in Equ. 3 with N = 4. The PSD of the signal decimated with the CIC filter is shown in the upper right part of Fig. 14. Notice that, as expected, the signal decimated of a factor D = 16 has bandwidth fc I 1 = D 2 ρ = On the lower leftmost side of Fig. 14 the power spectral density of the signal decimated with the proposed filter H 11 (z) is reported. Finally, the lower rightmost part of Fig. 14 shows the time domain signals obtained by decimating the oversampled sinusoid of a factor D = 16 (therefore still oversampled of a factor K = rho/d = 8) with both decimation filters. As we can observe, the output of H 11 (f d ) is less attenuated with respect to the output of H C4 (f d ) because of the lower pass-band drop of the proposed structure. C. Generalization to Higher Order Σ Modulators The validity of the proposed approach has been verified also for high order modulators, i.e. for decimation structures H ii (f d ) with i > 1. The results obtained for i = 2, 3 are briefly summarized, but the approach is valid also for higher orders. By evaluating Equ. 1 for n = m = 2 and n = m = 3 it is possible to obtain: H 22 (f d ) = H 3 p(f d )(10 15H p (f d ) 6H 2 p(f d )) (18) and H 33 (f d ) = H 4 p(f d )(35 84H p (f d ) 70H 2 p(f d ) 20H 3 p(f d )). (19) that lead to the following modified decimation filters: H 22 (f d ) = H p (f d )H q1 (f d )H q2 (f d )(10 15H p (f d ) 6H 2 p(f d )) (20)

14 14 and H 33 (f d ) = H p (f d )H q1 (f d )H q2 (f d )H q3 (f d )(35 84H p (f d ) 70H 2 p(f d ) 20H 3 p(f d )). (21) According to the approach described in Section III, the optimal q i factors can be determined, obtaining the following results: q 1 = 0.6, q 2 = 0.9 for H 22 (f d ). The extra gain obtained with these configurations ranges between 17 db (for D = 4, = 0.001) and 21.9 db (for D = 32, = ). q 1 = 0.5, q 2 = 0.8, q 3 = 0.9 for H 33 (f d ). The extra gain obtained with these configurations ranges between 24.8 db (for D = 4, = 0.001) and 32.9 db (for D = 32, = ). These results show that better results in terms of extra quantization noise rejection can be achieved for higher order modulators. Obviously, the pass-band drop of the modified decimation filters H ii (f d ) decreases as i increases (at the price of increased complexity), since higher order null derivatives are imposed by Equ. 1. VI. CONCLUSIONS This paper is focused on the design of a novel decimation filter, especially suitable for Σ A/D converters, which shows better performances in terms of both pass-band drop and quantization noise rejection with respect to conventional CIC decimation filters. In particular, we proposed a combination of the sharpened filters and the modified CIC filters with the goal of increasing the rejection of the Σ quantization noise around the folding bands and reducing the pass-band drop of the designed decimation filters with respect to CIC structures. Furthermore, we discussed design criteria leading to optimized filters, compared novel and CIC structures, and described the proposed method through a design example. VII. ACKNOWLEDGEMENT We wish to thank Dr. John S. Thompson, and the anonymous reviewers for many useful suggestions that have improved the quality of our paper.

15 15 REFERENCES [1] R. M. Gray, Oversampled Sigma-Delta Modulation, IEEE Transactions on Communications, Vol. COM-35, pp , No. 5, May [2] P. W. Wong, R. M. Gray, Two-Stage Sigma-Delta Modulation, IEEE Transactions on Acoustic, speech and Signal Processing, Vol. 38, pp , No. 11, November [3] S. R. Norsworthy, R. Schreier, G. C. Temes, Delta-Sigma Data Converters, Theory, Design, and Simulation, IEEE Press, [4] R. E. Crochiere, L. R. Rabiner, Multirate Digital Signal Processing, Prentice-Hall PTR, [5] A. Y. Kwentus, Z. Jiang, A. N. Willson, Application of Filter Sharpening to Cascaded Integrator-Comb Decimation Filters, IEEE Transactions on Signal Processing, Vol. 45, pp , No. 2, February [6] L. Lo Presti, Efficient Modified-Sinc Filters for Sigma-Delta A/D Converters, IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing, Vol. 47, pp , No. 11, November [7] H. Aboushady, Y. Dumonteix, M. Louërat, H. Mehrez, Efficient Polyphase Decomposition of Comb Decimation Filters in Σ Analog-todigital Converters, IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing, Vol. 48, pp , No. 10, October [8] Y. Gao, J. Tenhunen, H. Tenhunen, A Fifth-Order Comb Decimation Filter for Multi-Standard Transceiver Applications, In Proceedings of ISCAS 2000, IEEE International Symposium on Circuits and Systems, May 28-31, 2000, Geneva, Switzerland, pp. III-89-III-92. [9] E. B. Hogenauer, An Economical Class of Digital Filters for Decimation and Interpolation, IEEE Transactions on Acoustic, Speech and Signal Processing, Vol. ASSP-29, pp , No. 2, April [10] H. J. Oh, K. Sunbin, C. Ginkyu, On the Use of interpolated Second-Order Polynomials for Efficient Filter Design in Programmable Downconversion, IEEE Journal on selected areas in Communications, Vol. 17, pp , No. 4, April [11] H. Samueli, An Improved Search Algorithm for the Design of Multiplierless FIR Filters with Powers-of-Two Coefficients, IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing, Vol. 36, pp , No. 7, July [12] Q. Zhao, Y. Tadokoro, A Simple Design of FIR Filters with Powers-of-Two Coefficients, IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing, Vol. 35, pp , No. 5, May [13] R. J. Hartnett, G. F. Boudreaux-Bartels, On the Use of Cyclotomic Polynomial Prefilters for Efficient FIR Filter Design, IEEE Transactions on Signal Processing, Vol. 41, pp , No. 5, May [14] H. J. Oh, Y. H. Lee, Design of Discrete Coefficient FIR and IIR Digital Filters with Prefilter-Equalizer Structure Using Linear Programming, IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing, Vol. 47, pp , No. 6, June [15] J. Kaiser, R. Hamming, Sharpening the Response of a Symmetric Nonrecursive Filter by Multiple Use of the Same Filter, IEEE Transactions on Acoustic, Speech and Signal Processing, Vol. ASSP-25, pp , October [16] J. A. Wepman, Analog-To-Digital Converters and Their Applications in Radio Receivers, IEEE Communications Magazine, Volume. 33, pp , No. 5, May [17] T. Ritoniemi, T. Karema, H. Tenhunen, Design of Stable High Order 1-bit Sigma-Delta Modulators, In Proceedings of IEEE International Symposium on Circuits and Systems, 1-3 May 1990, vol. 4, pp

16 16 [18] A. Kwentus, O. Lee, A. N. Willson, Jr., A 250 Msample/sec Programmable Cascaded Integrator-Comb Decimation Filter, In Proceedings of Workshop on VLSI Signal Processing, IX, 1996, 30 Oct-1 Nov, pp

17 17 LIST OF FIGURES 1 General architecture of a decimation chain Transfer Function (TF) of the filter H 11 (f d ) (dotted line) compared with a classical 4th-order CIC filter (dashed line) in different frequency intervals. H 11 (f d ) has been designed with α = q 2π, where q = 0.85 and = The decimation factor D is equal to 8. In particular, the lower leftmost curve represents the frequency behavior around f d = 1 D = 1 8 where the first zeroes are located. The positions of the rotated zeros are evident in this subplot, while they are not visible in the upper leftmost subplot because of the wider frequency range Pass-band drops of the filter H 11 (f d ) as a function of the decimation factor D. The curves have been parameterized with respect to the normalized maximum frequency of the Σ -modulated signal and have been obtained for q = 0.85, the optimal choice of the rotated zeros inside the bandwidth [0, ] as discussed in Section III Pass-band drops of the filter H 11 (f d ) as a function of the decimation factor D. Curves have been parameterized with respect to the normalized frequency of the Σ -modulated signal and have been obtained for q = Gain (in terms of extra attenuation of P qn ) achieved by rotating the two couples of zeroes in the transfer function H 11 (f d ) of an angle α = q 1 2π with respect to the case α = q 1 = 0, for 0 q 1 1, for = (in all subplots) and decimation factors D = 4, 8, 16, 32. In particular, D = 4 in the upper leftmost subplot, D = 8 in the upper rightmost subplot, D = 16 in the lower leftmost subplot and D = 32 in the lower rightmost subplot Gain (in terms of extra attenuation of P qn ) achieved for q = 0.85 as a function of the normalized maximum frequency in the signal bandwidth and for decimation factors D = 4, 8, 16, Recursive Implementation of a second-order generalized CIC filter H q (z). In the figure, b = 2 cos(α), c = 2 cos(αd), and α = q2π Recursive Implementation of the filter cell H r (z) in Equ

18 18 9 Number of extra bits required to represent the real coefficients b and c, implementing the corresponding multipliers in the decimation filter H q (z) with shift registers. Upper figure shows the number of bits L 1 required to represent b = 2 cos(q2π ), while the lower subplot represents the number of bits L 2 required to represent c = 2 cos(q2π D), for all the combination of the parameters q, and D such that c (1, 2]. Both figures have been obtained in the optimal case q = Example of practical implementation of the filter H 11 (z), where b = 2 cos(α) and c = 2 cos(αd) Oversampling factor ρ required by a N-th order Σ modulator with one-bit quantizer (M =1) followed by decimation in order to guarantee an output bit resolution R Drop saving d scic ( ) between the filter H 11 (f d ) and a fourth-order CIC filter H C4 (f d ), as a function of the decimation factor D. Curves are parameterized with respect to the normalized maximum frequency of the input signal Drop saving d sm ( ) between the filter H 11 (f d ) and a fourth-order decimation filter H m (f d ) proposed in [6], as a function of the decimation factor D. Curves are parameterized with respect to the normalized maximum frequency of the input signal Results obtained by simulating a sinusoidal input signal, oversampled by a third-order Σ A/D converter of a factor 128 and then decimated by D = 16 with the proposed filter H 11 (f d ) and with a classical fourth-order CIC filter H C4 (f d )

19 19 x(t) A/D Converter Σ Decimation Filter D Decimation Filter K FIR x(n) Overall Decimation Factor =D K Fig. 1. General architecture of a decimation chain.

20 20 0 TF of H 11 (f d ) for 0 f d TF of H 11 (f d ) for 0.1 f d 0.5 Frequency Response db Frequency Response db Digital frequency f d TF of H 11 (f d ) around the first zero 0 Digital frequency f d Pass band behaviour of H 11 (f d ) Frequency Response db Frequency Response db H 11 (f d ) 4th order CIC Digital frequency f d Digital frequency f d Fig. 2. Transfer Function (TF) of the filter H 11 (f d ) (dotted line) compared with a classical 4th-order CIC filter (dashed line) in different frequency intervals. H 11 (f d ) has been designed with α = q 2π, where q = 0.85 and = The decimation factor D is equal to 8. In particular, the lower leftmost curve represents the frequency behavior around f d = 1 D = 1 where the first zeroes are located. The positions 8 of the rotated zeros are evident in this subplot, while they are not visible in the upper leftmost subplot because of the wider frequency range.

21 Pass band drop [db] =0.002 =0.003 =0.004 =0.005 =0.006 =0.007 =0.008 =0.009 =0.01 =0.011 = Decimation factor, D Fig. 3. Pass-band drops of the filter H 11 (f d ) as a function of the decimation factor D. The curves have been parameterized with respect to the normalized maximum frequency of the Σ -modulated signal and have been obtained for q = 0.85, the optimal choice of the rotated zeros inside the bandwidth [0, ] as discussed in Section III.

22 22 0 x Pass band drop [db] = = = = Decimation factor, D Fig. 4. Pass-band drops of the filter H 11 (f d ) as a function of the decimation factor D. Curves have been parameterized with respect to the normalized frequency of the Σ -modulated signal and have been obtained for q = 0.85.

23 23 0 Denoising Gain for D=4 0 Denoising Gain for D=8 2 2 Gain db fc= fc= fc= fc= q 1 q 1 0 Denoising Gain for D=16 Gain db Denoising Gain for D=32 Gain db q 1 Gain db q 1 Fig. 5. Gain (in terms of extra attenuation of P qn ) achieved by rotating the two couples of zeroes in the transfer function H 11 (f d ) of an angle α = q 1 2π with respect to the case α = q 1 = 0, for 0 q 1 1, for = (in all subplots) and decimation factors D = 4, 8, 16, 32. In particular, D = 4 in the upper leftmost subplot, D = 8 in the upper rightmost subplot, D = 16 in the lower leftmost subplot and D = 32 in the lower rightmost subplot.

24 H 11 (f d ) Gain db D= D=8 D=16 D= fc Fig. 6. Gain (in terms of extra attenuation of P qn ) achieved for q = 0.85 as a function of the normalized maximum frequency in the signal bandwidth and for decimation factors D = 4, 8, 16, 32.

25 25 x[i] D 1/D 2 y[k] b c z -1 Fig. 7. Recursive Implementation of a second-order generalized CIC filter H q (z). In the figure, b = 2 cos(α), c = 2 cos(αd), and α = q2π.

26 26 1/D 2 scale D D 3 D 2 scale Fig. 8. Recursive Implementation of the filter cell H r (z) in Equ. 13.

27 27 18 number of bits for b number of bits for c D=4 D=8 D=16 D=32 D= Fig. 9. Number of extra bits required to represent the real coefficients b and c, implementing the corresponding multipliers in the decimation filter H q (z) with shift registers. Upper figure shows the number of bits L 1 required to represent b = 2 cos(q2π ), while the lower subplot represents the number of bits L 2 required to represent c = 2 cos(q2πd), for all the combination of the parameters q, and D such that c (1, 2]. Both figures have been obtained in the optimal case q = 0.85.

28 28 M 1/D 4 scale b z -1 1/D 2 scale D D 3 D 2 scale scale - - scale R bits c Fig. 10. Example of practical implementation of the filter H 11 (z), where b = 2 cos(α) and c = 2 cos(αd).

29 N=1 N=2 N=3 N=4 N=5 70 oversampling factor bit resolution, R Fig. 11. Oversampling factor ρ required by a N-th order Σ modulator with one-bit quantizer (M=1) followed by decimation in order to guarantee an output bit resolution R.

30 30 Drop Saving [db] =0.001 =0.002 =0.003 =0.004 =0.005 =0.006 =0.007 =0.008 =0.009 =0.01 = Decimation factor, D Fig. 12. Drop saving d scic ( ) between the filter H 11 (f d ) and a fourth-order CIC filter H C4 (f d ), as a function of the decimation factor D. Curves are parameterized with respect to the normalized maximum frequency of the input signal.

31 31 Drop Saving [db] =0.001 =0.002 =0.003 =0.004 =0.005 =0.006 =0.007 =0.008 =0.009 =0.01 = Decimation factor, D Fig. 13. Drop saving d sm ( ) between the filter H 11 (f d ) and a fourth-order decimation filter H m (f d ) proposed in [6], as a function of the decimation factor D. Curves are parameterized with respect to the normalized maximum frequency of the input signal.

32 32 50 PSD at the output of the Σ A/D converter PSD after decimation by D=16 with a 4 th order CIC 50 Spectrum [db] Spectrum [db] Digital frequency f d 4 th order CIC Digital frequency f d PSD after decimation by D=16 with the filter H 11 (f d ) Decimated signals in the time domain Spectrum [db] H 11 (f d ) Digital frequency f d Amplitude th order CIC H 11 (f d ) time [s] Fig. 14. Results obtained by simulating a sinusoidal input signal, oversampled by a third-order Σ A/D converter of a factor 128 and then decimated by D = 16 with the proposed filter H 11 (f d ) and with a classical fourth-order CIC filter H C4 (f d ).