On Achieving Micro-dB Ripple Polyphase Filters with Binary Scaled Coefficients

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1 Second International Symposium on DSP for Communication Systems, SPRI, South Australia, April 994 On Achieving Micro-dB Ripple Polyphase Filters with Binary Scaled Coefficients Abstract I. Kale, A. Krukowski and N. P. Murphy University of Westminster, School of Electronics and Manufacturing Systems Engineering, 5 New Cavendish Street, London WM 8JS, UK kale@cmsa.westminster.ac.uk tel. (+44) , fax. (+44) This paper reports on the establishment of the coefficient bounds for half-band fixed-point polyphase filters, having all their zeroes on the unit circle. The paper also reports on the algebraic details of the established bounds, exposing pictorially the coefficient interactions and the corresponding polo-zero-patterns (pzp) for the 5 th and 7 th order cases. Potential applications as well as sample filters, designed through the use of these bounds are also reported. Half-band filters having extremely stringently specified frequency domain magnitude response attributes, such as µdb passband ripple, with stopband attenuation in excess of db can be effectively composed of parallel combinations of digital allpass recursive filter constructs in a polyphase environment. Furthermore, the coefficient accuracy requirements for such filters is moderately relaxed, rendering them suitable to economical hardware realisations. However, finding the correct combination of minimum word-length coefficients to meet a given specification in the lowest order is not a trivial task. Design algorithms, techniques and examples for this class of filter, employing an elliptic approximation have been reported in depth []-[3]. However, restricting ourselves to the elliptic approximation will not deliver the flexibility in trading transition-band width for stopband attenuation on a fixed point grid, and the coefficient word-lengths for the elliptic approximation will by their nature be longer. As the elliptic approximation is not effective in these circumstances, a non-analytic approach will provide a feasible design route. Having apriori knowledge of the filter coefficient bounds paves the way to employing almost any optimisation approach within the determined search space ensuring that zeros of the possible filters are always on the unit circle. As the search space is bounded the convergence of the chosen algorithm is also enhanced. Furthermore the unit circle zeroes will ensure maximum null depths being achieved at the zero frequencies (as is the case with the elliptic approximation). This fact is reinforced by observations of pole-zero patterns from many filter design runs. The stopband depth to transition band trade-off is also most effectively achieved when the zeroes are on the unit circle. To have full flexibility the zeroes' frequencies should not be restricted to the mathematical constraints imposed by the elliptic approximation, thus, giving full freedom in meeting given magnitude response requirements, which may not necessarily be optimal in any sense, but complying to a specification acceptability criterion. These effects have been studied and examined in depth in [4]-[6]. A typical application of such filters is in the first decimation stages of a high resolution oversampled Σ Analog-Digital Converter (ADC) [7]-[8]. Where a binary scaled coefficient set of a=.25 and b=.5625, yields a pass-band ripple of approximately.5µdb and 7dB stopband attenuation for a 5th order filter.

2 Second International Symposium on DSP for Communication Systems, SPRI, South Australia, April 994 The two-coefficient fifth-order filter The two coefficient filter is an attractive structure from the design as well as realisation point of view. With the correct starting point the filter can be designed to completely avoid the use of multipliers by finding suitable binary scaled coefficients with few bits. Avoiding non-trivial multiplication opens the way to simple shift and add operation intensive, high-speed and compact custom filter implementations, in comparison to a multiplier with the same bit width. Although the use of this polyphase arrangement lends itself to highly efficient filter structures, there exists a gap between sophisticated design methods []-[3] and realities of implementation [8]-[9]. Addressing the design issues with minimal complexity implementation kept in mind is the main driving thrust of this and the following sections. The transfer function of the two coefficient filter employing the second-order all-pass section, having the structure shown in Figure is given by (). Figure : The half-band filter, (a) The basic second-order all-pass. (b) The two-path halfband lowpass filter. a + z b + z H( z) = z + az + bz () where 'a' and 'b' are the branch coefficients. As the prime objective is on placement of zeroes, only the numerator H n (z) of the transfer function H(z) is considered, b b Hn( z) =. a ( z + ) z + z z b z b a a a (2) To establish the bounded area, two extreme cases are considered. The first of these employs a pair of conjugate zeroes which are moved along an arc of the unit circle from Nyquist to half- Nyquist frequency, with three fixed zeroes at the Nyquist frequency as seen in Figure 2. 2

3 Second International Symposium on DSP for Communication Systems, SPRI, South Australia, April 994 Im z n (3) m Re z Figure 2: PZP of Conjugate Zero Movement from the Nyquist to Half-Nyquist frequency. The numerator of the transfer function for Figure written in terms of the zeroes' Cartesian coordinates is: 3 H ( z) = 5. ( z + ) ( z m jn)( z m + jn) n [ ] =. 5( z + ) z + ( 2 2m) z + ( 2 4m) z + ( 2 2m) z + ( Q m + n = ) (3) where, 'm' and 'n' are the magnitudes of the zeroes' real and imaginary parts respectively. Elimination of 'm', using (2) and (3) results in a closed-form relationship between the branch coefficients 'a' and 'b', as seen in (4). b = 5a + 3 a (4) The second case considered is where a conjugate pair of double zeroes traverse the unit circle in a similar fashion to the previous case. In this instance only one zero is fixed at the Nyquist frequency. Again, considering the numerator of the transfer function for this case, 2 [ ] [ ] H ( z) = 5. ( z + ) ( z m jn)( z m + jn) 2n = 5. ( z + ) z + 4mz + 2( 2m + ) z 4mz + ( Q m + n = ) (5) and comparing coefficients of (2) and (5), yields (6), which is a second relationship between coefficients 'a' and 'b', when 'm' is eliminated, b = 2 + a ( a), a [, ] (6) The trajectories for the two cases above can be seen in Figure 3 and are as confirmed in [5]. Each point (a, b) in the shaded region enclosed by these trajectories guarantees that all the zeroes of () are on the unit circle, within the frequency range half-nyquist to Nyquist. 3

4 Second International Symposium on DSP for Communication Systems, SPRI, South Australia, April 994 Figure 3: The Fifth Order Coefficient Interrelationship Plot, for Zeroes on the Unit Circle Having established the analytical interrelationship between these coefficients, it is used to good effect in providing a good starting point as well as confining the search space for a combinatorial, binary weighted coefficient optimisation algorithm. Figure 4, shows the magnitude, group delay responses and the PZP of a typical filter obtained by, this technique, which is the result of running the algorithm within the bounds defined by (4) and (6). A typical application for such filters is in the first decimation stages of a high resolution oversampled Σ Analog-to-Digital Converter (ADC) [7],[9]. The filter whose response is given in Figure 3 is an example of an ADC decimation filter design having binary scaled coefficients a=.25 and b=.5625, and having passband ripple of approximately ±.25µdB and 7dB stopband attenuation. 4

5 Second International Symposium on DSP for Communication Systems, SPRI, South Australia, April 994 Figure 4: Magnitude response, Group Delay response and PZP of a First Stage Decimation Filter With 4-Bit Coefficients. The Three Coefficient Seventh Order Filter As can be seen from Figure 4, the price of having a high stopband attenuation for a given order (in this case fifth order) is a wide transition-band width. The need to have sharper transitionband widths, as the decimation process nears the desired base-band is obvious. One way of accomplishing this is by increasing the order of the filter. The polyphase two-branch approach results in minimal increase in complexity, as it delivers an order increase of two for each additional coefficient. Although the hardware implementation overhead is kept to a minimum, the search-space for the binary scaled coefficients increases from the simple two-dimensional area of Figure 3, to a search-space whose dimensionally is directly proportional to the number of branch coefficients. For example, in the case of the three-coefficient, seventh-order filter the search space is a three-dimensional volume bounded by the intersections of four surfaces. This is again established in a similar fashion to that of the two-coefficient situation; by manipulating (7) to yield the canonical form of the numerator, as given in (8). In addition the three extreme cases in which all the zeroes are forced to lie on the unit circle are considered. ( a + z )( a + z ) 2 ( + a z )( + a z ) H( z) =. 5 + z ( b + z ) 2 ( + b z ) H ( z ) = b a a z + a a z + b z a + a + a a b a a z n a a b a a z b b z + a a a a z (7) (8) The first case has a conjugate pair of zeroes moving around the unit circle from the Nyquist frequency to the half-nyquist frequency with the remaining five zeroes fixed at the Nyquist frequency. The numerator of the transfer function, expressed in terms of its zeroes' Cartesian locations is: 5

6 Second International Symposium on DSP for Communication Systems, SPRI, South Australia, April 994 [ [ ] 5 H ( z) = 5. ( z + ) ( z m jn)( z m + jn) 4n =. 5 z + (5 2m) z + ( m) z + ( 5 2m) z ( 5 2m) z + ( m) z + (5 2m) z + ] (9) Comparing coefficients of (8) and (9) yields the parametric relationships ()-(2): b a a = 5 2 m + + b = m a a () () (2) + b 5 2m a a + a a = Extensive efforts were made to solve these equations analytically, first manually and then with the aid of the symbolic computation tool Mathematica [], in which an attempt was made to eliminate 'm' and express any one branch coefficient in terms of the other two. This attempt was futile. This difficulty in establishing the three-dimensional trajectory, owing to the interdependency of the branch coefficients and 'm', led to the adoption of a new strategy for solving these non-linear equations. This involved the use of parametric techniques that proved to be of paramount importance in establishing the bounded volume. This approach was undertaken for all the three cases in which either a single, double or triple pair of conjugate zeroes was rotated around the unit circle from the Nyquist frequency to the half-nyquist frequency. Here, the set of three equations ()-(2) were parametrically solved, generating a further set of three new equations whose coefficients b, a and a were expressed in terms of 'm'. Again these solutions were achieved using Mathematica. The complexity of the solutions demanded run-times of the order of hours on an Apple SuperMac, to produce trajectories containing approximately a thousand points. These were plotted for 'm' varying in the range ( + j) z ( + j ). For each value of 'm', the corresponding values of b, a and a were calculated giving a co-ordinate in the three-dimensional coefficient space. Careful study of the trajectories reveals that there is a unique volume encompassed by these which assures placement of the zeroes on the unit circle. To further define this volume, intermediate cases between the primary trajectories were investigated. Finally the three primary and all the generated intermediate trajectories were superimposed and plotted as shown in Figure 5. This reveals the view of the surfaces and hence the volume encompassed by them. The thick bold line at the top represents the special case of the single coefficient third-order filter, and the darkest shaded plane represents the special case for the two-coefficient fifth-order filter (the same as the shaded area of Figure 3). 6

7 Second International Symposium on DSP for Communication Systems, SPRI, South Australia, April 994 Figure 5: Coefficient Interrelationship Plot, for zeroes on the unit circle for the Seventhorder Half-band filter As can be clearly seen from Figure 5, the valid coefficient space for unit-circle zeroes is tightly confined within the overall coefficient space a, a, b. Using these trajectories in conjunction with bicubic-spline interpolation, to approximate the volume's outer surface in more detail, enabled us to again reduce the search space and hence the convergence time for a combinatorial algorithm as was the case with the fifth order filter. A closer scrutiny of Figure 5 which comprises the one, two and three coefficient cases, reveals how the boundaries for higher-order filters (more coefficients) are interlinked and originate from the lower-order ones. The one coefficient case is the line for b = a =. and varying a. The two coefficient case is consequently the plane for a =., when a and b are varying. By inspection the coefficients at the zero delay (z ) and the maximum delay (z -7 ) positions in (8) are equal. This ensures that the product of all the vectors from the origin on the z-plane to each zero position (root position) is equal to unity, implying that the product of all two-path filter zero magnitudes is also unity (3). If a zero pair moves off the unit circle, another pair must follow to the reciprocal locations, as dictated by (3). This situation will arise if coefficients are outside the established boundaries. Each shaded plane on Figure 5 corresponds to a family of filter coefficients having the same transition band, but varying attenuation 7

8 Second International Symposium on DSP for Communication Systems, SPRI, South Australia, April N + z i = i= (3) For any number of coefficients the space of valid coefficients for which zeros are on the unit circle is always bounded by the planes of repetitive zeros on the unit circle. This implies that there exists single and double zero trajectories for the two-coefficient filter (Figure 3); single, double and triple zero trajectories for the three-coefficient filter (Figure 5). Hence, the boundaries for the N coefficient case can be alternatively found from the (N-)th case by determining the trajectories for which a pair of conjugate zeroes start from the half-nyquist frequency and moves along the unit circle down to the Nyquist frequency as illustrated in Figure 2. The starting points of these trajectories should be the convergent points. These points only exist when all the zeroes are at the Nyquist frequency. The volume of the space of coefficients for which zeroes are on the unit circle and to the left of the imaginary axis decreases with increasing number of coefficients. It is 2/3 for one coefficient, approximately /6 for two and goes down to /3 for the case of three, and is expected to fall below /5 for the four coefficient one. As we know coefficients of the polyphase filter come in ascending order, hence + the volume of the search space can be limited to 2 N, to a first approximation. These values show the importance of establishing the bounded search space for the optimisation algorithms in order to find an appropriate set of coefficients without loosing much computational time, in searching through the whole unit space. Remarks The establishment of the valid coefficient space for unit-circle zeroes, speeds up the convergence of the combinatorial algorithm, by providing a starting point local to the possible acceptable solutions and excluding those not meeting the "zeroes on the unit-circle" criterion. The availability of information on the bounds for the search space of the fifth and seventh order filters provides an efficient pathway to establishing effectively implementable, front end decimation filters for oversampled Σ ADCs. Potentially useful practical data may also be obtained if fixed transition bandwidths or stopband attenuations are localised to coefficient sets within the volume, possibly in the form of nomographs or slices through the volume. Furthermore, insight into the three and higher order coefficient cases may be gained if a detailed study of the established volume is carried out. The potential gains in finding the bounds for higher order cases of n-path (not only two-path!) filters are large, but the nature of the problem necessitates larger computer memory and execution time requirements if our current approach is to be employed. This as well as the afore mentioned ideas are currently under scrutiny. 8

9 Second International Symposium on DSP for Communication Systems, SPRI, South Australia, April 994 References. Valenzuela, R. A. and A. G. Constantinides, "Digital Signal Processing Schemes for Efficient Interpolation and Decimation", IEE Proceedings, vol. 3, Part G, no. 6, pp , December, harris, f., M. d'oreye de Lantremange and A. G. Constantinides, "Digital Signal Processing With Efficient Polyphase Recursive All-Pass Filters", International Conference on Signal Processing, (Florence, Italy), September, Lawson, S. and T. Wicks, "Improved Design of Digital Filters Satisfying a Combined Loss and Delay Specification", IEE Proceedings, vol. 4, Part G, no. 3, pp , June, Krukowski A. "Decimation Filter Design for A/D Converters", Project Report, for MSc in DSP Systems, University of Westminster, London, Patel M. V., "Digital Filter Chip Design for Sigma Delta Modulator A/D Converter", Project Report, for BEng. Honours in Electronic Engineering, University of Westminster, London, June Kale I., N. P. Murphy and M. V. Patel, "On Establishing the Bounds for Binary Scaled Coefficients of Fifth and Seventh Order Polyphase Half-Band Filters", accepted for presentation at ISCAS'94, London, May Hejn, K., N. P. Murphy and I. Kale, "Measurement and Enhancement of Multistage Sigma Delta Modulators", Proc. IEEE, IMTC'92, New York, USA, May 2-4, Curtis, T. E. and A. B. Webb, "High Performance Signal Acquisition Systems for Sonar Applications", IEE Conference on A/D and D/A Conversion, Swansea, U.K., September 7-9, Kale I., R. C. S. Morling, A. Krukowski, D. Devine, Architectural Design and Simulation and Silicon Implementation of a Very High Fidelity Decimation Filter for Sigma-Delta Data Converters, accepted for presentation at IMTC/94, Hamamatsu, Japan, April 994. Wolfram S., "Mathematica- A System for Doing Mathematics by Computer", Addison- Wesley, Second Edition, ISBN , 99. 9

DIGITAL SIGNAL PROCESSING UNIT III INFINITE IMPULSE RESPONSE DIGITAL FILTERS. 3.6 Design of Digital Filter using Digital to Digital

DIGITAL SIGNAL PROCESSING UNIT III INFINITE IMPULSE RESPONSE DIGITAL FILTERS Contents: 3.1 Introduction IIR Filters 3.2 Transformation Function Derivation 3.3 Review of Analog IIR Filters 3.3.1 Butterworth