Design of linear-phase FIR filters with minimum ... - ResearchGate

where cmωT ( , ) is a proper trigonometric function dependingon if the filter order is odd or even and if the impulseresponse is symmetric or anti-symmetric.Let the specifications of the filter be– δωT ( ) ≤ H R ( ωT) – D( ωT)≤ δωT ( )where D( ωT) is the desired magnitude and δ( ωT)is themaximum allowed ripple for a given angle ωT. Typically,D( ωT)is one in the passband and zero in the stopband ofthe filter.Note that (3), for each angle ωT, describes an inequalitythat is linear in the filter coefficients.When fixed-point coefficient is considered it is oftenuseful to allow a non-unity passband gain [7]. This can bedone by introducing a scaling factor, s, as– δωT ( ) ≤ H R ( ωT) ⁄ s – D( ωT)≤ δωT ( )The scaling factor is always larger than 0, so (4) can berewritten as(3)(4)sDωT [ ( ) – δ( ωT)] ≤ H R ( ωT) ≤ sDωT [ ( ) + δ( ωT)] (5)to obtain a linear inequality.3. PROPOSED DESIGN METHODThe objective is to form an MILP formulation of linearphaseFIR filter design that minimizes the Hamming distancebetween adjacent coefficients. This will be done byfirst introducing a linear expression for the coefficientbits, where the bits corresponds to binary variables, x.Then, a second set of binary variables, y, will be introducedthat corresponds to if a bit position is switched betweentwo coefficients. As we want to minimize thenumber of switches, the optimization problem can be statedas minimizing the sum of all y variables.3.1. Two’s Complement CoefficientsTwo’s complement representation is commonly used inDSP applications. A coefficient, h m , of wordlength B expressedusing two’s complement can be written asB – 1h m = – x m,0 + x m, i 2 – ii = 1where x m, i ∈ { 0,1}. Thus, the coefficients of the FIR filtercan be described using a linear expression with binaryvariables.The Hamming distance between two coefficients isequal to the number of bits that are different. Introduce abinary variable, y m, i ∈ { 0,1}, as∑⎧1,x mi , ≠ x m + 1,iy m, i = ⎨0 , x x , m = 1 , …,M – 1⎩ mi , = m + 1 , i(6)(7)i.e., if bit i changes value between h m and h m + 1 , y m,i is oneotherwise it is zero. This can be expressed using linear expressionsas(8)Only one half of the impulse response needs to be considereddue to the symmetry of the magnitude of the coefficients.3.2. Signed Magnitude RepresentationAn alternative to two’s complement is signed magnituderepresentation. A B bit signed magnitude coefficient canbe written asHence, the first bit corresponds to the sign, while the remainingbits corresponds to the magnitude. Signed magnituderepresentation has a low switching probabilitywhen the numbers are small and with varying sign. Multiplicationis straight-forward using signed magnitude representation.The magnitudes can be multiplied usingunsigned multiplication, while the sign of the result is determinedwith a simple xor-operation of the sign bits ofthe multiplier and multiplicand. However, addition andsubtraction is more complex. It is possible to use two’scomplement representation of the data and signed magnituderepresentation of the coefficients. The multiplicationis then performed in two’s complement, with the coefficientalways being positive. The sign bit of the coefficientcontrols if the result of the multiplication should be addedor subtracted to the accumulator.Equation (9) can not be used directly in a linear formulation,instead for each coefficient a binary sign variables m is introduced along with B – 1 binary variables, x+m,i ,for positive coefficients and B – 1 binary variables, x–m,i ,for negative coefficients. The coefficient can now be writtenaswith the constraintsandy m, i ≥ x mi , – x m + 1,iy m, i ≥ x m + 1,i – x mi ,⎧m = 1 ,…,M – 1, ⎨⎩ i = 0 , …,B – 1B – 1∑h m = (–1) x m, 0x mi , 2 – ii = 1h m=B – 1( x+ m,i – x–∑mi , )2 – ii = 1B – 1∑ x +m,i ≤ ( B – 1) ( 1 – s m ),1 ≤ m≤Mi = 1(9)(10)(11)

B – 1x–∑ m,i ≤ ( B – 1)s m , 1 ≤ m ≤Mi = 1(12)These constraints make sure that if the coefficient h m ispositive, s m = 0 and x– m,i = 0 for all i, and similarly, ifthe coefficient h m is negative, s m = 1 and x+ m,i = 0 for alli. The new variables are related to the formulation in (9) asx m, 0 = s mx m, i = x+ m,i + x –mi , , i = 1 , …,B – 1(13)The number of switched bits can now be expressed as in(8).3.3. Problem FormulationThe resulting optimization problem can for two’s complementrepresentation be formulated asminimizesubject toM – 1∑m = 1B – 1∑i = 0y m,isDωT [ ( ) – δ( ωT)] ≤ h m cmωT ( , )M∑m = 1(14)h m cmωT ( , ) ≤ sDωT [ ( ) + δ( ωT)]m = 1y m, i ≥ x m, i – x m + 1,iy m, i ≥ x m + 1,i – x m,iB – 1For signed magnitude representation the last row of (14)can be exchanged for (10)–(13).As the constraints are continuous in ωT it is necessaryformulate the constraint for discrete values of ωT. This iseasily done by selecting a number of values in the passbandand the stopband to obtain a grid of values whicheach leads to one constraint. As the specifications arechecked only in these points it is necessary to check the resultingtransfer function with a much finer grid to verifythat a valid coefficient set is obtained. If not, additionalpoints must be added to the constraints.If a prescribed passband gain is required the value of scan be fixed before the optimization starts. Note also thats must be larger than 0 as this otherwise would yield anoptimal solution with all variables equal to 0. In general itis possible to constrain s to be larger than 0.1, say, without∑M∑h m = – x m,0 + x mi , 2 – ii = 1losing optimality in the solution. This is due to the factthat scaling s with 2 is equal to shifting each coefficientone step.4. REDUCING THE NUMBER OF VARIABLESIn the optimization problem in (14) the number of 0/1-variablesis (2M – 1)B and 3MB – M – B for two’s complementand signed magnitude representation, respectively.This number will be high for filters with stringent specificationsand the execution time will therefore be long.Hence, it is of interest to find ways to decrease the numberof variables before the actual optimization starts. We willapply a method similar to that proposed in [3].To obtain the ranges for the coefficients it is possibleto use linear programming. The problem can be stated asfirst a separate maximization of each variable h m , subjectto (3). These values are assigned to the variables h ( ub)m .Then a minimization of each variable h m is performed andthe result is assigned the variables h( lb)m . However, thesevalues are for s = 1, so the possible range of s must befound to obtain the true bounds for h m .For a both a two’s complement and a signed-magnitudenumber with wordlength B, the maximal value forany positive coefficient is bounded approximately by one.A negative coefficient is bounded by minus one. To usethe complete range of the coefficients the absolute valuefor the coefficient with the largest magnitude should be atleast 1/2.The minimal value for the scaling factor is obtainedwhen the maximal absolute value for the upper or lowerbound times the scaling factor is 1/2. Hence(15)The maximal value for the scaling factor is when themaximal value of the coefficients minimum absolutebounds are 1. This is for coefficients that do not changesign. The upper bound is then(16)where the coefficients concerned are those that do notchange sign. This reasoning is only valid when at least onecoefficient have the same sign on its maximum and minimumvalue. However, this is the case for most filters unlessthe design margin is very large.New bounds on the coefficients can now be derived asands ( lb)1= -----------------------------------------------------------------------2max{ max{ h ( ub) m , h( lb)m }}s ( ub)1= -------------------------------------------------------------------max{ min{ h( ub) m , h ( lb)m }}ĥ ( ub)mmin( 1,h( ub )m s ( lb)) h ( ub) ⎧, m ≥ 0= ⎨max( – 1,h ( ub )m s ( ub)),h( ub) ⎩m < 0(17)

ĥ( lb)mmin( 1,h( lb )m s ( ub)) h ( lb) ⎧, m ≥ 0= ⎨max( – 1,h ( lb )m s ( lb)),h ( lb) ⎩m < 0(18)If the passband gain is specified we have s (ub) = s (lb) = s s ,where s s is the specified passband gain. Finally, the followinginequality is obtainedĥ ( lb ) m ≤ h ≤ ĥ ( ub )m m(19)This inequality can either be added to the constraints ofthe optimization or used to preevaluate possible values ofthe variables. Here, (19) is utilized to determine which bitsthat do have a fixed value and remove these from the optimizationproblem. This can be done by searching all possiblevalues for the filter coefficient and keep track ofwhich values the bits assume. All variables that only assignone value can be fixed and thereby removed from theoptimization problem.As will be shown in the example below, the number ofvariables that can be removed depends on the availabledesign margin and the representation used. The number ofvariables that can be removed are in general not dependenton the coefficient wordlength, as typically it is the mostsignificant bits of a coefficient that are fixed. However,the relative savings will be smaller when the wordlengthis increased.5. EXAMPLETo show the usefulness of the proposed idea we will considerone filter design example. The filter under considerationis a lowpass filter with passband edge at 0.3π andstopband edge at 0.5π. The allowed ripple is 0.001 in boththe passband and the stopband. A filter order of 33 is used.Hence, there are 17 coefficients to be designed. Using theRemez exchange algorithm and rounding, 15 coefficientbits are required to satisfy the specification. For two’scomplement representation the sum of the Hamming distancesbetween subsequent coefficients are 118. Using theproposed design method a solution with 13 bits can befound with a total Hamming distance of 79. The originaloptimization problem has 448 variables, while after theproposed preprocessing it has 292 variables. Thus, a significantspeed-up was obtained for the solution. The problemwas solved on an AMD Athlon 1800+ with 1.5 GB ofRAM. The pre-processing took approximately 30 s, whilethe MILP stage took 55 minutes. For signed magnituderepresentation, the rounded solution has a total Hammingdistance of 78. The original optimization problem has 652variables, while after preprocessing 255 variables remain.The optimized solution has a total Hamming distance of60. For this problem the MILP stage took approximately150 minutes. The magnitude responses for the three designedfilters are shown in Fig. 2.|H(e jωT )| [dB]0−10−20−30−40−50−60−70−800 0.2π 0.4π 0.6π 0.8π πωT [rad]Figure 2: Magnitude responses for example filterdesigned using (solid) Remez and rounding,(dashed) proposed approach withtwo’s complement representation, and(dotted) proposed approach with signedmagnitude representation.6. CONCLUSIONSIn this paper a mixed integer linear programming problemfor design of linear-phase FIR filters with minimum Hammingdistance between adjacent coefficients was formulated.It has been shown in earlier work that the Hammingdistance is a good measure of the power consumption fora programmable FIR filter. A preprocessing technique toremove 0/1-variables, and, thus, decrease the solutiontime, was discussed. Both two’s complement and signedmagnitude representation were considered. A design exampleshowed the usefulness of the proposed method.7. REFERENCES[1] T. Saramäki and J. Yli-Kaakinen, “Design of digital filtersand filter banks by optimization: applications,” in Proc. XEuropean Signal Processing Conf., Tampere, Finland,Sept. 4–8, 2000.[2] H. Samuli, “An improved search algorithm for the designof multiplierless FIR filters with powers-of-two coefficients,”IEEE Trans. Circuits Syst., vol. 36, pp. 1044–1047, July 1989.[3] O. Gustafsson, H. Johansson, and L. Wanhammar, “AnMILP approach for the design of linear-phase FIR filterswith minimum number of signed-power-of-two terms”, inProc. European Conf. Circuit Theory Design, Espoo, Finland,Aug. 28–31, 2001.[4] M. Mehendale, S. D. Sherlekar, and G. Venkatesh, “Coefficientoptimization for low power realization of FIR filters,”in Proc. IEEE Workshop VLSI Signal Processing,1995, pp. 352–361.[5] P. K. Merakos et al, “A novel transformation for reductionof switching activity in FIR filters implementation,” inProc. Int. Conf. Digital Signal Processing, 1997, vol. 2,pp. 653–656.[6] ILOG, http://www.ilog.com/, June 2002.[7] Y. C. Lim, “Design of discrete-coefficient-value linearphase FIR filters with optimum normalized peak ripplemagnitude,” IEEE Trans. Circuits Syst., vol. 37, pp.1480–1486, Dec. 1990.