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)