Design and implementation of finite impulse response(FIR) filters have been an important area in digital signal processing. In this dissertation, we consider the design of 1-D and 2-D FIR filters: specifically, 1-D and 2-D $\{sparse}$ FIR filters are designed and the methods for designing 2-D fan filters and diamond-shaped filters based on newly proposed closed-form formulas for $\{McClellan}$ transformparameters are proposed.
Sparse filters, which have intentionally zeroed tap weights, can lead to either reduction of multipliers or additional stopband suppression at the expense of increased delays. We propose a new search technique for determining optimal zeroed tap weights using the branch-and-bound method. This technique is combined with the quadratic programming or the linear programming and leads to flexible designs under several error criterions such as the least squares, the minimax, and the least squares subject to ripple constraints. For two kinds of sparse filters, which are performance optimized and arithmetic complexity minimized, the use of efficient branch-and-bound methods is proposed. The proposed sparse filter design methods are successfully applied to 2-D FIR filter design, beamformer design, and two-channel perfect-reconstruction linear-phase filter bank design. In 2-D filter and filter bank designs, to which any previous sparse filter designs have not been applied, the designed sparse filters show the substantial gains in the arithmetic complexities as compared with the nonsparse filter designs. For the beamformer design, in spite of its restricted search because of computational burden, the proposed method outperformed any other suboptimal methods.
Finally, the closed-form formulas for the McClellan transforms which are useful for designing 2-D fan filters and diamond-shaped filters are derived. By imposing respectively the constraints $F(0,0)=cosω_c$ and $F(0.6495ω_d,0.3505ω_d)=cosω_c$, where F(ㆍ) is the McClellan transform, $ω_c$ is the cutoff frequency of the 1-D prototype filter, and $ω_d$ is the constant of the diamond-shaped filter, the integral squared error is directly minimized and, as a consequence, closed-form formulas for the McClellan transform parameters are obtained. The closed-form formulas obtained are very simple, and lead to fast filter design. The McClellan transforms obtained by using these formulas exhibit comparable errors as compared with the other transforms designed by exhaustive search.
