Optimization Of FRM Filters Using The WLS–Chebyshev Approach

Abstract

This paper presents efficient methods for designing linear-phase finite impulse response filters by combining the frequency-response masking (FRM) approach and the weighted least-squares (WLS)-Chebyshev method. We first use the WLS-Chebyshev method to design quasi-equiripple FRM filters, achieving better performances with respect to the passband ripple or the stopband attenuation, when compared with the standard FRM design. Then, by exploiting the concept of critical bands, introduced in this paper, we present a method for designing modified FRM filters with a further reduction in the computational complexity. This is achieved by properly relaxing the specifications for the FRM base and masking filters and using the ability of the WLS-Chebyshev method to trade off the minimum attenuation and the total energy in the filter's stopband. Computational savings are in the order of 10%-15% of the original number of coefficients of the standard FRM design (using the concept of "don't care" bands for the masking filters).

FRM-Based FIR Filters with Minimum Coefficient Sensitivities

“…Let E{x} denotes the expected value of x. Taking the expected values for both sides of (3) and taking note that E{∆H a (e jωM )∆H Ma (e jω )} = E{∆H a (e jωM )∆H Mc (e jω )} = E{∆H Ma (e jω )∆H Mc (e jω )} = 0, we have E{(∆H(e jω )) 2 } = {H Ma (e jω ) − H Mc (e jω )} 2 E{(∆H a (e jωM )) 2 } + (H a (e jωM )) 2 E{(∆H Ma (e jω )) 2 } + {e − jω • −H a (e jωM )} 2 E{(∆H Mc (e jω )) 2 } (4) Depending on the parity of N x , the frequency response H x (e jω ) of a symmetrical impulse response FIR filter with length N x and coefficient values h x (n), n = 0, …, N x − 1 is given by either (5a) or (5b) ( ) ) ( cos( 2 ) ( x e H (9) From (6), (7), (8) and (9), we have ||∆H x (e jω )|| 2 = N x E{(∆h x (n)) 2 } = N x ε 2 (10) Although ∆H x (e jω ) is a function of ω for a given filter, E{(∆H x (e jω )) 2 } for a large number of independent filters is a constant independent of ω if ∆h x (i) has flat spectrum. Thus, E{(∆H x (e jω )) 2 } = N x ε 2 (11) Applying the result of (11), (4) becomes E{(∆H(e jω )) 2 (15) and (16), we have ||∆H(e jω )|| 2 = S 2 ε 2 (17) From (17), and since ε 2 = E{(∆h x (i)) 2 } by definition, it is clear that S 2 is a coefficient sensitivity measure.…”

“…HE frequency response masking (FRM) technique [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] has received much attention for the synthesis of very sharp digital filters with very sparse coefficients. It has found applications in diverse fields including the synthesis of various types of filters such as half-band filters [21]- [23], 2D filters [24], IIR filters [25]- [28], filter banks [29]- [34], decimators and interpolators [35], [36], and Hilbert transformers [37], [38], FPGA implementations [39][40][41], transmultiplexer design [42], ECG signal processing [43], hearing aids [44], digital audio [45]- [49] application and analysis, speech recognition [50], array beamforming [51], software radio [52], and noise thermometer [53].…”

Design Equations for Jointly Optimized Frequency-Response Masking Filters

“…The frequency-response masking (FRM) approach is one of the most computationally efficient techniques for the design of linear-phase arbitrary bandwidth sharp FIR digital filters [1], [3]- [10], [12]- [29], [31]- [38]. A basic FRM filter contains four subfilters, i.e., a pair of complementary bandedge shaping filters H a (z) and H c (z) and two masking filters H Ma (z) and H Mc [20], [21].…”

Hybrid Genetic Algorithm for the Design of Modified Frequency-Response Masking Filters in a Discrete Space

“…It was reported in [1], [8], [12], [15], [20], [22]- [27], [31], and [40] that the frequency-response masking (FRM) technique is one of the most computationally efficient ways to synthesize arbitrary bandwidth sharp linear phase finite impulse response (FIR) digital filters. A great benefit of the FRM approach is a significant reduction in the number of multiplications, which can be as high as 98%, as reported in [26].…”