Design of a low-pass filter by multi-scale even order Gaussian derivatives

In this paper, a new low-pass FIR filter design technique for achieving variable stopband attenuation without altering the passband and stopband edges, is proposed. The filter function is a linear combination of multi-scale Gaussian derivatives. In the frequency domain, the spectral modes corresponding to the continuous Gaussian derivatives have monotonic tails. In discrete domain, this amounts to variable stopband attenuation which depends upon the truncation length of the continuous kernel, the ‘scale’ and the order of Gaussian derivatives. The proposed algorithm consists of derivation of some useful mathematical relations between the basic design parameters and the parameters of the Gaussian derivatives and an optimization technique that finally produces an equiripple passband lowpass FIR filter with a higher falloff rate at the beginning of the transition band. Such a filter with variable stopband attenuation may be effective at the time of reconstruction of the desired passband signal in a Σ–Δ modulator output.