Linear phase FIR filter based on function approximation

Filtering is a key application of signal processing. FIR filter with linear phase has an important role in filtering. Normally, the FIR filtering design involves, taking an ideal low pass filter and finding its impulse response. This impulse response which is infinite in length is made finite by multiplying using suitable window function. Here we propose a method based on the filter response based on the function approximation instead of an ideal filter with step transition in frequency domain. This has the advantage of removing the windowing step from the design process. It also has the advantage of removing the Gibbs' phenomenon. Butterworth squared function is taken as the low pass filter function. Explicit formulae for impulse response are derived based on the theory of complex variable. A low pass filter design example is shown to demonstrate the proposed method.