The key of realizing convolution filter

which processes discrete signals

is to design expectant frequency transformation of filters. Although the ideal lowpass filters in theory can restrain noise and improve signal-noise ratio of the system

its time function is an infinite long sequence. In practice

this sequence must be truncated to a finite number of terms

and then gibbs phenomenon will arise and lead to the frequency spectrum distortion. This paper discribes an optimal frequency transformation of filter which has minimum gibbs phenomenon after truncated by time window. The truncation error corresponding different time window length is analysed. The ability of restraining noise of the filter is also discussed. Finally

the block diagram of doing digital filter is given out

and the system is verified by experiment.