The sine cardinal function, abbreviated sinc function, denoted , is defined as

.

A variation of the sinc function that is usually used in signal processing and used in this work for producing one kind of FIR filters, termed the normalized sinc function, has the form

.

The design of a Sync FIR filter takes place in the time domain. Three parameters are used for this purpose: (1) the filter type (i.e. low pass, band pass, etc.), (2) the cutoff frequency, or frequencies, and (3) the filter’s length. The filter's impulse response, , is then given by the following set of equations:

(1)	Let , where is the cutoff frequency, measured at the ½ amplitude level, and specified in normalized form (a value between 0 and 0.5).

(2)	Let be the length of the filter.

will affect the transition bandwidth, , according to the approximation , where also depends on whether the filter is windowed or not, and if so, on the window type that was used.

(3)	Let , then,

(4)	Low Pass Filter: . If  is odd, we use at

(5)	High Pass Filter: . If is odd, we use at

(6)	Band Pass Filter: . If is odd, we use at

(7)	Band Stop Filter: . If is odd, we use at

Where and are the low and high cutoff frequencies, respectively. Finally, a window function may be applied to the resulting filter coefficients in order to improve the filter's impulse response, since we have truncated the normalized sinc function to a finite length, when in reality it extends from to in representing an ideal filter kernel.