the digital time domain for video frame k where typically u03b4t u2155 seconds to create the recursive form of a filter with this time response it is necessary to take the uni lateral z transform as shown in equation 8 where z is a general complex number f k u03c3 p 1 8 p e u03bd p u03c9 k u03b4t 7 f z u03c3 k 0 u221e f k u00d7 z u2212 k 8 since the z transform is simply 8 infinite geometric summations with common factors e u03bd i u03c9 u03b4t z u2212 1 all lt 1 the z domain impulse response of the bessel filter may be written as equation 9 similar to the laplace domain the filtered result s z in the z domain may be found from the product of the signal s z and the filter impulse response f z this is shown in equation 10 expansion of the representation of f z in equation 9 shows it to be a ratio between a 7 th and an 8 th order polynomial of z as shown in equation 11 advantage is then taken of the property of the z transform shown in equation 12 re arranging equation 11 before performing an inverse z.