Discrete IIR Filters Versus Discrete FIR Filters

A simple RC (resistor-capacitor) filter without any switches is considered a continuous-time filter because its output is continuously updated according to its input. With a discrete filter, the output is updated only at discrete time instants.

Discrete filters can be divided into two groups:

  • Infinite-impulse-response (IIR) filters
  • Finite-impulse-response (FIR) filters

Discrete IIR filter

A discrete IIR filter is similar to a continuous-time filter because, given an impulse at its input, the response of its output theoretically takes an infinite amount of time to decay back to zero if it decays back to zero at all. For example, a very simple low-pass discrete IIR filter can be described by the following difference equation:

???MATH???O_n = I_{n-1} + 0.5 \times O_{n-1}???MATH???

where n is an integer.

  • If ???MATH???I_0 = 1???MATH??? and ???MATH???I_n = 0???MATH??? for ???MATH???n \neq 0???MATH???, the input is considered an impulse input for a discrete filter.
  • If ???MATH???O_n = 0???MATH??? for ???MATH???n ≤ 0???MATH???, the output response will be ???MATH???O_n = 0.5^{n-1}???MATH??? for $n \geq 1$, which will take an infinite amount of time before the output decays to zero.

Discrete FIR filter

A discrete FIR filter, on the other hand, produces an output that occupies only a finite amount of time from an impulse input. For example, a simple average FIR filter can be described by the following equation:

???MATH???O_n = 0.5 \times ( I_n + I_{n-1} )???MATH???

  • Given the same scenario of ???MATH???I_0 = 1???MATH??? and ???MATH???I_n = 0???MATH??? for ???MATH???n \neq 0???MATH???, this simple average FIR filter produces an output sequence where ???MATH???O_0 = O_1 = 0.5???MATH??? and ???MATH???O_n = 0???MATH??? for ???MATH???n \neq 0,1???MATH???.
  • The response of this filter to the impulse input is a very short sequence, and the output drops back to zero very quickly.

The SIMPLIS first-order and second-order discrete time filters are typically used to model IIR filters. These first-order and second-order discrete time filters can easily be turned into simple FIR filters if the denominator coefficients D0 and D1 are set to zero.