The all-zero system:

The all-zero system If ak = 0 for 1 ≤ k ≤ N, we have H(z) =   . Either take a₀ = 1 or consider that a₀ is absorbed in the br coefficients, such that 

In this case, H(z) has M zeros and an Mth order pole at origin z = 0. As the system contains only trivial poles (at z = 0) and M non-trivial zeros, it is called as all-zero system. This type of a system has a finite-duration impulse response (FIR), and is called as FIR system or a moving average (MA) system. The corresponding deference equation is

a₀ y(n) = b₀ x(n) + b₁ x(n-1) + ... + b_M x(n-M)

The all-pole system Alteratively if b_k = 0 for 1 ≤ k ≤ M, we have

Here again, take a₀ = 1 or imagine that it is absorbed in the other coefficients which are b₀, a₁, a₂, ..., a_N. Thus

Here H(z) has N poles and an Nth order zero at origin z = 0. We generally do not make reference to these trivial zeros. As a result this system function has only non-trivial poles and the corresponding system is called as all-pole system. Because of the presence of the poles, the impulse response of this type of system is infinite in duration, and therefore it is an IIR system. (We can divide numerator into denominator and thereby expand H(z) into an infinite series from which it is evident that h(n) is of infinite duration). Note that corresponding deference equation is given by

a₀ y(n) + a₁ y(n-1) + ... + a_N y(n-N) = b₀ x(n)

The pole-zero system The general form, contains both poles and zeros and the system is called as pole-zero system with N poles and M zeros,
 
Poles and/or zeros at z = 0 and z = ∞ are implied but are not counted explicitly. Due to the presence of poles, the pole-zero system is an IIR system.

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