State LCCDE and system function

Linear constant-coefficient difference equation (LCCDE)

An LTI discrete-time system of order N is described by

Σ(k=0 to N) aₖ y[n − k] = Σ(k=0 to M) bₖ x[n − k], a₀ = 1

or y[n] = −Σ(k=1 to N) aₖ y[n − k] + Σ(k=0 to M) bₖ x[n − k].

The aₖ and bₖ are constants (independent of n). The first sum is the feedback (recursive) part, the second the feed-forward part. The equation is solved with initial rest conditions for an LTI, causal system.

System function

Take the z-transform of both sides. Since x[n − k] ↔ z⁻ᵏ X(z):

Y(z) [1 + a₁z⁻¹ + … + a_N z⁻ᴺ] = X(z) [b₀ + b₁z⁻¹ + … + b_M z⁻ᴹ]

H(z) = Y(z)/X(z) = (Σ bₖ z⁻ᵏ) / (1 + Σ aₖ z⁻ᵏ)

The roots of the numerator are the zeros, and the roots of the denominator are the poles.

Example: y[n] = 0.5y[n−1] + x[n] + 2x[n−1] gives H(z) = (1 + 2z⁻¹)/(1 − 0.5z⁻¹), pole at z = 0.5, zero at z = −2.