2.2e LTI LCCDE

LTI Linear Constant Coefficient Difference Equation (LCCDE) Representation

A linear constant coefficient difference equation is a mathematical expression that describes the relationship between the input and output of an LTI system. In discrete-time, is defined as the relationship
${\sum }_{k=0}^N{a}_{k}y(n-k)={\sum }_{m=0}^M{b}_{m}x(n-m),$
where $N$ is the order of the equation, and $M$ is the delay of the input terms.

Similarly, in continuous-time,

Properties

(Transfer Function) $H(\omega )=\frac{{\sum }_{m=0}^M{b}_m{e}^{-j\omega m}}{{\sum }_{k=0}^N{a}_k{e}^{-j\omega k}}.$

Example

(Discrete-Time) Finding the Transfer Function Using the LCCDE

In 2.2d, we found that the system with a recursive LCCDE $y(n)=\alpha y(n-N)+x(n)$ has a frequency response of $H(\omega )=\frac{1}{1-\alpha e^{-j\omega N}}$. However, there is a simpler way to find this. The LCCDE for an eigenfunction input $x(n)=e^{j\omega n}$ becomes
$H(\omega )e^{j\omega n}=\alpha H(\omega )e^{j\omega (n-N)}+e^{j\omega n}.$
Dividing both sides by $e^{j\omega n}$ and factoring out $H(\omega )$, we get
$H(\omega )=\frac{1}{1-\alpha e^{-j\omega N}}.$
It’s as simple as that.