First Order and Second Order Discrete Temporal Systems

First Order

$$y[n]-ay[n-1]=x[n]$$

⇒

$$H(\omega )=\frac{1}{1-a\exp (-j\omega )}$$

⇒

$$h[n]=a^{n}u[n],s[n]=\frac{1-a^{n+1}}{1-a}u[n]$$

Unlike the continuous case, if $a<0$ we can detected oscillation in first order discrete systems

Impulse Response: Step Response:

Second Order

$$y[n]-2r\cos \theta y[n-1]+r^{2}y[n-2]=x[n]$$

⇒

$$H(\omega )=\frac{1}{1-2r\cos \theta \exp (-j\omega )+r^2\exp (-2j\omega )}$$

⇒

$$H(\omega )=\frac{A}{1-r\exp (j\theta )\exp (-j\omega )}+\frac{B}{1-r\exp (-j\theta )\exp (-j\omega )}$$

where

$$A=\frac{\exp (j\theta )}{2j\sin \theta },B=-\frac{\exp (-j\theta )}{2j\sin \theta }$$

⇒

$$h[n]=r^n\frac{\sin \left[(n+1)\theta \right]}{\sin \theta }u[n]$$

Properties

• Higher $r$ leads to slower converge (but is better selective in frequency domain)
• $\theta =0$ is the critical damped case (best)
• $\theta >0$ leads to overshoot and oscillation