Steady-state and transient responses for a first order system

Although the presentation is only for a 1st order system, the relationship established for steady-state response in the terms of the transfer function of the system generally result for stable systems and sinusoidal inputs.

The system is

y(n) = a y(n-1) + x(n),             n ≥ 0

with initial condition y(-1) and the input x(n) = cos w0 n u(n). (We have considered time-domain behaviour of this system in Unit I). Suppose |a| < 1in order to have a stable system. The system function can be obtained with zero initial conditions,

The solution of difference equation is obtained by taking z-transform and by using the given initial condition 

The solution of difference equation can be obtained by taking z-transform and by using the given initial condition

 


Here Y₁(z) is zero-input response because of the initial condition(s)

   


and Y₂(z) is forced response because of the input x(n)


 
 Y₁(z) is already in a convenient form for taking inverse, but Y₂(z) should be expanded into

partial fractions as below.

 
 
By taking inverse z-transform we get

 
 


Since |a| < 1 transient term will gradually go to zero as n → ∞. Even if the initial condition is zero, y(-1) = 0, there is a transient response still Aaⁿu(n) which eventually dies down.

  If there exists a nonzero initial condition, y(-1), but input x(n) = 0, the solution becomes

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