Question

The difference equation for an N^th order discrete-time system is ___________

I have been asked this question in a national level competition.

I need to ask this question from Discrete-Time Systems in the Time-Domain topic in chapter Signals and Systems Basics of Signals and Systems

Answer 1

The difference equation for an \( N \)-th order discrete-time system is generally expressed as:

\[

y(n) + a_1 y(n-1) + a_2 y(n-2) + \dots + a_N y(n-N) = b_0 x(n) + b_1 x(n-1) + b_2 x(n-2) + \dots + b_M x(n-M)

\]

where:

- \( y(n) \) is the output signal,

- \( x(n) \) is the input signal,

- \( a_1, a_2, \dots, a_N \) are the coefficients for the output terms (feedback terms),

- \( b_0, b_1, \dots, b_M \) are the coefficients for the input terms (feedforward terms),

- \( N \) represents the order of the system, which corresponds to the highest number of previous output terms \( y(n-k) \) needed.

This equation captures the relationship between current and past values of the input and output, which is key to analyzing the behavior of discrete-time systems.