Question

Consider a causal discrete-time LTI system whose input signal x[n] and output signal y[n]

are related by the following difference equation: y[n]−0.25y[n−1]+0.5y[n−2]=x[n−1]

Answer 1

The question deals with a difference equation for a discrete-time LTI system in signal processing, which requires techniques like the z-transform to solve for the output signal or system function.

Step-by-step explanation:

The question involves analyzing a difference equation which describes the relationship between an input signal and an output signal in a discrete-time Linear Time-Invariant (LTI) system. The equation provided is a second-order linear difference equation, which indicates that the system's current output, y[n], depends on the previous outputs y[n-1] and y[n-2], and the previous input x[n-1]. To solve this difference equation, we would typically apply methods such as the z-transform to find the system's transfer function or use techniques like recursion or iteration to compute the output for given initial conditions and input signal.

It's important to treat this question strictly in the context of signal processing or control systems within the field of electrical engineering. The relationship provided does not imply any simplification regarding the magnitude of the input signal x[n]; thus, replacing 0.25 with x in the equation would not be appropriate unless specific assumptions about the signal's behavior or scale are stated.

Answer 2

The given difference equation represents a causal discrete-time LTI (Linear Time-Invariant) system. To solve this difference equation, you can use techniques such as Z-transform or time-domain analysis to find the frequency response, impulse response, or stability of the system.

Step-by-step explanation:

The given difference equation represents a causal discrete-time LTI (Linear Time-Invariant) system. LTI systems are often used to model the behavior of various physical systems and signal processing applications.

In this case, the equation relates the current output value y[n] to the past output values y[n-1] and y[n-2] as well as the past input value x[n-1]. It is a recursive equation that shows how the output values depend on the previous output and input values.

To solve this difference equation, you can use techniques such as Z-transform or time-domain analysis to find the frequency response, impulse response, or stability of the system.