The total response of any discrete time system can be decomposed as _____. 

Explanation:

Decomposition of Total Response in Discrete Time Systems

Definition: In the context of discrete-time systems, the total response of the system to an input can be decomposed into specific components that help in analyzing and understanding the system's behavior. This decomposition is crucial for simplifying the analysis and design of discrete-time systems, especially when dealing with linear time-invariant (LTI) systems.

Correct Option Analysis:

The correct option is:

Option 2: Total Response = Zero-state response + Zero-input response

This option correctly describes the decomposition of the total response of a discrete-time system. The total response (y[n]) of any discrete-time LTI system can be broken down into two primary components: the zero-state response (y_zs[n]) and the zero-input response (y_zi[n]).

Zero-State Response (y_zs[n])

The zero-state response is the part of the system's response that results from the input signal alone, assuming that the initial conditions of the system are zero. It is determined by the convolution of the input signal (x[n]) with the system's impulse response (h[n]). Mathematically, it can be expressed as:

y_zs[n] = x[n] * h[n]

Where '*' denotes the convolution operation.

Zero-Input Response (y_zi[n])

The zero-input response is the part of the system's response that results from the initial conditions alone, with no external input applied to the system. It captures the natural behavior of the system as it responds to its own initial energy. This response is governed by the system's characteristic equation and can be found by solving the homogeneous difference equation associated with the system.

y_zi[n] = Homogeneous solution of the difference equation

Combining the Responses

The total response of the system is the sum of the zero-state response and the zero-input response:

y[n] = y_zs[n] + y_zi[n]

This decomposition is beneficial because it allows engineers and system designers to analyze the influence of the input signal and the initial conditions separately. By understanding these components, one can gain deeper insights into the system's behavior and design more effective control strategies.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: Total Response = Zero-output response

This option is incorrect because the term "zero-output response" is not a standard term used in the analysis of discrete-time systems. The correct components are the zero-state response and the zero-input response.

Option 3: Total Response = Impulse response + Ramp response

This option is incorrect because the impulse response and ramp response are specific types of responses to particular input signals (an impulse signal and a ramp signal, respectively). They are not components of the total response decomposition in the general sense.

Option 4: Total Response = Impulse response + Step response

This option is incorrect for similar reasons as option 3. The impulse response and step response are specific responses to impulse and step inputs, respectively. They do not represent the general decomposition of the total response of a discrete-time system.

Conclusion:

Understanding the decomposition of the total response of discrete-time systems into the zero-state response and the zero-input response is fundamental for analyzing and designing these systems. This decomposition allows for a clear separation of the effects of the input signal and the initial conditions, facilitating a more comprehensive understanding of the system's behavior. By correctly identifying these components, engineers can better predict and control the performance of discrete-time systems in various applications.