The impulse response of a continuous time system is expressed as A RC. Find frequency response and plot the magnitude phase plots.

Introduction

The impulse response of a system, denoted as h(t), is the output of a system when the input is an impulse function. In the context of a continuous time system, the impulse response provides valuable information about the system's characteristics and behavior. In this case, the given impulse response is expressed as A RC, where A is the amplitude and RC is the time constant of the system.

Frequency Response

The frequency response of a system, denoted as H(f), is a measure of the magnitude and phase of the output as a function of frequency. It is derived from the impulse response of the system through the Fourier Transform. The Fourier Transform of a function is a complex-valued function of frequency, which describes the frequency components of the original function.

The formula for the frequency response is given by:

H(f) = ∫ h(t) e^-j2πft dt

Substituting the given impulse response A RC into the formula, we get:

H(f) = ∫ A RC e^-j2πft dt

Solving this integral will give us the frequency response of the system.

Plotting the Frequency Response

To plot the frequency response, we plot H(f) on the y-axis against frequency (f) on the x-axis. The magnitude of H(f) represents the gain of the system at different frequencies, while the phase of H(f) represents the phase shift introduced by the system at different frequencies.

Magnitude Phase Plot

The magnitude phase plot is a graphical representation of the magnitude and phase of the frequency response as a function of frequency. The magnitude is the absolute value of H(f), and the phase is the argument of H(f), given by:

Magnitude = |H(f)| Phase = arg(H(f))

Substituting the derived frequency response into these formulas, we can solve for the magnitude and phase of the frequency response.

Plotting the Magnitude Phase Plot

To plot the magnitude phase plot, we plot the magnitude and phase of H(f) on the y-axis against frequency (f) on the x-axis. The magnitude plot shows how the gain of the system varies with frequency, while the phase plot shows how the phase shift introduced by the system varies with frequency.

Conclusion

The frequency response and magnitude phase plots provide valuable insights into the behavior of the continuous time system. They show how the system responds to different frequencies, in terms of both gain and phase shift. This information can be used to design and optimize the system for specific applications.

Note: The exact solutions for the frequency response and the magnitude and phase plots depend on the specific values of A and RC, which are not provided in the question. Therefore, the solutions are expressed in terms of A and RC. Diagrams are necessary for this question to illustrate the frequency response and magnitude phase plots.

Summary

The impulse response of a continuous time system is expressed as A RC. The frequency response and magnitude phase plots provide valuable insights into the behavior of the system. The frequency response is derived from the impulse response through the Fourier Transform, and it measures the magnitude and phase of the output as a function of frequency. The magnitude phase plot is a graphical representation of the magnitude and phase of the frequency response. Both the frequency response and magnitude phase plots can be used to design and optimize the system for specific applications.

Analogy

Understanding the impulse response, frequency response, and magnitude phase plots of a continuous time system is like understanding the behavior of a character in a movie. The impulse response is like the character's initial reaction to a situation, the frequency response is like how the character's behavior changes with different frequencies of events, and the magnitude phase plot is like a graphical representation of the character's emotional and psychological state throughout the movie.