Frequency Domain Analysis - the math

• phiby
In summary, the conversation is about a math question on Frequency Domain Analysis and the evaluation of the constant a. The person is struggling to understand how the value of a is obtained and asks for help. Another person provides an explanation and mentions that the limits are evaluated to obtain the coefficients.

phiby

I am studying Control Theory from Ogata. My math is a little rusty, so this is a math question about Frequency Domain Analysis.

I get everything upto Equation 8-4

However I don't get the line after that.

The line "where the constant a can be evaluated from Equation (8-2) as follows"

And they write the value of a.

How is
a = G(s) (ωX) /(s^2 + ω^2).

How do they arrive at this value of a?

Can someone help?

And why did they substitute s = -jω after that?

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here is another way

http://img41.imageshack.us/content_round.php?page=done&l=img41/4643/12112011130.jpg [Broken]

i put s = - jw to get value of a, if you put s = jw you will get the value of a bar

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reddvoid said:
here is another way

http://img41.imageshack.us/content_round.php?page=done&l=img41/4643/12112011130.jpg [Broken]

i put s = - jw to get value of a, if you put s = jw you will get the value of a bar

Thank you. That's a little clearer.

You get values of a & abar by evaluating the limit as s = -jω & s = jω

However, I still don't get why you evaluate these limits?

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Its simply a mathematical way of obtaining the coefficients. You can simply factorize the denominator into complex factors and proceed to obtain the inverse laplace transform.

The value of a is derived from Equation (8-2), which is the transfer function of a system in the frequency domain. This transfer function is defined as G(s) = Y(s)/X(s), where Y(s) is the output of the system and X(s) is the input. In this case, X(s) is the input signal ωX and Y(s) is the output signal a(s^2 + ω^2).

To find the value of a, we can substitute s = jω into the transfer function G(s). This is because in the frequency domain, s represents the complex frequency jω. By substituting s = jω, we get G(jω) = a(jω^2 + ω^2). This can be simplified to G(jω) = aω^2 + ajω^2.

Now, we can compare this to the given expression for a = G(s) (ωX) /(s^2 + ω^2). By substituting s = jω, we get a = G(jω) (ωX) /(jω^2 + ω^2). Since G(jω) = aω^2 + ajω^2, we can substitute this into the equation and simplify to get a = G(jω) (ωX) /(jω^2 + ω^2) = (aω^2 + ajω^2) (ωX) /(jω^2 + ω^2) = a (ωX) /(jω^2 + ω^2). This is the same as the given expression, so we can conclude that a = G(jω) (ωX) /(jω^2 + ω^2).

To answer your second question, they substituted s = -jω because this is the negative frequency of s = jω. In control theory, it is common to analyze systems in the frequency domain using complex frequencies s = jω and s = -jω. By substituting s = -jω, we can analyze the system's response at negative frequencies and gain a more complete understanding of its behavior.

1. What is frequency domain analysis?

Frequency domain analysis is a mathematical technique used to analyze signals or data in the frequency domain. It involves converting a signal from the time domain to the frequency domain using mathematical transformations like the Fourier transform.

2. Why is frequency domain analysis important?

Frequency domain analysis is important because it allows us to understand the frequency components of a signal. This can help us identify patterns, trends, and anomalies that may not be apparent in the time domain. It is widely used in fields such as signal processing, image processing, and control systems.

3. What are the main mathematical tools used in frequency domain analysis?

The main mathematical tools used in frequency domain analysis include Fourier series, Fourier transform, Laplace transform, and z-transform. These transformations allow us to convert a signal from the time domain to the frequency domain and vice versa, making it easier to analyze and manipulate the signal.

4. How is frequency domain analysis different from time domain analysis?

The main difference between frequency domain analysis and time domain analysis is that the former focuses on the frequency components of a signal, while the latter focuses on the amplitude and time values of the signal. In frequency domain analysis, a signal is represented by a series of frequencies and their respective magnitudes, while in time domain analysis, a signal is represented by its amplitude as a function of time.

5. What are some practical applications of frequency domain analysis?

Frequency domain analysis has many practical applications in various fields, such as audio and video processing, communication systems, image and signal compression, and control systems. It is also used in medical imaging to analyze and interpret data from MRI and CT scans, and in geological studies to analyze seismic data.