Need this derivation to solidify understanding

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In summary, the Fourier Transform of x(t) is given by z(t), where z(t) is a complex variable defined as z(t) = A[cos(w0t) + i sin(w0t)] in the complex plane. This is in relation to the differential equation d2x/dt2 + w02 x = 0, where the solution is x = A ei w0 t. Here, ei w0 t represents a rotating unit-amplitude vector in the complex plane, with i = sqrt(-1).
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HasuChObe
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Can someone please give a concise and complete derivation of the following Fourier Transform:

x(t) = e^(j*w0*t) (w0 is a constant)

Please explain all parts. Thank you very much!
 
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Consider the relation

z(t) = A[cos(w0t) + i sin(w0t)] in the complex plane

Where z(t) is a complex variable z(t) = x(t) + i y(t)

For the differential equation d2x/dt2 + w02 x = 0
the solution is x = A ei w0 t

where ei w0 t as a rotating unit-amplitude vector in the complex plane and i = sqrt(-1):

z(t) = A ei w0 t
 
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  • #3


I am happy to provide a concise and complete derivation of the Fourier Transform for the given function x(t) = e^(j*w0*t), where w0 is a constant.

The Fourier Transform is a mathematical tool used to analyze signals and functions in the frequency domain. It allows us to decompose a function into its individual frequency components.

To begin, we will start with the definition of the Fourier Transform:

X(w) = ∫x(t)e^(-j*w*t)dt

where X(w) represents the frequency spectrum of the function x(t).

Substituting the given function into the above equation, we get:

X(w) = ∫e^(j*w0*t)e^(-j*w*t)dt

Using the properties of exponents, we can simplify the equation to:

X(w) = ∫e^(j*t*(w0-w))dt

Next, we can use the integral property:

∫e^(a*t)dt = (1/a)e^(a*t) + C

where C is a constant.

Applying this property to our equation, we get:

X(w) = (1/(j*(w0-w)))e^(j*t*(w0-w)) + C

Now, we need to evaluate the integral limits. Since we are integrating with respect to t, we need to have limits in terms of t. We can choose any arbitrary limits, but for simplicity, we will choose t = -∞ to t = ∞.

Substituting these limits into our equation, we get:

X(w) = (1/(j*(w0-w)))(e^(j*∞*(w0-w)) - e^(j*(-∞)*(w0-w))) + C

Since e^(j*∞) = cos(∞) + j*sin(∞) is undefined, we can assume that it goes to 0 as t approaches ∞. Similarly, e^(j*(-∞)) = cos(-∞) + j*sin(-∞) is also undefined, but it goes to 0 as t approaches -∞.

Therefore, our equation becomes:

X(w) = (1/(j*(w0-w)))(0 - 0) + C

which simplifies to:

X(w) = C

Since C is a constant, we can rewrite it as a function of w:

X(w) = C(w
 

1. What is a derivation in science?

A derivation in science is a logical sequence of steps that are used to arrive at a conclusion or explain a phenomenon. It is a way of proving or demonstrating a concept or theory using established principles and known facts.

2. How is a derivation different from an experiment?

A derivation is a theoretical approach to understanding a concept, while an experiment is a practical approach to testing a hypothesis or theory. Derivations are typically used in math and physics, while experiments are more commonly used in biology and chemistry.

3. Why is it important to have a solid understanding of a derivation?

Having a solid understanding of a derivation allows scientists to apply the concept to other related problems and make predictions based on established principles. It also helps to ensure accuracy and reliability in scientific research and experiments.

4. How can I improve my understanding of a derivation?

To improve your understanding of a derivation, it is important to practice and apply the concept to different scenarios. You can also seek additional resources such as textbooks, online tutorials, or consult with other experts in the field.

5. Can a derivation be wrong?

While derivations are based on established principles and known facts, they can still be incorrect if there are errors in the logic or assumptions used. It is important for scientists to constantly review and critique derivations to ensure their accuracy and validity.

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