# Mathematically solving fourier transform

• Robismyname
In summary, the conversation is about someone trying to understand the real-world applications of Fourier Transform by working on problems from a signals and systems book. They are specifically trying to find the Fourier Transform of a given signal and are using a step-by-step approach to solve it. However, they are having trouble understanding how the book arrived at the answer, and are advised to use u-substitution to come to the correct answer.
Robismyname
Since I lack the understand of real world applications of Fourier Transform in the real world I decided to buy a signals and systems book (Lathi) do some Fourier Transform problems and them do the same problem in Matlab.

The question in the book wants me to find the Fourier Transform of signal f(t) = e^-at; from 0 to T

I know in order to find FT I have to do the following:
T
step 1: F(w) = ∫ f(t) * e^-jwt dt
0

T
step 2: ∫ e^-(a+jw)t dt [combine like terms]
0

step 3: [ e^-(a+jw)0 ] - [ e^-(a+jw)T ] [integrate over 0 to T]

step 4: 1 - e^-(a+jw)T [solve]

The book says the answer is:

1-e^-(jw+a)T
----------------------------
jw+a

How did the book get the denominator section of jw+a? I can't get from step 4 to the book answer. What am I missing here?

## 1. What is the Fourier transform?

The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies. It converts a signal from its original domain (usually time or space) to a representation in the frequency domain.

## 2. What is the purpose of solving the Fourier transform?

The purpose of solving the Fourier transform is to analyze and understand the frequency components of a signal. It can be used to identify specific frequencies present in a signal, filter out unwanted frequencies, and even reconstruct a signal from its frequency components.

## 3. How is the Fourier transform calculated?

The Fourier transform is calculated using complex numbers and integration. It involves breaking down a signal into an infinite sum of sine and cosine functions with different frequencies and amplitudes.

## 4. What is the difference between the Fourier transform and the inverse Fourier transform?

The Fourier transform converts a signal from the original domain to the frequency domain, while the inverse Fourier transform converts it back from the frequency domain to the original domain. They are essentially inverse operations of each other.

## 5. In what fields is the Fourier transform commonly used?

The Fourier transform is commonly used in fields such as signal processing, image processing, quantum mechanics, and engineering. It has applications in audio and video compression, medical imaging, and solving differential equations, among others.

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