# Fourier Transform question work shown

• johnq2k7
In summary, a Fourier Transform is a mathematical tool used to break down a complex signal into its individual frequency components. It works by representing a signal as a sum of sine and cosine waves of different frequencies, making it easier to analyze and manipulate. Some applications of Fourier Transform include signal and image processing, data compression, and solving differential equations. The main difference between Fourier Transform and Inverse Fourier Transform is that the former converts a signal from the time domain to the frequency domain, while the latter converts it back. Fast Fourier Transform (FFT) is an algorithm used to compute the Fourier Transform efficiently, with a much faster time complexity than the traditional method.
johnq2k7

The "Free Induction Decay signal" (FID) is a particular type of NMR signal observed in both MRI and MRS. An idealized representation of the signal S(t) is given by:

S(t)= S(0)exp (i*w_0*t)exp(-t/T2*), t>=0
S(t)=0 , t<0

You showed the spectrum G(w) corresponding signal is given by:

G(w)= S(0) {((T2* + (w-w_0)(T2*)^2))/ (1+[(w-w_0)T2*]^2}

However, the exact form of the spectrum above is obtained assuming that the signal is acquired for an infinite period of time. In reality the signal is acquired over a finite time that will be denoted here as T. In that case the shape of the spectrum is obtained by assuming that S(t)=0, for t>T and that leads to the following spectrum:

G(w)= S(0){(T2* + i*(w-w_0)(T2*)^2 / (1 + [(w-w_0)T2*]^2} (1- X)

where X is a complex parameter that depends on T,T2*,w, and w_0

a.) Determine the value of the magnitude of X for the case where T=T2*

b.) if T=a*T2*, determin the min. value of the parameter 'a' such that |X|<0.01
(|X| rep. the magnitude of 'X')

Work shown:

For a.)

do ignore the imaginary part for the exact eqn. and sub in T for T2* and solve for 'X'?

If so,

I get X= 1- S(0){(T/ (1 + [(w-w_0)T]^2}

For b.)

if |X|<0.01 and T= aT2* then, T2*= T/a

then i get 1- S(0){((T/a)/ (1 + [(w-w_0)(T/a)]^2} < 0.01

how do i solve for a.

!Solution:a) To determine the value of the magnitude of X for the case where T = T2*, we can simply substitute T = T2* into the equation for X and solve for X. This gives us: X = 1 - S(0){(T2* / (1 + [(w-w_0)T2*]^2)} The magnitude of X is then simply |X| = |1 - S(0){(T2* / (1 + [(w-w_0)T2*]^2)}|.b) To determine the minimum value of the parameter 'a' such that |X| < 0.01, we need to rearrange the equation for X so that it is written in terms of 'a'. We can do this by substituting T2* = T/a into the equation for X and rearranging to give: 1 - S(0){(T/(a^2)) / (1 + [(w-w_0)(T/a)]^2} < 0.01We can now solve for a by rearranging the equation and solving for a: a > (T * sqrt(S(0)) * sqrt(1 - 0.01)) / (sqrt(1 + [(w-w_0)(T/a)]^2) * sqrt(T))Thus, the minimum value of the parameter 'a' such that |X| < 0.01 is given by the expression above.

## What is a Fourier Transform?

A Fourier Transform is a mathematical tool used to analyze and decompose a complex signal into its individual frequency components. It is commonly used in fields such as signal processing, image processing, and data analysis.

## How does a Fourier Transform work?

A Fourier Transform works by representing a signal as a sum of sine and cosine waves of different frequencies. By breaking down the signal into its frequency components, it allows for easier analysis and manipulation of the signal.

## What are some applications of Fourier Transform?

Fourier Transform has a wide range of applications in various fields such as audio and image processing, signal filtering, data compression, and pattern recognition. It is also used in solving differential equations and in quantum mechanics.

## What is the difference between Fourier Transform and Inverse Fourier Transform?

Fourier Transform converts a signal from the time domain to the frequency domain, while Inverse Fourier Transform converts it back from the frequency domain to the time domain. Essentially, Fourier Transform decomposes a signal, while Inverse Fourier Transform synthesizes it.

## How is Fourier Transform related to the Fast Fourier Transform (FFT)?

FFT is an algorithm used to compute the Fourier Transform efficiently. While Fourier Transform has a time complexity of O(n^2), FFT has a time complexity of O(nlogn), making it much faster for larger signals. FFT is commonly used in practical applications due to its efficiency.

Replies
6
Views
2K
Replies
3
Views
1K
Replies
5
Views
847
Replies
1
Views
1K
Replies
3
Views
833
Replies
1
Views
1K
Replies
9
Views
2K
Replies
3
Views
1K
Replies
2
Views
1K
Replies
0
Views
635