Fourier Transform question work shown

[johnq2k7]

Fourier Transform question... please help.. work shown!

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.


Please help me with this problem... work is shown!

Please help me with this problem.. I need a lot of help here

 

[lippyka]

!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.