johnq2k7
Dec20-09, 02:40 PM
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
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
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
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution