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 Sf(t) is given by(adsbygoogle = window.adsbygoogle || []).push({});

Sf(t)= Sf(0) exp (-i2pi(f_0)(t))*exp(-t/T2*) t>=0

Sf(t)= 0

it was proven that Gf(f) corresponding to this signal is given by:

Gf(f)= Sf(0) { [(T2*)/ (1+(2pi(f-f_f0)T2*)^2)] + [i2pi(f-f0)(T2*)^2/(1+ (2pi(f-f0)T2*)^2)]}

a.) Show that the spectrum of the echo is given by

Ge(f)= Se(0) { 2T2*/ (1+ (2pi(f-f0)T2*)^2}

b.)using properties of even and odd func. and shift theorem, show that img. part of spect. must equal zero for any signal of form:

S(t)= S(0) exp (i(2pi)(f0)t)Fe(t) , where Fe(t) is an even func. of t, Fe(t) does not need to be exp.

Work shown:

for part a.) if you ignore the img. part

then Gf(f) is S(0) {{ T2*/ (1+ (2pi(f-f0)T2*)^2) + { T2*/ (1+ (2pi(f-f0)T2*)^2)}

therefore, Ge(f)= Se(0) { 2T2*/ (1+ (2pi(f-f0)T2*)^2}

however, i'm not sure my proof here is correct

for part b.)

i'm not sure how to use the even and odd func. to prove the signal equals zero. especial for img section

Please help!!!

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Another fourier transform nmr question work shown need help!

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