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Here are my mathematica command and output:

FourierTransform[((a*Cos[2*c*\[Pi]*x]*Sin[b*\[Pi]*x])/(\[Pi]*x))^3, x,2*Pi*f, FourierParameters -> {-1, 1}]

(1/(256 \[Pi]))a^3 (-9 (2 b + c - 2 f)^2 Sign[2 b + c - 2 f] - 
   3 (6 b + c - 2 f)^2 Sign[6 b + c - 2 f] - 
   3 (2 b + 3 c - 2 f)^2 Sign[-2 b - 3 c + 2 f] - 
   3 (2 b - 3 c + 2 f)^2 Sign[
     2 b - 3 c + 2 f] - (6 b - 3 c + 2 f)^2 Sign[6 b - 3 c + 2 f] + 
   9 (-2 b + c - 2 f)^2 Sign[2 b - c + 2 f] + 
   3 (-6 b + c - 2 f)^2 Sign[6 b - c + 2 f] - 
   3 (-6 b + c + 2 f)^2 Sign[-6 b + c + 2 f] - 
   9 (-2 b + c + 2 f)^2 Sign[-2 b + c + 2 f] - 
   9 (2 b + c + 2 f)^2 Sign[2 b + c + 2 f] - 
   3 (6 b + c + 2 f)^2 Sign[
     6 b + c + 2 f] + (-6 b + 3 c + 2 f)^2 Sign[-6 b + 3 c + 2 f] + 
   3 (-2 b + 3 c + 2 f)^2 Sign[-2 b + 3 c + 2 f] + 
   3 (2 b + 3 c + 2 f)^2 Sign[
     2 b + 3 c + 2 f] + (6 b + 3 c + 2 f)^2 Sign[
     6 b + 3 c + 2 f] - (6 b + 3 c - 2 f)^2 Sign[-6 b \[Pi] - 
      3 c \[Pi] + 2 f \[Pi]])

for example for first statement (Sign[b + 2 c - 2 f]) for a=1, b=3 , c=20 , we should have a sign function like this Sign[43 - 2 f] but as I put this number in first command I don't get any Sign[43 - 2 f] in my output

(or for Sign[2 b + 3 c + 2 f] for a=1, b=3 , c=20 => Sign[66+ 2 f])

FourierTransform[((1*Cos[2*20*\[Pi]*x]*Sin[3*\[Pi]*x])/(\[Pi]*
      x))^3, x, 2*Pi*f, FourierParameters -> {-1, 1}]

(1/(256 \[Pi]))(-(129 - 2 f)^2 Sign[-129 + 2 f] + 
  3 (123 - 2 f)^2 Sign[-123 + 2 f] - 
  3 (117 - 2 f)^2 Sign[-117 + 2 f] + (111 - 2 f)^2 Sign[-111 + 2 f] - 
  3 (49 - 2 f)^2 Sign[-49 + 2 f] + 9 (43 - 2 f)^2 Sign[-43 + 2 f] - 
  9 (37 - 2 f)^2 Sign[-37 + 2 f] + 3 (31 - 2 f)^2 Sign[-31 + 2 f] - 
  3 (31 + 2 f)^2 Sign[31 + 2 f] + 9 (37 + 2 f)^2 Sign[37 + 2 f] - 
  9 (43 + 2 f)^2 Sign[43 + 2 f] + 
  3 (49 + 2 f)^2 Sign[49 + 2 f] - (111 + 2 f)^2 Sign[111 + 2 f] + 
  3 (117 + 2 f)^2 Sign[117 + 2 f] - 
  3 (123 + 2 f)^2 Sign[123 + 2 f] + (129 + 2 f)^2 Sign[129 + 2 f])

what is my mistake?

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    $\begingroup$ Try restarting he kernel and doing it again (you probably have old definitions around)... I do get the Sign[43 - 2 f] for {a -> 1, b -> 3, c -> 20} $\endgroup$ – bill s Oct 18 '16 at 22:53
  • $\begingroup$ Unfortunately it doesn't work again , I restart my kernel with Evaluation->Quit Kernal and here is my result -> obrazki.elektroda.pl/6442679300_1476858409.png $\endgroup$ – Ehsan Zakeri Oct 19 '16 at 6:28
  • $\begingroup$ What symbol are you using to get that little x -- multiplication can be done with space or with * $\endgroup$ – bill s Oct 19 '16 at 13:16
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Copy this line:

fft = FourierTransform[((a*Cos[2*c*\[Pi]*x]*Sin[b*\[Pi]*x])/(\[Pi]*x))^3, x, 2*Pi*f, FourierParameters -> {-1, 1}]

Now evaluate at the points:

fft /. {a -> 1, b -> 3, c -> 20}

You will see that the very first Sin term is Sin[43 - 2 f].

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