Starting with the DTFT transform pair
F
sinc ( n / w ) ←⎯
→ w rect ( wF ) ∗ δ1 ( F )
make the change of variable F = Ω / 2π to verify the transform pair
F
sinc ( n / w ) ←⎯
→ w rect ( wΩ / 2π ) ∗ δ 2 π ( Ω ) .
We are making a change of variable in both functions that are being convolved.
Therefore we must use the convolution property
If y ( t ) = x ( t ) ∗ h ( t ) then y ( at ) = a x ( at ) ∗ h ( at ) (Chapter 5)
Using that property
F
=Ω /2 π
F
sinc ( n / w ) ←⎯
→ w rect ( wF ) ∗ δ1 ( F ) ⎯F⎯⎯⎯
→ sinc ( n / w ) ←⎯
→ (1 / 2π ) w rect ( wΩ / 2π ) ∗ δ1 ( Ω / 2π )
Now, using the scaling property of the periodic impulse
δ a ( bt ) = 1 / b δ a /b ( t ) (Chapter 2)
we get
F
sinc ( n / w ) ←⎯
→ w rect ( wΩ / 2π ) ∗ δ 2 π ( Ω ) .
Now, just to prove that this result is correct, we can derive the omega-form
transform directly from the definition of the inverse transform.
F −1 ( w rect ( wΩ / 2π ) ∗ δ 2 π ( Ω )) =
1
2π
∫
2π
⎡⎣ w rect ( wΩ / 2π ) ∗ δ 2 π ( Ω ) ⎤⎦ e jΩn dΩ
There is only one rectangle centered at zero in the range −π < Ω < π and its full width is
2π / w .
w
F ( w rect ( wΩ / 2π ) ∗ δ 2 π ( Ω )) =
2π
−1
=
π /w
π /w
∫
− π /w
e
jΩn
w ⎡ e jΩn ⎤
w e jπ n /w − e− jπ n /w
dΩ =
=
2π ⎢⎣ jn ⎥⎦ − π /w j2π
n
1 e jπ n /w − e− jπ n /w sin (π n / w )
=
= sinc ( n / w )
j2
πn / w
πn / w