Question 7.3: Inverse DTFT of two periodic shifted rectangles Find and gra......
Inverse DTFT of two periodic shifted rectangles
Find and graph the inverse DTFT of
X(F) = [rect(50(F − 1/4)) + rect(50(F + 1/4))] ∗ δ_1(F)(Figure 7.13).
We can start with the table entry sinc(n/w)\overset{\mathcal{F}}{\longleftrightarrow }wrect(wF) ∗ δ_1(F) or, in this case, (1/50)sinc(n/50)\overset{\mathcal{F}}{\longleftrightarrow }rect(50F) ∗ δ_1(F). Now apply the frequency-shifting property e^{2 \pi F_0 n} x[n] \stackrel{\mathcal{F}}{\longleftrightarrow} \mathrm{X}\left(F-F_0\right),
e^{jπn/2}(1/50)sinc(n/50)\overset{\mathcal{F}}{\longleftrightarrow }rect(50(F-1/4)) ∗ δ_1(F) (7.18)
and
e^{-jπn/2}(1/50)sinc(n/50)\overset{\mathcal{F}}{\longleftrightarrow}rect(50(F+1/4)) ∗ δ_1(F) (7.19)
(Remember, when two functions are convolved, a shift of either one of them, but not both, shifts the convolution by the same amount.) Finally, combining (7.18) and (7.19) and simplifying,
(1/25)sinc(n/50)cos(πn/2) \overset{\mathcal{F}}{\longleftrightarrow }[rect(50(F-1/4))+ rect(50(F+1/4)) ∗ δ_1(F).