For the given signal h(t)

The Fourier transform is/are

2 S a (ω) + e^{- j ω} S a (ω) + e^{j ω} S a (ω)

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\frac{1}{j ω} [e^{+ 2 j ω} - e^{- 2 j ω} + e^{j ω} - e^{- j ω}]

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2 S a (ω) + 2 e^{- j ω} S a (ω) - 2 e^{j ω} S a (ω)

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4 S a (2 ω) + 2 S a (ω)

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Solution

The correct option is D $4 S a (2 ω) + 2 S a (ω)$

option (a)
$h (t) = r e c t (\frac{t}{4}) + r e c t (\frac{t}{2})$

$H (ω) = 4 S a (\frac{ω \times 4}{2}) + 2 S a (\frac{ω \times 2}{2})$

$H (ω) = 4 S a (2 ω) + 2 S a (ω)$

option (c)
$h (t) = r e c t (\frac{t}{4}) + r e c t (\frac{t}{2})$

$= \frac{1}{2} [s g n (t + 2) - s g n (t - 2) + s g n (t + 1) - s g n (t - 1)]$

$= \frac{1}{2} [e^{2 j ω} \times \frac{2}{j ω} - e^{- 2 j ω} \times \frac{2}{j ω} + e^{- j ω} \times \frac{2}{j ω}]$

$= \frac{1}{j ω} [e^{2 j ω} - e^{- 2 j ω} + e^{j ω} - e^{- j ω}]$

Only options A,C are correct

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