Shifted Two-Sided Exponential

Correct Answer

A — 4e^(−jω)/(4 + ω²)

Concept and Reasoning

Use the pair e^{-a|t|}↔2a/(a²+ω²). For a=2 the base transform is 4/(4+ω²). Replacing t by t−1 shifts the signal right by 1 s and multiplies the spectrum by e^{-jω}. Thus X(jω)=4e^{-jω}/(4+ω²), option A.

Calculation / Key Relation

e^{-2|t−1|} ↔ 4e^{-jω}/(4+ω²)

Exam Takeaway

Right time shift → negative spectral phase factor e^{-jωt0}.

Common Mistake

Do not change the magnitude denominator because of the time shift.