GATE 2014 EC (EC4) – Question 27
A Fourier-transform pair is printed as $(2/3)^nu[n+3]\leftrightarrow A e^{-j6\pi f}/[1-(2/3)e^{-j2\pi f}]$, where u[n] is the unit step. Find A.
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Correct answer: 3.36 to 3.39
Explanation
The sequence begins at n=−3. Summing the geometric series gives X(f)=(2/3)⁻³e^(+j6πf)/[1−(2/3)e^(−j2πf)], so the amplitude coefficient is **A=27/8=3.375**, matching the official range. However the printed numerator has a negative phase sign; an advance u[n+3]requires the positive sign. No constant A makes the displayed pair exact for all f.