Question

Analytic signal, \(A(\theta)\), for \(f(\theta) = \cos\theta\) is equal to

• A. \(A(\theta) = -\cos\theta\)
• B. \(A(\theta) = -\sin\theta\)
• C. \(A(\theta) = e^{i\theta}\)
• D. \(A(\theta) = -e^{i\theta}\)

Hint:

The analytic signal is \(f(\theta)+i\hat f(\theta)\); the Hilbert transform of \(\cos\theta\) is \(\sin\theta\), giving Euler's identity.

Answer 1

The Correct Option is C

Reach the same result from the frequency-domain definition of the analytic signal rather than the Hilbert-transform-pair table.

Step 1: Write \(\cos\theta\) as a sum of exponentials.
\(\cos\theta = \dfrac{e^{i\theta}+e^{-i\theta}}{2}\), i.e. it has equal-amplitude components at 'frequencies' \(+1\) and \(-1\).

Step 2: Apply the analytic-signal rule in the frequency domain.
The analytic signal is formed by doubling the positive-frequency component and discarding the negative-frequency component entirely (this is exactly what multiplying by \(i\) and adding the Hilbert transform accomplishes): \(A(\theta) = 2\times\dfrac{e^{i\theta}}{2} = e^{i\theta}\).

This one-sided-spectrum construction gives the same answer, \(A(\theta) = \boxed{e^{i\theta}}\) - option (C), confirming the Hilbert-transform-pair method above.