 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1/4/g/solution

sin (A t) / (π t)

has Fourier transform equal to one for

∣ ω ∣ < A

and zero for

∣ ω ∣ > A

. Thus the two bandwidths add to give

z (ω) = ⎩ ⎨ ⎧ 2, 1, 0, ∣ ω ∣ < 1, 1 < ∣ ω ∣ < 2, ∣ ω ∣ > 2.

The requested squared-magnitude spectrum therefore has heights four and one:

∣ z (ω) ∣^{2} = ⎩ ⎨ ⎧ 4, 1, 0, ∣ ω ∣ < 1, 1 < ∣ ω ∣ < 2, ∣ ω ∣ > 2.

Values at the jump points depend on the chosen Fourier convergence convention and do not affect any integral. The graph is shown below.

The Parseval identity with this Fourier transform normalization gives

E = \frac{1}{2 π} \int_{- \infty}^{\infty} ∣ z (ω) ∣^{2} d ω = \frac{1}{2 π} (4 \cdot 2 + 1 \cdot 2) = \frac{5}{π} .

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