Question

The Fourier series of the given signal is _______________

(a) -4/π sin x

(b) 4/π sin x

(c) 4/π cos x

(d) -4/π cos x

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Question is from Fourier Series Analysis using Circuits in division Fourier Analysis on Complex Spaces of Signals and Systems

Answer 1

Correct choice is (b) 4/π sin x

To elaborate: We know that the Fourier series is given by a0 + \(∑_{n=1}^∞\) (an  cos nx + bn  sin⁡ nx)

This can be also written as a0 + a1cos x + b1 sin x + a2 cos 2x + b2 sin 2x ……

Here, c0 = a0 = 0

So, cn = bn = \(2∫_0^t f(t).sin⁡ωt  \,dt\)

∴ c1 = b1 = \(2∫_0^t sin⁡ωt \,dt\)

= \(– \frac{2}{ω} [cos⁡ωt]_0^t\)

= \(-\frac{2}{π}\)[-1-1] = 4/π

And a1 = \(2∫_0^t f(t).cos⁡ωt \,dt\)

= \(\frac{2}{ω}[sin⁡ωt]_0^t\)

= 0

So, the Fourier series can be written as 0 + 0.cos x + 4/π sin x

= 4/π sin x.