Question

Suppose we are given the following information about a signal ():

a) () is a real signal. B) () is periodic with period = 4, and it has Fourier series coefficients. C) = 0 for || > 1. D) The signal with Fourier coefficients = −/2− is odd. E). Analyze this information and deduce the signal () to within a sign factor

Answer 1

Let's break down the given information to deduce the signal:

a) () is a real signal.

This tells us that the signal is not complex.

b) () is periodic with period = 4, and it has Fourier series coefficients.

This means that the signal can be represented by its Fourier series.

c) = 0 for || > 1.

This means that the signal is zero outside the interval [-1, 1].

d) The signal with Fourier coefficients = −/2− is odd.

This tells us that the signal is odd, meaning that () = −(), where () is the Fourier series coefficients.

Putting all the information together, we can write the Fourier series of the signal as:

() = /2 + ∑ (−1)^n+1 /n sin(nπ/2) e^(j nω_0 t)

where ω_0 = 2π/4 = π/2.

Now, we can deduce the signal within a sign factor:

() = -1 + sin(π/2)t - sin(3π/2)t + sin(5π/2)t - ...

Therefore, the signal () is a square wave with period 4 and amplitude 2, alternating between -2 and 0 at the odd half-integers.