Question

Derive the Fourier series expansion for each of the following continuous-time signals:

(a) x(t)= sin (2 π t)+cos (2 π t)+sin (πt)-cos (π t)

Answer 1

The Fourier series is usually used for non-sinusoidal periodic signals, but since the signal x(t) is already a sum of sinusoidal terms, it can be seen as its own Fourier series with specific harmonics and their corresponding coefficients.

Step-by-step explanation:

The question asks to derive the Fourier series expansion for the continuous-time signal x(t) = sin(2πt) + cos(2πt) + sin(πt) - cos(πt). However, as the signal is already in a simple sinusoidal form, it is not typical to express it as a Fourier series, since the Fourier series is generally used for periodic signals with non-sinusoidal waveforms. Nevertheless, it could be observed that the signal is composed of sinusoids with the following frequencies and harmonics: f1 = 1Hz for the sin(2πt) and cos(2πt) terms, and f2 = 0.5Hz for the sin(πt) and cos(πt) terms. The signal is essentially its own Fourier series representation with coefficients corresponding to the amplitudes of the sine and cosine components at these harmonics. For an actual derivation, one would integrate each term of x(t) against a sinusoid of the corresponding frequency over one period to determine the coefficients but, in this case, the coefficients correspond simply to the individual amplitudes of the sinusoids present in x(t).