Final answer:
The student is tasked with deriving the exponential Fourier series for a given signal and sketching its frequency spectrum, which includes discrete frequency lines corresponding to the complex amplitudes of the signal components.
Step-by-step explanation:
The question involves deriving the exponential Fourier series for a given periodic signal x(t) and sketching its frequency spectrum. For the signal x(t) = (2+j2)e−˟³ᵗ + j2e−˟ᵗ + 3 − j2e˟ᵗ + (2−j2)e˟³ᵗ, the exponential Fourier series is already given, and its coefficients are the complex amplitudes at different frequencies. The spectrum will consist of discrete lines at frequencies multiples of the fundamental frequency, indicating the presence of frequency components of the signal. The magnitudes of these lines correspond to the amplitudes of the complex coefficients. For illustration, the positive frequency terms are (2−j2) at 3ᵗ and −j2 at ᵗ, with the corresponding negative frequency terms at −ᵗ and − 3ᵗ. The spectrum will be symmetric due to the complex conjugate nature of the coefficients.