Question

A CT real and periodic signal x(t) has a fundamental period T₀=0.1 s and the following nonzero FS coefficients aₖ:a₀=−1,a₁=(1+j),a₋₁=a₁*,a₃=0.5e⁻ʲπ/²,a₋₃=a₃*

a) Derive and sketch the frequency-domain (FD) representation of the signal x(t).
b) Derive the time-domain (TD) representation of the signal x(t).

Answer 1

The frequency-domain and time-domain representations of the signal x(t) with given Fourier series coefficients are described.

Step-by-step explanation:

In the frequency-domain (FD) representation of the signal x(t), the nonzero Fourier series coefficients are used to determine the amplitudes and phase shifts of the sinusoidal components of the signal. The amplitude of the DC component (a₀) is -1. The amplitude of the sinusoidal component with frequency ω₁ = 2π/T₀ is a₁ = 1+j, and the amplitude of its complex conjugate is a₋₁ = a₁* = 1-j. The amplitude of the sinusoidal component with frequency ω₃ = -π/2 is a₃ = 0.5e⁻ʲπ/², and the amplitude of its complex conjugate is a₋₃ = a₃∗ = 0.5e⁻ʲπ/².

In the time-domain (TD) representation of the signal x(t), the signal can be expressed as a sum of sinusoidal components with different frequencies and amplitudes. The signal x(t) = a₀ + Σ(aₖe^(jωₖt) + a₋ₖe^(-jωₖt)), where the summation is taken over all nonzero Fourier series coefficients. In this case, the time-domain representation of the signal x(t) is x(t) = -1 + (1+j)e^(jω₁t) + (1-j)e^(-jω₁t) + 0.5e^(jω₃t) + 0.5e^(-jω₃t).