Final answer:
To determine the exponential Fourier series of a periodic signal, one must analyze the signal's sinusoidal characteristics using a wave function and then apply the Fourier transform to break down the signal into a series of sinusoidal components.
Step-by-step explanation:
Fourier Series of a Periodic Signal
To determine the exponential Fourier series of a periodic signal, it is important to understand the signal's behavior in terms of sinusoidal waves. The signal can be analyzed using the wave function which is a representation of the sinusoidal wave's characteristics such as amplitude A, wave number k, angular frequency ω, and phase shift φ. The general form of the wave function is y(x, t) = A sin(kx - ωt + φ) or using Euler's formula, y(x, t) = Aei(kx-ωt) for complex exponentials. In the context of superposition of waves, individual waves with different phase shifts can be combined to determine the resulting wave's characteristics.
The workflow to find the Fourier series starts with writing down the signal's wave function. Next, one applies the Fourier transform to decompose the signal into a sum of sine and cosine functions (or equivalently into exponential functions with imaginary exponents). This yields coefficients that describe the amplitude and phase of each sine and cosine term in the series, collectively representing the original signal over one period.
Note: For more complex signals like square or step functions, it is crucial to remember that Fourier's theorem allows us to represent these as well by the sum of sines and cosines.