Which statement correctly describes the Complex Exponential Fourier Series representation?

• A. It represents a signal as a sum of sines and cosines.
• B. It represents a signal as a sum of cosines only.
• C. It represents a signal as a sum of complex exponentials e^{jkω0 n}.
• D. It cannot represent phase information.

Answer: (C)

The main idea is that the Complex Exponential Fourier Series represents a periodic discrete-time signal as a sum of complex exponential basis functions at multiples of a fundamental frequency. Specifically, a periodic signal x[n] is written as x[n] = sum_k X[k] e^{j k ω0 n}, where ω0 is the fundamental angular frequency and X[k] are complex coefficients that capture both how strong each harmonic is and its phase.

This form is powerful because the complex exponentials e^{j k ω0 n} simultaneously encode sine and cosine behavior; the coefficients X[k] contain the necessary phase information. If you look at real signals, you can see real-valued cosines and sines as combinations of these exponentials (for example, cos(ω0 n) = (e^{j ω0 n} + e^{-j ω0 n})/2). So the CEFS is the natural, most general way to express the spectrum: it handles both amplitude and phase in a compact way and aligns with how linear time-invariant systems operate, where these exponentials are eigenfunctions and filtering simply scales each spectral component by a complex factor.

The statement that it is a sum of cosines only is not correct, since the complex form uses exponentials that carry both sine and cosine content. The claim that it cannot represent phase information is also false, because the complex coefficients X[k] include phase, and the exponential basis functions themselves advance in phase with n. While cosines-and-sines forms are related to the complex exponential form via Euler’s formula, the complex exponential representation is the complete language for the spectrum.