Which statement correctly describes the Complex Exponential Fourier Series representation?

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 Complex Exponential Fourier Series uses complex exponentials as the building blocks: a periodic signal is written as a sum of terms like X[k] e^{j k ω0 n}, with complex coefficients X[k]. This form is preferred because complex exponentials provide a compact, natural basis that encodes both magnitude and phase in the coefficients, and they are fundamental for analyzing linear systems. While you can relate sines and cosines to exponentials via Euler’s formula, the explicit complex-exponential representation is about summing e^{jkω0 n} terms with their own complex weights. That’s why the statement describing the signal as a sum of complex exponentials is the correct one. Phase information is captured in the complex coefficients, not lost, so the option claiming phase cannot be represented is not correct.