The Complex Exponential - Free MCQ Practice Test with solutions, GATE EE

This is the corollary of DeMoivre/Euler’s Theorem.

exp(a+b) =exp(a) * exp(b), and |exp(3i)| = 1.

Fundamental period = 2pi/w, hence fundamental frequency will be w.

Energy possessed by a periodic signal is the integral of the square of the magnitude of the signal over a time period.

The decaying exponentials dampen the amplitudes of sinusoids. Hence, the term damped sinusoids.

The power = Energy/Time period for a periodic signal. Hence, Power = 1.

The fundamental period = 2pi/(pi/4) = 8.

Any two instants, t and t + 2pi will be equal, hence the signal will be periodic with period 2pi.

The first signal, will repeat itself after 3 cycles. The second will repeat itself after 8 cycles. Thus, both of them together will repeat themselves only after LCM(8,3) = 24 cycles.

Discrete exponentials are periodic only for a particular choice of the fundamental frequency.