What is exp(ja) equal to, where j is the square root of unity?
This is the corollary of DeMoivre/Euler’s Theorem.
What is the magnitude of exp(2+3j)?
exp(a+b) =exp(a) * exp(b), and |exp(3i)| = 1.
What is the fundamental frequency of exp(2pi*w*j)?
Fundamental period = 2pi/w, hence fundamental frequency will be w.
Total energy possessed by a signal exp(jwt) is?
Energy possessed by a periodic signal is the integral of the square of the magnitude of the signal over a time period.
Sinusoidal signals multiplied by decaying exponentials are referred to as
The decaying exponentials dampen the amplitudes of sinusoids. Hence, the term damped sinusoids.
What is the power possessed by a signal exp(jwt)?
The power = Energy/Time period for a periodic signal. Hence, Power = 1.
What is the period of exp(2+pi*j/4)t?
The fundamental period = 2pi/(pi/4) = 8.
exp(jwt) is periodic
Any two instants, t and t + 2pi will be equal, hence the signal will be periodic with period 2pi.
Define the fundamental period of the following signal x[n] = exp(2pi*j*n/3) + exp(3*pi*j*n/4)?
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.
exp[jwn] is periodic
Discrete exponentials are periodic only for a particular choice of the fundamental frequency.