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QUESTION: 1

What is exp(ja) equal to, where j is the square root of unity?

Solution:

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

QUESTION: 2

What is the magnitude of exp(2+3j)?

Solution:

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

QUESTION: 3

What is the fundamental frequency of exp(2pi*w*j)?

Solution:

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

QUESTION: 4

Total energy possessed by a signal exp(jwt) is?

Solution:

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

QUESTION: 5

Sinusoidal signals multiplied by decaying exponentials are referred to as

Solution:

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

QUESTION: 6

What is the power possessed by a signal exp(jwt)?

Solution:

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

QUESTION: 7

What is the period of exp(2+pi*j/4)t?

Solution:

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

QUESTION: 8

exp(jwt) is periodic

Solution:

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

QUESTION: 9

Define the fundamental period of the following signal x[n] = exp(2pi*j*n/3) + exp(3*pi*j*n/4)?

Solution:

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.

QUESTION: 10

exp[jwn] is periodic

Solution:

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

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