Basic Continuous Time Signals - Waveform Properties 2

Question 1

What is the definition of periodic signals?

Answer 1

A signal is periodic if it repeats itself after some interval ( T_0 )

The formal condition is ( x(t) = x(t + T_0) ), where ( T_0 ) is the fundamental period.

Question 2

What are the most important families of periodic signals?

Answer 2

• Sinusoids
• Complex exponentials

Question 3

What are examples of nonperiodic waveforms?

Answer 3

• Step function
• Ramp function
• Pulse waveform

Question 4

How can nonperiodic waveforms be represented?

Answer 4

Using the step function and the ramp function.

Question 5

True or False: A periodic signal is constant and maintains the same frequency.

Answer 5

False

A periodic signal varies but repeats.

Question 6

What are the three types of waveform properties?

Answer 6

• Even/Odd symmetry
• Periodic vs. Nonperiodic
• Power/Energy signals

Question 7

What is the definition of the unit impulse function (Dirac delta function)?

Answer 7

The unit impulse function ( delta(t) ) is 0 for ( t
eq 0 ), has unit area, and is infinitely narrow and infinite in height.

Question 8

What is the sampling property of the impulse function?

Answer 8

+∞ ∫−∞ x(t) δ(t − T) dt = x(T)

Question 9

What is the time-scaling transformation of the impulse function?

Answer 9

δ(at) = (1/|a|) δ(t) for ( a
eq 0 )

This means that when the time variable is scaled by a factor ‘(a)’, the impulse function is modified by the reciprocal of the absolute value of the scaling factor, while still remaining at the origin.

Question 10

How does the unit impulse function relate to the step function?

Answer 10

The integral of the impulse function results in the step function: ( u(t) = ∫−∞ δ(τ) dτ )

Question 11

What is the time constant for growing exponentials?

Answer 11

The time it takes to rise to 63% of the final value.

Question 12

What is the time constant for decaying exponentials?

Answer 12

The time it takes to fall to 37% of the initial value.

Question 13

What does the sinc function represent?

Answer 13

sinc(x) = sin(x)/x

It is heavily used in Fourier Transforms.

Question 14

What defines energy signals?

Answer 14

Energy signals have ( 0 < E_x < ∞ ) and ( P_x = 0 ).

Question 15

What defines power signals?

Answer 15

Power signals have ( P_x ) finite and ( E_x → ∞ ).

Question 16

What are the conditions for a signal to be classified as neither power nor energy?

Answer 16

If ( P_x → ∞ ) and ( E_x → ∞ ).

Question 17

What is the average power of a signal?

Answer 17

The average square of a signal: ( P_x = ∫ x(t)^2 dt )

Question 18

Fill in the blank: A signal is called an energy signal if ______.

Answer 18

0 < E_x < ∞ and P_x = 0.

Question 19

Fill in the blank: The average of a periodic signal is defined as the average over ______.

Answer 19

one period.

Question 20

What is the relationship between power signals and energy signals?

Answer 20

They are mutually exclusive.