Signal Representation and Analysis

Question 1

What are the two main categories of signal? (What-time?)

What two classifications do these then fall into?

Answer 1

Continuous-time and Discrete-time

Periodic or Non-periodic

Question 2

What length of time do we assume the four classes of signal all have?

Answer 2

Infinite-duration

Question 3

What does it mean if a continuous-time signal x(t) is said to be an energy signal?

Answer 3

If the total energy, E, dissipated by the signal over all time is both nonzero and finite.

Thus the magnitude of the signal must tend to 0 as time heads to infinity.

Question 4

What is the equation for E, the energy dissipated by a signal over all time?

Answer 4

Question 5

What does it mean for a signal to be a power signal?

Answer 5

If the average power delivered by the signal over all time is both nonzero and finite.

Question 6

What is the equation for P, the power of a signal?

Answer 6

Question 7

Name a type of signal which is not an example of an energy signal.

Answer 7

A periodic signal has finite energy over one period, so consequently has infinite energy. However has a finite average power and is therefore a power signal and not an energy signal

Question 8

How do we convert our abstract signal energy and power values to real values for use in circuits?

Answer 8

Divide by the resistance in Ohms

Question 9

What is the simplified equation for power for a periodic signal?

Answer 9

Question 10

Using the Fourier Series (trignometric form) how can we represent any finite-power periodic signal x(t) with period T ?

Answer 10

Question 11

What is the fundamental frequency of a periodic signal x(t), with relation to the Fourier series of that signal.

Answer 11

The fundamental frequency is the inverse of the fundamental period

Question 12

How do we find the fourier coefficients A_n and B_n in the trignometric Fourier Series of a signal?

Answer 12

Question 13

How can a complex phasor of amplitude A and frequency ω₀ be split into real and imaginary trignometric parts?

Answer 13

Question 14

Using the Complex Fourier Series how can we represent a periodic signal x(t)?

Answer 14

Question 15

How do we calculate X_n , the complex fourier coefficient?

Answer 15

Question 16

Define how the complex coefficient of the fourier series X_n relates to the trignometric coefficients A_n and B_n. Consider the cases:

(i) . n >0
(ii) . n=0
(iii) . n<0

Answer 16

Question 17

Write the equation for x_n(t), the n^th harmonic of a complex fourier series

Answer 17

Question 18

If the product of two periodic signals is integrated over one period and the result is zero then what can we say about those signals?

Answer 18

They are orthogonal (at 90^o)

Question 19

How do we calculate the power of a signal from its complex fourier coefficients?

Answer 19

Use Parseval’s Theorem

Question 20

What is Parsevak’s Theorem for complex Fourier coefficients?

Answer 20

Question 21

What is Parseval’s theorem for trignometric fourier coefficients?

Answer 21

Question 22

How do we use the principles of the Fourier Series for a non-periodic signal; ie: T→∞

Answer 22

We instead use the forward Fourier Transform

Question 23

Give the formula for the forward Fourier transform

Answer 23

Question 24

Give the formula for the inverse Fourier transform

Answer 24

Question 25

What is the energy is a signal x(t) using the **fourier transform** and **parseval's theorem**.

Answer 25

Question 26

Describe the **duality** property of the fourier transform.

Answer 26

Question 27

Describe the **shift in time** property of the Fourier Transform

Answer 27

Question 28

Describe the **scale in time** property of the Fourier Transform.

Answer 28

Question 29

Describe the **shift in frequency** property of the Fourier Transform

Answer 29

Question 30

Describe the **linearity** property of the Fourier Transform

Answer 30

Question 31

Describe the **Heaviside Step Function** u(t)

Answer 31

Question 32

Describe the **unit impulse** function (a.k.a. Dirac Delta)

Answer 32

Question 33

What do we call a periodic signal with impulses centered at integer multiples of a **sampling period** T

Answer 33

An **impulse train**

Question 34

What is the **sifting theorem** and what does it mean?

Answer 34

The area under the product of a function with an impulse is equal to the value of that function at the instance at which the impulse is located.

Question 35

What happens when we take a **fourier series** of an **impulse train** of sampling period T. (What are the coefficients?)

Answer 35

All the coefficients are identical and equal to 1/T

Question 36

What is the **Fourier transform** of a **Fourier series?**

Answer 36

The Fourier transform of a Fourier Series is an **infinite summation** of **weighted impulses** centered at integer multiples of the **fundamental frequency**. The weights of the impulses are 2π multiplied by the Fourier coefficients. Hence, the vertical lines in the plots of Fourier coefficients are actually **impulses**!