Chapter 1 and Complex numbers

Question 1

What is a signal?

Answer 1

A function of one or more variables that conveys information
about some (usually physical) phenomenon.

Question 2

What is a one dimensional (1D) signal?

Answer 2

A signal with one independent variable

Question 3

What is a multi-dimensional signal?

Answer 3

A signal with more than one independent variable

Question 4

What is a continuous time (CT) signal?

Answer 4

A signal with continuous independent variables

Question 5

What is a discrete time (DT) signal?

Answer 5

A signal with discrete independent variables

Question 6

What is a continuous valued signal?

Answer 6

A signal with a continuous dependent variable

Question 7

What is a discrete valued signal?

Answer 7

A signal with a discrete dependent variable

Question 8

What is a continuous-valued, CT signal called?

Answer 8

analogue

Question 9

What is a discrete-valued, DT signal called?

Answer 9

digital

Question 10

What is a system?

Answer 10

an entity that processes one or more input signals in order to
produce one or more output signals.

Question 11

What is a single input (SI) system?

Answer 11

A system with one input

Question 12

What is a multi input (MI) system?

Answer 12

A system with more than one input

Question 13

What is a single output (SO) system?

Answer 13

A system with one output

Question 14

What is a multi output (MO) system?

Answer 14

A system with more than one output

Question 15

Cartesian form of complex number?

Answer 15

z = x+ jy

Question 16

Polar form of complex number?

Answer 16

z = r(cosθ+ jsinθ) or equivalently z = re^(jθ)

Question 17

What are the magnitude and argument of a complex number in polar form?

Answer 17

|z|= r, arg(z) = θ

Question 18

What is the principal argument?

Answer 18

denoted Arg(z), is the
particular value θ of arg(z) that satisfies −π < θ ≤ π.

Question 19

How to get the argument of a complex number in cartesian form?

Answer 19

arctan(y/x).
If x is negative, need to add π

Question 20

How to convert from cartesian to polar

Answer 20

r = √(x^2 + y^2). θ = arctan(y/x), If x is negative, need to add π

Question 21

What is the conjugate of a complex number in cartesian form

Answer 21

z* = x - jy

Question 22

identity: |z*| =

Answer 22

|z|

Question 23

identity: arg(z*) =

Answer 23

-arg(z)

Question 24

identity: zz* =

Answer 24

|z|^2

Question 25

identity: Re(z) =

Answer 25

(1/2)(z+z*)

Question 26

identity: Im(z) =

Answer 26

(1/(2j))(z-z*)

Question 27

identity: (z1 + z2)* =

Answer 27

z1* + z2*

Question 28

identity: (z1/z2)* =

Answer 28

z1* - z2*

Question 29

identity: (z1z2)* =

Answer 29

z1*z2*

Question 30

Multiplication or division in polar form?

Answer 30

Multiplication: r1r2e^(j(θ1+θ2)) Division: (r1/r2)e^(j(θ1- θ2))

Question 31

Multiplication or division in cartesian form?

Answer 31

Multiplication: Regular (x1+jy1)(x2+jy2) with FOILing, taking into account i^2 = -1 Division: for z1/z2, Multiply top and bottom by z2* and simplify. Bottom should be a single number

Question 32

identity: |z1z2| =

Answer 32

|z1||z2|

Question 33

identity: |z1/z2| =

Answer 33

|z1|/|z2| for z2 not equal to 0

Question 34

identity: arg(z1z2) =

Answer 34

arg(z1) + arg(z2)

Question 35

identity: arg(z1/z2) =

Answer 35

arg(z1) - arg(z2) for z2 not equal to 0

Question 36

What is Euler's relation?

Answer 36

e^(jθ) = cosθ+ jsinθ

Question 37

identity from Euler's relation: cos(θ) =

Answer 37

(1/2)(e^(jθ) + e^(−jθ))

Question 38

identity from Euler's relation: sin(θ) =

Answer 38

(1/(2j))(e^(jθ) - e^(−jθ))

Question 39

What is De Moivre's Theorem?

Answer 39

e^(jnθ) = (e^(jθ))^n

Question 40

nth roots of polar complex number formula?

Answer 40

n√(r)(e^((jθ+2πk)/n)) for k = 0,1,...,n−1. where n√(r) is the nth root of r

Question 41

differentiability, continuity, and analyticity of a polynomial function

Answer 41

A polynomial function is differentiable, continuous, and analytic everywhere

Question 42

differentiability, continuity, and analyticity of a rational function

Answer 42

A rational function is differentiable, continuous, and analytic everywhere except where the denominator is zero

Question 43

zeros and nth order zeros?

Answer 43

Zero of a function: The point z0 where F(z0) = 0 nth order zeros: F(z0) = 0, F^(1)(z0) = 0,..., F^(n−1)(z0) = 0 where F^(k) denotes the kth order derivative of F), F is said to have an nth order zero at z0.

Question 44

what is a singularity and how does it relate to polynomial and rational functions?

Answer 44

A point at which a function fails to be analytic is called a singularity. Polynomials do not have singularities. Rational functions can have a type of singularity called a pole.

Question 45

What is a pole

Answer 45

A type of singularity that rational functions can have. Occurs where the denominator is zero. The pole is nth order following the same rules as nth order zeros. Ex: F(z) = 1/(x-1)^3 has an a 3rd order pole at x=1

Question 45

what are simple and repeated poles and zeros?

Answer 45

simple poles and zeros are of 1st order. Repeated poles and zeros are of nth order

Question 46

How to plot zeros and poles

Answer 46

Plot them on a graph with the x axis being the Re axis and the y axis being the Im axis. Plot zeros as O's and poles as X's. Put a number in brackets beside the plotted pole or zero that corresponds to the order of that pole or zero.