Contents
1 Fundamentals of Signals 5
1.1 What is a Signal? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Review on Complex Numbers . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Basic Operations of Signals . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Even and Odd Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Impulse and Step Functions . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 Continuous-time Complex Exponential Functions . . . . . . . . . . . 22
1.8 Discrete-time Complex Exponentials . . . . . . . . . . . . . . . . . . 24
2 Fundamentals of Systems 27
2.1 System Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 System Properties and Impulse Response . . . . . . . . . . . . . . . . 37
2.4 Continuous-time Convolution . . . . . . . . . . . . . . . . . . . . . . 41
3 Fourier Series 43
3.1 Eigenfunctions of an LTI System . . . . . . . . . . . . . . . . . . . . 43
3.2 Fourier Series Representation . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Properties of Fourier Series Coefficients . . . . . . . . . . . . . . . . . 54
4 Continuous-time Fourier Transform 57
4.1 Insight from Fourier Series . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Relation to Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Properties of Fourier Transform . . . . . . . . . . . . . . . . . . . . . 66
4.6 System Analysis using Fourier Transform . . . . . . . . . . . . . . . . 69
3
,4 CONTENTS
5 Discrete-time Fourier Transform 73
5.1 Review on Continuous-time Fourier Transform . . . . . . . . . . . . . 73
5.2 Deriving Discrete-time Fourier Transform . . . . . . . . . . . . . . . . 74
5.3 Why is X(ejω ) periodic ? . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 Properties of Discrete-time Fourier Transform . . . . . . . . . . . . . 77
5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.6 Discrete-time Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Sampling Theorem 83
6.1 Analog to Digital Conversion . . . . . . . . . . . . . . . . . . . . . . 83
6.2 Frequency Analysis of A/D Conversion . . . . . . . . . . . . . . . . . 84
6.3 Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4 Digital to Analog Conversion . . . . . . . . . . . . . . . . . . . . . . 90
7 The z-Transform 95
7.1 The z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2 z-transform Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3 Properties of ROC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4 System Properties using z-transform . . . . . . . . . . . . . . . . . . 104
,Chapter 1
Fundamentals of Signals
1.1 What is a Signal?
A signal is a quantitative description of a physical phenomenon, event or process.
Some common examples include:
1. Electrical current or voltage in a circuit.
2. Daily closing value of a share of stock last week.
3. Audio signal: continuous-time in its original form, or discrete-time when stored
on a CD.
More precisely, a signal is a function, usually of one variable in time. However, in
general, signals can be functions of more than one variable, e.g., image signals.
In this class we are interested in two types of signals:
1. Continuous-time signal x(t), where t is a real-valued variable denoting time,
i.e., t ∈ R. We use parenthesis (·) to denote a continuous-time signal.
2. Discrete-time signal x[n], where n is an integer-valued variable denoting the
discrete samples of time, i.e., n ∈ Z. We use square brackets [·] to denote a
discrete-time signal. Under the definition of a discrete-time signal, x[1.5] is not
defined, for example.
5
, 6 CHAPTER 1. FUNDAMENTALS OF SIGNALS
1.2 Review on Complex Numbers
We are interested in the general complex signals:
x(t) ∈ C and x[n] ∈ C,
where the set of complex numbers is defined as
√
C = {z | z = x + jy, x, y ∈ R, j = −1.}
A complex number z can be represented in Cartesian form as
z = x + jy,
or in polar form as
z = rejθ .
Theorem 1. Euler’s Formula
ejθ = cos θ + j sin θ. (1.1)
Using Euler’s formula, the relation between x, y, r, and θ is given by
( ( p
x = r cos θ r = x2 + y 2 ,
and
y = r sin θ θ = tan−1 xy .
Figure 1.1: A complex number z can be expressed in its Cartesian form z = x + jy, or in its polar
form z = rejθ .
1 Fundamentals of Signals 5
1.1 What is a Signal? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Review on Complex Numbers . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Basic Operations of Signals . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Even and Odd Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Impulse and Step Functions . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 Continuous-time Complex Exponential Functions . . . . . . . . . . . 22
1.8 Discrete-time Complex Exponentials . . . . . . . . . . . . . . . . . . 24
2 Fundamentals of Systems 27
2.1 System Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 System Properties and Impulse Response . . . . . . . . . . . . . . . . 37
2.4 Continuous-time Convolution . . . . . . . . . . . . . . . . . . . . . . 41
3 Fourier Series 43
3.1 Eigenfunctions of an LTI System . . . . . . . . . . . . . . . . . . . . 43
3.2 Fourier Series Representation . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Properties of Fourier Series Coefficients . . . . . . . . . . . . . . . . . 54
4 Continuous-time Fourier Transform 57
4.1 Insight from Fourier Series . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Relation to Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Properties of Fourier Transform . . . . . . . . . . . . . . . . . . . . . 66
4.6 System Analysis using Fourier Transform . . . . . . . . . . . . . . . . 69
3
,4 CONTENTS
5 Discrete-time Fourier Transform 73
5.1 Review on Continuous-time Fourier Transform . . . . . . . . . . . . . 73
5.2 Deriving Discrete-time Fourier Transform . . . . . . . . . . . . . . . . 74
5.3 Why is X(ejω ) periodic ? . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 Properties of Discrete-time Fourier Transform . . . . . . . . . . . . . 77
5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.6 Discrete-time Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Sampling Theorem 83
6.1 Analog to Digital Conversion . . . . . . . . . . . . . . . . . . . . . . 83
6.2 Frequency Analysis of A/D Conversion . . . . . . . . . . . . . . . . . 84
6.3 Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4 Digital to Analog Conversion . . . . . . . . . . . . . . . . . . . . . . 90
7 The z-Transform 95
7.1 The z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2 z-transform Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3 Properties of ROC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4 System Properties using z-transform . . . . . . . . . . . . . . . . . . 104
,Chapter 1
Fundamentals of Signals
1.1 What is a Signal?
A signal is a quantitative description of a physical phenomenon, event or process.
Some common examples include:
1. Electrical current or voltage in a circuit.
2. Daily closing value of a share of stock last week.
3. Audio signal: continuous-time in its original form, or discrete-time when stored
on a CD.
More precisely, a signal is a function, usually of one variable in time. However, in
general, signals can be functions of more than one variable, e.g., image signals.
In this class we are interested in two types of signals:
1. Continuous-time signal x(t), where t is a real-valued variable denoting time,
i.e., t ∈ R. We use parenthesis (·) to denote a continuous-time signal.
2. Discrete-time signal x[n], where n is an integer-valued variable denoting the
discrete samples of time, i.e., n ∈ Z. We use square brackets [·] to denote a
discrete-time signal. Under the definition of a discrete-time signal, x[1.5] is not
defined, for example.
5
, 6 CHAPTER 1. FUNDAMENTALS OF SIGNALS
1.2 Review on Complex Numbers
We are interested in the general complex signals:
x(t) ∈ C and x[n] ∈ C,
where the set of complex numbers is defined as
√
C = {z | z = x + jy, x, y ∈ R, j = −1.}
A complex number z can be represented in Cartesian form as
z = x + jy,
or in polar form as
z = rejθ .
Theorem 1. Euler’s Formula
ejθ = cos θ + j sin θ. (1.1)
Using Euler’s formula, the relation between x, y, r, and θ is given by
( ( p
x = r cos θ r = x2 + y 2 ,
and
y = r sin θ θ = tan−1 xy .
Figure 1.1: A complex number z can be expressed in its Cartesian form z = x + jy, or in its polar
form z = rejθ .
