21
Continuous-Time
Second-Order Systems
The properties of the Laplace transform make it particularly useful in analyzing LTI systems that are represented by linear constant-coefficient differential equations. Specifically, applying the Laplace transform t
20
The Laplace
Transform
Since we first introduced Fourier analysis in Lecture 7, we have relied heavily
on its properties in the analysis and representation of signals and linear, timeinvariant systems. The Fourier transform was developed from the concep
10
Discrete-Time
Fourier Series
In this and the next lecture we parallel for discrete time the discussion of the
last three lectures for continuous time. Specifically, we consider the representation of discrete-time signals through a decomposition as a li
22
The z-Transform
In Lecture 20, we developed the Laplace transform as a generalization of the
continuous-time Fourier transform. In this lecture, we introduce the corresponding generalization of the discrete-time Fourier transform. The resulting
transfo
16
Sampling
The sampling theorem, which is a relatively straightforward consequence of
the modulation theorem, is elegant in its simplicity. It basically states that a
bandlimited time function can be exactly reconstructed from equally spaced
samples prov
7
Continuous-Time
Fourier Series
In representing and analyzing linear, time-invariant systems, our basic approach has been to decompose the system inputs into a linear combination of
basic signals and exploit the fact that for a linear system the response
6
Systems Represented by
Differential and
Difference Equations
An important class of linear, time-invariant systems consists of systems represented by linear constant-coefficient differential equations in continuous
time and linear constant-coefficient di
17
Interpolation
In developing the sampling theorem, we based the reconstruction procedure
for recovering the original signal from its samples on the use of a lowpass filter. This follows naturally from the interpretation of the sampling process in
the fr
2
Signals and Systems:
Part I
In this lecture, we consider a number of basic signals that will be important
building blocks later in the course. Specifically, we discuss both continuoustime and discrete-time sinusoidal signals as well as real and complex
18
Discrete-Time
Processing of
Continuous-Time
Signals
One very important application of the concept of sampling is its role in processing continuous-time signals using discrete-time systems. Specifically, the
continuous-time signal, which either is assum
9
Fourier Transform
Properties
The Fourier transform is a major cornerstone in the analysis and representation of signals and linear, time-invariant systems, and its elegance and importance cannot be overemphasized. Much of its usefulness stems directly f
1
Introduction
This first lecture is intended to broadly introduce the scope and direction of
the course. We are concerned, of course, with signals and with systems that
process signals. Signals can be categorized as either continuous-time signals,
for wh
13
Continuous-Time
Modulation
In this lecture, we begin the discussion of modulation. This is an important
concept in communication systems and, as we will see in Lecture 15, also provides the basis for converting between continuous-time and discrete-time
3
Signals and Systems:
Part II
In addition to the sinusoidal and exponential signals discussed in the previous
lecture, other important basic signals are the unit step and unit impulse. In
this lecture, we discuss these signals and then proceed to a discu
19
Discrete-Time
Sampling
In the previous lectures we discussed sampling of continuous-time signals. In
this lecture we address the parallel topic of discrete-time sampling, which has
a number of important applications. The basic concept of discrete-time
25
Feedback
The tools that we have developed throughout this set of lectures provide the
basis for a thorough understanding and analysis of linear feedback systems.
Feedback is a process that arises naturally in many practical situations, and it
is import
1.1
Discrete-Time
Fourier Transform
The discrete-time Fourier transform has essentially the same properties as
the continuous-time Fourier transform, and these properties play parallel
roles in continuous time and discrete time. As with the continuous-tim
8
Continuous-Time
Fourier Transform
In this lecture, we extend the Fourier series representation for continuoustime periodic signals to a representation of aperiodic signals. The basic approach is to construct a periodic signal from the aperiodic one by p
5
Properties of Linear,
Time-Invariant
Systems
In this lecture we continue the discussion of convolution and in particular explore some of its algebraic properties and their implications in terms of linear,
time-invariant (LTI) systems. The three basic pr
15
Discrete-Time
Modulation
The modulation property is basically the same for continuous-time and discrete-time signals. The principal difference is that since for discrete-time signals the Fourier transform is a periodic function of frequency, the convol
12
Filtering
In discussing Fourier transforms, we developed a number of important properties, among them the convolution property and the modulation property.
The convolution property forms the basis for the concept of filtering, which
we explore in this
14
Demonstration of
Amplitude Modulation
In this lecture, we demonstrate many of the concepts and properties discussed and developed in several of the previous lectures. The demonstration
centers around a number of signals displayed in both the time domai
4
Convolution
In Lecture 3 we introduced and defined a variety of system properties to
which we will make frequent reference throughout the course. Of particular
importance are the properties of linearity and time invariance, both because
systems with the
26
Feedback Example:
The Inverted Pendulum
In this lecture, we analyze and demonstrate the use of feedback in a specific
system, the inverted pendulum. The system consists of a cart that can be
pulled foward or backward on a track. Mounted on the cart is
24
Butterworth Filters
To illustrate some of the ideas developed in Lecture 23, we introduce in this
lecture a simple and particularly useful class of filters referred to as Butterworthfilters. Filters in this class are specified by two parameters, the cu
23
Mapping ContinuousTime Filters to
Discrete-Time Filters
In Lecture 22 we introduced the z-transform. In this lecture we discuss some
of the properties of the z-transform and show how, as a result of these properties, the z-transform can be used to anal