Signals and Systems
Dr. Mohamed Bingabr
University of Central Oklahoma
Some of the Slides For Lathi’s Textbook Provided by
Dr. Peter Cheung
Course Objectives
• Signal analysis (continuous-time)
• System analysis (mostly continuous systems)
• Time-domain analysis (including convolution)
• Laplace Transform and transfer functions
• Fourier Series analysis of periodic signal
• Fourier Transform analysis of aperiodic signal
• Sampling Theorem and signal reconstructions
Outline
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Size of a signal
Useful signal operations
Classification of Signals
Signal Models
Systems
Classification of Systems
System Model: Input-Output Description
Internal and External Description of a System
Size of Signal-Energy Signal
• Signal: is a set of data or information collected
over time.
• Measured by signal energy Ex:
• Generalize for a complex valued signal to:
• Energy must be finite, which means
Size of Signal-Power Signal
•If amplitude of x(t) does not  0 when t  ",
need to measure power Px instead:
•Again, generalize for a complex valued signal
to:
Useful Signal Operation-Time Delay
Find x(t-2) and x(t+2) for the signal
1 t  4
2
x(t )  
 0 elsewhere
x(t)
2
t
1
4
Useful Signal Operation-Time Delay
Signal may be delayed by time T:
(t) = x (t – T)
or advanced by time T:
(t) = x (t + T)
Useful Signal Operation-Time Scaling
Find x(2t) and x(t/2) for the signal
1 t  4
2
x(t )  
 0 elsewhere
x(t)
2
t
1
4
Useful Signal Operation-Time Scaling
Signal may be compressed in time (by a
factor of 2):
(t) = x (2t)
or expanded in time (by a factor of 2):
(t) = x (t/2)
Same as recording played back at
twice and half the speed
respectively
Useful Signal Operation-Time Reversal
Signal may be reflected about the vertical axis (i.e. time
reversed):
(t) = x (-t)
Useful Signal Operation-Example
We can combine these three operations.
For example, the signal x(2t - 6) can be obtained in two ways;
• Delay x(t) by 6 to obtain x(t - 6), and then time-compress this
signal by factor 2 (replace t with 2t) to obtain x(2t - 6).
• Alternately, time-compress x(t) by factor 2 to obtain x(2t), then
delay this signal by 3 (replace t with t - 3) to obtain x(2t - 6).
Signal Classification
Signals may be classified into:
1. Continuous-time and discrete-time signals
2. Analogue and digital signals
3. Periodic and aperiodic signals
4. Energy and power signals
5. Deterministic and probabilistic signals
6. Causal and non-causal
7. Even and Odd signals
Signal Classification- Continuous vs Discrete
Continuous-time
Discrete-time
Signal Classification- Analogue vs Digital
Analogue, continuous
Analogue, discrete
Digital, continuous
Digital, discrete
Signal Classification- Periodic vs Aperiodic
A signal x(t) is said to be periodic if for some positive constant To
x(t) = x (t+To)
for all t
The smallest value of To that satisfies the periodicity condition of
this equation is the fundamental period of x(t).
Signal Classification- Deterministic vs Random
Deterministic
Random
Signal Classification- Causal vs Non-causal
Signal Classification- Even vs Odd
Signal Models – Unit Step Function u(t)
Step function defined by:
Useful to describe a signal that begins at t = 0 (i.e. causal signal).
For example, the signal e-at represents an
everlasting exponential that starts at t = -.
The causal for of this exponential e-atu(t)
Signal Models – Pulse Signal
A pulse signal can be presented by two step functions:
x(t) = u(t-2) – u(t-4)
Signal Models – Unit Impulse Function δ(t)
First defined by Dirac as:
Multiplying Function  (t) by an Impulse
Since impulse is non-zero only at t = 0, and (t) at t = 0
is (0), we get:
We can generalize this for t = T:
Sampling Property of Unit Impulse Function
Since we have:
It follows that:
This is the same as “sampling” (t) at t = 0.
If we want to sample (t) at t = T, we just multiple (t)
with
This is called the “sampling or sifting property” of the
impulse.
Examples
Simplify the following expression
 1 

 (  3)
 j  2 
Evaluate the following

t

(
t

3
)
e
dt


Find dx/dt for the following signal
x(t) = u(t-2) – 3u(t-4)
The Exponential Function est
This exponential function is very important in signals &
systems, and the parameter s is a complex variable
given by:
The Exponential Function est
If  = 0, then we have the function ejωt, which has a
real frequency of ω
Therefore the complex variable s =  +jω is the
complex frequency
The function est can be used to describe a very large
class of signals and functions. Here are a number of
example:
The Exponential Function est
The Complex Frequency Plane s= + jω
A real function xe(t) is said to be an even function of t if
A real function xo(t) is said to be an odd function of t if
HW1_Ch1: 1.1-3, 1.1-4, 1.2-2(a,b,d), 1.2-5, 1.4-3, 1.4-4, 1.4-5, 1.4-10 (b, f)
Even and Odd Function
Even and odd functions have the following properties:
• Even x Odd = Odd
• Odd x Odd = Even
• Even x Even = Even
Every signal x(t) can be expressed as a sum of even and
odd components because:
Even and Odd Function
Consider the causal exponential function
What are Systems?
•Systems are used to process signals to modify or extract
information
•Physical system – characterized by their input-output
relationships
•E.g. electrical systems are characterized by voltage-current
relationships
•From this, we derive a mathematical model of the system
•“Black box” model of a system:
Classification of Systems
Systems may be classified into:
1. Linear and non-linear systems
2. Constant parameter and time-varying-parameter systems
3. Instantaneous (memoryless) and dynamic (with memory)
systems
4. Causal and non-causal systems
5. Continuous-time and discrete-time systems
6. Analogue and digital systems
7. Invertible and noninvertible systems
8. Stable and unstable systems
Linear Systems (1)
•A linear system exhibits the additivity property:
if
and
then
•It also must satisfy the homogeneity or scaling property:
if
then
•These can be combined into the property of superposition:
if
and
then
•A non-linear system is one that is NOT linear (i.e. does not
obey the principle of superposition)
Linear Systems (2)
Linear Systems (3)
Linear Systems (4)
Is the system y = x2 linear?
Linear Systems (5)
A complex input can be represented as a sum of simpler inputs
(pulse, step, sinusoidal), and then use linearity to find the
response to this simple inputs to find the system output to the
complex input.
Time-Invariant System
Which of the system is time-invariant?
(a) y(t) = 3x(t)
(b) y(t) = t x(t)
Instantaneous and Dynamic Systems
Causal and Noncausal Systems
Which of the two systems is causal?
a) y(t) = 3 x(t) + x(t-2)
b) y(t) = 3x(t) + x(t+2)
Analogue and Digital Systems
Invertible and Noninvertible
Which of the two systems is invertible?
a) y(t) = x2
b) y= 2x
System External Stability
Electrical System
R
i(t)
i(t)
+
v(t)
-
i(t)
+ v(t) -
+ v(t) -
v(t )  R i(t )
i (t )  C
dv
dt
v (t )  L
di
dt
Mechanical System
2
d
y
x(t )  M y (t )  M 2
dt
x(t )  k y(t )
x(t )  B y (t )  B
dy
dt
Linear Differential Systems (1)
Linear Differential Systems (2)
Find the input-output relationship for the transational
mechanical system shown below. The input is the force x(t),
and the output is the mass position y(t)
Linear Differential Systems (3)
Linear Differential Systems (4)
HW2_Ch1: 1.7-1 (a, b, d), 1.7-2 (a, b, c), 1.7-7, 1.7-13, 1.8-1, 1.8-3