Signals and Systems
Dr. Mohamed Bingabr
University of Central OklahomaSlides 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
• Size of a signal• Useful signal operation• 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
• 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
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
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 attwice and half the speedrespectively
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 thissignal 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 signals2. Analogue and digital signals3. Periodic and aperiodic signals4. Energy and power signals5. Deterministic and probabilistic signals6. Causal and non-causal7. 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:
Signal Models – Unit Impulse Function
May use functions other than a rectangular pulse. Here are three example functions:
Exponential Triangular Gaussian
Note that the area under the pulse function must be unity
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.
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 andodd 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 systems2. Constant parameter and time-varying-parameter systems3. Instantaneous (memoryless) and dynamic (with memory)
systems4. Causal and non-causal systems5. Continuous-time and discrete-time systems6. Analogue and digital systems7. Invertible and noninvertible systems8. 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)
Linear Systems (5)
Time-Invariant System
Instantaneous and Dynamic Systems
Causal and Noncausal Systems
Analogue and Digital Systems
Invertible and Noninvertible
Invertible and Noninvertible
Linear Differential Systems (1)
Linear Differential Systems (2)
Linear Differential Systems (3)
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