signals and systems dr. mohamed bingabr university of central oklahoma some of the slides for...
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Signals and Systems
Dr. Mohamed Bingabr
University of Central OklahomaSome of the Slides For Lathi’s Textbook Provided by
Dr. Peter Cheung
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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
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Outline
• 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
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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
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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:
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Useful Signal Operation-Time Delay
Find x(t-2) and x(t+2) for the signal
elsewhere
ttx
0
412)(
1 4
2
t
x(t)
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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)
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Useful Signal Operation-Time Scaling
Find x(2t) and x(t/2) for the signal
elsewhere
ttx
0
412)(
1 4
2
t
x(t)
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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
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Useful Signal Operation-Time Reversal
Signal may be reflected about the vertical axis (i.e. time reversed):
(t) = x (-t)
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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).
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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
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Signal Classification- Continuous vs Discrete
Continuous-time
Discrete-time
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Signal Classification- Analogue vs Digital
Analogue, continuous
Analogue, discrete
Digital, continuous
Digital, discrete
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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).
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Signal Classification- Deterministic vs Random
Deterministic
Random
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Signal Classification- Causal vs Non-causal
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Signal Classification- Even vs Odd
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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)
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Signal Models – Pulse Signal
A pulse signal can be presented by two step functions:
x(t) = u(t-2) – u(t-4)
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Signal Models – Unit Impulse Function δ(t)
First defined by Dirac as:
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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:
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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.
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Examples
Simplify the following expression
)3(2
1
j
Evaluate the following
dtet t)3(
Find dx/dt for the following signal
x(t) = u(t-2) – 3u(t-4)
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The Exponential Function est
This exponential function is very important in signals & systems, and the parameter s is a complex variable given by:
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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:
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The Exponential Function est
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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)
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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:
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Even and Odd Function
Consider the causal exponential function
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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:
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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
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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)
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Linear Systems (2)
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Linear Systems (3)
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Linear Systems (4)
Is the system y = x2 linear?
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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.
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Time-Invariant System
Which of the system is time-invariant? (a) y(t) = 3x(t) (b) y(t) = t x(t)
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Instantaneous and Dynamic Systems
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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)
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Analogue and Digital Systems
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Invertible and Noninvertible
Which of the two systems is invertible?a) y(t) = x2
b) y= 2x
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System External Stability
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Electrical System
)()( tiRtv dt
dvCti )(
dt
diLtv )(
Ri(t)
+ v(t) -
i(t) + v(t) -
+ v(t) -i(t)
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Mechanical System
2
2
)()(dt
ydMtyMtx )()( tyktx
dt
dyBtyBtx )()(
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Linear Differential Systems (1)
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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)
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Linear Differential Systems (3)
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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