1. Signals and Systems Dr. Mohamed Bingabr University of Central Oklahoma
Some of the Slides For Lathi’s Textbook Provided by
Dr. Peter Cheung

2. 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

3. 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

4. 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

5. 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:

6. Useful Signal Operation-Time Delay
Find x(t-2) and x(t+2) for the signal
x(t) 2 t 1 4

7. 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)

8. Useful Signal Operation-Time Scaling
Find x(2t) and x(t/2) for the signal
x(t) 2 t 1 4

9. 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

10. Useful Signal Operation-Time Reversal
Signal may be reflected about the vertical axis (i.e. time reversed):
(t) = x (-t)

11. 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).

12. 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

13. Signal Classification- Continuous vs Discrete
Continuous-time
Discrete-time

14. Signal Classification- Analogue vs Digital
Analogue, continuous
Digital, continuous
Analogue, discrete
Digital, discrete

15. 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).

16. Signal Classification- Deterministic vs Random

17. Signal Classification- Causal vs Non-causal

18. Signal Classification- Even vs Odd

19. 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)

20. Signal Models – Pulse Signal
A pulse signal can be presented by two step functions:
x(t) = u(t-2) – u(t-4)

21. Signal Models – Unit Impulse Function δ(t)
First defined by Dirac as:

22. 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:

23. 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.

24. Examples Simplify the following expression Evaluate the following
Find dx/dt for the following signal
x(t) = u(t-2) – 3u(t-4)

25. The Exponential Function est
This exponential function is very important in signals & systems, and the parameter s is a complex variable given by:

26. 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:

27. The Exponential Function est

28. 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, , (a,b,d), 1.2-5, , , , (b, f)

29. 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:

30. Even and Odd Function
Consider the causal exponential function

31. 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:

32. Classification of Systems
Systems may be classified into:
Linear and non-linear systems
Constant parameter and time-varying-parameter systems
Instantaneous (memoryless) and dynamic (with memory) systems
Causal and non-causal systems
Continuous-time and discrete-time systems
Analogue and digital systems
Invertible and noninvertible systems
Stable and unstable systems

33. 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)

34. Linear Systems (2)

35. Linear Systems (3)

36. Linear Systems (4)
Is the system y = x2 linear?

37. 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.

38. Time-Invariant System
Which of the system is time-invariant?
(a) y(t) = 3x(t) (b) y(t) = t x(t)

39. Instantaneous and Dynamic Systems

40. 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)

41. Analogue and Digital Systems

42. Invertible and Noninvertible
Which of the two systems is invertible?
y(t) = x2
y= 2x

43. System External Stability

44. Electrical System
+ v(t) -
i(t) R +
v(t) -
i(t)
i(t)
+ v(t) -

45. Mechanical System

46. Linear Differential Systems (1)

47. 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)

48. Linear Differential Systems (3)

49. Linear Differential Systems (4)
HW2_Ch1: (a, b, d), (a, b, c), 1.7-7, , 1.8-1, 1.8-3