1. COSC 3451: Signals and Systems
Course Instructor: Amir Asif
Teaching Assistant: TBA
Contact Information: Instructor: Teaching Assistant:
Office: CSB Durwas, Neha
(416) X70128
URL:
Text: A. V. Oppenheim and A. S. Willsky with S. H. Nawab, Signals and
Systems, NY: Prentice Hall, 1997.
Class Schedule: TR 13: :30  (CB 115)
Assessment: Assignment / Quiz: 20%
Projects: 15%
Mid-term: 25%
Final: 40%
Office Hours: Instructor: CSB 3028, TR 12:00 – 13:00 TA: TBA

2. Course Objectives
Introduce CT and DT signals and systems in terms of physical phenomena
Review elementary signals and see how these can be represented in domains other than time
Understand different transforms (Fourier, Laplace, Z) used to analyze signals
Complete analysis of a linear time invariant (LTI) systems in time and other domains
Design filters (systems) to process a signal for different applications
Present state-of-art technologies used in communication systems
Use computers for digital signal processing
Demystify terminology !!!!!!

3. What is a Signal? Signal is a waveform that contains information
System is a model for physical phenomena that generates, processes, or receive signals.
Speech Waveform
x(t): Continuous Time

4. Examples of Signals Image I[m,n]: 2D DT signal (a) 1D CT signal x(t)
(b) 1D DT signal x[n]
Image
I[m,n]: 2D DT signal
Activity 1: For each of the representations: (a) z[m,n,k] (b) I(x,y,z,t), establish if the signal is CT or DT. Specify the independent and dependent variables. Also, think of an example from the real world that will have the same mathematical representation.

5. Power vs. Energy Signals (1)
Activity 2:
a. Consider the sinusoidal signal x(t) = cos(0.5pt). Choosing T = 4, determine the average power of x(t).
b. Consider the signal x(t) = 5 sin(2pt) for the interval -1 <= t <= 1 and is 0 elsewhere. Calculate the energy of x(t).
c. Calculate the energy of the signal x[n] = (0.8)n for n >= 0 and is 0 elsewhere.

6. Power vs. Energy Signals (2)
Energy Signals: have finite total energy for the entire duration of the signal. As a consequence, total power in an energy signal is 0.
Power Signals: have non-zero power over the entire duration of the signal. As a consequence, the total energy in a power signal is infinite.
Activity 3:
Classify the signals defined in Activity 2 as a power or an energy signal.

7. Linear Transformations
There are two types of linear transformations that we will consider
1. Time Shifting:
2. Scaling:

8. Linear Transformations (2)
Scaling for DT Signals:
Note that y[n] = x[n/2] is not completely defined

9. Linear Transformations (3)
Precedence Rule:
Establishes the order of shifting and scaling in relationships involving both shifting and scaling operations.
Time-shifting takes precedence over time-scaling. In the above representation, the time shift b is performed first on x(t), resulting in an intermediate signal v(t) defined by
v(t) = x(t + b)
followed by time-scaling by a factor of a, that is,
y(t) = v(at) = x(at + b).
Example: Sketch y(t) = x(2t + 3) for x(t) given in (a)

10. Linear Transformations (4)
For the DT signal x[n] illustrated below:
Activity 4: Draw the following:
(a) x[-n]; (b) x[2n]; (c) x[n + 3 ]; (d) x[2n + 3]

11. Periodic Signals
Periodic signals: A periodic signal x(t) is a function of time that satisfies the condition
x(t) = x(t + T) for all t
where T is a positive constant number and is referred to as the fundamental period of the signal.
Fundamental frequency (f) is the inverse of the period of the signal. It is measured in Hertz (Hz =1/s).
Nonperiodic (Aperiodic) signals: are those that do not repeat themselves.
Activity 5: For the sinusoidal signals (a) x[n] = sin (5pn) (b) y[n] = sin(pn/3), determine the fundamental period N of the DT signals.

12. Even vs. Odd Signals (1)
Even Signal: A CT signal x(t) is said to be an even signal if it satisfies the condition
x(-t) = x(t) for all t.
Odd Signal: The CT signal x(t) is said to be an odd signal if it satisfies the condition x(-t) = -x(t) for all t.
Even signals are symmetric about the vertical axis or time origin.
Odd signals are antisymmetric (or asymmetric) about the time origin.

13. Even vs. Odd Signals (2)
Signals that satisfy neither the even property nor the odd property can be divided into even and odd components based on the following equations:
Even component of x(t) = 1/2 [ x(t) + x(-t) ]
Odd component of x(t) = 1/2 [ x(t) - x(-t) ]
Activity 6: For the signal
do the following:
(a) sketch the signal (b) evaluate the odd part of the signal (c) evaluate the even part of the signal.

14. CT Exponential Signals (1).
CT exponential signals are of the form
where C and a can both be complex numbers.
Characteristic C a
Real Exponential Signals
real
Periodic Complex Exponential
imaginary
General Complex Exponential
complex

15. CT Exponential Signals: Real (2).
Real CT exponential signals:
where C and a are both positive numbers.

16. CT Exponential Signals: Periodic Complex (3).
Periodic complex exponential signals:
where C is a positive number but (a = j wo) is an imaginary number.
Difficult to draw. Magnitude and phase are drawn separately.
Activity 7: Show that the periodic complex signal x(t) = C exp(jwot) has a fundamental period given by:

17. CT Exponential Signals: Sinusoidal (4).
Sinusoidal Signals:
where C is real and a is complex.
Types of Sinusoidal Signals:

18. CT Exponential Signals: Sinusoidal (5).
Activity 8: Sketch the following sinusoidal signal
What is the value of the magnitude, fundamental frequency, fundamental phase, and power for the sinusoidal signal.
Activity 9: Show that the power of a sinusoidal signal
is given by A2 / 2.

19. CT Exponential Signals: Complex (6).
General Complex Exponential signals:
where C and a are both complex numbers.
By substituting C = |C|ejq and a = (r + jwo), the magnitude and phase of x(t) can be expressed as

20. CT Exponential Signals (7).
Activity 10: Derive and plot the magnitude and phase of the composite signal

21. DT Exponential Signals (1).
DT exponential signals are of the form
where C and a can both be complex numbers.
Characteristic C a
Real Exponential Signals
Real
Sinusoidal Signals
Imaginary (a = jw)
General Complex Exponential
Complex

22. DT Exponential Signals: Real (2).
For real exponential signals, C and a are both real
Depending upon the value of a, a different waveform is produced.
a > 1 gives a rising exponential
a = 1 gives a constant line
0 < a < 1 gives a decaying exponential
-1 < a < 0 gives an alternating sign decaying exponential
a < -1 gives an alternating sign rising exponential

23. DT Exponential Signals: Sinusoidal (3).
Sinusoidal Signals:
where C is real and a is complex.
Types of Sinusoidal Signals:
Not all DT sinusoidal signals are periodic

24. x[n] =cos(pn/3) sin(pn/6).
DT Exponential Signals: Sinusoidal (4) .
Condition for periodicity: A DT sinusoidal signal
is periodic if (wo / 2p) is a rational number with the period given by
Activity 11: Consider the signal
x[n] =cos(pn/3) sin(pn/6).
Determine if the signal is periodic. If yes, calculate the period N.

25. Unit Step Function DT domain: CT domain
Activity: For the discrete time signal
Describe x[n] as a function of two step functions. Ans: U[n + 5] - U[n - 10]

26. Unit Sample (Impulse) Function
DT domain:
CT domain: Impulse function is defined as

27. Gate Function
DT domain:
CT domain

28. Ramp Function
DT domain:
CT domain