(section 1471x) discrete-time signals and systems · discrete-time signals and systems lecture 2...

EEL 3135EEL 3135(Section 1471X)(Section 1471X)

DiscreteDiscrete--Time Signals and SystemsTime Signals and Systems

Lecture 2Lecture 2

Prof. Jian Li

Dept. of Electrical and Computer EngineeringUniversity of Florida,Gainesville, FL 32611

Office: 437 EBPhone: 352-392-2642 Email: li@dsp.ufl.edu

Dept. of ECE 2

Topics

Signals

Transformations

Even and Odd

Complex Numbers

Cartesian and Polar Forms

Sinusoids

Properties

Phasors

Dept. of ECE 3

Signals

Continuous-Time Signal x(t)

with t being a continuous variable.

Discrete-Time Signals x[n] = x(nTs)

with n being a discrete value.

Dept. of ECE 4

Transformations of Time Variable

x(t-t0) means shifting to right t0 units

Dept. of ECE 5

More Transformations (2)

x(-t) means reflection about t = 0

Dept. of ECE 6

More Transformations (3)

x(2t) means compressing

x(t/2) means stretching

Dept. of ECE 7

More Transformations (4)

Dept. of ECE 8

Periodic Signals

x(t) = x(t-T)

T = period, so is 2T, 3T, ….

Fundamental period T0 is the smallest possible period.

Dept. of ECE 9

Even and Odd SignalsEven and Odd Signals

Even Signal:

x(t) = x(-t)

Odd Signal:

x(-t) = -x(t)

Dept. of ECE 10

More Even and Odd SignalsMore Even and Odd Signals

Any signal can be

broken into:

Dept. of ECE 11

SinusoidsSinusoids

Sinusoids occur in: music, phone, radar, communication, ….

(a) Before (b) After

ISAR autofocusing example

Dept. of ECE 12

SinusoidsSinusoids

AmplitudeRadian freq

Phase

Cycles per second (Hz)

Dept. of ECE 13

Complex NumbersComplex Numbers

Cartesian form:

Polar form:

Relationships:

Dept. of ECE 14

Euler Euler Formulas and Formulas and PhasorsPhasors

Euler Formula:

Inverse Euler Formulas:

Sinusoid:

Phasor (Complex Amplitude):

Positive freq Negative freq

Dept. of ECE 15

Phasor Phasor Addition RuleAddition Rule

Sum of sinusoids with frequency

is another sinusoid with same frequency

Dept. of ECE 16

Phasor Phasor Addition ExampleAddition Example

Dept. of ECE 17

DiscreteDiscrete--Time SinusoidsTime Sinusoids

Larger does not mean faster oscillation for

Dept. of ECE 18

DiscreteDiscrete--Time SinusoidsTime Sinusoids

Discrete-Time Sinusoids are not always periodic!

Dept. of ECE 19

AssignmentsAssignments

Finish Lecture 1 Assignments

Read Chapter 1

Read Appendix A on Complex Numbers

Read Chapter 2 as much as possible

Implement Lab. C.2 in Appendix C as much as possible

Dept. of ECE 20

THANK YOU

And

Happy Learning!!