Lecture #2. EE 313 Linear Systems and Signals

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1 Lecture #2 EE 313 Linear Systems and Signals

2 Preview of today s lecture What is a signal and what is a system? o Define the concepts of a signal and a system o Why? This is essential for a course on Signals and Systems Signal transformations o Time-shift, reflection, and compression o Why? Allows us to describe signals and build other signals Important signal properties o Periodicity, even, and odd o Why? Special signals have special properties Exponential and sinusoidal signals o Why? The signals we will use most often in the course, and which often describe the response of real-world systems 2

3 What is a signal and what is a system? Learning objectives o o Explain what is meant by the term signal and give examples Explain what is meant by the term system and give examples

4 What is a signal? u Representation of a value/info. relative to an independent variable ª Often a time variable (EE313) but could be something else u Two main classes of signals in EE313: ª Continuous-time (CT) signals x(t) x(t) t ª Discrete-time (DT) signals x[n] Often obtained by sampling CT signal Taken at values nt x[n] n 4

5 Signal example: Temperature x(t) x[n] 5 a.m. 4 p.m. 9 p.m. t n u Computing the average temperature integral vs sum 5

6 Signal example: Voltage received by a cell phone antenna v(t) P(t) t ª t u Average power ª CT: DT: P 1 t 2 t 1 ÿ ª t2 ` t 1 P ptqdt P 1 N K`N 1 ÿ n K P rns (units not same) P[n] = p(nt) 6

7 ÿ ÿ Complex signals u Signals generally take complex values (equivalently they have an amplitude and a phase) p q t p qu ` t p qu xptq Retxptqu ª ` jimtxptqu r s t r su ` t r su xrns Retxrnsu ` jimtxrnsu u Complex signals ÿ are found in many practical problems ª Most digital signal processors support complex operations u Example: in phase (real) and quadrature (imag) electromagnetic signals. rides a cosine Many connections between xptq Retxptqu ` jimtxptqu complex signals, Maxwell s r s t r su ` t r su rides a sine equations, phasors, etc. 7

8 What is a system? p q input x(t) system yptq output mathematical description of how the input is transformed into the output convolution + - x(t) input system + y(t) - output transfer function frequency response 8

9 System example: filters u Optical filter Light Red u Digital filter Lens Audio Equalizer Better sounding signal 9

10 Systems can be complicated with many sub-systems u Cell phone Voice/ Audio Vocoder Modulation /Coding Digital- to- Analog CT signal (speech) DT signal DT signal CT signal (voltage) RF CT signal (Emag wave) 10

11 In summary u A signal is a function ª Representation of a value as a function of some index (usually time) ª May be in continuous or discrete time ª Discrete-time often created from sampling continuous-time ª Derived from measurements or models u A system is an operation on an input signal to generate an output ª Abstracted by block diagrams ª Each block represents a certain functions/process on its input ª Characterized (later) based on their properties 11

12 Basic signal transformations Learning objectives o o Apply different transformations on continuous and discrete signals Create new continuous and discrete signals from these transformations

13 Moving mountains (function is over space, here) height? mass? location? u Use mountain description to build a map ª Where are the mountains located? ª What are their size? Signal transformations are ways to describe and manipulate signals from clipartpanda.com 13

14 Example: time shift A x(t) right shift A 0 T t 0 t x[n] x[n+3] x[n+3] n left shift n 14

15 Example: reflection or time reversal A x(t) x(-t) A x(-t) 0 T t -T 0 t x[n] x[-n] x[-n] n n 15

16 Example: time compression / downsampling A x(t) x(2t) A x(2t) 0 T t 0 T/2 t x[n] x[2n] x[2n] n -1 n 16

17 Example: time expansion (upsampling) A x(t) x(t/2) A x(t/2) 0 T t 0 2T t x[n] (complicated operation, but this is the correct notion) x[n] after upsampling n n 17

18 Example u Let x[n] be a signal with x[n]=0 for n < -2 and n > 4. For the signal below, determine the values of n for which it is guaranteed to be zero u Solution u Hint. Always do operations in this order: ª Shift ª Flip (or not) ª Scale 18

19 In summary u There are different ways to transform a signal ª This section focused on transformations of the independent variable u Transformations ª Time shifting changes the starting point of a signal ª Time scaling changes how fast the signal is played ª Time reversal flips a signal ª Compression / expansion change the speed of a signal 19

20 Signal characteristics: periodic, even, and odd Learning objectives o o o Distinguish between periodic and aperiodic signals Compute the period of a periodic signal Compute even and odd parts of an arbitrary signal

21 Periodic signals u Periodic signals satisfy for some finite non-zero T or N xptq xpt T q p q p ` q r s r s xrns xrn ` Ns periodic with period T periodic with period N (integer) u Period is the smallest non-zero value for which the signal is periodic ª This is called the fundamental period ª Normally period means fundamental period Periodic signals can be treated with special mathematical tools 21

22 Examples of periodic signals Dots indicate that the signal continues x(t) A t x[n] n 22

23 Solution u Determine whether or not the following signal is periodic? If it is periodic, determine its fundamental period. xptq rcosp2t {3qs 2 u Recall that cos 2 (x) = ½ (1 + cos 2x) u By showing this simplifies to a known periodic function, we can also conclude it is period and find the frequency 23

24 Related definition: even signals u Even signals satisfy u Examples (need not be periodic) x(t) x[n] A -T T t n 24

25 Related definition: odd signals u Odd signals satisfy u Examples x[n] x(t) 1 A -T T t -1 x(t)=sgn(t) -A 1 n -1 t 25

26 Interesting facts about even and odd functions u Any function can be written in terms of its even and off parts 26

27 Interesting facts about even and odd functions u Integration properties u Multiplications of two odd or two even functions à even function 27

28 Example u Determine and sketch the even and odd parts of the signal 28

29 Sketching the even solution plus e(t) = ½ (x(t) + x(-t)) 29

30 Sketching the odd solution o(t) = ½ (x(t) - x(-t)) minus 30

31 In summary u Periodic signals ª Special type of signals that repeat ª Need to determine if a signal is periodic and its period u Even and odd signals ª Signals with symmetry about the y axis ª Fact used to simplify computations and derive intuition u These special signal structures will be used in Fourier analysis 31

32 CT exponential and sinusoidal signals Learning objectives o o Explain the properties of exponential and sinusoidal signals Analyze problems that include exponential and sinusoidal signals

33 CT complex exponential: general case u The constants C and a are complex in general ª Can be decomposed to provide more intuition C c ` jc!, C e j, a r ` j! 0, cartesian polar cartesian p q 33

34 Familiar special case: real exponential signal C and a are real Increasing exponential (a > 0) Decaying exponential (a<0) 34

35 Special case: constant amplitude complex sinusoid u Using Euler s identity xptq e jp!t` q! ª is the frequency of the sinusoid ª is the phase ª is the period u Larger is the p T 2! pulse and! cosp!t ` q`j sinp!t ` q leads to higher frequency and shorter period 35

36 Visualizing CT complex exponentials Increasing or decaying exponential envelope Complex sinusoid r < 0 r > 0 36

37 In summary u Exponentials & sinusoids ª Intimately related through Euler s identity ª Important in both continuous and discrete-time ª Will become close friends by the end of the course J u You should be able to ª Determine the period ª Plot, shift, and scale real and complex exponentials 37

Continuous time and Discrete time Signals and Systems

Continuous time and Discrete time Signals and Systems 1. Systems in Engineering A system is usually understood to be an engineering device in the field, and a mathematical representation of this system