Communication Theory II

Transcription

1 Cmmunicatin Thery II Lecture 3: Review n Furier analysis (cnt d) Ahmed Elnakib, PhD Assistant Prfessr, Mansura University, Egypt Febraury 15 th,

2 Sectin plicy Gal: slve assciated prblems related t the tpic f the lectures Assistant lectures: help yu in slving the prblem Each student will slve the assigned questins and the assistant will help yu if yu have any questins Needed materials: 1) The sheet, Furier tables (sfcpy r hardcpy) 2) A ntebk, a pencil, and an eraser 2

3 Curse Website The site cntains the lectures, quizzes, hmewrk, and pen frums fr feedback and questins Lg in using yur name and passwrd Passwrd fr quizzes: third 3

4 Circuit switching vs. packet switching (a) (b) Circuit switching (link sharing is based n fixed allcatin) vs. packet switching (message is divided int segments r packets if necessary, sharing is per demand) * * Image curtesy f 4

5 Lecture Outlines Review n Furier analysis f signals and systems The Furier series The Furier transfrm Relatin between time and frequency representatins The Dirac delta functin Furier transfrm f peridic signals Transmissin d signals thrugh LTI systems Hilbert transfrm 5

6 The Furier series Let dente a peridic signal, where the subscript T 0 dentes the duratin f peridicity. By using a Furier series expansin f this signal, we are able t reslve it int an infinite sum f sine and csine terms: Where is the fundamental frequency represent the amplitudes f the csine and sine terms, respectively are called basis functins 6

7 Basis functins ( ) Frm an rthgnal set ver the perid T 0 In that they satisfy three cnditins: 7

8 Mean value (a 0 ) and ther cefficients Integrate ver a cmplete perid bth sides f the equatin: yields: Other cefficients: 8

9 Existence f Furier series A peridic signal can be expanded in a Furier series if the signal satisfies the Dirichlet cnditins: Frm an engineering perspective, hwever, it suffices t say that the Dirichlet cnditins are satisfied by the peridic signals encuntered in cmmunicatin systems 9

10 Cmplex expnential Furier series Where Inner prduct f the signal with the basis functin [ ] 10

11 The Furier Transfrm Representatin f nn-peridic signals Derivatin is based n Furier series (text bk: page 16,17) 11

12 Existence f Furier Transfrm Physical realizability is a sufficient cnditin fr the existence f a Furier transfrm (e.g., all energy signals are Furier transfrmable ). Cnditin fr energy signal: 12

13 Energy vs. pwer signals 13

14 Furier Transfrm Prvides mathematical tls fr measuring the frequency cntent r spectrum f a signal G(f) is the Furier transfrm r the spectrum f the signal Simple evaluatin f magnitude spectrum G(f) and phase spectrum G(f) 14

15 Tables 15

16 16

17 Relatin between time and frequency respnse If the time-dmain descriptin f a signal is changed, the frequency-dmain descriptin f the signal is changed in an inverse manner, and vice versa. If a signal is strictly limited in frequency, then the time-dmain descriptin f the signal will trail n indefinitely, even thugh its amplitude may assume a prgressively smaller value. A signal is strictly limited in frequency (i.e., strictly band limited) if its Furier transfrm is exactly zer utside a finite band f frequencies. If a signal is strictly limited in time (i.e., the signal is exactly zer utside a finite time interval), then the spectrum f the signal is infinite in extent, even thugh the magnitude spectrum may assume a prgressively smaller value. A signal cannt be strictly limited in bth time and frequency Time bandwidth prduct: duratin bandwidth = cnstant 17

18 The bandwidth dilemma The bandwidth f a signal prvides a measure f the extent f significant spectral cntent f the signal fr psitive frequencies. When the signal is strictly band limited, the bandwidth is well defined. Fr example, the sinc pulse sinc(2wt) has a bandwidth equal t W. If a signal is lw-pass (i.e., its spectral cntent is centered arund the rigin f = 0), the bandwidth is defined as ne-half the ttal width f the main spectral lbe, since nly ne-half f this lbe lies inside the psitive frequency regin rect t f) B.W=1/T T -0.5T 0.5T -1/T 1/T 18

19 The bandwidth dilemma (cn d) If the signal is band-pass with main spectral lbes centered arund ±fc where fc is large enugh, the bandwidth is defined as the width f the main lbe fr psitive frequencies. This definitin f bandwidth is called the null-t-null bandwidth. B.W=2/T Magnitude spectrum f the RF pulse, shwing the null-t-null bandwidth t be 2/T, centered n the mid-band frequency fc 19

20 3 db bandwidth The 3 db bandwidth f a lw-pass signal is defined as the separatin between zer frequency, where the magnitude spectrum attains its peak value, and the psitive frequency at which the amplitude spectrum drps t f its peak value. 20

21 The Dirac delta functin Evlutin f the sinc functin 2W sinc(2wt) tward the delta functin as the parameter W prgressively increases 21

22 Furier transfrm f peridic signals Can Furier transfrm wrks fr peridic signals? 22

23 Furier transfrm fr analyzing signals and systems Prvides the mathematical link between the time dmain f a signal (wavefrm) and its frequency dmain (spectrum) Time and frequency respnse f a linear time-invariant (LTI) system defined in terms f its impulse respnse and frequency respnse, respectively F F F -1 23

24 LTI system If and x 1 (n) h(n) y 1 (n) x 2 (n) h(n) y 2 (n) If x 1 (n) h(n) y 1 (n) Then x 1 (n-k) h(n) y 1 (n-k) Then a 1 y 1 (n)+a 2 y 2 (n)=h{a 1 x 1 (n)+a 2 x 2 (n)} Where a1, a2 are scalars (a) (b) A (a) linear and (b) a time invariant system 24

25 Questins 25