6.071 Spring 2006, Chaniotakis and Cory 1

Transcription

1 Signals and Sysems: Maerial for he classes on: //6 /4/6 /6/6 The goals of he following hree classes are: Define and explore various ypes of signals Explore he concep of a sysem and define LTI sysems Explore ime and frequency domain represenaion of signals Review Fourier series/ransform. Focus on heir physical/pracical significance Sampling and Nyquis raes. The phenomenon of aliasing. Numbering sysems Conversion beween ypes of signals 6.7 Spring 6, Chanioakis and Cory

2 A signal represens a se of one or more variables and is used o convey he characerisic informaion (or he aribues) of a physical phenomenon. The world around us is full of signals. Indeed our connecion wih he world is hrough he various signals ha our senses can inerpre for heir corresponding physical phenomena: he human voice, he sounds of naure, he ligh we see, he hea we feel, are all signals. The classificaion of a signal is based on: () how is i represened in ime and () how is is ampliude allowed o vary. There are four basic ypes of signals based on he above classificaion. They are: Coninuous ime, coninuous value. Defined for each insan of ime and is ampliude may vary coninuously wih ime and assume any value. o Signals from ransducers o Analog signals Discree ime, coninuous value. Defined a discree insans of ime and is ampliude may vary coninuously wih ime and assume any value Coninuous ime, discree value. Defined for each insan of ime and is ampliude may assume discree values. o Signal is sampled a discree imes and he oupu assumes discree values Discree ime, discree value. Defined a discree insans of ime and is oupu may assume discree values o Digial signals 6.7 Spring 6, Chanioakis and Cory

3 In general we will use ime as he independen variable when we represen a signal. This is appropriae in he sudy of elecrical and elecronic sysems bu here are many oher cases in which signals depend on some oher variable. For example, in some engineering applicaions he signal may be he pressure along a pipe or i migh be he pressure profile on an airplane wing or i migh be he emperaure profile across he cross secion of a fuel rod of a nuclear reacor. In his class we will focus on elecrical signals (volage, curren, energy) ha vary in ime. An imporan class of ime-varying signals is he periodic signal. Mahemaically, a periodic signal x() is one ha saisfies he equaion x () = x ( + nt), forn=,,3, (.) Where T is he period of he signal x(). In our sudy of elecronic sysems we will encouner periodic signals of various ypes. Some of he mos common are shown schemaically on Figure (a) (c) (b) (d) (e) (f) Figure. (a) sine wave signal, (b) square wave signal, (c) pulse rain signal, (d) riangular wave signal, (e) sawooh signal, (f) arbirary periodic signal wih noise one period shown 6.7 Spring 6, Chanioakis and Cory 3

4 Before proceeding le s define and calculae some of he mos relevan parameers describing a signal.. The mos frequenly encounered signal, he generic sinusoidal signal, is given by he funcion, x () = Α sin( ω+ φ) (.) In he sudy of elecronics we encouner his signal very frequenly where x() may represen a volage, a curren or energy. The parameers describing he signal of Eq. (.) are: A - he ampliude, ω - he radian frequency, and φ - he phase. The radian frequency ω is given in unis of radians/sec and is relaed o he frequency f given in cycles/sec. or Hz by The period T of he signal is ω = π f (.3) π T = = (.4) f ω The phase φ represens a shif of he signal relaive o origin ( = ). Figure illusraes he various parameers jus described in a graphical fashion. - - φ= φ=π/3.5 (sec). Figure. Sinusoidal signal wih a phase of degrees and 6 degrees. In many applicaions involving ime-varying signals, he relevan measuremen parameers migh be an average values of he signal. The elecrical signal delivering he sandard Vol household elecriciy is a good example. The household elecrical signal is a sinusoid wih a frequency of eiher 6 or 5 Hz depending on locaion. The Vols correspond o an average value of he signal and no o is ampliude. Figure 3 shows he ypical Vol signal measured a a wall oule. Noe ha he ampliude of he signal is 7 Vols, no Vols. So where does his number Vols come form? I is cerainly no a simple average since ha would be zero for a signal symmeric abou zero. Vols is a number which gives an indicaion of he flucuaions of he signal abou he average value. I is called he roo-mean square value of he signal and as we 6.7 Spring 6, Chanioakis and Cory 4

5 will see laer when we sudy elecrical signals in deail, i is imporan since i is relaed o energy conen of he signal. The roo-mean square value of a signal V() is defines as Vrms T T = V () d (.5) V rms For V() = cos( ω), is calculaed as follows. V rms T = T Α π ω ω = π Α π ω ω = Α + cos ( ω) d cos ( ωd ) cos( ω ) d π π ω ω Α = Α + cos( ω) d π zero Α = (.6) For our elecriciy example, V rms = Vols and hus he ampliude of he corresponding sinusoidal signal is 7 Vols as indicaed on Figure 3. RMS - average (sec).5 Figure 3. Vol elecrical signal In some siuaions cerain signals may preven ohers from been received and undersood. For example, our abiliy o lisen o a conversaion may be compromised by he engine noise of a low flying airplane or a by a passing rain. In hese siuaions he signals are sill ransmied and received by our audiory sysem bu we are unable o exrac he 6.7 Spring 6, Chanioakis and Cory 5

6 useful informaion conained in hem. The signal of ineres o us is corruped by he noise of he airplane engine. The signal o noise raio (SNR)describes he relaive amouns of informaion and noise in a signal. SNR = Informaion in signal Informaion in noise Since signals usually have a very wide dynamic range (can vary over many order of magniude) he SNR is given in decibels (db) defined as follows. A s SNR( db) = log (.7) A where A s is he ampliude of he signal and A n is he ampliude of he noise. Figure 4 shows a sinusoidal wih various values of SNR n SNR=6dB SNR=4dB SNR = -6dB SNR=-dB Figure 4. Signals wih noise of various SNR. 6.7 Spring 6, Chanioakis and Cory 6

7 Sysems. Signals are always associaed wih one or more sysems. For example, a cerain sysem may generae he signal while anoher may operae on i in order o process i or o exrac relevan informaion from i. The represenaion of a sysem wih is associaed inpu and oupu signals is shown on Figure 5. The inpu signal is also called he exciaion signal and he oupu is also called he response signal. The sysem may hus be represened by an operaor F which may be designed o perform any desirable operaion on he inpu signal x() resuling in he oupu signal y. ( ) In elecronics, for example, he sysem may be an amplifier where he exciaion inpu volage v ( in ) is operaed on by he operaor F o produce he oupu vou ( ) wih an amplificaion A such ha F v () v () = Av () in ou in x() y() Sysem x() Inpu Oupu F y() Figure 5. Block diagram of a sysem Some common forms of he operaor F are shown on he following able. Inegral Amplifier Muliplier Adder x () y () y () = x( τ ) dτ x () y () A y () = Ax () y () = x() x() x () y () y () = x() + x() x () y () The characerisics of he Sysem operaor F are fundamenal in sysem analysis. We are paricularly ineresed in linear, ime invarian (LTI) sysems. A linear sysem is one which is boh homogeneous and addiive. A homogeneous sysem is one for which a scaled inpu volage produces an equally scaled oupu volage. Figure 6 illusraes he principle of homogeneiy where m can be any consan. mx() F my() Figure 6. Homogeneous sysem 6.7 Spring 6, Chanioakis and Cory 7

8 An addiive sysem is one for which, x() x() x ()+ x () F F F y() y() y()+y () Figure 7. Demonsraion of sysem addiiviy. The general definiion of a linear sysem is one ha can be homogeneous and addiive. If y () is he response of a sysem o an inpu x ( ) and y ( ) is he response of a sysem o an inpu x () hen if he sysem is linear he response o he signal ax() + bx(), where a and b are any consans is ay() + by(). This very imporan propery of linear sysems is called he principle of superposiion which we may represen mahemaically as F ax () + bx () ay () + by () (.8) In ou sudy of elecronic sysems we will make exensive use of his propery in order o obain soluions of wha seemingly appear difficul problems. A ime invarian sysem is one for which a delay τ in he applicaion of he exciaion signal (inpu) resuls in he same delay in he response signal (oupu). For example if an inpu signal, x( ), o a sysem described by he operaor F resuls in he oupu y () like, x() F y() (.9) Then he sysem is ime-invarian if F x ( τ ) y ( τ ) (.) The inerconnecions beween sysems is also a very imporan consideraion for heir overall behavior. In elecronic sysems special aenion is paid o heir inpu and oupu characerisics. When sysems are conneced ogeher he oupu characerisics of a sysem mus mach he inpu characerisics of he sysem ha i connecs o. As an 6.7 Spring 6, Chanioakis and Cory 8

9 example consider wo sysems represening waer sorage anks. The inpu of he sysem is characerized by is abiliy o receive a cerain flow rae of waer. The oupu represens a pump wih he capabiliy o supply a cerain flow rae of waer. The block diagram of hese inerconneced sysems is shown on Figure 8. For opimal sysem operaion, he rae a which he pump a he oupu of he sysem -ank- supplies he waer mus be compaible wih he rae a which he sysem -ank- can accep he waer. In elecronics we have an analogy where he inpu and oupu characerisics of he sysem refer o he resisance seen by he signals a he inpu and oupu of he sysem. In he case of elecronics we mus mach he wo resisances for opimal operaion of he elecronic sysem. We will explore hese principles in deail as we design and invesigae elecronic devices and sysems. Inpu of ank ank y() ank Oupu of ank Figure 8. Block diagram of an inerconneced sysem In pracical sysems, he Sysem block indicaed on Figure 5 is usually made up of various subsysems, componens or devices each performing a specific ask. In general, he componens and devices incorporaed in a Sysem may hemselves be considered as subsysems. For example, he block diagram of a digial sound recording sysem, comprised of a microphone, elecronics for amplificaion and filering, an analog o digial converer (ADC), a compuer, a digial o analog converer (DAC), an amplifier, and a speaker is shown on Figure 9. The doed recangle represens he complee sysem which is comprised of various oher subsysems. Analog signal X() Inpu Sound Microphone Eleronics ADC V() V() V3() Digial signal Compuer DAC Amplifier V4() V5() V6() Speaker Oupu Sound Y() Figure 9. Block diagram of sound recording sysem 6.7 Spring 6, Chanioakis and Cory 9

10 The microphone is a ransducer which may be considered as a sysem ha convers he pressure variaions in he inpu signal X() o he volage signal V(). In urn V() is processed by he elecronics module resuling in he signal V(). The elecronics module may perform such operaions as amplificaion, filering and offseing. Boh signals V() and V() are ime coninuous analog signals. Signal V() is in urn operaed by module ADC resuling in signal V3() which is now a digial signal (discree ime) ha may be furher processed by he compuer. The conversion of he analog signal V() o he digial signal V3() involves hree very imporan operaions: () sampling, () quanizaion and (3) encoding. Sampling is he process by which he signal values are acquired a discree poins in ime. This is a non-linear process since informaion is irrevocably los. Quanizaion is he process by which he coninuum of ampliude values is convered o a finie number of values (quanized values). This is a non-linear process since informaion is irrevocably los. Encoding is he process of convering each quanized value o a binary number represened by a binary bi paern. No informaion is los in his ranslaion. We will explore hese operaions in laer secions. For now le s esablish he framework for signal represenaion and analysis. 6.7 Spring 6, Chanioakis and Cory

11 Time and frequency domain Physical signals, such as he volage oupu of a microphone or he elecrical signal oupu of a srain or a pressure gage, are usually represened as funcion of ime. These signals may be manipulaed (amplified, filered, offse ec.) in he ime domain and many applicaions deal wih signals solely in he ime domain. However, i is ofen convenien and frequenly necessary, when signal analysis and processing is required, o represen he signal in he frequency domain. A signal in he frequency domain shows how much of he signal is associaed wih a cerain frequency. Figure shows he ime domain and he frequency domain represenaion of a sinusoidal signal wih a frequency of khz. Since his is a signal wih a single frequency of khz, he frequency domain represenaion of he signal is a single line a a frequency of khz. The heigh of he line a he frequency of khz corresponds o he magniude or srengh of he signal a ha frequency. As anoher example consider he signal given by he funcion x( ) = + cos(π ) + sin(6π ) (.) This signal is ploed on Figure. The wo frequencies presen in he resuling signal are 5Hz and 3Hz.Therefore, in he frequency domain represenaion only hese wo frequencies conain signal informaion as shown on Figure. Noe he srengh of he signal as represened in he frequency domain (sec) Frequency (Hz) Figure. Time and frequency domain represenaion of a sinusoidal signal (sec) Frequency (Hz) Figure. Time and frequency domain represenaion of he signal 6.7 Spring 6, Chanioakis and Cory

12 x( ) = + cos(π ) + sin( 6π ) Signals may in general conain a large number of frequencies and in his case he frequency domain represenaion of he signal becomes very useful. A signal wih large variaions in is rae of change in he ime domain conains proporionally larger number of frequencies. Compare he wo signals shown on Figure. The signal on Figure (a) appears o be smooher han he signal on Figure (b). Indeed he frequency conen of he signal in (b) is higher han ha of he signal in (a). In he frequency domain represenaion of he signals, informaion exiss only a he frequencies of he sinusoids comprising he signals. Furhermore he frequency domain represenaion conain deails abou he relaive srengh of he various frequency componens as can be seen by comparing he mahemaical expression of he signals o heir corresponding frequency domain represenaions. We will explore his concep furher in he following secions. In he case of he square wave signal where he slope a he ransiions becomes infinie, he frequency conen of he signal is also infinie. Signals wih finie frequency conen are called band-limied signals E+ 5E-3 (sec) E- (a) Frequency E+ 5E-3 (sec) E- (b) Frequency 3 Figure. Comparing signals in he frequency and he ime domain. (a) x( ) = sin(65( π ) ) + cos(8( π ) ) (b) x( ) = sin(6( π ) ) + sin(8( π) ) + sin(4( π) ) + cos(( π ) ) Spring 6, Chanioakis and Cory

13 The graphical represenaion of signals in he frequency domain jus presened will be enhanced by he appropriae mahemaical represenaion of signals in he frequency domain. The heory of complex numbers is essenial in undersanding frequency domain represenaion. In he following secion he conceps of Fourier analysis will provide us wih a very powerful ool for he general ransformaion of a signal from he ime domain o he frequency domain and equivalenly from he frequency domain o he ime domain. 6.7 Spring 6, Chanioakis and Cory 3

14 Complex number arihmeic: A review A complex number may be represened in recangular form as follows: c = a+ jb Recangular forma of complex number (.) The number j =. a is he real par of he complex number and b is he imaginary par of he complex number. The complex conjugae of a complex number is obained by replacing j wih j. For he number given by Eq. (.) is c = a jb The magniude of he complex number is * = cc = a + jb a jb = r = a + b (.3) magniude ( )( ) And he phase is b phase = θ = an a The graphical represenaion of he complex number in he complex plane is: Im (.4) b r θ a Re Euler s ideniy is an imporan relaionship in he heory of complex numbers. I saes: e jφ = cosφ + jsinφ (.5) From he graphical represenaion of a complex number and Euler s ideniy we may represen he complex number in polar form as j c= r(cosθ + jsin θ) = re θ Polar forma of complex number (.6) 6.7 Spring 6, Chanioakis and Cory 4

15 j Example: Conver he number c = 5 j6 o polar form c = re θ. Firs le s calculae he magniude. r = = = o The phase is θ = an = 5.9 = 5.4 radians 5 And he complex number in polar form is c = 7.8e j5.4. The graphical represenaion of his number is j e π 3 Example: Conver he number c = o recangular form. The magniude of he number is and he phase is π/3. The recangular form is c = a+ jb and hus we need o evaluae a and b. π From Euler s ideniy we know ha a= rcosθ = cos 3 = / and π b= rsinθ = sin = 3 / 3 And he number in recangular form is c = + j Spring 6, Chanioakis and Cory 5

16 Impulse Funcion. A review In science and engineering here are many examples when an acion occurs a an insan in ime or a cerain poin in space. For example he force exered on a baseball when i is hi by a ba is of very shor duraion. Also, he poin es used in maerials esing applies a very localized force on a maerial. The mahemaical represenaion of his ype of acion is ε < ε ε δε () = < τ < (.7) ε ε > For which we also impose he condiion: + δ ε () d= (.8) The funcion may be hough of as a recangular pulse of widh ε and heigh /ε as shown on Figure 3(a). In he limi ε, he heigh /ε increases in such a way ha he oal area is. This leads o he definiion δ () = lim δ () (.9) ε ε The funcion δ () is called he uni impulse funcion which is also known as he Dirac Dela funcion or simply as he Dela funcion. The graphical represenaion of he Dela funcion is shown on Figure 3(b) δ() δ() /ε -ε/ ε/ (a) (b) Figure 3. Dela funcion (a) visualizaion and (b) symbol For a more general represenaion, he funcion δ ( ) represens is shifed Dela funcion and represens an impulse cenered a =. The graphical and mahemaical represenaions of his general Dela funcion is, 6.7 Spring 6, Chanioakis and Cory 6

17 δ() + δ( τ) d = (.) δ ( τ ) = for τ The usefulness of he Dela funcion resuls no from wha i represens bu raher from wha i can do. The wo fundamenal properies, and defaul definiions, of he Dela funcion are: jπ fτ jπ fτ e e df δτ ( τ) (.) f δ τ d = f () ( ) ( τ ) (.) Equaion (.) is referred o as he sampling propery of he Dela funcion and i is a very imporan propery used exensively in signal analysis. 6.7 Spring 6, Chanioakis and Cory 7

18 Fourier Transform and he Fourier Series. The Fourier ransform (FT) is a mahemaical funcion ha ransforms a signal from he ime domain, x(), o he frequency domain, X ( f ). The ime o frequency domain ransformaion is given by: + jπ f X( f) x( ) e d = (.3) Equivalenly, he inverse Fourier ransform may be used o conver a signal from he frequency domain o he ime domain as follows: + () X( f) e j π f x = df (.4) When he Fourier ransform is o be expressed in erms of he angular frequency ω (rad/sec raher han he frequency f (Hz) he conversion is achieved by leing ) dω = π df. Therefore Eqs. (.3) and (.4) when wrien in erms of ω ake he form + j X( ω) = x( ) e ω d (.5) + j x() = X( ω) e ω dω π (.6) The Fourier ransform is he mos used mahemaical funcion in signal processing and daa analysis. I gives he ools o visualize, by looking ino he frequency domain, signal characerisics ha are no direcly observable in he ime domain. An illusraive example is he signal associaed wih sound. Figure 4(a) shows he volage signal as a funcion of ime corresponding o he sound of he middle C noe of a piano. The imporan informaion of a sound signal is is frequency conen. This informaion is revealed when we ransform he signal o he frequency domain as shown on Figure 4(b). The frequency domain represenaion of he signal clearly shows us ha he signal has, besides he fundamenal frequency of 6 Hz, addiional frequency componens. These addiional frequencies (he harmonics) ell us abou he sound characerisics of he piano and indeed hey are he reason for he richness and he uniqueness of each insrumen. 6.7 Spring 6, Chanioakis and Cory 8

19 (a) (b) Figure 4. (a) he ime domain signals of a middle C noe of a piano represened as a volage from a microphone. (b) Fourier ransform of he signal represens he same signal in he frequency domain. Before proceeding wih he physical and hus he pracical significance of FT le s become more familiar wih he process by calculaing he ransform for various pracical signals. We will look a periodic as well as non-periodic signals. Le s sar wih he calculaion of he Fourier ransform of he signal v ( ) = sin( ω ) (.7) This is our familiar sine wave characerized by a frequency of πω. Since his signal represens - by definiion - a single frequency, we anicipae ha in he frequency domain, all informaion will be conained a ha frequency. So le s proceed wih he calculaion o deermine he Fourier ransform of v () which is given by jω V( ω) = sin( ω ) e d (.8) By using Euler s ideniy, Eq. (.5), we obain, jω jω e e jω V( ω) = e d j j ( ( ω+ ω ) j ( ω ω ) = j e e ) d (.9) According o Eq. (.), And, πδ( ω ) j( ω ω ) + ω e + = d (.3) 6.7 Spring 6, Chanioakis and Cory 9

20 Therefore, Eq. (.9) becomes j( ω ω ) πδ( ω ω ) = e d (.3) [ ] V( ω) = π j δ( ω+ ω ) δ( ω ω ) (.3) The graphical represenaion of V ( ω ) is shown on Figure 5. Vj ( ω) π/j -ω ω ω π/j Figure 5. Fourier ransform of a sine wave. Similarly he Fourier ransform of he signal Is calculaed as follows v () = cos( ω ) (.33) jω V( ω) = cos( ω ) e d jω jω e + e jω = e d = + = π δ ω ω + δ ω+ ω j( ω ω) j( ω+ ω) ( e e ) [ ( ) ( )] d (.34) Figure 6 shows he Fourier ransform of he cosine signal. V (ω) π π -ω ω ω Figure 6. Fourier ransform of a cosine wave. 6.7 Spring 6, Chanioakis and Cory

21 The recangular pulse funcion given by, x < / yx ( ) =, x > / represens anoher very useful signal in elecronics and engineering in general. The ransform Y ( ω ) of he pulse funcion is + jω Y( ω) = y( ) e d = / / / / e [ jω d / sin = cos( ) d = [ ωτ ] ] ω ( ) = cos( ωτ ) jsin( ω) d ω (.35) (.36) Figure 7 shows he plo of he recangular funcion and is Fourier ransform. yx) ( - ½ ½ x Figure 7. recangular pulse and is Fourier ransform Similarly, he Fourier ransform of he shifed recangular pulse, < x < 3 yx ( ) =, x<, x> 3 (.37) 6.7 Spring 6, Chanioakis and Cory

22 yx) ( 3 x Y( ω) = y( ) e = + 3 e jω jω d d (.38) Le s simplify he above inegral by changing variables asξ =. jωξ ( + ) Y( ω) = e d = e = 4e jω jωξ e dξ (.39) j ω ω ω sin ξ Uni impulse funcion The Fourier ransform of he Dela funcion, given by Eq. (.), is jω ( ω) = δ( ) e d = e + jω (.4) For = he graphical represenaion of δ () and ( ω) is shown on Figure 8. δ() (ω) ω Figure 8. Dela funcion and is Fourier ransform. 6.7 Spring 6, Chanioakis and Cory

23 So now le s explore furher he physical significance of he Fourier ransform by invesigaing how he energy conen of a signal is represened in he ime domain and he frequency domain. From fundamenal conservaion principles we should expec ha he esimaion of global parameer such as energy should be he same regardless of how he signal is represened. The oal energy conen of signal x() is given by The expression E x() d = x( x ) ( d ) where x( ) is he complex conjugae of x ( ) jω = x () X ( ω) e dω d π jω = X ( ω) x( ) e d dω π = π = π X ( ω) π X ( ω) X( ω) dω X( ω) dω (.4) represens he energy per uni frequency and hus he expression X ( ω) dω is he oal energy conen of he signal ( ) X ω in he frequency π domain. Therefore we have shown ha he Fourier ransformaion is an energy conservaion ransformaion. x() d = X( ω) dω π (.4) This is a very imporan resul since i enables us o exrac global signal parameers such as energy by looking a eiher he ime or he frequency domain. 6.7 Spring 6, Chanioakis and Cory 3

24 The Fourier ransform resuls may be presened in a variey of ways. I may be represened as: The ampliude: plo he ampliude of he sinusoidal componen a he appropriae frequency. The RMS ampliude: plo he RMS ampliude of he sinusoidal componen a he appropriae frequency. Power specrum: plos values ha are proporional o he square of RMS ampliude. The plos of Figure 9 show a sine wave wih a frequency of 3 Hz and he corresponding frequency domain represenaion (sec) Frequency (Hz) Frequency (Hz) Frequency (Hz) Figure 9. Sine wave signal and various forms of is frequency domain represenaion. As anoher example le s consider he signal shown on Figure (a) and is calculaed Fourier ransform on (b). From he FT we see ha here are hree idenifiable frequency componens in our signal: 6 Hz, 3 Hz and 5 Hz. In he laboraory environmen many sysems pick up an undesirable 6 Hz noise from fluorescen lighs and oher devices, including wiring, ha are powered by a 6 Hz wall power. Our example is a simplified bu represenaive case of such a scenario. In order o deal wih hese ype of undesirable signals we firs have o idenify heir exisence and ascerain heir relaive energy conribuion o he signal of ineres. The Fourier ransform gives he ool o make his deerminaion. 6.7 Spring 6, Chanioakis and Cory 4

25 (sec) (a). 6 Hz Frequency (Hz) (b) 6 Figure. Signal conaining a 6 Hz noise and is ampliude specrum. The composie signal is given by x( ) = + cos(π ) + sin(6π ) + sin(π ) 6 Hz noise Summary: Some of he fundamenal properies of Fourier ransform are: Check all hese. Time domain Frequency domain Lineariy ax() + bx() ax( ω) + bx ( ω) Produc x() x() X( ω) X( ω) Differeniaion dx() d jω X( ω ) Inegraion X ( ω) Χ() δω ( ) x() τ dτ + jω Time delay x ( τ ) jωτ e X( ω) jωo Frequency shif e x() X ( ω ωο ) Energy conservaion + + x() d (Parseval s heorem) X ( ω) π jωo Frequency shif e x() X ( ω ω ) Time scaling x( a ) ω X π a a Dualiy X () x( ω) Convoluion x() x() X( ω) X( ω ) ο dω 6.7 Spring 6, Chanioakis and Cory 5

26 Fourier series and is relaion o Fourier ransform. Fourier series is jus a special case of Fourier ransform. In fac he Fourier series is associaed wih periodic signals, while he Fourier ransform is a more general represenaion of non-periodic signals in he frequency domain. Periodic signal may be represened by a linear combinaion of sinusoids whose frequencies vary by a consan ineger value. Since we may also represen a sinusoid wih complex exponenials, by using Euler s formula, he funcional form of his linear combinaion of complex exponenials is known as he Fourier series of he periodic signal and i is given by + jk x() = c e ω (.43) k = k The coefficiens ck are in general a complex numbers, ck = ak + jb k, and are given by ck T jk π Τ = x() e T d (.44) wheret is he period of x() and he inegraion is performed over one period. The coefficiens c k are called he Fourier series coefficiens or he specral coefficiens of he funcion x() and hey represen a measure of how much signal (he srengh of he signal) here is a each frequency kω. Therefore, he ask in deermining he Fourier series represenaion of a cerain signal is ha of deermining he complex coefficiens c k. If he signal x() is real hen is Fourier series represenaion is reduced o where, a ( k ω k ω ) (.45) x() = a + a cos( k ) b sin( k ) = T T T k = x() d π ak = x()cos ( k T ) d k,,3, T = (.46) T π bk = x()sin ( k T ) d k =,,3, T The coefficien a is jus he average value of he signal x(). In calculaing he inegrals of Eqs. (.46) i is useful o keep in mind he orhogonaliy properies of funcions. For example 6.7 Spring 6, Chanioakis and Cory 6

27 T m n sin mπ sin nπ d = / m= n T T T m= n= T m n cos mπ cos nπ d = / m= n T T T m= n= T sin mπ cos mπ d = T T T (.47) As an example le s calculae he Fourier series of he periodic square wave shown on Figure..5 A τ T-τ Τ+τ (sec) Figure. Square wave signal The period of he square wave is T and is frequency, he fundamenal frequency, is τ ω = π /T. Furhermore, he duy facor of he signal is defined as, df T and i is arbirary. Since he signal is an even funcion of, he Fourier coefficiens b k = and a are given by k 6.7 Spring 6, Chanioakis and Cory 7

28 A τ jkω ak = e d T τ ja jkω = e k kω T τ jkωτ jkωτ A e e = kωt j A = sin( kωτ ) kω T τ (.48) For k = we have τ A τ = T = (.49) T τ a d A where a is he average value of he signal. The Fourier series of he square wave is x () = a + a cos( kω) k = k τ A = A + kω τ s( kω) sin( )co T k = kωt (.5) Le s consider he case of a 5% duy facor square wave signal, shown on Figure (a) for which τ = T /4. The firs 5 non-zero coefficiens are: a =, a = 4, a3, a5, a7 π = 3π = 5π = 7 π Plos (b), - (f) of Figure show he Fourier series represenaion for a number of harmonics, saring wih he firs and ending wih he fifh. As he number of harmonics used in he approximaion increases he approximaion becomes closer and closer o he square wave signal. 6.7 Spring 6, Chanioakis and Cory 8

29 (sec).5 (a) (sec).5 (b) (sec).5 (c) (sec).5 (d) (sec) (e) (sec) (f). Figure. 5% duy facor square wave (a) and is s five Fourier harmonics (b) (f) For a deeper undersanding le s explore he significance of he coefficiens. A plo of he magniude of he coefficiens ak as a funcion of k is shown on Figure 3. Each value of k corresponds o a frequency called a harmonic which are ineger muliples of he frequency of he square wave also called he fundamenal frequency. The magniude of he coefficiens is relaed o he relaive srengh of he signal a he corresponding frequencies. The k a k dependence of he magniude is an indicaion of he relaively slow rae of convergence of he series. This implies ha a large number of harmonics is required in order o reproduce a square wave; a direc consequence of he disconinuiies associaed wih he square wave signal. The magniude plo of he Fourier coefficiens is direcly relaed o he Fourier ransform of he square pulse given by Eq. (.36). a k 6.7 Spring 6, Chanioakis and Cory 9

30 A τ Figure 3. Plo he values ak = sin kπ kπ T as a funcion of k for τ / Τ= / 4 I is also insrucive o plo he frequency specrum of he Fourier coefficiens for various values of he duy facor τ / Τ. Figure 4 shows a plo of a k for ( a) τ / Τ= /4,( b) τ / Τ= /,( c) τ / Τ= /6,( d) τ / Τ= /3,( e) τ / Τ= /64,( f) τ / Τ= /8 The plo shows he ampliude of a as a funcion of k, he mode number. Our firs k observaion is ha he frequency specrum of a k has an oscillaory behavior wih a slowly decreasing envelope. The decrease is proporional o / k. We also noice ha he spacing beween hese harmonics is a funcion of he so called duy facor. As τ he square wave signal approaches a series of Dela funcions. We noice ha as he pulses become narrower in he ime domain he Fourier series coefficiens is disribued over a wider range in he frequency domain. Therefore we see ha narrow ime signals require many harmonics in order o reproduce he original signal. Broader ime signals require fewer harmonics for he reproducion since he ampliude of he higher harmonics end o decrease more rapidly. In fac as τ / Τ, he firs crossing of he coefficien goes o and here is a very broad specrum conaining many harmonics which all essenially have he same ampliude. 6.7 Spring 6, Chanioakis and Cory 3

31 (a) (b) (c) (d) (e) (f) Figure 4. Fourier series coefficiens for square waves of various duy facors. ( a) τ / Τ= / 4,( b) τ / Τ= /,( c) τ / Τ= /6,( d) τ / Τ= / 3,( e) τ / Τ= / 64,( f) τ / Τ= /8 A he disconinuiies of he signal here are cerain imporan observaions o be made. Firs, noe ha he approximaion passes hrough he average value of he signal. This is given by he coefficien a which for he signal used on Figures 5 is zero. We also observe from he resuls shown on Figure 5 ha he error of he approximaion, ε = (real signal) - (approximaed signal), shown by he rippled hick solid line in he curves of Figure 5, decreases as he number of erms used in he approximaion increases. As he number of erms increases he ripple concenraes in he viciniy of he disconinuiies. Closer observaion indicaes ha, as he number of erms increases, he maximum ampliude of he error remains unchanged and is locaion moves closer and closer o he disconinuiies. The maximum ripple can be shown o be abou % of he signal value for all finie values of k. The ripple a he disconinuiies and is properies jus described is called Gibbs phenomenon. 6.7 Spring 6, Chanioakis and Cory 3

32 (sec).5 (a) (sec).5 (b) (sec).5 (c) (sec).5 (d) (sec) (e) (sec) (f) Figure 5: Gibbs phenomenon The FT of any periodic signal is always composed of jus impulses. The area of hese impulses are he FS coefficiens for he exponenial form. 6.7 Spring 6, Chanioakis and Cory 3

33 Fourier series expansion of: Triangular wave. Figure (sec) (sec) (sec) (sec) Figure 6. 8A sin( ) y () = sin π kπ k=,3,5, k ( kf) Sawooh wave Figure (sec) (sec) (sec) Figure (sec) 6.7 Spring 6, Chanioakis and Cory 33

34 y() = Average sin k f π k k = Half wave recified signal ( π ) A Α A y ( ) = + sin( f) cos( kf) π π k =,4,6, k Full wave recified signal A 4A y () = cos( kf) π π k = 4k 6.7 Spring 6, Chanioakis and Cory 34

35 Sampling Transducers generae coninuous ime signals bu compuers and microprocessors, ha are used o process hese signals, operae a discree imes. These discree ime signals are generaed by sampling he coninuous ime signal a regular inervals. Sampling is hus he process which generaes a discree ime signal from a coninuous ime signal. The fundamenal quesion herefore is how o sample a coninuous ime signal so ha he resuling sampled signal reains he informaion of he original signal. The sampling process is depiced graphically on Figure 8. 8(a) shows a signal x() and Figure 8(c) he corresponding sampled signal sampled a inervals τ s Time (s) τ s (a) Time (s) (b) Time (s) (c) Figure 8. (a) original signal, (b) sampling waveform, (c) resuling discree ime signal Figure 8(b) depics he sampling wave form which may be hough of as a series of narrow periodic pulses wih period τ s. From he analyical perspecive hese pulses may be hough of as dela funcions. In pracice hese are narrow pulses produced by some ype of clocking device in he circui of ineres Spring 6, Chanioakis and Cory 35

36 Inuiively we know ha in order o reconsruc a cerain signal he number of samples per period of he sampled signal mus be above a cerain minimum value. The signals shown on Figure 9 are sampled wih he same rae. The sampled poins are indicaed wih he solid do. I is inuiively apparen ha he plo in 9 (b) is sampled frequenly enough for reconsrucion, while he plo on 9(a) can no be reconsruced wih he sampled signal Time (s) (a) Time (s) (b). Figure 9. Sampling of a fas varying (a) and a slow varying (b) signal. In order o be able o reconsruc he original signal from he sampled signal he following wo relaed consrains mus be saisfied.. The original signal mus be band-limied (i.e. mus have a finie frequency conen). The samples mus be aken wih a sampling frequency ( f = / τ ) which is higher han wice he highes frequency ( f H ) presen in he original signal. These saemens form he famous Sampling Theorem or he Nyquis-Shannon Sampling Theorem. The criical frequency ( fh ) which mus be exceeded by he sampling frequency is called he Nyquis rae. The frequency ( fh ) ha corresponds o one-half he Nyquis rae is also called he Nyquis Frequency fnyquis = fh. s s In erms of he Fourier ransform, he original coninuous ime signal can be recovered from he sampled signal if he frequency specrum of he original signal can be exraced from he frequency specrum of he sampled signal. For a mahemaical proof of his heorem see Alan Oppenheim, Signals and Sysems, Prenice hall or go o he original aricles. H. Nyquis, Cerain Topics in Telegraph Transmission heory, AIEE Transacions, 98, p. 67 C. E. Shannon, Communicaion in he presence of noise Proceedings of IRE, January 949, pp Spring 6, Chanioakis and Cory 36

37 Aliasing When he sampling frequency is less han wice he bandwidh of a signal he ime coninues signal can no reconsruced from he samples. As we saw in our Fourier series analysis when he pulses are spaced furher apar in ime he Fourier harmonics ge closer ogeher. A some poin here is an overlap of he impulse specra and reconsrucion of he original signal becomes impossible. This is called aliasing. The mahemaics of his is given in he accompanying noes. Here we presen an inuiive graphical represenaion of he phenomenon on Figure 3. On Figure 3(a) we see he generic Fourier ransform of a cosine signal of frequency ω. On Figure 3(b) we presen a scenario where he sampling frequency ωs = 4ω. Noe now ha he frequency of ineres ω remains wihin he recangle defined by he ω s / regions. Figure 3(c) shows anoher case for which ωs = 5/ω. Here again he frequency of ineres remains wihin he recangle defined by he ω s / regions. Finally on Figure 3(d) ωs = 3/ω he frequency ω has moved ouside he ω s / regions. In he ω s / regions now appears he lower frequency ωs ω. 6.7 Spring 6, Chanioakis and Cory 37

38 Χ( j ω) π π -ω ω ω (a) Χ( j ω) -ω s -ω ω ω s ωs/ ω - ω s + ω - ω s / (b) Χ( j ω) ω s -ω -ω s -ω ω ω s ωs/ ω - ω s + ω - ω s / (c) Χ( j ω) ω s -ω -ω s -ω ω ω s ωs/ ω - ω s / - ω s + ω ω s -ω (d) Figure 3. Oversampling and undersampling showing aliasing. (a) ransform of he cosine wave. (b) sampling he cosine signal wih ωs = 4ω (No aliasing). (c) sampling he cosine signal wih ωs = 5/ω (No aliasing). (d) sampling wih ωs = 3/ω (Aliasing) 6.7 Spring 6, Chanioakis and Cory 38

39 Numbering sysems: A review Before proceeding wih he las wo seps quanizaion and encoding in he process of convering an analog signal o a digial signal le s review he fundamenal rules ha govern he represenaion of numbers in he various numbering sysems. We primarily ineresed in he conversion of analog signals o digial signals. Binary Code. In digial elecronics he signals are formed wih only wo volage values, HI and LOW, or level and level and i is called binary digial signal. Therefore, he informaion conained in he digial signal is represened by he numbers and. In mos digial sysems he sae corresponds o a volage range from V o 5V while he sae corresponds o a volage range from a fracion of a vol o vols. Digial operaions are performed by creaing and operaing on binary numbers. Binary numbers are comprised of he digis and and are based on powers of. Each digi of a binary number, or, is called a bi. Four bis ogeher is a nibble, 8 bis is called a bye. (8, 6, 3, 64 bi arrangemens are also called words) The lef mos bi is called he Leas Significan Bi (LSB) while he righmos bi is called he Mos Significan Bi (MSB). The schemaic below illusraes he general srucure of a binary number and he associaed labels. MSB nibble bye word LSB In addiion o binary digial sysems and is associaed binary logic, mulivalued logic also exiss bu we will no consider i in our discussion. 6.7 Spring 6, Chanioakis and Cory 39

40 Binary o Decimal Conversion. The conversion of a binary number o a decimal number may be accomplished by aking he successive powers of and summing for he resul. For example le s consider he four bi binary number. The conversion o a decimal number (base ) is illusraed below. 3 x + x + x + x = 5 For his four bi binary number he range of powers of goes from, corresponding o he LSB, o 3, corresponding o he MSB. The number 5 is shown as 5 o indicae ha i is a decimal number (power of ). The signal represened on Figure 3(a) has a value of 5 V a ime=6τ. The binary represenaion of ha value is and i is shown on Figure 3(b) replacing Level 4. We will see more of his laer when we consider he fundamenals of he device which convers he analog signal o a digial signal. Signal (V) Signal (V) τ τ 3τ 4τ 5τ 6τ 7τ (a) Time Level 7 Level 6 Level 5 Level 3 Level Level τ τ 3τ 4τ 5τ 6τ 7τ (b) Time Figure 3. Digiizaion process. In he nex few examples we will use he subscrip o indicae a binary number bu he subscrips will be omied afer ha. 6.7 Spring 6, Chanioakis and Cory 4

41 Examples: Verify he Binary o Decimal conversion = 5 = 4 = 55 = 9 = 347 = 395 Decimal o Binary Conversion. The conversion of a decimal number o a binary number is accomplished by successively dividing he decimal number by and recording he remainder as or. Here is an example of he conversion of decimal number 5 o binary = 6 + LSB = 3 + = 5 + = 7 + = 3 + = + = + MSB Pracice number conversion by verifying he conversions from decimal o binary: Decimal Binary Spring 6, Chanioakis and Cory 4

42 Represenaion of fracions and signed numbers. A fracional number may be represened as a binary fracion by simply exending he procedure used in represening ineger numbers. For example, 3.75 =. The procedure is clearly visualized by considering he following mapping Signed binary numbers may be represened by assigning he MSB o indicae he sign. A is used o indicae a posiive number and a is used o indicae a negaive number. For example, an 8 bi signed binary number represens he decimal numbers from -8 o +7. Two s complemen is used o represen negaive numbers. The use of s complemen simplifies he operaion of subracion since he circui is only required o perform he operaion of addiion. The s complemen of a binary number is obained by subracing each digi of he binary number from digi. This is equivalen o replacing all s by s and all s by s. Negaive numbers of s complimen can hen be found by adding o he complemen of a posiive number. For example, he s complemen of he 8 bi binary number is = The negaive number of his s complemen represenaion is = - The procedure is oulined in he following binary number ( ) 's complemen Spring 6, Chanioakis and Cory 4

43 The able below shows he s complemen represenaion of a few numbers. Fill in he empy spaces. Decimal s complemen Spring 6, Chanioakis and Cory 43

44 Quanizaion and Encoding Analog o Digial Conversion The elecrical signals (volage or curren) generaed by a ransducer is an analog signal. The ampliude of he signal corresponds o he value of he physical phenomenon ha he ransducer deecs. The signal values are coninuous in ime. The processing of he signal by a digial sysem requires he conversion of he analog signal o a digial signal. The analog o digial conversion is no a coninuous process bu i happens a discree ime inervals. Furhermore he magniude of he digial signal a he ime of conversion corresponds o he magniude of he analog signal. The analog o digial converer (ADC) is a device ha receives as is inpu he analog signal along wih insrucions regarding he sampling rae (how ofen is a conversion going o be performed 3 ) and scaling parameers corresponding o he desired resoluion of he sysem. The oupu of he ADC is a binary number a each sampling ime. In he preceding secion on Sampling we explored he condiions on he sampling rae. The following schemaic shows he basic srucure of an 8 bi ADC. Reference Volage Signal (V) Analog Signal 8 Bi ADC Sampling signal D7 D6 D5 D4 D3 D D D Pulses LSB MSB The selecion of an 8 bi ADC ses he resoluion of our conversion and he selecion of he scale for he analog signal deermines he measuremen resoluion for our ADC. In ou example he 8 bi ADC implies 8 = 56 differen levels wihin he maximum signal range. 3 The sampling frequency mus be larger han he highes frequency of he analog signal o be convered. In fac as saed by he Sampling Theorem The sampling frequency mus be a greaer han imes he bandwidh of he inpu signal. 6.7 Spring 6, Chanioakis and Cory 44

45 Since we are measuring a volage wih possible values beween V and V, our 8 bi ADC is no able o resolve volages smaller han mv = 39mV. 8 If our ADC has a resoluion of 6 bis, like he one ha you have in your laboraory, he resoluion, for he same measuremen range, would be 6 mv=.5mv. The able below summarizes he conversion process Pulse Signal Value Level Binary number 56 = = = = = = = = Spring 6, Chanioakis and Cory 45