Modulation Introduction

Transcription

1 Modulaion Inroducion -8 F. Dellsperger

2 Conen. Inroducion o Modulaion.... ccess mehods, bundling o signals Useul Signal-heory and Mahemaical Mehods Classiicaion o signals Deerminisic signals Random signals Causal signals Energy signals Power signals nalyic signal and Hilber ransormaion Coninuous signals Discree signals and sampling Mahemaical descripion and graphical represenaion o signals Sine/Cosine signal Discree cosine signal Recangular Pulse Dirac Impulse (Uni Impulse) Sinc uncion (cardinal sine uncion) Fourier Series and Fourier ransormaion Fourier Series and Fourier ransormaion o coninuous signals Discree Fourier ransormaion DF Examples: Convoluion Convoluion o coninuous signals Convoluion o discree signals Correlaion Energy and power specrum densiy Reerences F. Dellsperger

3 Fig. -: Reasons or modulaion... Fig. -: Modulaion o pulse carrier... 3 Fig. -3: Modulaion o sinusoidal carrier... 3 Fig. -4: Frequency allocaion in FDM sysems... 4 Fig. -5: ime allocaion in DM sysems... 4 Fig. -6: Code allocaion in CDM sysems... 5 Fig. -7: Modulaed carriers in OFDM sysem... 6 Fig. -8: Phase shi o a Hilber-ransormer... 7 Fig. -9: a) Real signal, b) nalyic signal... 8 Fig. -: Coninuous or analog signal... 9 Fig. -: Coninuous and discree signals... 9 Fig. -: nalog signal and sampling uncion... Fig. -3: Sampled analog signal... Fig. -4: Sampled analog signal, convoluion in requency domain... Fig. -5: liasing, specrum overlapping... Fig. -6: Cosine signal in ime domain,... Fig. -7: Cosine signal in requency domain, magniude specrum o a cosine signal... 3 Fig. -8: Complex sinusoidal signal in phase diagram, phasor o complex sinusoidal signal.. 3 Fig. -9: Recangular pulse... 4 Fig. -: Dirac impulse... 4 Fig. -: Sample a... 5 Fig. -: Sampling uncion... 5 Fig. -3: Sinc uncion... 6 Fig. -4: mpliude and Phase Specrum o a discree signal... 9 Fig. -5: Unipolar square wave, ime domain and specrum... Fig. -6: pproximaion o square wave using Fourier series... Fig. -7: Bipolar square wave, ime domain and specrum... Fig. -8: Pulse rain, ime domain and specrum... Fig. -9: Pulse rain, ime domain and specrum... Fig. -3: Sampling uncion, ime domain and specrum... 3 Fig. -3: Cosine signal, ime domain and specrum... 3 Fig. -3: Sine signal, ime domain and specrum... 4 Fig. -33: Recangular pulse, ime domain and specrum... 4 Fig. -34: sinc-pulse, ime domain and specrum... 5 Fig. -35: riangle pulse, ime domain and specrum... 6 Fig. -36: Convoluion in ime domain o an arbirary signal wih a Dirac pulse... 7 Fig. -37: rbirary signal and cosine signal: a) Muliplicaion in ime domain, b) convoluion in requency domain... 8 Fig. -38: Sampling o an arbirary signal: a) Muliplicaion in ime domain, b) convoluion in requency domain... 9 Fig. -39: uocorrelaion uncion o a PN-Code wih lengh Fig. -4: Power specrum densiy o a PN-Code wih lengh F. Dellsperger

4 . Inroducion o Modulaion In modern communicaion sysems, he signal generaed by he inormaion source can normally no be ransmied direcly. In he majoriy o cases, an adjusmen o he physical channel, i.e. a coax cable, a wireless channel or an opical iber, is necessary. For he purpose o he economical uilizaion o he channel, he bundling o dieren signals is common. For ransmission in wireless communicaion sysems, he use o sinusoidal carriers is common, wih he inormaion inluencing heir ampliude, requency, phase or combinaions o hese characerisics. he erm modulaion means he changing o one or more signal parameers (ampliude, requency or phase) o a carrier as a uncion o he inormaion. Due o his, he inormaion is imprined ono he carrier signal. Modulaion allows or achieving he ollowing objecives: djusmen o he requency range o he respecive ransmission channel wih low-pass, band-pass or high-pass behavior (physical reasons). he modulaed signal is shied o a requency range, which allows or an easy implemenaion o communicaion sysems, e.g. emission by means o anennas, use o iber opics, ec. Muliplex ransmission, i.e. bundling o he signals Combinaion o muliple primary signals using he appropriae modulaion mehod o creae a secondary signal. In his process, he same ransmission channel is used several imes by means o muliplexing in order o increase he channel capaciy. In he case o radio broadcasing, or example, he modulaion is used or he selecion and separaion o numerous communicaions signals, i.e. several VHF or V saions wih dieren carrier requencies. In his process, or each o he inormaion channels a racion o he enire requency band is made available or he enire ime (requency muliplex). Increased inererence proecion: ranslaion o Shannon s law in order o adjus i o a cerain channel, e.g. by means o coded modulaion mehods or spread specrum echniques, or example, immuniy o noise and inererences can be signiicanly increased by exchanging bandwidh or signal-o-noise raio. Modulaion Wire Line Lowpass dapion o ransmission Pah dapion o Frequency Range Bandpass Radio Pah Bundling o Signals Frequency Division Muliple ccess FDM ime Division Muliple ccess DM Code Division Muliple ccess CDM Orhogonal Frequency Division Muliplexing OFDM Noise Immuniy Enhancemen Coded Modulaion Spread Specrum Coaxial Line Waveguide Fiberopic Cable Fig. -: Reasons or modulaion -8 F. Dellsperger

5 Modulaion and demodulaion also serve or ranslaing inormaion ino a signal orm, which guaranees he ransmission o he inormaion over he larges possible disance or any given disance while mainaining he required signal-o-noise raio. For his process, he basic condiions wih regard o he channel capaciy and he speciic characerisics o he ransmission channel mus be aken ino accoun (requency-dependen aenuaion and phase, ime- and requency-selecive channels). Modulaion ransorms he inormaion ino a dieren orm, usually in a higher requency range (radio requency, RF). In heory, any ype o signal is possible or he carrier signal, even noise. echnically, however, only wo signal ypes are used: a) Pulse carrier (periodic pulse sequence) or low-pass or base-band ransmission and ime division muliplexing as well as b). Harmonic (sinusoidal) carrier or band-pass or high-pass ransmission and requency division muliplexing For wireless communicaions echnology, modulaion mehods wih pulse carriers canno be used due o heir unavorable specral behavior. In his case, only sinusoidal carriers are used. he modulaion or inormaion signal, which is also called baseband signal (Baseband, BB), can be analog or digial. nalog signals have coninuous imes and values, e.g. he audio signal rom a microphone. Digial modulaion signals have discree imes and values in he orm o discree symbols, which can have only a inie se o values, e.g.,,-. Binary symbols which can only have wo values (e.g. and, or + and -) are oen used. Examples o digial modulaion signals include serial bi sequences a he inerace o a compuer or he oupu signals o an /D converer. Modulaion ypes can hereore be divided ino our groups:. nalog modulaion o a pulse carrier. Digial modulaion o a pulse carrier 3. nalog modulaion o a sinusoidal carrier 4. Digial modulaion o a sinusoidal carrier he dierence beween analog and digial modulaion mehods is no heir consiuion, bu only he orm o he modulaion or inormaion signal. he mos imporan basic mehods are illusraed in Fig. - and Fig. -3. In summary, his mehod has hree characerisics:. ype o modulaion carrier (harmonic carrier or pulse carrier),. ype o modulaion or inormaion signal (analog or digial) 3. Choice o he carrier s signal parameer ha is o be inluenced (ampliude, requency or phase). -8 F. Dellsperger

6 Pulse Carrier nalog Modulaion Signal Pulse ampliude modulaion PM Pulse requency modulaion PFM Pulse phase modulaion PPM Digial Modulaion Signal Pulse code modulaion PCM Dela modulaion DM Dierence pulse code modulaion DPCM Pulse duraion modulaion PDM Fig. -: Modulaion o pulse carrier Sinusoidal Carrier nalog Modulaion Signal mpliude modulaion M Digial Modulaion Signal mpliude shi keying SK Complex Modulaions Frequency modulaion FM Phase modulaion PM Frequency shi keying FSK Phase shi keying PSK MSK, GMSK QPSK, DQPSK, ec QM Fig. -3: Modulaion o sinusoidal carrier. ccess mehods, bundling o signals he goal o modern wireless communicaion sysems is o give a large number o users access o he limied requency or ime range, wihou inerering wih each oher. here are he ollowing dieren muliplex mehods: Frequency Division Muliple ccess FDM ime Division Muliple ccess DM Code Division Muliple ccess CDM Orhogonal Frequency Division Muliplexing OFDM Space Division Muliple ccess SDM (Picocells) ll o he mehods lised here are commonly used oday, someimes even several o hem a he same ime. -8 F. Dellsperger 3

7 FDM Frequency Division Muliple ccess: he available requency range is divided ino several dieren channels (users). he individual channels are made available a he same ime and can be operaed absoluely independen rom one anoher (e.g. or broadcasing). However, i is also possible ha one ransmier generaes all o he channels and hereore makes muliple channels available (e.g. wih older microwave sysems). he bandwidh o he channel is subjec o he used modulaion mehod. he unused requency band beween channels prevens inererence caused by adjacen channels. I signals in muliple channels are presen simulaneously, nonlineariies in he sysem resul in inermodulaion producs, and he channel allocaion mus be careully planned. mpliude ime Frequency Fig. -4: Frequency allocaion in FDM sysems ypical applicaions: Broadcas FM, V-Channels DM ime Division Muliple ccess One channel is assigned o dieren users during subsequen periods o ime. he channel is made available o each o he users during a deined ime slo. proecion period beween he individual slos is necessary in order o preven inererences beween adjacen slos due o diering signal propagaion imes. Since one channel is available o a user only or a racion o he ime, he daa mus be ransmied in ime-compressed orm. his resuls in high daa hroughpu on he ransmission channel. he individual ime slos can be assigned dieren ransmission direcions. his allows or a ull-duplex mode, wihou complex ilers. In many cases, DM and FDM are used in combinaion: he oal available bandwidh is divided ino individual channels (FDM) which are hen accessed using he DM mehod. mpliude ime Frequency Fig. -5: ime allocaion in DM sysems ypical applicaions: GSM, Blueooh -8 F. Dellsperger 4

8 CDM Code Division Muliple ccess Wih he CDM mehod, all users are allowed he uilizaion o he enire available specrum a he same ime. In order o be able o disinguish he individual signals, hey are assigned dieren codes. demonsraive example or his: a pary, dieren languages (= codes) are being spoken a he same ime and in he same room, bu each language only once. I a paricipan undersands a cerain language, he can ollow his paricular conversaion. Due o he coding, spreading o he individual signals occurs so ha subsequenly, hey occupy several imes heir original bandwidh. Pracical spreading acors: Usually as a power o. CDM is also reerred o as spread specrum echnology. mpliude ime User 5, Code 5 User 4, Code 4 User 3, Code 3 User, Code User, Code Frequency Fig. -6: Code allocaion in CDM sysems In mobile communicaions, mainly he ollowing wo mehods are used: - Direc Sequence (DS-) CDM and - Frequency Hopping (FH-) CDM. For he DS mehod, spreading is done wih a code, which does no depend on he daa o be sen. he receiver operaes synchronously o he code sequence o he ransmier and hereore reverses he spreading. Due o he despreading, discree noise signals are spread, resuling in an improvemen o he signal-o-noise raio a he receiver. dvanages o spread specrum mehod: - Low suscepibiliy o mulipah propagaion eecs. Due o he large ransmiing bandwidh, only a small racion is inluenced by he requency-selecive Rayleigh ading only weak signal dips (ading) occur. - Minimal eecs rom inererence signals. - Low power specral densiy. - he spreading code also serves as encrypion. he inormaion can only be deeced i he code is known (mached iler). - he inluence o spread specrum signals o oher convenional sysems is a deerioraion o he signal-o-noise raio only. he same also applies vice-versa. ypical applicaions: D-MPS, W-CDM (UMS), Blueooh (FH-CDM) -8 F. Dellsperger 5

9 OFDM Orhogonal Frequency Division Muliplexing OFDM is a muli-carrier modulaion echnique ha uses muliple carrier requencies o ransmi a digial signal. Each carrier is modulaed using only a racion o he inormaion. s a modulaion resuls in he generaion o sidebands wih zeros spaced by muliples o he daa rae, his requency spacing is chosen or he carrier signals. Carrier Power Frequency Fig. -7: Modulaed carriers in OFDM sysem ypical applicaions: Digial Video Broadcas DVB, Digial udio Broadcas DB, WLN IEEE 8.g, LE. Useul Signal-heory and Mahemaical Mehods nalog and digial modulaions use dieren orms o signals or heir descripions. For analog modulaions, he modulaion signal consiss o a coninuous signal and inluences a cosine carrier in is ampliude, requency and phase. I boh signals are described in he ime domain, he modulaion can primarily be explained by means o he muliplicaion o cosine signals. Digial modulaions use discree (digial) modulaion signals o inluence one or several cosine carriers. For he descripion and analysis o hese signals, urher procedures and mehods relaing o signal descripion and signal processing are useul. hese procedures and mehods will be considered in he ollowing o an exen required or he undersanding o analog and digial modulaion. Generally, derivaions and proos are no provided. Please reer o he comprehensive specialis lieraure [9], []... Classiicaion o signals... Deerminisic signals Deerminisic signals are signals whose uncional values may be compleely described wih a mahemaical erm. Examples: Periodic signals, sine and cosine signals... Random signals Random signals are arbirary signals which may only be described wih saisic mehods. Examples: Noise, inerering signals, voice signals -8 F. Dellsperger 6

10 ...3 Causal signals Causal signals are signals whose value is zero on he negaive ime scale s scs hey include he swich-on issue. causal signals whose values are no zero even or < are mahemaically easier o process han causal signals....4 Energy signals Energy signals have a inie energy: E s d E (.) E is he energy normalized o wih he uni Ws. Examples: ime-limied pulses such as he recangular pulse, riangular pulse, Gaussian pulse...5 Power signals Power signals have a inie power: (.) P lim s d P Examples: ll periodic signals such as sine and cosine signals and random signals such as noise...6 nalyic signal and Hilber ransormaion he ideal Hilber ransormer has a consan ampliude response wih he value and a phase o o shi o 9 or negaive requencies (<) and 9 or posiive requencies (>). hereore, o he Hilber ransormer is also reerred o as 9 phase shier. I may be approximaed very well wih a FIR iler. he Hilber ransormaion is deined as j S : S j sgn S : j S : (.3) Signum: : x sgn x : x : x Or inerpreed as a phase shier: o 9 : o 9 : (.4) S o 9 o 9 Fig. -8: Phase shi o a Hilber-ransormer -8 F. Dellsperger 7

11 n analyic signal is deined as a signal whose specrum has he value zero a negaive requencies and has only specral componens a posiive requencies. S S : : (.5) n analyic signal may only have complex values since real signals always have specral componens a negaive requencies. ˆ s Re s jim s s js n analyic signal may be ormed by means o he Hilber ransormaion rom a real signal by aking he real signal as real par and orming he imaginary par wih he Hilber ransormaion. he real par s and he imaginary par ŝ are linked o each oher via he Hilber ransormaion: ˆ ˆ ˆ s s js s s S S js (.6) s j ŝ s ˆ s s js analyic signal he consideraion in he requency domain resuls in he ollowing: Wih S Ŝ j S : j S : S j j S S S jjs (.7) = analyic signal (.8) S S a) b) Fig. -9: a) Real signal, b) nalyic signal -8 F. Dellsperger 8

12 Discree ime mpliude mpliude mpliude mpliude he ollowing applies as well: s s s = envelope (.9) s s arcan = insananeous phase (.) d = insananeous requency (.) d...7 Coninuous signals coninuous signal (analog signal) has coninuous ampliude and a coninuous ime. s Fig. -: Coninuous or analog signal Examples: Microphone signals, sensor signals, sine and cosine signals...8 Discree signals and sampling discree signal (digial signal) is obained rom he analog signal by means o sampling and quanizaion. he sampling and quanizaion order is arbirary. he quanizaion is he breakdown o he ampliude domain in discree values, e.g. wih an n-bi analog-digial converer. In his case, n he number o quanizaion seps is. Coninuous mpliude Discree mpliude Coninuous ime Fig. -: Coninuous and discree signals -8 F. Dellsperger 9

13 he ideal sampler muliplies he analog signal s() wih he sampling uncion s. (.) s s s n s n n s s s s s n n s s s s Fig. -: nalog signal and sampling uncion is he sampling period and s / he sampling requency or sampling rae. s he resul is a value-coninuous, ime-discree signal consising o a Dirac pulse sequence whose s n. weighs correspond o he sampling values s s ss s s n s s Fig. -3: Sampled analog signal he specrum o he sampled signal is obained wih he convoluion as shown in chap...4. is he Fourier ransorm o he sampling uncion S is he Fourier ransorm o s() and. s S S B S S s s s s s s s s Fig. -4: Sampled analog signal, convoluion in requency domain For he correc signal reconsrucion, he specrums o he sampled signal may no overlap. his resuls in he ollowing sampling heorem: s (.3) max max is he maximum requency or bandwidh B o he analog signal. I his condiion is no me and he specrums overlap, his eec is called "aliasing" and he original analog signal may no be reconsruced correcly. Signal pars o he analog signal above may be suppressed wih an ani-aliasing iler (low-pass iler) beore sampling. max -8 F. Dellsperger

14 liasing S liasing s s s s Fig. -5: liasing, specrum overlapping For band-pass signals wih a requency range rom min o max, wih min, as hey occur, or example, as inermediae requency in a receiver, he sampling heorem may also be ormulaed dierenly: min k k whereas k max s k,,,... max min min (.4) (.5) In he case o band-pass signals, i is hereore possible o manage wih lower sampling requencies. Discree (digial) signals are represened as a sequence o ime-discree values in he orm x n...,x,x,x,x,x,... n : n is an ineger in he range n. In he ollowing, discree signals and operaions wih discree signals are lised as compleion... Mahemaical descripion and graphical represenaion o signals... Sine/Cosine signal In general, a coninuous cosine signal in he ime domain is described by s cos (.6) c c Peak ampliude ngular requency = = Phase = ime c Wih Fourier ransormaion, he coninuous signal resuls in he requency domain j j j S s se d e c e c (.7) -8 F. Dellsperger

15 cosine carrier inluenced by he modulaion signal is described by ˆ s V cos c c c M FM PM ˆV c c Carrier peak volage Carrier angular requency = = Carrier phase a = ime c (.8) here are hree opions or inluencing he carrier by means o he modulaion signal: ˆV,, c c mpliude modulaion: mpliude modulaion is modiying he carrier ampliude ˆV c wih he modulaion conen. c and remain consan. is assumed o be in mos cases. Frequency modulaion: Frequency modulaion is modiying he carrier requency he carrier ampliude remains consan. Due o he relaion also resuls in a phase change. Phase modulaion: c wih he modulaion conen. d, a requency change Phase modulaion is modiying he carrier phase wih he modulaion conen. d he carrier ampliude remains consan. Due o he relaion, a phase change also d resuls in a requency change. Frequency and phase modulaion are boh covered by he erm angle modulaion. Boh inluence he argumen (angle) o he cosine. Represenaion opions o he sine signal For describing he signals, dieren represenaions may be used. he represenaions are explained wih he cosine signal. a) Represenaion in he ime domain s c() ˆ s V cos (.9) c c c he insananeous ampliude o he signal is represened as a uncion o ime. his represenaion corresponds o he represenaion wih an oscilloscope. ime is represened linearly on he horizonal axis and he ampliude generaes he verical delecion. ˆV c c Fig. -6: Cosine signal in ime domain, c c -8 F. Dellsperger

16 b) Represenaion in he requency domain (magniude specrum) he Fourier ransormaion o he cosine signal resuls in a specral line, a boh + c and c : Vˆ Vˆ s V cos S e e Vˆ c Vˆ c Sc c c ˆ c j c j c c c c c c Sc (.) ˆV c ˆV c - c c Fig. -7: Cosine signal in requency domain, magniude specrum o a cosine signal In mos cases i is suicien o only consider he posiive requency axis o he specrum since he magniude specrum o a real signal is mirror-symmerical. he signals a negaive requencies resul rom he correc mahemaical descripion and do no include any addiional inormaion. his represenaion corresponds o he represenaion wih a specrum analyzer. he requency axis is divided linearly or logarihmically. he ampliude produces he verical delecion. c) Represenaion in he phase sae diagram he complex sine signal is obained rom (.9) according o Euler jc c c c c c c I s Vˆ e Vˆ cos jvˆ sin (.) Q he signal is represened in he polar diagram as a phasor a a cerain poin in ime. he phasor lengh corresponds o he ampliude ˆV and he angle o he insananeous phase. is he angle a. he posiive I-axis (x-axis) corresponds o an angle o degree. c For he signal represenaion wih digial modulaions, he horizonal (real) axis is oen called I-axis (in-phase componen) and he verical (imaginary) axis is called Q-axis (quadraure phase componen). Q c ˆV c c c I I = In-Phase-Componen Q = Quadraure-Phase-Componen Fig. -8: Complex sinusoidal signal in phase diagram, phasor o complex sinusoidal signal -8 F. Dellsperger 3

17 ... Discree cosine signal n N Number o samples in xn cos c N n=,,,...,n-...3 Recangular Pulse In he digial modulaion, he recangular signal is requenly presen. s : / rec : / (.) (.3) s Fig. -9: Recangular pulse...4 Dirac Impulse (Uni Impulse) I, or he recangular pulse, he widh and he heigh / (area = ) is chosen and approaches zero, an ininiely high and ininiely hin pulse wih he area o is generaed. he pulse deined in his way is called Dirac pulse. he Dirac pulse is used o analyze he pulse response o a sysem and or sampling coninuous signals. he Dirac pulse is graphically represened by a verical arrow. he area is speciied wih a number nex o he arrowhead. In communicaion echnology, he area indicaion regarding an area o is mosly omied or simpliicaion purposes. Fig. -: Dirac impulse I he signal s is muliplied wih a Dirac pulse a signal value o s is achieved a ime. displaced by and hen inegraed, s d s (.4) he signal s is sampled a by he Dirac pulse. -8 F. Dellsperger 4

18 s s s Fig. -: Sample a series o Dirac pulses wih equal disance is he imporan sampling uncion or he analog-digial converer. n n,,, 3, (.5) n Fig. -: Sampling uncion...5 Sinc uncion (cardinal sine uncion) his uncion is requenly used in modulaion and signal processing. I is used in wo slighly dieren deiniions. In mahemaics as si x sin x (.6) x In signal processing and inormaion heory as sinc x sin x x (.7) -8 F. Dellsperger 5

19 he uncion s sinc muliples o. sin has a maximum value o and all zeros a ineger.5 sinc( ) Fig. -3: Sinc uncion..3 Fourier Series and Fourier ransormaion..3. Fourier Series and Fourier ransormaion o coninuous signals While sudying hea propagaion, he French mahemaician and physician Jean Bapise Joseph Fourier discovered in 87 ha each periodic signal wih he periodic ime may be represened as a sum o sine and cosine signals. he lowes requency is /, all oher signals are ineger muliples (harmonics) o his requency. he periodic signal s() wih he periodic ime may be represened by he Fourier series: a s an cosn bn sinn n a acos acos... ancosn bsin b sin... bn sinn n,,3,... (.8) is he undamenal requency Wih he Fourier coeiciens a n and b n a s d a s cos n d n b s sin n d n (.9) (): Inegraion over an arbirary periodic inerval o lengh -8 F. Dellsperger 6

20 By combining he sine and cosine erms o he same requency, an ampliude phase orm is obained. a cos n b sin n cos n (.3) n n n n n n n n s a cos n b sin n cos n n n cos cos... cos n n n (.3) Wih h n n n n a DC a b mpliude o n Harmonic b n h arcan Phase o n Harmonic (+ i an ) an For even uncions s s : (.3) a n n s d a scosn d (.33) b For odd uncions s s : a a n b s sin n d n (.34) Wih he relaions cos n e e sin n e e j jn jn jn jn he complex Fourier series wih he complex Fourier coeiciens c n is obained: s n c e n jn c s e d c c jn * n n n c c c n n n n (.35) -8 F. Dellsperger 7

21 For convering he coeiciens, he ollowing applies: a n jb n : n a a n jb n : n c n : n n n n a Re c n b Im c n n (.36) he periodic signal resuls in a line specrum wih specral lines a,,3...n. he power P s o a periodic signal normalized o a load resisance o Ohm corresponds o he sum o he powers o he individual harmonics (heorem o Parseval): n a an bn s n n N n n n (.37) P P c non-periodic (aperiodic) signal s() may be described by means o he Fourier ransormaion in he requency domain: j S s s e d (.38) nd he inverse ransormaion in he ime domain: s S S e j d (.39) S and s s orm a ransormaion pair, which is expressed wih a symbol S S speciies he disribuion o ampliude versus requency. I he Fourier ransorm s has he uni o a volage, S has he uni Vs or V/Hz and represens an ampliude densiy. n aperiodic signal has a coninuous ampliude densiy specrum. he Fourier ransorm imaginary par or magniude and phase: S is a complex uncion and may be represened as a real and S Re S jim S S e Wih Im S S Re S Im S arcan Re S j (.4) -8 F. Dellsperger 8

22 ..3. Discree Fourier ransormaion DF he discree Fourier ransormaion represens a ime-discree, periodic signal on a periodic requency specrum. leas one period wih N samples mus be used. N n N n n X k x n e x n cos k jsin k n n N N j k N (.4) k,,,...,n Requires N muliplicaions + N(N-) addiions Example: N MC s (Muliply and ccumulae) 5 ˆx Hz xdc.5 N 3 n,,...,n.8.6 X( k).4. 3 k k Posiive Frequenzen N/- N/ Negaive Frequenzen N- N Fig. -4: mpliude and Phase Specrum o a discree signal Or or a complex signal jn x n e mi n x n n n N j kn N X k x n e (.4) X k Re X k Im X k k arg X k Inverse Discree Fourier ransormaion IDF N n N n n x n X k e X k cos k jsin k N k N k N N j k N (.43) n,,,...,n x n Re x n Im x n n arg x n -8 F. Dellsperger 9

23 Fas Fourier ransormaion FF and IFF FF and IFF are special algorihms or calculaing he discree Fourier ransormaion and requires signiicanly less compuer resources han DF and IDF. Condiion: Requires q N q,,... N N MC s (Muliply and ccumulae)..3.3 Examples: Periodic unipolar square wave s S s s s Fig. -5: Unipolar square wave, ime domain and specrum s he Fourier coeiciens o his signal are: s s a sd d s S 3 5 s n sin s an s cosn d cos(n )d sin n n s n bn (even uncion) Insered in (.3) -8 F. Dellsperger

24 n sin s cos n a cos n cos n n n n n cos cos3 cos n n n he specrum o he ideal square wave wih a duy cycle o : ( s / ) has only odd harmonics. heir ampliudes have a behavior proporional o / n. n sin he envelope o he ampliudes o he specral lines ollows he sinc uncion. n he zeros are a even harmonics. he undamenal requency is also called he s harmonic..5 N=5 3.5 N=99 9% s() s.5 s() s.5 N: Number o Harmonics Fig. -6: pproximaion o square wave using Fourier series he overshoo o approx. 9% is independen rom he number o considered harmonics and is called Gibbs phenomenon. Periodic bipolar square wave s 4 S Fig. -7: Bipolar square wave, ime domain and specrum 4 s cos cos3 cos Wih he excepion o he missing DC par and he double ampliude o he coeiciens, he specrum is idenical o he specrum o he unipolar square wave. -8 F. Dellsperger

25 S Periodic unipolar pulse rain s S n sin n Fig. -8: Pulse rain, ime domain and specrum s,v log S,dB 5ns s Fig. -9: Pulse rain, ime domain and specrum n sin s cos n n n 3 sin cos sin cos sin cos he envelope o he ampliudes o he specral lines ollows he sinc uncion he zeros are a n. n sin. n -8 F. Dellsperger

26 Sampling uncion he specrum o a periodic Dirac pulse is also a periodic Dirac pulse Fig. -3: Sampling uncion, ime domain and specrum he Fourier coeiciens are n n cosk k cos cos cos 3... he Fourier ransormaion o he cosine erms resuls in he ollowing (.44) n k (.45) n Due o o, he disance o he Dirac pulses becomes larger in he requency domain, he smaller he disance o he Dirac pulses is in he ime domain. k Sine and cosine signals he ollowing applies as a general rule: j j s cosc S e c e c (.46) Cosine signal : s cosc S c c (.47) s S c c Fig. -3: Cosine signal, ime domain and specrum -8 F. Dellsperger 3

27 Sine signal 9 o : s cosc sin c S j c j c (.48) s js c c Fig. -3: Sine signal, ime domain and specrum Recangular pulse he Fourier ransormaion o he aperiodic recangular pulse resuls in a coninuous specrum conaining requencies up o ininiy. j j j e e sin j S e d si sinc (.49) s rec S si sinc (.5) s S( ) db S( ).5 log S( ) Fig. -33: Recangular pulse, ime domain and specrum he shorer he pulse, he larger he bandwidh o he main lobe in he specral domain is and vice versa. -8 F. Dellsperger 4

28 I he bandwidh B is deined as a requency range beween he -6dB poins o he main g lobe and he bandwidh muliplied by he ime duraion o he pulse, you will ind: B B g g n imporan inding o his is: B consan (.5) Sinc pulse Wih he symmery and similariy heorems o he Fourier ransormaion, he ollowing is obained or he recangular pulse: s si sinc S rec (.5) S sinc Fig. -34: sinc-pulse, ime domain and specrum riangular pulse he Fourier ransormaion o he aperiodic riangular pulse resuls in a coninuous specrum conaining requencies up o ininiy. j j S e d e d j j j e d e d e d (.53) sin si sinc : S s si sinc (.54) : -8 F. Dellsperger 5

29 s S( ) db log S( ) Fig. -35: riangle pulse, ime domain and specrum..4 Convoluion..4. Convoluion o coninuous signals For wo coninuous signals s and s, he convoluion is deined as (.55) y s s d s s ` is a ormal inegraion variable wih he dimension o ime. For he convoluion operaion, he abbreviaed orm is used y s s (.56) Due o he Fourier ransormaion o he convoluion, he ollowing applies: j Y s s d e d j s s e d d (.57) -8 F. Dellsperger 6

30 he erm in he square brackes is he Fourier ransormaion o he signal j and hereore according o he ime delay heorem S e. s delayed by j Y s S e d hus, j S s e d Y S S S S S (.58) s s (.59) and wih he dualiy S S s s (.6) he convoluion heorem speciies ha he normal produc and he convoluion produc o wo uncions orm a Fourier ransormaion pair. he produc o wo ime uncions resuls in he convoluion o he wo associaed specrums and vice versa. Examples: Convoluion o an arbirary signal s wih a Dirac pulse (in he ime domain): s s s s s s (.6) s s s s s s Fig. -36: Convoluion in ime domain o an arbirary signal wih a Dirac pulse -8 F. Dellsperger 7

31 Convoluion o an arbirary signal s wih a cosine signal (in he requency domain): Due o he dualiy o he convoluion in he ime domain and he convoluion in he requency domain, he above procedure may also be applied in he requency domain. I he signal s is a cosine signal s cos c S, he associaed specrum and he convoluion wih an arbirary signal c c S S S S S S c S c s c c S : (.6) s S s s S a) S S S S S c S c S c c S c c c c b) Fig. -37: rbirary signal and cosine signal: a) Muliplicaion in ime domain, b) convoluion in requency domain Due o he convoluion, he signal S is displaced o he requencies c and c. -8 F. Dellsperger 8

32 Inerpreaion: I an arbirary signal convoluion o he specrums resul is s is muliplied wih a cosine signal s S and S requency displacemen o he signal in he ime domain, a is perormed in he requency domain. he S o he requencies c and c double side band ampliude modulaion wih suppressed carrier s = carrier) ( s =modulaion signal, Sampling o a signal wih a Dirac pulse series s S s a) 3 3 S S B S S s s s s s s s s b) Fig. -38: Sampling o an arbirary signal: a) Muliplicaion in ime domain, b) convoluion in requency domain Here, i becomes clear ha he specrums overlap and a rue reconsrucion o he signal is no longer possible, i he bandwidh B o he signal o be sampled becomes larger han / s ( s sampling requency). his eec is called aliasing. -8 F. Dellsperger 9

33 ..4. Convoluion o discree signals discree signals xn and hn y n x i h n i x n h n (.63) i h n x n H k X k h n x n H k X k..5 Correlaion Correlaion is an operaion which is used in modulaion echnology or he deerminaion o he power specral densiy o a signal and or he comparison o digial codes (e.g. or CDM). I is closely relaed o he convoluion. For energy signals he correlaion is deined as: xy (.64) R s s d (.65) and or power signals / R lim s s d (.66) xy / he correlaion uncion s and s Rxy o which he second signal represens a measure or he similariy or relaion o boh signals in uncion o he displacemen ime. is he displacemen ime by means ) in respec o he irs signal s. s is shied o he le hand ( ) or o he righ hand side ( For periodic power signals wih he periodic ime, he ollowing applies R s s d (.67) xy Rxy is a periodic uncion as well provided ha applies o I s and s I s s s (CF): s and s. are wo dieren signals, a cross-correlaion uncion (CCF) is given. a signal is correlaed wih isel which is called auo-correlaion uncion / Rxx lim ss d (.68) / In pracice, i is necessary o se a limied ime inerval wih he resul / Rxx ss d (.69) / -8 F. Dellsperger 3

34 Or or a periodic power signal wih he periodic ime Rxx ss d (.7) Characerisics o xx Rxx : xxmax : R R P P = signal power, R xx R xx Rxxmax (even uncion) nd or a periodic signal xx xx xx = maximum value o he CF R R n R n,,3,... For discree signals, or example, binary random bi sequences (PN codes), he correlaion uncion may be deermined wih P R v a b xy k kv P k P R v a a xx k kv P k a k,b k = sequences o bis k = bi number =,,, P- P = number o bis o a period o he bi sequence = lengh o he PN code v = displacemen in bis (.7) Example o a PN code wih he lengh 7: a k k v a k P k Rxx v ak ak /7 ak /7 ak v ak /7 6-8 F. Dellsperger 3

35 Rxx v / 7 v Fig. -39: uocorrelaion uncion o a PN-Code wih lengh 7 he associaed power densiy specrum is according o (.75) and he consideraion o he riangular pulse in (.53): G() S(.5 b b Birae b b Bi-ime mpliude b Fig. -4: Power specrum densiy o a PN-Code wih lengh 7..6 Energy and power specrum densiy he Fourier ransormaion (.38), describes he disribuion o he ampliude o a signal versus requency. For energy signals and power signals, he disribuion o he energy or power versus requency may be described. his resuls in he energy specrum densiy and power specrum densiy. ccording o he Parseval heorem, he oal energy in he ime domain is equal o he oal energy in he requency domain. Wih (.), he normalized energy wih he uni Ws is: (.7) E s d S d he energy specral densiy E (in Ws/Hz) is hereore E S S S R xx (.73) S is he conjugae complex value o S. -8 F. Dellsperger 3

36 For he power specral densiy (PSD, in W/Hz), he power specral densiy G() and he auo-correlaion uncion R are Fourier ransorms o each oher according o Wiener and Khinchine: Rxx G xx (.74) j G R R e d xx xx j Rxx G G e d he oal normalized power resuls in (.75) P G d (.76) -8 F. Dellsperger 33

37 .3 Reerences [] aub, H., Schilling, D.L.: Principles o communicaion sysems. McGraw-Hill, nd Ediion 986 [] Kammeyer, K.D. : Nachrichenüberragung. Vieweg+eubner, 4. ulage 8 [3] Roppel, C.: Grundlagen der digialen Kommunikaionsechnik. Carl Hanser Verlag, 6 [4] Ohm, J-R., Lüke, H.D.:Signalüberragung. Springer Verlag Berlin,. ulage 7 [5] Schwarz, M.: Inormaion, ransmission, Modulaion, and Noise. McGraw-Hill, 98 [6] Zinke, O., Brunswig, H.: Hochrequenzechnik, Springer Verlag Berlin, 5. ulage 999 [7] Sumpers, F.L.M.H.: heory o requency modulaion noise. Proc. Ins. Radio Engrs. 36, 948, 8-9 [8] Rice, S.O. : Saisical properies o a sine wave plus random noise. Bell Sys. echn.j. 7, 948, 9-57 [9] von Grünigen, D.Ch.: Digiale Signalerarbeiung mi einer Einührung in die koninuierlichen Signale und Syseme. Carl Hanser Verlag, 5. ulage 4 [] von Grünigen, D.Ch.: Digiale Signalerarbeiung: Bauseine, Syseme, nwendungen. Fooroar Prin und Media, 8 [] Dellsperger, F.: Passive Filer der Hochrequenz- und Nachrichenechnik. Lecure Scrip, -8 F. Dellsperger 34