Communicaion Sysems, 5e Chaper : Signals and Specra A. Bruce Carlson Paul B. Crilly The McGraw-Hill Companies
Chaper : Signals and Specra Line specra and ourier series Fourier ransorms Time and requency relaions Conoluion Impulses and ransorms in he limi Discree Fourier Transorm (new in 5 h ed.) The McGraw-Hill Companies
Fourier Transorm Time o requency domain V exp j d Frequency o ime domain V expj Condiion E d d Table T. on pages 78-78 3
Table T.: Fourier Transorm Pairs
Fourier Transorm Properies Lineariy Superposiion Time Shiing Scale Change Conjugaion Dualiy Frequency Translaion Conoluion Muliplicaion Modulaion Table T. on pages 78-78 Secion.3, pp. 44-5 Secion.4, pp. 5-58 5
Fourier Transorm Properies 6
Two Mos Imporan Properies Conoluion w V W Mixing w V W * 7
Conoluion Conoluion in he ime domain The Fourier Transorm Pair is: 8 d w d w w W V w W V d e W d e W d d e w d e d w w F j j j j
Mixing Conoluion in he requency domain The Fourier Transorm Pair is: 9 d V W d W V W V W V w * w d e V w d e w d V d e W d e d W V W V F j j j j
Periodic Signals Relaionship m or m an ineger and T o he period Aerage/mean (compued rom one period) T Aerage Power P T T T T d d
Fourier Series (Periodic Signals) Fourier Series Coeiciens T cn T Signal Represenaion exp j n d c expj n n where n T
Table T.: Fourier Series
Fourier Series: The Fourier Transorm o Periodic Signals The conoluion o one period wih a comb uncion 3 rec A A V sinc k T k T comb n T n T T comb k T k rec A n T n T A V sinc
Periodic Signal Transorm Relaionship Equialen m T Fourier Transorm p comb k T V comb T n V p T n k enelope muliplied by scaled, spaced impulses 4 T T
Fourier Transorms and Series For a periodic signal: Take he Fourier Transorm o one period o he waeorm and plo he specrum This is he enelope o he Fourier Series specrum The Fourier Series specrum is hen a line specrum wih coeiciens spaced a n x 5
Fourier Transorms and Series Fourier Transorm o one res pulse Fourier Series o an ininie rec pulse rain 6
Fourier Series o Time Signal I you know he Fourier Transorm or Fourier Series, you can ind he ime waeorm approximaely. Inerse Fourier Transorms and series do no correcly represen disconinuiies. I he original ime domain cure makes a discree change, The inerse Fourier Cure gies he mid-poin beween he wo cures The inerse Fourier Series has he Gibbs Phenomenon Ininie energy and ininie power limiaions means ha he uncions are approximaed, bu no direcly soled. 7
Copyrigh The McGraw-Hill Companies, Inc. Permission required or reproducion or display. Gibbs phenomenon: sep disconinuiy e lim c expj d N T N nn n 8
Copyrigh The McGraw-Hill Companies, Inc. Permission required or reproducion or display. Fourier-series reconsrucion o a recangular pulse rain: (Gibb s) N cn expj nn Malab Gibbs 9
Copyrigh The McGraw-Hill Companies, Inc. Permission required or reproducion or display. Fourier-series reconsrucion o a recangular pulse rain See MATLAB example
Symmery in Transorms In working wih ransorms, here are many properies o ime signals (and specra) ha can simpliy aking orward and inerse ransorms! Purely real signals Purely imaginary signals Een symmeric signals Odd symmeric signals Conjugae symmeric signals The properies can be used o check your resuls! I he resul doesn hae he propery, you did he mah wrong!
Useul Signal Properies: Symmery Symmery (een) Ani-Symmery (odd) Conjugae (real is een, imag is odd) *
Symmery Examples () Symmery (een) cos cos Ani-Symmery (odd) sin sin Conjugae Symmery (real is een, imaginary is odd) exp j conjexp 3
Real Signals The Fourier Transorm o Real becomes V V j The real par o V() has een symmeric The imag. Par o V() has odd symmeric For an Imaginary Signal he properies lip Real par o V() has odd symmery Imag. par o V() has een symmery d cos d j sin Alernaely: i you calculae he posiie requencies, you can use hem o deine he negaie requencies using symmery. e d 4
Een-Symmeric and Real Signals The Fourier Transorm becomes 5 ' ' cos ' ' ' ' d d e e d e d e d e d e d e d e d e V j j j j j j j j j
Conjugae Symmeric Signals The Fourier Transorm becomes 6 * * ' * ' sin Im cos Re sin cos ' ' ' ' d j d d j d d e d e d e d e d e d e d e V j j j j j j j Noe ha boh erms are real; hereore, he resul is purely real!
Rayleigh s Energy Theorem E Energy in he ime domain is equal o energy in he requency domain! E V E d d j * * j e d d e d V conj * j * e d V d V V E d V d d 7 d
Imporan Signal Propery: Causaliy Signals ha haen happened ye are no known! Usual applicaion For a single signal analysis Signals sar a ime =, and ()= or < Laplace ransorm signals Filers are ypically deined as saring a = or For signal processing The signal exis up o a ime =, and ()= or > We don know wha comes nex 8
Causaliy and Filering Conoluion Form z h x d The iler, h, can be deined or posiie ime only The signal, x, is deined or all pas ime up o ime Then, when limied by he iler impulse response: z T h x d 9
The RC Filer y() () V s Y s H s R sc sc s RC RC h RC exp, RC Using Rayleigh s Energy E V d d 3
3 Homework.-8: An RC Filer Percen o oal Energy based on he bandwidh o an exponenial ime signal impulse response a W= b/π and W = 4b/π = b/π,, b exp A b A b b A d b A E E exp exp j b A V Toal Energy is:
The exponenial ime signal Time Response A exp, b, W=b/π Frequency Response V A b j E A E b 3
33 Homework.-8 (con) Percen o oal Energy a arious Bandwidhs W W= b/π and W = 4b/π = b/π b A d b A E E W W W percen b W b b b E E W W W an an 4 percen W W W V d b A d b j A b j A E
Homework.-8 (con) Percen o oal Energy a W= b/π and W = 4b/π = b/π Wp Wp Wp b an 4 4 b an 4.4. 844 b an.468. 937 For a simple RC iler w co is he 5% power poin b w co RC A RC 34
Realizable Filers, RC Nework Noes and igures are based on or aken rom maerials in he course exbook: Bernard Sklar, Digial Communicaions, Fundamenals and Applicaions, Prenice Hall PTR, Second Ediion,. 35
The Use o Percen Toal Energy I you wan o receie a inie ime signal (inie energy) signal, wha bandwidh perec iler should you use? For a decaying exponenial signal 5% o he energy receied a iler = co =/RC 84.4% receied a iler =4 x co =4/RC 93.7% receied a iler = x co =/RC We usually wan 9%-99% o a pulses energy This also has implicaion or digial sampling raes! Nyquis Theory or a specrum wih ininie ails?! 36
Anoher Example Wha bandwidh ideal iler should be use i we wan o iler a bipolar square wae and receie 9% o he power? Use he approach jus shown Percen energy in requency or one period o he periodic square wae The general resul is he inegral o he sinc^ uncion rom = o =W? (no easily perormed, numerical inegraion useul) 37
Modulaion (Mixing Prop) Frequency ranslaion due o real or complex mixing producs z x cos x expj exp j j e X Z e j Using rig uncions, ry cos x cos mixing Table T.3 on pages 85-853 38
Table T.3
Copyrigh The McGraw-Hill Companies, Inc. Permission required or reproducion or display. Frequency Translaion o a Bandlimied Specrum s exp j c S V c (a) Iniial Signal Specrum (b) Oupu Specrum 4
Muliplicaion-Conoluion Conoluion in he ime domain w w d w The Fourier Transorm Pairs are: w V W w V W * d 4
Copyrigh The McGraw-Hill Companies, Inc. Permission required or reproducion or display. Frequency Translaion: Complex Mixing s exp j S c V j* c c (a) Iniial Signal Specrums (b) Oupu Specrum 4
Copyrigh The McGraw-Hill Companies, Inc. Permission required or reproducion or display. RF Pulse Mixing s A cos c (a) RF Pulse Is conoluion easier? S A A sinc sinc c c (b) Magniude specrum 43
Anoher Inerpreaion A limied ime duraion cosine waeorm window sample o an ininie periodic signal S A s A cos A sinc sinc c As he window becomes longer, he sinc ges narrower going o an impulse as τ This is criically imporan when we alk abou inie ime sample lenghs o signals. c c 44
Mixing RF A cos RF RF RF IF RF LO RF Inpu IF Oupu LO Local Oscillaor A cos LO LO LO IF A cos A cos IF RF RF RF A A cos cos LO RF RF RF LO LO LO LO LO 45
Trigonomery Ideniies sin sin cos cos Table T.3 on pages 85-853 sin cos cos sin sin cos cos sin cos cos sin sin cos cos sin sin sin sin cos sin sin cos sin sin sin cos cos cos cos sin cos cos 46
Mixing () Resaing IF A A cos cos IF LO RF Using an Ideniy A A cos A LO LO A RF RF cos RF Ideal Low Pass Filering IF A A cos LO RF RF RF RF RF LO LO LO LO RF RF RF LO LO LO LO 47
Specral Equialen Real Mixing Real Signal Cosine Mixing Producs Real Signal Specrum Mixing Cos Signal Specrum Conoled Signal Specrum LPF Low Pass Specrum The mixing o a real RF inpu wih a real Cosine local oscillaor Real Signal and Cosine LO specrum Pos mixer sum and dierence specrum Pos Low Pass Filer (LPF) resul 48
Specral Equialen Complex Mixing Real Signal Complex Oscillaor Mixing Producs Real Signal Specrum Mixing Exp Signal Specrum Conoled Signal Specrum LPF The mixing o a real RF inpu wih a Complex local oscillaor Real Signal and Complex LO specrum Pos mixer sum specrum (conoluion in req.) Pos Low Pass Filer (LPF) resul Low Pass Specrum 49
Higher Order Mixing Mixers in Microwae Sysems (Par ) Auhor: Ber C. Henderson WJ Tech-noe hp://www.riquin.com/prodser/ech_ino/docs/wj_classics/ol7_n.pd Par o WJ Comm. Technical Publicaions hp://www.riquin.com/prodser/ech_ino/wj_ech_publicaions.cm The old Wakins Johnson and laer WJ is now owned by Triquin hp://www.riquin.com/ 5
Conoluion Filering o unwaned specral componens is perormed by ilering. Conoluion in he ime domain Muliplicaion in he requency domain 5
Copyrigh The McGraw-Hill Companies, Inc. Permission required or reproducion or display. Graphical inerpreaion o conoluion w w Ae d Skeching a Conoluion Where does i sar? Where does i change? Where does i end? Wha is he general shape enering a region? Wha is he shape in he region? Wha is he shape leaing he region? u T w rec T T 5
Copyrigh The McGraw-Hill Companies, Inc. Permission required or reproducion or display. Resul o he conoluion Skeching a Conoluion Where does i sar? Where does i change? T, riangle oerlaps exponenial Where does i end? neer Wha is he general shape enering a region? More han linear increase Wha is he shape in he region? Exponenial decrease Wha is he shape leaing he region? Always inside, doesn happen Now derie equaions: enering and decreasing And one alue: he maximum a T 53
Impulses and Transorms in he Limi When dealing wih discree, inherenly disconinuous message daa we require appropriae mahemaical mehods o derie and describe he modulaed waeorms. Signal descripions or impulses (in ime and requency), sep uncions, ec. are required. Deine a coninuous ime, parameerized uncion ha approaches an impulse/sep uncion as one o he parameers approaches ininiy or zero. Wha are some o hese uncions. 54
Dela Funcion Approximaions Rec rec Sinc sinc Gaussian exp 55
Copyrigh The McGraw-Hill Companies, Inc. Permission required or reproducion or display. Rec and Sinc impulses as 56
Impulse Properies Coninuous sampling is equialen o discree samples d d d Scaling d d d a a 57
Impulses in Frequency Transorm pairs A W lim sinc W W F W lim AsincW A W A F W A A F A W 58
Signal Smoohing Signal approximaions ha proide rounding or smoohing o rapid ransiions in ime. Inheren in ransmied signals due o componen and channel eecs 59
Smoohing he Edges A more pracical requency domain iler: The raised Cosine iler Cosine band edge roll-o is oen used Easy o implemen in MATLAB A nice explanaion o inding he order o iler roll-o is proided in he ex. 6
Copyrigh The McGraw-Hill Companies, Inc. Permission required or reproducion or display. Raised cosine pulse. (a) Waeorm (b) Deriaies (c) Ampliude specrum Figure.5-7 6
Using Rec o Make a Filer Shape in he Time Domain Any coninuous ime signal can hae a possibly desirable par isolaed o creae a iler. Raised Cosine Cosine squares Sine (or odd-symmeric signals) Gaussian Main-Lobe Sinc 6