Pulse amplitude modula.on Baseband to passband and back
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1 Pulse ampliude modula.on Baseband o and back message inpu ) Today s opics concern wih he modulaor and demodulaor ransminer acharapan Suwansan.suk Sampler Quan.zer Source Channel Modulaor analog sequence symbol sequence inerface Channel #7 EIE 7 Digial Communica.ons King Mongku s Universiy of Technology Thonburi Chaper : Channels, modula.on, and demodula.on Inerpolaor message oupu ) Table lookup Source Channel receiver Demodulaor Topics Pulse ampliude modula.on PAM;.) Modula.on: o and back.) AXer his lecure, you will be able o Conver a sequence of bis ino a, and vice versa, using he pulse ampliude modula.on Iden.fy pulses ha are bandlimied and do no creae he inersymbol inerference i.e., he shix-orhonormal pulses) Describe a mehod o conver a o a, and vice versa Plo he and s in.me- and frequency-domains for he noiseless channel
2 Recall ha a modulaor urns a sequence of bis ino a analog) Also recall ha a demodulaor urns a ino a sequence of bis value value... Modulaor value... Demodulaor value 5 7 ime sec) 5 7 ime sec).me sec).me sec) 5 This lecure ignores emporarily he effec of noise ha comes from he channel In our design, he modulaor and demodulaor are divided ino blocks inpu Modulaor x) inpu modulaor Modulaor o oupu Demodulaor sequence of real numbers channel The modulaor is conneced direcly o he demodulaor as if he channel were noiseless In nex lecure, we sudy random processes and add back he noise oupu demodulaor o Demodulaor Each pair of he opposing block performs a reversed role 7
3 Modulaor Overview: PAM works as follows inpu oupu sequence of real numbers modulaor demodulaor o o channel Amplifies he ) b,b,b,... inpu u k = Builds a vecor sum) u,u,u,... modulaor a, if b k = a, if b k = u) = X u k p k= kt) Demodulaor OVERVIE AND BASEBAND MODULATION Reads coefficiens on axes) Projecs a vecor ono each axis) c,c,c,... v,v,v,... demodulaor v) oupu c k =, if v k, if v k < Z v k = v)p kt)d 9 Baseband modula.on and demodula.on involve hree design parameers. Symbol inerval T>. ShiX-orhonormal pulse. Amplifica.on facor a> p) For simpliciy of drawing, we will use he square wave o demonsrae he concep of modula.on and demodula.on p) PAM uses he ampliude of a pulse o convey bi or Bi maps o ap) Bi maps o ap) a ap) ap) a ap)
4 To covey many bis, shix he scaled pulses by mul.ples of T and add hem ap) The u) can be inerpreed as a vecor sum a ) ap T ) a ) a ap T ) u) =ap) ap T ) a T ) u) =a ) a ) Binary inpu a u = a a Binary inpu Here is a for he inpu a u) Example a: Use he square roo raised cosine filer o modulae he inpu Suppose he uni of.me is second Suppose T =sec, he rolloff facor is =., and a = Plo u) for he inpu is 5 Answer: u) =ap) ap T ) ap T ) =p) p ) p ) a where p) is he square roo raised cosine filer wih =. Binary inpu 5
5 Solu.on: plo of. u) con.nued) Solu.on: Malab code u) Baseband u) ime sec) clear all; % clear variables! T = ; % symbol inerval! alp =.; % rolloff facor! a = ; % amplificaion! = -:.:5; % range of ime o plo!! %, modulaed of bis! u = a*src,t,alp) - a*src-t,t,alp) - a*src-*t,t,alp);!! % plo! figure);! clf; % clear figure! plo, u, 'Linewidh', );! grid on;! xlabel'ime sec)');! ylabel'u)');! ile'baseband u)');! 7 Fourier ransform ûf) of he is easy o obain u,u,u,...,u N modulaor In his elemenary course, we will only have a finie number of symbols, says u,u,u,...,u N, o ransmi Then, he is a finie sum: u) = NX u k p k= And he Fourier ransform is kt) u) = NX u k p k= kt) " N # X ûf) = u k e ifkt ˆpf) k= con.nued) From he Fourier ransform, we ge ûf) = In oher words, bandwidh of NX u k e ifkt ˆpf) k= ûf) = bandwidh of ˆpf) For a square roo of raised cosine filer, his will be + T 9
6 To derive he expression of ûf), apply hese proper.es of he Fourier ransform All func.ons in hese proper.es are ransform may no even exis) Oherwise, he Fourier. Lineariy: Suppose func.on x) is a linear combina.on of func.ons g) and h), i.e., x) =ag)+bh) where a and b are complex numbers. Then, he Fourier ransform ˆxf) is a linear combina.on of he Fourier ransforms and, i.e., ĝf) ˆxf) =aĝf)+bĥf) L ĥf) con.nued). Time shif: Suppose func.on x) is a #me shi) of func.on g), i.e., where x) =g ) is a real number eiher nega.ve, zero, or posi.ve). Then, he Fourier ransform Fourier ransform ĝf), i.e., ˆxf) ˆxf) =ĝf)e if Example: The Fourier ransform of ˆxf) =ˆpf)e ift is a phase shi) of he x) =p T ) is Ex b: Plo he magniude and phase of he Fourier ransform. ûf) Solu.on: plos of ûf) and. Magniude uhaf) of he Fourier ransform uhaf) \ûf) Recall from example a ha he is u) =ap) ap T ) ap T ) In his example, a= and T = Magniude Answer: ûf) =aˆpf) aˆpf)e ift aˆpf)e ift ) h = a e ift e ift iˆpf) angle radian) Angle of he Fourier ransform uhaf)
7 Solu.on: Malab code clear all; % clear variables! T = ; % symbol inerval! alp =.; % rolloff facor! a = ; % amplificaion! = +alp)/*t); % bandwidh! f = -.5:.:.5; % range of frequency o plo! uha = a*f_srcf,t,alp)...!!!.* - exp-*pi*i*f*t) - exp-*pi*i*f*t) );! % plo magniude of uha! subplo,,);! plo f, absuha), 'Linewidh', );! xlabel'');! ylabel'magniude');! ile['magniude uhaf) of he ',...!!! 'Fourier ransform uhaf) ]);! se gca, 'xick', [ ] ); % mark he x-axis! grid on;! con.nued) % plo phase of uha! subplo,,);! plo f, angleuha), 'Linewidh', );! xlabel'');! ylabel'angle radian)');! ile'angle of he Fourier ransform uhaf)');! se gca, 'xick', [ ] ); % mark he x-axis! grid on;! 5 Example c: ha is he bandwidh of u)? Solu.on: The bandwidh of u) equals he bandwidh of he pulse he square roo raised cosine filer): = + T = +. =.5 Hz The answer is consisen wih he drawing of ûf) inpu oupu sequence of real numbers modulaor demodulaor Modulaor o o Demodulaor BASEBAND DEMODULATION channel 7
8 Le s move on o he demodula.on Amplifies he ) Builds a vecor sum) X b,b,b,... u,u,u,... u) = u k p k= inpu modulaor a, if b k = u k = a, if b k = kt) Example : Demodula.on using he square wave v) noiseless inpu o he demodulaor Reads coefficiens on axes) Projecs a vecor ono each axis) c,c,c,... v,v,v,... demodulaor v) oupu c k =, if v k, if v k < Z v k = v)p kt)d PAM parameers:, square wave p) =s), T = Ques.on: ha is he oupu? Answer: The oupu is a = 9 con ) Solu.on: he firs bi is v) con ) Solu.on: he second bi is v) ) ) v) ) Z v) )d = Guess ha bi was sen - v) ) Z v) )d = < Guess ha bi was sen
9 d d con ) Geomerically in a vecor space, we projec vecor v ono each axis s axis Example 5: Demodula.on using square roo raised cosine pulse inpu a a u) = ap)+ap T ) modulaor h axis oupu v v demodulaor The coefficiens from he projec.on and ) are ranslaed ino bis and ) v Suppose T =, A =, and he pulse is he square roo of raised cosine filer wih =. The inpu is Verify in Malab ha he demodulaor gives Approxima.ons because Malab compues he inegrals numerically v a, v a Solu.on: Plo of s needed o compue. v = R u)p)d Solu.on: Plo of s needed o compue. v = R u)p T )d ) =p) u) u) ) =p T ) u)p T ) v is he area under his graph u)p) v is he area under his graph 5
10 Answer: Yes, he approxima.ons hold clear all;! T = ; % sampling inerval! a = ; % amplificaion! alp =.; % rolloff facor! = linspace-*t, 5*T ); % range of imes: -T o 5T!! % inpu =! %! u = -a*src, T,. ) + a*src -T, T, alp); Z 5T! Numerically inegrae u)p)d T % demodulaion:! v = rapz, u.* src, T, alp) ) % oupu: v = -.997! v = rapz, u.* src-t, T, alp) ) % oupu: v =.997! inpu oupu sequence of real numbers modulaor demodulaor Modulaor o o Demodulaor BASEBAND TO PASSBAND channel Trapezoidal numerical inegra.on Numerically inegrae Z 5T T u)p T )d 7 Passband conversion needs a bandlimied and a large enough carrier freq This propery of Fourier ransform is cenral o modula.on All func.ons in his propery are L Symbol Meaning AssumpIon for o- conversion Bandwidh of u) Also equals he bandwidh of he shix-orhonormal pulse p) Carrier frequency < The pulse is bandlimied) ModulaIon propery: if frequency, hen g) =h) cos f ) f ĝf) = ĥf + f )+ ĥf f ), for any consan Inerprea#on: if we mul.ple a func.on by a cosine erm of frequency f, hen is Fourier ransform shix o he lex and o he righ by f 9
11 To cover a o a one, mul.ply by. cos ) ˆxf) consiss of wo hills cenered a. ûf) ± b,b,b,... u,u,u,... inpu modulaor u) o x) \ûf) frequency f The is x) =u) cos ) carrier frequency frequency f In he frequency-domain, ˆxf) =ûf + )+ûf ) For example, suppose he magniude and phase for he are above con ). ˆxf) =ûf + ) + ûf fc) The bandwidh is wice he bandwidh ˆxf) Baseband: ûf) bandwidh + + frequency f \ˆxf) frequency f Passband: ˆxf) bandwidh + + frequency f ˆxf) is a sum of wo iden.cal copies of ûf), lex- and righshixed by he amoun frequency f
12 u) Baseband u) ime sec) Magniude uhaf) of he Fourier ransform uhaf) Angle of he Fourier ransform uhaf) Example a: o Solu.on: Time-domain plo. u) Passband x) x) =? angle radian) Magniude ûf) \ûf) o ˆxf) =? x) Suppose u) is he in example, and he carrier frequency is = Plo he in.me domain frequency domain ˆxf) x) and in ime sec) 5 con ) frequency-domain plo Magniude Magniude xhaf) of he Fourier ransform xhaf) Example b: Bandwidh of he Ques.on: ha is he bandwidh of x)? Answer: Twice bandwidh of he : angle radian) Angle of he Fourier ransform xhaf) =.5 Hz.5 Hz from example b)
13 inpu sequence of real numbers modulaor Modulaor o channel To cover from o, mul.ply by cos ) and low-pass filer c,c,c,... oupu v,v,v,... demodulaor v) o y) oupu demodulaor o Demodulaor PASSBAND TO BASEBAND The is In he frequency-domain, v) =g h LP )) g) =y) cos ) convolu.on = bandwidh of he pulse h LP ) = sinc ) Ideal low-pass filer of bandwidh ) ˆvf) = ŷf )+ ŷf + ), f apple, oherwise 9 5 The y) is frequency-shixed and low-pass filered In frequency domain, he a each sage is below v) ideal low-pass filer: impulse response. h LP ) g) y) ĝf) = ŷf + )+ ŷf ) ŷf) Magniude ghaf) of he Fourier ransform ghaf) Magniude yhaf) of he Fourier ransform yhaf) v) Baseband before LPF g) g) cos ) Passband x) y) Magniude.5.5 Magniude Angle of he Fourier ransform ghaf) Angle of he Fourier ransform yhaf) angle radian) angle radian) ime sec) ime sec)
14 con.nued) ĥ LP f) Ideal low-pass filer Here is a summary of PAM Amplifies he ) Builds a vecor sum) ShiX he cener frequency from o ) angle radian) Magniude ˆvf) Magniude vhaf) of he Fourier ransform vhaf).5.5 Angle of he Fourier ransform vhaf).5.5 angle radian) Magniude ĝf) Magniude ghaf) of he Fourier ransform ghaf).5.5 Angle of he Fourier ransform ghaf).5.5 b,b,b,... u,u,u,... inpu u k = a, if b k = a, if b k = Reads coefficiens on axes) Projecs a vecor ono each axis) c,c,c,... v,v,v,... oupu c k =, if v k, if v k < modulaor u) = demodulaor o X u k p kt) k= v) x) =u) cos ) o Channel ShiX he cener frequency from o ) y) 5 5 For he noiseless channel, PAM achieve reliabiliy and frequency consrains inpu b,b,b,... u,u,u,... = modulaor u) o x) = = = Laer we sudy channel and learn ha indeed v) is a noisy version of u). u) v) oupu c,c,c,... demodulaor v,v,v,... v) o The inpu a each sage = he corresponding oupu y) The mehods for PAM mod. and demod. are unchanged ih noise, he oupu may no equal he inpu 55 5
15 Here are acronym and symbols Acro/Symbol Meaning PAM pulse ampliude modula.on T symbol inerval, T> sec) p) shix-orhogonal pulse,,,... orhonormal func.ons basis elemens): k) =p kt) bandwidh of p), i.e., < < Hz) A u) amplifica.on facor of PAM, A> a he modulaor v) a he demodulaor Here are Fourier-ransform proper.es Lineariy: Time shix: Modula.on: x) =ag)+bh) $ ˆxf) =aĝf)+bĥf) x) =g ) $ ˆxf) =ĝf)e if g) =h) cos f ) $ ĝf) = ĥf + f )+ ĥf f ) h LP ) carrier frequency, Hz) ideal low-pass filer of bandwidh : h LP ) = sinc ) and ĥ LP f) =, f apple, oherwise 57 5 Srucure of he PAM / Summary modulaor / demodulaor -> / -> Mehods of he PAM Amplifica.on facor Baseband modula.on forming a sum of vecors) Bandwidh of he Baseband demodula.on projec.on) Baseband o Passband o Examples plos, Malab) in.me- and frequency-domains 59
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