Wrap Up. Fourier Transform Sampling, Modulation, Filtering Noise and the Digital Abstraction Binary signaling model and Shannon Capacity
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1 Wrap Up Fourier ransorm Sampling, Modulaion, Filering Noise and he Digial Absracion Binary signaling model and Shannon Capaciy Copyrigh 27 by M.H. Perro All righs reserved. M.H. Perro 27 Wrap Up, Slide
2 Cosines and Sines as Basis Funcions Periodic uncions can be approximaed by he addiion o weighed cosine and sine waveorms wih progressively increasing requency Even Funcion Odd Funcion cos(ω o ) sin(ω o ) cos(2ω o ) sin(2ω o ) cos(3ω o ) sin(3ω o ) M.H. Perro 27 Wrap Up, Slide 2
3 Fourier Series and Fourier ransorm he Fourier Series deals wih periodic signals he Fourier ransorm deals wih non-periodic signals M.H. Perro 27 Wrap Up, Slide 3
4 M.H. Perro 27 Wrap Up, Slide 4 2A π B A A π 2A 3π -2A π -2A 3π B A Graphical View o Fourier Series /4 x() /4 A -A /2 x() A -A
5 Graphical View o Fourier ransorm A x() - 2A X() his is called a sinc uncion - 2 M.H. Perro 27 Wrap Up, Slide 5 2
6 Modulaion ransmission Recepion Voice ransmier Receiver Voice X() Y() Voiceband requencies ransmission requencies Voiceband requencies recepion requencies Modulaion is used o change he requency band o a signal Enables communicaion in dieren requency bands Used in cell phones, AM/FM radio, WLAN, cable V,. M.H. Perro 27 Wrap Up, Slide 6
7 Filering ransmission Recepion Voice ransmier Inererer Receiver Voice X() Y() Voiceband requencies ransmission requencies Voiceband requencies recepion requencies Filering is used o remove undesired signals ouside o he requency band o ineres Enables selecion o a speciic radio, V, WLAN, cell phone, cable V channel Undesired channels are oen called inererers M.H. Perro 27 Wrap Up, Slide 7
8 AM Modulaion and Demodulaion ransmier Oupu Receiver Oupu x() y() y() w() Lowpass Filer H() r() 2cos(2π o ) 2cos(2π o ) Muliplicaion (i.e., mixing) operaion shis in requency Also creaes undesired high requency componens a receiver Lowpass ilering passes only he desired baseband signal a receiver M.H. Perro 27 Wrap Up, Slide 8
9 Frequency Domain Analysis X() Y() H() W() - o o -2 o - o o 2 o x() y() ransmier Oupu y() w() H() r() Receiver Oupu 2cos(2π o ) 2cos(2π o ) Lowpass Filer - o o - o o When ransmier and receiver local oscillaors are mached in phase: Demodulaed signal consrucively adds a baseband M.H. Perro 27 Wrap Up, Slide 9
10 I () Q () i () - o o 2cos(2π o ) 2sin(2π o ) q () j o - o -j I/Q Modulaion y() ransmier Oupu Y i () j - o o Y q () o - o -j 2cos(2π o ) 2sin(2π o ) Receiver Oupu M.H. Perro 27 Wrap Up, Slide y() j - o o o - o -j H() Lowpass Lowpass Modulae wih boh a cosine and sine wave I and Q channels can be broadcas over he same requency band I/Q modulaion allows wice he amoun o inormaion o be sen compared o basic AM modulaion wih same bandwidh H() i r () q r () 2 2 I r () Q r ()
11 Energy ranser in Wireless Communicaion x() ransmier ransmission r Recepion Anenna Area Receiver rx() Lowpass H() Voice X() RX() Signal Noise khz 45 MHz khz 45 MHz Receiver anenna is limied in is abiliy o capure ransmier energy according o is area and disance, r, rom ransmier Noise in he receiver causes corrupion Amoun o corrupion depends on signal-o-noise raio M.H. Perro 27 Wrap Up, Slide
12 he Need or Sampling Real World (USRP Board) x c () x c () A-o-D Converer x[n] Malab x[n] n he boundary beween analog and digial Real world is illed wih coninuous-ime signals Compuers (i.e. Malab) operae on sequences Crossing he analog-o-digial boundary requires sampling o he coninuous-ime signals M.H. Perro 27 Wrap Up, Slide 2
13 he Sampling heorem x c () p() x p () A X c () P() A X p () - bw bw bw bw bw bw Overlap in requency domain (i.e., aliasing) is avoided i: We reer o he minimum / ha avoids aliasing as he Nyquis sampling requency M.H. Perro 27 Wrap Up, Slide 3
14 he Discree-ime Fourier ransorm Allows us o deal wih non-periodic, discree-ime signals Frequency domain signal is periodic in his case x[n] Where: n X(e j2πλ ) - λ Noe: uncion in Malab used o compue DF M.H. Perro 27 Wrap Up, Slide 4
15 Digial Processing o Analog Signals x c () A-o-D Converer x[n] Downsample by N / Sample/s /(N) Sample/s r[n] Digial Signal Processing Operaions /(N) Sample/s u[n] Upsample by M M/(N) Sample/s D-o-A Converer M/(N) Sample/s y c () Digial circuis can perorm very complex processing o analog signals, bu require Conversion o analog signals o he digial domain Conversion o digial signals o he analog domain Downsampling and upsampling o mach sample raes o A-o-D, digial processor, and D-o-A M.H. Perro 27 Wrap Up, Slide 5
16 Advanages o Digial Processing Digial componens correc small analog errors a each processing sep We can build large, reliable sysems despie non-ideal componens and he presence o bounded noise We can accommodae more precision by represening inormaion wih longer sequences o symbols Excep or he conversion seps, we can use simple digial componens do achieve arbirary precision in processing We absrac ou he noion o real ime when convering o sequences o discree values he speed o inervening digial processing seps is independen o he speed o conversion seps (e.g., we can combine many analog sreams ino a single high-speed digial sream). M.H. Perro 27 Wrap Up, Slide 6
17 Baseband Inpu i () q () 2cos(2π o ) 2sin(2π o ) Send discree-valued symbols across an analog communicaion channel Mach I/Q samples o heir corresponding symbols based on decision regions Provides noise margin Digial Modulaion Receiver Noise i r () H() Lowpass 2cos(2π o ) 2sin(2π o ) q r () H() Lowpass Decision Boundaries Receiver Oupu M.H. Perro 27 Wrap Up, Slide Q Decision Boundaries Sample imes I Q I
18 Impac o SNR on Receiver Consellaion Received Peak Ampliude Decision Boundaries Q I ransmission Recepion x() ransmier r Receiver rx() X() RX() Signal Noise khz 45 MHz khz 45 MHz SNR inluenced by ransmied power, disance beween ransmier and receiver, and noise M.H. Perro 27 Wrap Up, Slide 8
19 Impac o Increased Signal on Consellaion Received Peak Ampliude Q Decision Boundaries I ransmission Recepion x() ransmier r Receiver rx() X() RX() Signal Noise khz 45 MHz khz 45 MHz Increase in received signal power leads o increased separaion beween symbols SNR is improved i noise level unchanged M.H. Perro 27 Wrap Up, Slide 9
20 Quaniying he Impac o Noise Q Received Peak Ampliude Decision Boundary Probabiliy Densiy Funcion or Noise (I-Componen) d min I d min Probabiliy Densiy Funcion or Noise (Q-Componen) Decision Boundary Minimum separaion beween symbols: d min PDF o noise: zero mean Gaussian PDF Variance o noise ses he spread o he PDF Bi errors: occur when noise moves a symbol by a disance more han dmin/2 M.H. Perro 27 Wrap Up, Slide 2
21 Q IN he Binary Symmeric Channel Model Communicaion Channel or Q Channel -P e P e Bi error -P e P e Q OU PDF o Received Q Sample ransmied ransmied Q (y) Q (x) Probabiliy o Bi Error = P e x d min d min 2 2 x Communicaion Channel or Q Channel Q IN d min Q received Decision Boundary Slicer Q OU d min /2 -d min /2 I signal I noise M.H. Perro 27 Wrap Up, Slide 2 PDF o Noise Variance X (x) = σ 2 Provides a binary signaling model o channel x
22 Q IN Resuling Bi Error Rae Versus SNR Communicaion Channel or Q Channel PDF o Received Q Sample -P e P e Bi error P e -P e Q OU Bi Error Rae versus SNR or Q Channel ransmied ransmied Q (y) Q (x) Probabiliy o Bi Error = P e d min 2 x d min 2 Noe: x Bi Error Rae Bi Error Rae = P e SNR (db) = SNR (db) Gaussian PDF or noise M.H. Perro 27 Wrap Up, Slide 22
23 Shannon Capaciy Communicaion Channel Q I IN Q IN Received Peak Ampliude Decision Boundary d min I d min I OU Q OU Decision Boundary In 948, Claude Shannon proved ha Digial communicaion can achieve arbirary low bi-errorraes i appropriae coding mehods are employed he capaciy o a Gaussian channel wih bandwidh BW o suppor arbirary low bi-error-rae communicaion is: M.H. Perro 27 Wrap Up, Slide 23
24 Summary he Fourier ransorm provides a powerul ool or analysis o sampling, modulaion, and ilering he digial absracion provides a pracical implemenaion ramework or complicaed sysems Analog signaling is highly suscepible o noise Digial signaling provides noise margin We can represen a digial communicaion channel wih a binary signaling model Bi errors are quaniied in erms o he signal-o-noise raio o he overall channel Claude Shannon inroduced he concep o using coding mehods o achieve arbirarily low bi error raes across pracical communicaion channels M.H. Perro 27 Wrap Up, Slide 24
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