Spectra of Digital Waveform

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Nickolas Walters
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1 Specta o Digital Waveom time-domai ad equecy-domai Po. Tzog-Li Wu Depatmet o Electical Egieeig ad Gaduate Istitute o Commuicatio Egieeig Natioal Taiwa Uivesity Basic Popeties o Fouie Tasom 1. peiodic xt ( ) F x t ( ( ) ) 1/ T T jωt () xt ( ) = = x te dt = C = - Ce jωt 1

2 Basic Popeties o Fouie Tasom 2. C = C o eal x( t) * - 3. F ( x ( t) + x ( t) ) = F ( x ( t)) + F ( x ( t))

3 Basic Popeties o Fouie Tasom jωα 4. F ( xt ( α) ) = F( xt ( ) ) e Basic Popeties o Fouie Tasom 5. ( (t ± ) ) C = (1/ ) k =, ± 1, ± 2, ± 3... j F δ kt T e ω α 3

4 Basic Popeties o Fouie Tasom k k ( k) k 6. ( ( )/ ) = ( ω) F d x t dt C j C Basic Popeties o Fouie Tasom 4

5 Specta o digital cicuit waveoms a. The peiodic, tapezoidal pulse tai epesetig clock ad data sigals o digital systems. A A/2 T t A: Amplitude, : pulse ise time, : pulse all time : pulse width ( 5 % ) c. To obtai the complex expoetial Fouie seies o this waveom (1) x A/ x A/ -A/ + ( )/2 A/ + ( )/2 T T Next peiod t t -A/ -A/ 5

6 (2) F( x'') = A / T - A e / T - A e / T (3) + Ae - jω jω ( + ( )/2) jω ( + ( + )/2) / T jω /2 jω /2 jω /2 jω /2 jω jω /2 jω /2 = A / T * [ ( e / )*( e - e ) ( e e / ) * ( e - e ) ] 2 jω ( + )/2 jω /2 jω /2 ω = j ( A / 2 π )*( ω ) * e *[si( ω / 2)* e /( ω / 2) - si( ω / 2)* e /( / 2) ] F x F x j 2 ( ) = ( '') / ( ω), i = = si( ω/ 2) si( ω / 2) jω ( + )/2 C A e T ω /2 ( ω /2) d. Expessio as oe-side spectum + = 1 xt () = C + C cos( ω t+ C) + whe C = 2 C = 2A si( π/ T)/( π/ T) si( π / T)/( π / T), o with =, C = A /T, ad ω = 2 π/ T Note : (1) 5% duty cycle /T =1/2 o eve hamoics (2) slight vaiatios om a 5% eve hamoics vay widely 6

7 Spectal bouds o tapezoidal waveoms 1, o small x a. si x/ x 1/ x, o small x 2 log1/ x = 2dB/decade, o small si x / x x 2dB/decade 1 π 2π 3π x Eect o ise/all time o Spectal Cotet b. let = / T evolope = 2A* si( π)/( π) si( π )/( π ) / T 2log(evolope) = 2log(2A / T ) + 2log si( π )/( π ) +2log si( π )/( π ) 2A /T 2 db / decade 4 db / decade 1/π 1/ π 7

8 Eect o ise/all time o Spectal Cotet 1v = =5 s 6.37MHz 15.9MHz 63.66MHz Eect o ise/all time o Spectal Cotet 2s 5s 11th hamoic 11MHz 68dBuV 86.1dBuV 8

9 Eect o ise/all time o Spectal Cotet c. o example : 1MHz, = =2 s ad = =5 s 1v = =5 s 6.37MHz 15.9MHz 63.66MHz High equecy pats Eect o ise/all time o Spectal Cotet Spectal boud pedictio 2s ise time Level11MHz 15.9MHz 11MHz = 2log 1 (1 V ) 2log 1 ( ) 4( ) 6.37MHz 15.9MHz = 78.45dBµ V 5s ise time Level11MHz 63.7MHz 11MHz = 2log 1 (1 V ) 2log 1 ( ) 4( ) 63.7MHz 63.7MHz = 9.5dBµ V 9

10 Eect o ise/all time o Spectal Cotet The exact value calculated by sic uctio 2s 73.8dBuV 5s 9.4dBuV Vey close to the spectal boud pedictio The measued values ae 2s 68.8dBuV 5s 86.1dBuV Why ae they lowe tha the value by the exact calculatios? What is the eect o the duty cycle o the spectal boud? Eect o Repetitio ate ad Duty Cycle What is the eect o the duty cycle o the spectal boud? Duty cycle: D = T Oe-sided spectum: si( π D) π c + = 2AD π D π DC level 1st beakpoit 1

11 Eect o Rigig (Udeshoot/Oveshoot) Assume = T /2 T /2 1 αt jωt jωt /2 1 αt jωt c = c( squaewave) + Ke si( ωt θ) e dt e Ke si( ωt θ) e dt T + + T = c + (1 e T /2 1 ) Ke T si( ωt+ θ) e dt jω /2 α t jω T t ( squaewave) jωt /4 = e V si( ω T) si( ω T) 4 K 4 jωt /4 pω e ω 2 1 T ωt p + 2αp+ α + ω si( ωt) V 4 jωt /4 p + (2 α + ( K/ V) ω) p+ ( α + ω ) = e ω p 2 p ( ) T + α + α + ω 4 Squae wave compoets Badpass whee p = jω Eect o Rigig (Udeshoot/Oveshoot) The spectum o a squae wave with udeshoot/oveshoot has a pat o its spectum ehaced o iceased about the igig equecy W. Quite ote we see i the adiated emissio poile a seemigly Resoat egio o ehaced emissios i a aow equecy bad. Oe possible explaatio o this is the udeshoot/oveshoot peset o the digital waveoms. 11

12 Use o Spectal Bouds i Computig Bouds o the Output Spectum o a Liea System Spectum Aalyze 12

13 Spectum Aalyze Badwidth o SA The displayed level at the cete equecy o the badwidth will be the sum o the spectal levels that all withi the badwidth o the ilte at that time. I ode to obtai the lowest possible level o the SA display, we should choose as small a badwidth as possible. Spectum Aalyze EMC eceive egulatio 13

14 Spectum Aalyze Spectum Aalyze Peak ad Quasi-peak Iequetly occuig sigals will esult i a measued Quasi-peak level that is cosideably smalle tha a peak detecto would give. That iequet evets may be o suiciet magitude distessigly lage eceived levels o a SA that is set to peak detecto uctio. The easo o the use o quasi-peak detecto uctio elates to the itet o the egulatoy limits, which is to pevet iteeece i adio ad wie commuicatio Receives. Iequet spikes ad othe evets do ot substatially pevet the listee om obtaiig the desied iomatio. 14

15 Repesetatio o Radom Sigals Digital data waveoms ae obviously adom sigals, ot detemiistic sigals 1 x() t = X[1 + m()] t 2 R ( ) = x( t) x( t+ ) x T /2 1 = lim xtxt () ( ) dt t T + T /2 statioay 1 2 = X [1 + m( t)][1 + m( t+ )] = X [1 + m() t + m( t+ ) + m() t m( t+ )] = X [1 + m( t) m( t+ )] 4 = 1 o < T T = o > T Repesetatio o Radom Sigals Powe Spectal desity jω Gx( ) = Rx( ) e d 15

Learning Objectives. Chapter 2 Pricing of Bonds. Future Value (FV)

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