King Fahd University of Petroleum & Minerals Computer Engineering g Dept

Transcription

1 Kig Fahd Uiversiy of Peroleum & Mierals Compuer Egieerig g Dep COE 4 Daa ad Compuer Commuicaios erm Dr. shraf S. Hasa Mahmoud Rm -4 Ex ashraf@kfupm.edu.sa 9/8/ Dr. shraf S. Hasa Mahmoud Lecure Coes. Fourier alysis a. Fourier Series Expasio b. Fourier rasform c. Ideal Low/bad/high pass filers 9/8/ Dr. shraf S. Hasa Mahmoud

2 Sigals sigal is a fucio represeig iformaio Voice sigal microphoe oupu Video sigal camera oupu Ec. ypes of Sigals alog coiuous-value coiuous-ime Discree discree-value coiuous-ime Digial predeermied discree levels much easier o reproduce a receiver wih o errors Biary oly wo predeermied levels: e.g. ad 9/8/ Dr. shraf S. Hasa Mahmoud 3 Example of Coiuous-Value Coiuous-ime sigal s () ad s () are wo example of aalog sigals s () s () 9/8/ Dr. shraf S. Hasa Mahmoud 4

3 Example of Discree-Value Coiuous-ime sigal d () ad d () are wo example of discree sigals d () akes more ha wo levels d () akes oly wo levels - biary d () d () 9/8/ Dr. shraf S. Hasa Mahmoud 5 Noe d () exeds o ± pplies o BOH aalog ad digial sigals ime Domai Represeaio ime domai represeaio we plo value (volage, curre, elecric field iesiy, ec.) versus ime Ca ifer rae of chage (speed or frequecy) iformaio e.g. s () seems faser ha s () Usig calculus erms: rae of chage for s () > rae of chage for s () s () d () s () d () 9/8/ Dr. shraf S. Hasa Mahmoud 6 3

4 Frequecy - Badwidh pplies o BOH aalog ad digial sigals s () faser ha s () s () coais higher frequecies ha hose coaied i s () s () ad s () coai more ha oe frequecy Miimum frequecy = f mi Maximum frequecy = f max Badwidh = Rage of frequecies coaied i sigal = f max f mi 9/8/ Dr. shraf S. Hasa Mahmoud 7 Frequecy Badwidh () For our example sigals, assume: S(): fmi = Hz, fmax = 5 Hz S(): fmi = 5 Hz, fmax = Hz his meas: BW for s () = 5 = 49 Hz BW for s () = 5 = 995 Hz pplies o BOH aalog ad digial sigals Noe ha: because s () is faser ha s () i should coai frequecies higher ha hose i s () E.g. s () coais frequecies (5,] which do o exis i s () 9/8/ Dr. shraf S. Hasa Mahmoud 8 4

5 Frequecy Badwidh (3) pplies o BOH aalog ad digial sigals Cosider he discree sigals d () ad d () he fucio plos have pois of ifiie slope rae of chage = frequecy = herefore for sigals ha look like d () ad d ( (), fmax = Furhermore, BW = Example: d () coais frequecies from some miimum fmi Hz o fmax = Hz 9/8/ Dr. shraf S. Hasa Mahmoud 9 Example of Sigal BW Cosider he huma speech ypically fmi ~ Hz fmax ~ 35 Hz BW of he huma speech sigal = 3 Hz 9/8/ Dr. shraf S. Hasa Mahmoud 5

6 Badwidh for Sysems For a sysem o respod (amplify, process, x, Rx, ec.) for a paricular sigal wih all is deails, he sysem should have a equal or greaer badwidh compared o ha of he sigal Example: he sysem required o process s () should have a greaer badwidh ha he sysem required o process s () 9/8/ Dr. shraf S. Hasa Mahmoud Badwidh for Sysems () Example : cosider he huma ear sysem Respods o a rage of frequecies oly fmi = Hz fmax =, Hz BW = 9,98 Hz I does o respod o souds wih frequecies ouside his rage Example 3: cosider he copper wire I passes (elecric) sigals oly bewee a cerai fmi ad a cerai fmax he higher he qualiy of he wire he wider he BW More o Sysems BW laer! 9/8/ Dr. shraf S. Hasa Mahmoud 6

7 Frequecy Represeaio pplies o BOH aalog ad digial sigals How o represe sigals ad idicae heir frequecy coe? he X-axis: frequecy (i Herz or Hz) Wha is he Y-axis he? he aswer will be pospoed! f max = 5 Hz BW for s () (): rage of ffrequecies f f mi = Hz f mi = 5 Hz BW for s (): rage of frequecies f max = Hz f 9/8/ Dr. shraf S. Hasa Mahmoud 3 Periodic Sigals pplies o BOH aalog ad digial sigals periodic sigal repeas iself every secods Period secods I calculus erms: s() is periodic if s() = s(+) for ay - < < For previous examples: s ( (), s ( (), ad d () are o periodic however, d () is periodic 9/8/ Dr. shraf S. Hasa Mahmoud 4 7

8 Periodic Sigals () pplies o BOH aalog ad digial sigals periodic sigal has a FUNDMENL FREQUENCY f f = / where is he period periodic sigal may also has frequecies oher ha he fudameal frequecy f fmi <= f <= fmax x x x x x x BW for periodic s() f mi f max 9/8/ Dr. shraf S. Hasa Mahmoud 5 f Periodic Sigals (3) pplies o BOH aalog ad digial sigals Examples of oher periodic sigals: s 3 () s 5 () ~ s 4 4( () 9/8/ Dr. shraf S. Hasa Mahmoud 6 8

9 Eergy/Power of Sigals Eergy for ay sigal is defied as E s s( ) d pplies o BOH aalog ad digial sigals where he iegral is carried over LL rage of I oher words, Es is he area uder he absolue squared of he sigal he ui of eergy is Joules 9/8/ Dr. shraf S. Hasa Mahmoud 7 Eergy/Power of Sigals () pplies o BOH aalog ad digial sigals Noe ha for periodic sigal E s is equal o ifiiy sice i is defied o (-, ) However power is FINIE for hese ype of sigals Power is defied as he average of he absolue squared of he sigal, i.e. P s he ui of power is Joules/sec or Wa s( ) 9/8/ Dr. shraf S. Hasa Mahmoud 8 d Noe he iegral ca be performed o [,], [-/, /], or ay coiuous ierval of legh 9

10 VERY SPECIL alog Sigal fucio of he form s() = cos(f + ) s() cos() - Period 9/8/ Dr. shraf S. Hasa Mahmoud 9 Characerisics of COSINE Compleely specified by: mpliude Phase - Frequecy f s( = ) = cos() Periodic sigal repeas iself every secods = / f ime o review your rigoomery!! E.g. si(x) = cos(x-/) 9/8/ Dr. shraf S. Hasa Mahmoud

11 Characerisics of COSINE () Eergy for his sigal, E s = ifiiy Power for his sigal, P g = / Noe P g is depede oly o he ampliude Exercise: Verify he above resuls usig he power formula I coais ONLY ONE frequecy f he pures form of aalog sigals Frequecy represeaio: x BW for s () f mi = f max = f Hz 9/8/ Dr. shraf S. Hasa Mahmoud f Characerisics of COSINE (3) Very Useful Properies (f = /) cos( f ) d cos (f ) d / cos( f ) d cos (f ) d / cos(f)cos(mf) d / m m 9/8/ Dr. shraf S. Hasa Mahmoud

12 v Example of Cosie Fucios Y () has a frequecy f of Hz ( = ½ sec) ampliude of 3 P Y = 3 / = 4.5 Was Y () - has a frequecy f of Hz ( =/ = sec) ampliude of P Y = / =.5 Was value Examples of aalog coieous-ime fucios y = 3 cos(*pi**) y = cos(*pi**) ime - 9/8/ Dr. shraf S. Hasa Mahmoud 3 ONLY FOR PERIODIC SIGNLS Fourier Series Expasio Ca we use he basic cosie fucios o represe periodic sigals? YES Fourier Series Expasio s 3 3( () facorizaio 9/8/ Dr. shraf S. Hasa Mahmoud 4

13 Fourier Series Expasio () For a periodic sigal s() ca be represeed as a sum of siusoidal sigals as i s( ) cos(f) B si(f ) where he coefficies are compued usig: s( ) d s( )cos(f ) d 9/8/ Dr. shraf S. Hasa Mahmoud 5 B f is he fudameal frequecy of s() ad is equal o / s( )si(f ) d Fourier Series Expasio (3) oher form for he series: C s( ) C cos(f ) where he coefficies are compued usig: C C B B B a 9/8/ Dr. shraf S. Hasa Mahmoud 6 C 3

14 Noes o Fourier Series Expasio he represeaio (he sum of siusoids) is compleely ideical ad equivale o he origial specificaio of s() I is applies o ay periodic sigal aalog or digial! Very powerful ool i reveals all frequecies coaied i he origial periodic sigal s() 9/8/ Dr. shraf S. Hasa Mahmoud 7 Noes o Fourier Series Expasio () I geeral, s() coais DC erm he zero frequecy erm = / (possibly ifiie) umber of harmoics (or siusoids) a muliples of he fudameal frequecy, f he coribuio of a harmoic wih frequecy f is proporioal o +B or C E.g. if C ~, he we say he harmoic a f (or higher does o coribue sigificaly owards buildig s() more o his whe we discuss oal power! 9/8/ Dr. shraf S. Hasa Mahmoud 8 4

15 Noes o Fourier Series Expasio (3) harmoic wih frequecy equal o f (>), has a period of /() I geeral he series expasio of s() coais INFINIE umber of erms (harmoics) However for less ha % accurae represeaio oe ca igore higher erms erms wih frequecies greaer ha cerai * f 9/8/ Dr. shraf S. Hasa Mahmoud 9 Noes o Fourier Series Expasio (4) Les defie he followig fucio: s_e(=k) o be he series expasio of s() up o ad icludig he = k erm I should be oed ha s_e(=k) is periodic wih period Examples: s _ e( ) / s _ e( ) / cos(f ) B si(f ) C cos(f ) / 9/8/ Dr. shraf S. Hasa Mahmoud 3 5

16 Noes o Fourier Series Expasio (5) Examples co d: s _ e( ) / cos( f ) B si( f ) ( cos( f ) B si( f ) / C cos(f ) C cos( f ) s _ e( ) ) cos(f) B si(f / C cos(f ) 9/8/ Dr. shraf S. Hasa Mahmoud 3 Noes o Fourier Series Expasio (6) I is obvious ha s() is % represeed by s_e(=) s_e( = * < ) produces a less ha % accurae represeaio of he origial s() For mos pracical periodic sigals s_e(=) provides a more ha eough accuracy i represeig s() No eed for ifiie umber of erms 9/8/ Dr. shraf S. Hasa Mahmoud 3 6

17 Example : Cosider he followig s() s() Over oe period, he sigal is defied as s() = -/4 < <= /4 = /4 < <= 3/4 -/4 /4 3/4 5/4 Fidig he Series Expasio: he DC erm s d ( ) 9/8/ Dr. shraf S. Hasa Mahmoud 33 Example : co d he erm : / 4 / 4 s( )cos(f ) d cos(f) d si(f) f si( ),4,6,...,5,9,... 3,7,,... Remember. f = /. cos(ax) dx = -/a si(ax) 3. si() = for ieger 4. si(/) = for =,5,9, 5. si(/) = - for =3,7,, 9/8/ Dr. shraf S. Hasa Mahmoud 34 7

18 Example : co d herefore is give by: ( ) /,4,6,...,3,5,7,... Remember (-) (-)/ = for =,5,9, = - for = 3,7,, 9/8/ Dr. shraf S. Hasa Mahmoud 35 Example : co d he erm B : B / 4 / 4 s( )si(f ) d si(f) d cos(f) cos( ) cos( ) f Remember. cos(ax)dx = -/a si(ax). cos(x) = cos(-x) 9/8/ Dr. shraf S. Hasa Mahmoud 36 8

19 Example : co d herefore, he overall series expasio is give by / ( ) s( ) cos(f),3,5 s( ) cos(f ) cos( 3 f ) 3 cos( 5 f) cos( 7 f) /8/ Dr. shraf S. Hasa Mahmoud 37 Example : co d Origial s() ad he series up o ad icludig = i.e. Comparig: vs. s() s_e(=) = / origial i s() up o = /8/ Dr. shraf S. Hasa Mahmoud 38 9

20 Example : co d Origial s() ad he series up o ad icludig = i.e. Comparig:..8 origial s() up o = vs. s().6.4. s_e(=) = / + -. / cos(f ) /8/ Dr. shraf S. Hasa Mahmoud 39 Example : co d Origial s() ad he series up o ad icludig = 3 i.e. Comparig:..8 origial s() up o = 3 vs. s().6.4. s_e(=3) = / + /cos(f ) /(3)cos(3f ) /8/ Dr. shraf S. Hasa Mahmoud 4

21 Example : co d Origial s() ad he series up o ad icludig = i.e. Comparig: s()..8 origial s() up o = vs. s_e(=) = / + /cos(f ) /(3)cos(3f ) + /(5)cos(5f ) /(7)cos(f ) +/()cos(f ) /8/ Dr. shraf S. Hasa Mahmoud 4 Example: co d clear all = ; = ; = -:.:; :; _max = ; s = (*square(*pi/*(+/4))+)/; figure() plo(, s); grid axis([ -..]); he malab code for ploig ad evaluaig he Fourier Series Expasio his code builds he series icremeally usig he for loop Make sure you sudy his code!! s_e = /*oes(size()); for =::_max s_e = s_e + (-)^((-)/) * */(*pi) * cos(*pi*/*); ed figure() plo(, s,'b-',, s_e,'r--'); axis([ -..]); leged('origial s()', 'up o = '); grid 9/8/ Dr. shraf S. Hasa Mahmoud 4

22 Noes Previous Example he more erms icluded i he series expasio he closer he represeaio o he origial s() i.e. comparig s() wih s_e(=*), he greaer he * he closer he represeaio is How o measure closeess? swer: Le s use power!! 9/8/ Dr. shraf S. Hasa Mahmoud 43 Power Calculaio Usig Fourier Series Expasio Rule: if s() is represeed usig Fourier Series expasio, he is power ca be calculaed usig: P s s( ) d 4 B C 4 9/8/ Dr. shraf S. Hasa Mahmoud 44

23 Power Calculaio Usig Fourier Series Expasio () he previous resul is based o he followig wo facs: () For f() = cosa power of f() = cosa Proof: power = / X cosa d = / X cosa X = cosa Was 9/8/ Dr. shraf S. Hasa Mahmoud 45 Power Calculaio Usig Fourier Series Expasio (3) he previous resul is based o he followig facs (coiued): () For f() = cos(f +) power of f() = / Proof: P f f ( ) d cos ( f ) d [ cos(4f )] d [ ] 9/8/ Dr. shraf S. Hasa Mahmoud 46 3

24 Example : Problem: Wha is he power of he sigal s() () used i previous example? d fid * such ha he power coaied i s_e(=*) is 95% of ha exisig i s()? Soluio: Le he power of s() be give by P s Ps s( ) d 9/8/ Dr. shraf S. Hasa Mahmoud 47.5 Example : co d Now i is desired o compue he power usig he Fourier Series Expasio Wha is he power i s_e(=) = /? s: we apply he power formula: P s _ e( ) s _ e( ) 4 4 9/8/ Dr. shraf S. Hasa Mahmoud 48 d.5 4

25 Example : co d Wha is he power i s_e(=) = / + / cos(f ) s: we ca use he resul o slide Power Calculaio Usig Fourier Series Expasio: P s _ e ( ) 4 s _ e ( ) d /8/ Dr. shraf S. Hasa Mahmoud 49 Example : co d Wha is he power i s_e(=3) = / + / cos(f ) /(3) cos(f ) s: we ca use he resul o slide Power Calculaio Usig Fourier Series Expasio: P s _ e ( 3) s _ e( 3) d /8/ Dr. shraf S. Hasa Mahmoud 5 5

26 Example : co d Wha is he power i s_e(=5) = / + / cos(f ) /(3) cos(f ) + /(5) cos(f ) s: we ca use he resul o slide Power Calculaio Usig Fourier Series Expasio: P s _ e ( 5) 4 s _ e ( 5) 9 d 9 5 9/8/ Dr. shraf S. Hasa Mahmoud Example : co d Wha is he power i / ( ) s _ e( ) cos(f ),3,5 s: we ca use he resul o slide Power Calculaio Usig Fourier Series Expasio: P s _ e( ) s _ e( ) d his he EXC SME power coaied i s() his is expeced sice s() is % represeed by s_e(=) 9/8/ Dr. shraf S. Hasa Mahmoud 5 6

27 Example : co d s_e(=k) Expressio Power % Power + k = /.5 (.5 )/(.5 ) = 5% k = / + /cos(f ).456 (.456 )/(.5 ) = 9.5% k = * / + /cos(f ) % k = 3 / + /cos(f ) % /(3)cos(3f ) k = 5 / + /cos(f ) /(3)cos(3f ) + /(5)cos(5f ) % + % power = power of s_e(=k) relaive o origial power i s() which is equal o.5 9/8/ Dr. shraf S. Hasa Mahmoud * 53 For k =, he expressio s_e(=k) is he same as ha for s_e(k=). Why? Example : co d herefore, s_e(=*) such ha 95% of power is coaied * = 3 9/8/ Dr. shraf S. Hasa Mahmoud 54 7

28 Power Specral Desiy Fucio Fourier Series Expasio: Specifies all he basic harmoics coaied i he origial fucio s() C / = ( +B ) / deermies he power coribuio of he h harmoic wih frequecy f he power Specral Desiy fucio is a fucio specifyig: how much power is coribued by a give frequecy 9/8/ Dr. shraf S. Hasa Mahmoud 55 Power Specral Desiy Fucio () ypical PSD fucio for periodic sigals: s( ) PSD(f) C / Periodic s() () Fourier Series Expasio C cos(f ) () Power calculaio ( /) C / Ps s( ) d 4 C f f C 3 / C 4 / (3) Ploig power versus frequecy f f 3f 4f frequecy 9/8/ Dr. shraf S. Hasa Mahmoud 56 8

29 Power Specral Desiy Fucio (3) mahemaical expressio for PSD(f) ca be wrie as / 4 f PSD( f ) C / f f oherwise oher way (more compac) of wriig PSD(f) is as follows: PSD( f ) ( f ) C ( f f) 4 where () is defied by ( f ) f f 9/8/ Dr. shraf S. Hasa Mahmoud 57 Power Specral Desiy Fucio (4) (f) is referred o as he dirac fucio or ui impulse fucio (f) f 9/8/ Dr. shraf S. Hasa Mahmoud 58 9

30 Noe o he PSD Fucio PSD fucio has uis of Was/Hz For periodic sigals PSD is a discree fucio - defied for ieger muliples of he fudameal frequecy Specifies he power coribuio of every harmoic compoe C / f he separaio bewee he discree compoes is a leas f I is exacly f if all C s are o zeros E.g. for he previous s() example, C = for eve separaio = f 9/8/ Dr. shraf S. Hasa Mahmoud 59 Noe o he PSD Fucio () o calculae he oal power of sigal Iegrae PSD over all coaied frequecies For discree PSD: iegraio = summaio herefore oal power of s(), P s = ( /) + C / i Was 9/8/ Dr. shraf S. Hasa Mahmoud 6 3

31 Example 3: Fid he PSD fucio of he periodic sigal s() cosidered i Example. From Example, s() is give by / ( ) s( ) cos(f),3,5 Usig Example : Power a he zero frequecy = (/) = /4 Power a he h harmoic ( odd) is equal o /() Power a he h harmoic ( eve) is zero herefore he PSD fucio is give by PSD( f ) 4 ( f ) 9/8/ Dr. shraf S. Hasa Mahmoud 6,3,5 ( f f ) Example 3: co d he PSD is ploed as show ( =, = ) PSD( f ) ( f ) 4,3,5 ( f f ).5. Power Specral Desiy fucio for s() - =, = frequecy - Hz 9/8/ Dr. shraf S. Hasa Mahmoud 6 3

32 Example 3: co d Malab Code o plo PSD clear all = ; = ; = -:.:; _max = ; Frequecy = [::_max]; PwrSepcralD = zeros(size(frequecy)); % Record he DC erm power a f = PwrSepcralD() = (/)^; % Record he h harmoic power a f = f for =::_max PwrSepcralD(+) = (*/(*pi))^ / ; ed figure() sem(frequecy, PwrSepcralD,'rx'); ile('power Specral Desiy fucio for s() - =, = '); xlabel('frequecy - Hz'); grid he sem fucio is ypically used o plo discree fucios 9/8/ Dr. shraf S. Hasa Mahmoud 63 Example 4: his is a ypical exam quesio Problem: Cosider he periodic half-wave recified sigal s() depiced i figure. Wrie a mahemaical expressio for s() Calculae he Fourier Series Expasio for s() Calculae he oal power for s() Fid * such ha s_e(*) has 95% of he oal power Deermie he PSD fucio for s() Plo he PSD fucio for s() s() - 9/8/ Dr. shraf S. Hasa Mahmoud 64 3

33 Example 4: co d swer: (a) o wrie a mahemaical expressio for s(), remember ha he geeral form of a siusoidal fucio is give by cos( X Freq X ), or cos( /Period X ) herefore s() is give by s() = cos(/ ) -/4 < /4 = /4 < 3/4 9/8/ Dr. shraf S. Hasa Mahmoud 65 Example 4: co d swer: (b) he F.S.E of s(): s ( ) cos( f ) B si( f ) he DC erm is give by s( ) d cos( / ) d si( / ) ( / 4 si( / ) si( / ) 9/8/ Dr. shraf S. Hasa Mahmoud 66 33

34 Example 4: co d Remember: si(ax+bx) si(ax-bx) [cos(ax)cos(bx)] dx = (a+b) (a-b) for a b si(a+/-b) = si(a)cos(b) +/- cos(a)si(b) swer: he erm is give by (remember / = f ) s( )cos(f ) d cos( / )cos(f) d si 4 cos / f si f f 4 f cos / For For 9/8/ Dr. shraf S. Hasa Mahmoud his meas: he = should be 67 special! Example 4: co d Bu cos(/) = = odd, = (-) (+/) = eve herefore ( ( / ) ( / ) ) ( )( ) For eve For odd, 9/8/ Dr. shraf S. Hasa Mahmoud 68 34

35 Example 4: co d he expressio for (for eve ) ca be furher simplified o ( ( / ) ( / ) ) ( )( ) ( / ) ( / ) ) ( )( ) ( ( ) ( / ) ( ) ( / ) ( ( ) / ) 9/8/ Dr. shraf S. Hasa Mahmoud 69 For eve Example 4: co d is sill o compleely specified we sill eed o calculae i for =; i oher words we eed o calculae : s( )cos( f) d cos( / )cos(f ) herefore: cos ( d f ) 4 f si4f si 4 9/8/ Dr. shraf S. Hasa Mahmoud 7 d si 8f 35

36 Example 4: co d his mea is equal o he followig: = / = odd, / = (-) (+/) =, 4, 6, ( -) he above expressio specifies for LL POSSIBLE values of specificaio is complee 9/8/ Dr. shraf S. Hasa Mahmoud 7 Example 4: co d We sill eed o compue B : B / 4 / 4 Remember: - cos(ax+bx) cos(ax-bx) [si(ax)cos(bx)] dx = (a+b) (a-b) for a b s( )si(f ) d cos( / )si(f) d cos 4 f f cos 4 / cos / f f cos cos For / cos / For 9/8/ Dr. shraf S. Hasa Mahmoud 7 his meas: he = should be special! 36

37 Remember: Example 4: co d si(ax) = cos(ax) si(ax) B is sill NO compleely specified we sill eed o calculae i for =; i oher words we eed o calculae B : B s( )si( f) d cos( / )si(f ) herefore: B cos( )si( d si(4 d f f ) f ) cos(4f ) cos( ) cos( ) 4 4 his meas B = for all 9/8/ Dr. shraf S. Hasa Mahmoud 73 d Example 4: co d o summarize: = / = odd, / = d (-) (+/) =, 4, 6, ( -) B = for all Havig compued ad B we are ow i a posiio o wrie he Fourier Series Expasio for s() 9/8/ Dr. shraf S. Hasa Mahmoud 74 37

38 Example 4: co d he Fourier Series Expasio for s() is give by s( ) ) cos(f) B si(f ( ) ( / ) cos(f ) cos(f ),4, 6 he C erms (here is a ypo i he exbook) are as follows: C = / C = / (-) (+/) C = , =, 4, 6, ( ) 9/8/ Dr. shraf S. Hasa Mahmoud odd, 75 Remember: Example 4: co d si(ax) = ½ cos(ax) si(ax) Plo for s() ad he Fourier Series Expasio for k=,,, ad 4..8 Origial sigal ad is Fourier Series Expasio origal s() s e (k=) s e (k=) s e (k=) s e (k=4) Noe: s k icreases s_e(=k) approaches he origial s() mpliude ime (sec) 9/8/ Dr. shraf S. Hasa Mahmoud 76 38

39 Example 4: co d he oal power of s() is give by: P s 3 s( ) d si(4 / ) 8 / cos ( / ) 4 herefore oal power of s() =.5 9/8/ Dr. shraf S. Hasa Mahmoud 77 Example 4: co d o fid * such ha power of s_e(=*) = 95% of oal power: s_e(=k) Expressio Power % Power + k = /.3 (.3 )/(.5 ) = 4.5% k = / + /cos(f ).63 (.6 )/(.5 ) = 9.5% k = / + / cos(f ) (.488 )/(.5 ) /(3) cos(f ) 99.5% herefore * = power of s_e(=) =.488 which is 99.5% of oal power of s() 9/8/ Dr. shraf S. Hasa Mahmoud 78 39

40 Example 4: co d he PSD fucio for s() is as follows: Power for DC erm = (/) Power for harmoic a f = f : (/) / = /8 Power for harmoic a f = f (=,4,6, ): [/(( -))] / = /(( -)) herefore PSD fucio equals o PSD( f ) ( f ) ( f 8 f ),4,6 ( f f ) 9/8/ Dr. shraf S. Hasa Mahmoud 79 Example 4: co d Plo of he PSD fucio for s().4.. Power specral desiy for s() Power - Was Muliples of fudameal freuqcy f (Hz) 9/8/ Dr. shraf S. Hasa Mahmoud 8 4

41 Fourier rasform Fourier Series Expasio aalysis is applicable for PERIODIC sigals ONLY here are impora sigals ha are o periodic such as Your voice waveform Pulse sigal p() used for modulaio ad rasmissio Examples: p () ad p () p () p () / / 9/8/ Dr. shraf S. Hasa Mahmoud / / 8 Fourier rasform () How o fid he frequecy coe of such sigals? Use FOURIER RNSFORM X ( f ) x( ) e jf jf x( ) X ( f ) e df 9/8/ Dr. shraf S. Hasa Mahmoud 8 d 4

42 Noes o Fourier rasform F. describes a wo-way rasformaio x() X(f) where x() is he ime represeaio of he sigal, while X(f) is he frequecy represeaio of he sigal X(f) is defied o a coiuous rage of frequecies ll frequecies wihi he rage of X(f) where X(f) is o zero coribue owards buildig x() 9/8/ Dr. shraf S. Hasa Mahmoud 83 Noes o Fourier rasform () he magiude of he coribuio of a paricular frequecy f* i x() is proporioal o X(f*) Example: Cosider he F.. pair show below clearly frequecies belogig o (-/,/) coribue sigificaly more compared o frequecies belogig o (/,) or (-,-/) x() X(f) F.. / / -/ / f 9/8/ Dr. shraf S. Hasa Mahmoud 84 4

43 Properies of Fourier rasform If x() is ime-limied X(f) is o frequecy-limied i.e. he rage of X(f) = (-, ) If x() is a real-valued symmeric X(f) is real-valued 9/8/ Dr. shraf S. Hasa Mahmoud 85 Relaio bewee Fourier Series Expasio ad Fourier rasform Cosider he followig wo sigals: s () x() is he same as s() excep ha is period = x() -/4 /4 3/4 5/4 / / S(f) For S(f) Separaio bewee he discree frequecies equal o f or / X(f) -3/ -/ / -/ / 3/ f f -4/ -/ 4/ Separaio become zero as ad he / discree fucio S(f) coiuous fucio X(f) 9/8/ Dr. shraf S. Hasa Mahmoud 86 43

44 Relaio bewee Fourier Series Expasio ad Fourier rasform () he separaio bewee specral lies for a periodic sigal is / s ifiiy ad s() becomes o periodic he separaio bewee specral lies zero (i.e. i becomes coiuous) 9/8/ Dr. shraf S. Hasa Mahmoud 87 Example 5: Problem: Cosider he square pulse fucio show i figure: Wrie a mahemaical expressio for p() Fid he Fourier rasform for p() Plo P(f) p() -/ / 9/8/ Dr. shraf S. Hasa Mahmoud 88 44

45 Example 5: co d swer: p() ca be expressed as p() = / = oherwise he F.. for p(), P(f) is give by P( f ) p( ) e jf d 9/8/ Dr. shraf S. Hasa Mahmoud 89 Example 5: co d Which is equal o P( f ) p( ) e jf f e d e jf jf jf e d jf e j si( f ) f jf d jf e jf e jf 9/8/ Dr. shraf S. Hasa Mahmoud 9 Remember: Euler ideiy : e jx = cos(x) + j si(x), OR cos(x) = (e jx +e -jx )/ si(x) = (e jx -e -jx )/(j) 45

46 Example 5: co d P(f) plo for = ad = Noe: P(f) is defie o (-, ) P(f) is coiuous P(f) = ZERO for f = / For pracical pulses P(f) approaches zero as f ± Fourier rasform of p() - =, aw = Mos of he eergy of p() is coaied i he period of (-/, / frequecy - Hz 9/8/ Dr. shraf S. Hasa Mahmoud 9 Example 6: Problem: If he FSE expasio for he fucio g() is as give below. Compue he FSE for he fucio s() wihou compuig he FSE coefficies for s(). g() s() -/ / -/ / 9/8/ Dr. shraf S. Hasa Mahmoud 46