Spectral Analysis. Joseph Fourier The two representations of a signal are connected via the Fourier transform. Z x(t)exp( j2πft)dt

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1 Specral Analysis Asignalx may be represened as a funcion of ime as x() or as a funcion of frequency X(f). This is due o relaionships developed by a French mahemaician, physicis, and Egypologis, Joseph Fourier(768-83). Boh he Fourier ransform and he closely associaed Fourier series are named in his honor. Even he elegraph hadn been invened in his lifeime and were he alive oday he would be asonished a he number of algorihms, sofware, and elecronic es insrumens ha bear his name. The fac ha he lived o accomplish he foundaion of specral analysis is miraculous since he was he las of eigheen children and escaped he guilloine several imes during he French Revoluion. Joseph Fourier The wo represenaions of a signal are conneced via he Fourier ransform X(f) =z{x()} = Z x()exp( j2πf)d Many of he signals of ineres in elecrical engineering are periodic funcions of ime. A periodic funcion is one for which x() =x( ± nt )

2 where n is any ineger and T is he smalles inerval of ime for which his relaionship is rue. The inerval T is called he period of he periodic funcion. If he periodic funcion saisfies consrains known as he Dirichle condiions (which are saisfied by any funcion produced by naure) i may be expanded in a Fourier series X x() = c n exp(jnω p ) n= where ω p =2πf p and f p =/T is he frequency in Herz of he periodic funcion. This is known as he complex Fourier series represenaion of a periodic funcion. The expansion coefficien c n are complex consans which can be deermined from x() as c n = Z α+t x()exp( jnω p )d T α where α is any real number. The erms in he Fourier series for which n is an even ineger are known as he even harmonics and he erms for which n is an odd ineger are known as odd harmonics. The erm for which n = ± are known as he fundamenal. Alernaive represenaion of he Fourier series is he real rigonomeric series where a n = 2 T Z α+t α x() =a o /2+ x()cos(nω p )d X [a n cos(nω p )+b n sin(nω p )] n= b n = 2 T Z α+t α x()sin(nω p )d These are no differen series; jus wo ways of expressing he same resul. The expansion coefficiens are relaed by c n = a n jb n 2 a n =2Re(c n ) b n = 2Im(c n ) The Fourier ransform of a periodic funcion is hen given by Z n= X(f) = z{x()} = X Z c n exp(jnω p )exp( j2πf)d = x()exp( j2πf)d = X n= c n δ(f nf p ) which is a line specra. The funcion δ is he Dirac dela which makes he specra zero everywhere exceps a frequencies which are inegral muliples of f p. The lines have ampliudes or weighs of c n. If a plo is made of he magniude of he specra for only posiive frequencies i would consiss of lines a f = nf p 2

3 and he heigh of each line would be 2 c n If he specra is o be ploed in rms each line would be 2 c n. A opic angenial o Fourier or Specral analysis is Toal Harmonic Disorion (THD) which measures how much a signal differs from a perfec sine wave. I is defined as (in percen) as v ux THD = c n 2 Sine Wave n=2 c A sine wave wih ampliude A and frequency f p =/T is given by x() =A sin(ω p ) x () Sine Wave is paricularly simple since exp(jθ) exp( jθ) sin θ = 2j so 2j n = c n = 2j n = n 6= ± and he specra is given by 3

4 2 dd k f k Specra of Sine Wave. where he frequencies and ampliudes have been normalized o uniy for simpliciy. So he Fourier series represenaion of a perfec sine wave is a perfec sine wave. Which makes he THD =which means ha here is no harmonic disorion or, anoher way of puing i, nohing looks like a sine wave more han a sine wave. Square Wave A symmeric square wave wih a dc level of zero is one which is +A half he ime and A he oher half. The choice of he ime origin is arbirary by a common one is A T/2 << x() = +A <<T/2 x( ± nt ) elsewhere 4

5 x () 2 3 Symmeric Square Wave. where A =and T =in he figure. The complex Fourier expansion coefficiens are c n = 4 π n n odd n even.5 2 ddk.5 The normalized specra is Specra of Symmeric Square Wave c n c = n n odd n even As a comparison of how well he Fourier series represens a square wave a plo can be made of he square wave and he firs five harmonics x() =a o /2+ fk 5X [a n cos(nω p )+b n sin(nω p )] n= 5

6 2 y () x () Square Wave and Fourier Approximaion using Firs 5 Terms. The ringing ha occurs where he square wave is swiching levels is known as he Gibbs phenomenon. Using he firs 9 componens he THD for he square wave is % which simply means a square wave doesn look very much like asinewave. Triangular Wave A symmeric riangular wave consiss of alernaing sraigh lines wih slopes of equal magniudes and a dc level of zero. 4A T T/4 4A x() = T +2A T/4 3T/4 4A T 4A 3T/4 T x( ± nt ) where A is he ampliude and T he period of he riangular wave. 6

7 x () 2 3 Symmeric Triangular Wave. The complex Fourier expansion coefficiens are nπ 4sin 2 c n = ja π 2 n 2 which are zero for n even and roll off as /n 2 for n odd. The specra for he riangular wave is dd k f k Specra for Symmeric Triangular Wave. A plo of he riangular wave and he firs 3 componens shows hey are almos indisinguishable 7

8 y () x () 2 3 Triangular Wave and Approximaion by Firs 3 Componens. The THD is only 2.48% which means ha a riangular wave is reasonable closeoasinewave. Ramp A ramp or sawooh wave is one for which 2A T T 2 < T 2 x() = x( ± nt ) n any ineger x () The expansion coefficiens are 2 3 Ramp Wave. c n = ja cos(nπ) nπ 8

9 The specra is given by.6 2 dd k f k Specra of Ramp. Using he firs 3 componens he approximaion and he ramp are y () x () 2 3 Ramp and Approximaion. The THD using he firs 9 componens is %. Recangular Pulse Train A recangular pulse rain is similar o a square wave in ha i swiches beween wo levels bu he duy cycle is no 5%. The duy cycle is he percenage of he ime he waveform is in he high sae. The pulse rain is A τ 2 τ x() = 2 << T x( ± nt ) n any ineger so he duy cycle is d = τ/t. 9

10 .5 p ().5.5 Pulse Train wih Duy Cycle.5. The Fourier expansion coefficiens are wih specra c n = Ad sin(πnd) πnd dd k f k Specra of Pulse Train which is, of course, a line specra bu he envelope of he specra has a sin(x)/x

11 behavior. The approximaion of he pulse rain as he firs 2 erms of he Fourier series is.5 y () x () Approximaion of pulse rain as firs 2 Terms of Fourier Series. for which he THD is 39 which means his really doesn look like a sine wave. If he duy cycle d =.5 his becomes a symmeric square wave. RF Pulse Train A rf pulse rain is a recangular pulse rain muliplied o a sinusoidal wih a frequency much higher han ha of he rain. Mahemaically i is given by A cos(ω c ) τ 2 x() = x( ± nt ) n is an ineger The duraion of he pulse is τ. I is assumed ha f c is an inegral muliple of /T. The number of cycles in he pulse N = τf c which is assumed o be an ineger. This is he sor of signal used in radar.

12 .5 x ( i ) i RF Pulse Train. The Fourier expansion coefficiens are given by c n = T Z T 2 T 2 x()exp(j2πf p n)d = T Z τ 2 τ 2 A cos(2πf c )exp(j2πf p n)d = Z τ 2 A cos(2πf c )cos(2πf p n)d = T τ 2 Z τ 2 A cos(2πf c )cos(2πf p n)d = Aτ T τ 2T 2 sin π(fc nf p ) π(f c nf p ) + sin π(f c + nf p ) π(f c + nf p ) which shows ha he specra of he rf pulse rain is jus ha of he recangular pulse rain shifed up o f = f c anddownof = f c, The specra is 2

13 db k f k RF Pulse Train Specra. The disance f c o he firs null is τ/2 where τ = N/f c where N is he number of cycles in he rf pulse. So he specra of all of hese signals is a line specra which is a direc consequence of heir periodiciy. However, he value of he expansion coefficiens is a funcion of he shape. The envelope of he specra has he ypical sin(u)/u shape. 3

14 Specra of RF Pulse Cenered abou High Frequency Carrier. The specra has nulls cenered abou he carrier a frequencies f = f c /τ f 2 = f c +/τ so he difference beween he firs wo nulls abou he cener of he sin(u)/u is given by f =2/τ =2f c /N where N is he number of cycles in he rf pulse. The specra of a signal is imporan for a number of reasons. Mos imporanly i deermines he bandwidh ha would have o be used o pass or ransmi he signal wihou disorion. I is fundamenal in signal processing and elecommunicaions. 4