The Fourier Transform. Consider an energy signal x(). Is energy is = E x( ) d 2 x() x () T Such signal is neiher finie ime nor periodic. This means ha we canno define a "specrum" for i using Fourier series. In order o ry o define a specrum, le us consider he following periodic signal x (): x ( ) = x( ) for inside an inerval of widh T. The signal repeas iself ouside of his inerval. We can define a Fourier series for his signal. Furhermore, i is eviden ha: We can wrie: x ( ) = n= x( ) = lim x ( ) c e n T jnω ω = x( ) e d e T n= T jn jnω As he period T goes o infiniy, he spacing beween he specral 2π lines decreases. This spacing is ω = ( n + ) ω nω = ω =. Since T
his value becomes small, he successive values of he frequencies nω can be replaced by he coninuous variable ω. So, we can rewrie he above expression as: jω jω x ( ) = x( ) e d e ω 2π T n= So, when we go o he limi T, he summaion will become an inegral and he spacing ω will become a differenial dω. So, we obain: j j x( ) = x( ) e d e d 2π ω ω ω The expression beween brackes is called he Fourier Transform of he signal x(). I is a funcion of he variable ω, so we can wrie: = jω X ( ω) x( ) e d jω x( ) = X ( ω) e dω 2π The firs expression is called he Forward Fourier Transform or he Analysis Formula. The second one expresses he signal as linear combinaion of phasors. I is called he Inverse Fourier Transform or he Synhesis Formula. The phasors now have frequencies ha belong o a coninuum of values. This is why he synhesis formula is now given by an inegral and no by a summaion. The inegrals are compued over infinie inervals. This implies ha we have o ake ino accoun convergence condiions. Wihou going ino deep mahemaical derivaion, we can affirm ha one sufficien condiion 2
for he exisence of he Fourier ransform is he fac ha he signal is an energy signal. In our course, we will find i easier o use he variable f raher han he variable ω. Since, ω = 2πf, we obain he following pair: = j2π f X ( f ) x( ) e d = 2 x( ) X ( f ) e j π f df The above relaions are more ineresing han he firs ones. They differ by only he sign of he exponen inside he inegral. Symmery relaions: The Fourier ransform X(f) = F[x()] of he signal x() is in general a complex funcion of he real variable f. We can express i as: X ( f ) X ( f ) e ϕ j ( f ) =, where he wo funcions X(f) and ϕ(f) have no paricular symmery if x() is complex. However, if x() is real, hen he Fourier ransform will have he same ype of symmery as he one we have seen in he sudy of Fourier series (Hermiian Symmery). * So: X ( f ) = X ( f ). This means ha: Example: The recangular pulse. Consider he signal x() = Π(). X ( f ) = X ( f ) (Even funcion) ϕ( f ) = ϕ ( f ) (Odd funcion) < Π ( ) = > 2 2 3
We have: X f e d f f 2 2 ( ) π f = 2 = sinπ π f = sinc.4 X.2.8.6.4.2-9 -8-7 -6-5 -4-3 -2-2 3 4 5 6 7 8 9 f The Sinc funcion Causal exponenial pulse: b Ae > x( ) = < -.2 b j2 f A = π = + 2 X ( f ) A e e d b j π f When he funcion is causal, he Fourier ransform can be seen as he evaluaion of he Laplace ransform on he imaginary (jω) axis. Properies of he Fourier ransform:. Lineariy: F[ax ()+bx 2 ()] = af[x ()] + bf[x 2 ()] 4
2. Dualiy: If x() and X(f) consiue a known ransform pair, hen F[X()] = x( f). For example: F[Π()] = sincf. Then F[sinc] = Π(f). 3. Time delay: F[x( τ)] = X(f)e j2πfτ 4. Scale change: F [ x( a) ] expands is specrum and vice versa. f = X. So, compressing a signal a a 5. Frequency ranslaion: ( ) j ω F x e = X ( f f ) ω = 2π f 6. Modulaion heorem: F x( )cos ω = X ( f f) + X ( f + f) 2 2 [ ] Example: The RF pulse: Consider x( ) = AΠ cosω. Using he τ above relaions, we obain: Aτ Aτ X ( f ) = sinc( f f) τ + sinc( f + f) τ 2 2 5
7. Differeniaion: d F x ( ) = j 2 π fx ( f ) d 8. Convoluion: The convoluion of wo funcions is defined as: z( ) = x( )* y( ) = x( λ) y( λ) dλ. We obain: F [ ] x( )* y( ) = X ( f ) Y ( f ) Example: Consider he riangular funcion Λ(). < < Λ ( ) = elsewhere I is easy o show ha Λ() = Π()*Π() F Λ ( ) = sinc So [ ] 2 f 9. Muliplicaion: [ ] F x( ) y( ) = X ( f )* Y ( f ) = X ( λ) Y ( f λ) dλ. The Rayleigh Energy heorem: 2 2 x( ) d = X ( f ) df The Dirac Impulse Funcion The Dirac impulse funcion or uni impulse funcion or simply he dela funcion δ() is no a funcion in he sric mahemaical sense. I is defined in advanced exs and courses using he heory of disribuions. In our course, we will suffice wih he following much simpler definiion. b a x( ) δ ( ) d x() a < < b = oherwise 6
where x() is an ordinary funcion ha is coninuous a =. If x() =, he above expression implies: ε δ ( ) d = δ ( ) d = for ε > ε We can inerpre he above resul by saying ha he impulse funcion has a uni area concenraed a he poin =. Furhermore, we can deduce from he above ha δ() = for. This also means ha he dela funcion is an even funcion. The defining inegral can also be used o compue he following inegral: > δ ( u) du = < The above is nohing bu he definiion of he uni sep funcion or he Heaviside funcion. So, we obain he following relaionship beween he wo funcions: = u( ) δ ( λ) dλ and δ ( ) = Properies of he dela funcion:. Replicaion: du( ) d x( )* δ ( τ ) = δ ( λ τ ) x( λ) dλ = x( τ ) 2. Sifing: x( ) δ ( τ ) d = x( τ ) 7
3. We can use he fac ha he dela funcion is even in he above inegral o show ha: x( λ) δ ( λ) dλ = x( ) The above relaion is a convoluion. 4. Since he expressions conaining he impulse funcion mus be inegraed, he following properies can be easily deduced: x( ) δ ( ) = x( ) δ ( ) and δ ( a) = δ ( ) a a Fourier Transform of he impulse: δ j2π f ( ) e d = Using he dualiy propery, we deduce ha: 2 [ ] j π f F = e d = δ ( f ) even hough he consan (dc) is a power and no an energy signal. In fac, using he frequency ranslaion propery, we can compue he Fourier ransform of he phasor: jω F e = δ ( f f) This allows us o compue he Fourier ransform of periodic signals. If x() is periodic wih fundamenal period T, we can develop i in j2 nf Fourier series: x( ) = c e π. Using he above, we obain: n= n j2π nf F [ x( ) ] = F cne = cnδ ( f nf) n= n= 8
Example: Consider he following impulse rain: x( ) = δ ( nt ). This funcion is a repeiion of he dela funcion n= T T every T seconds. In he inerval, 2 2, we have x() = δ() and i is periodic. Is Fourier series coefficiens are: T jnω cn = δ ( ) e d = T 2 T 2 T This implies ha he Fourier ransform of he impulse rain is: X ( f ) = δ ( f nf ) = f δ ( f nf ) T n= n= From hese relaions, we can now relae Fourier ransforms and Fourier series. In order o do so, le us consider he signal s() buil by a repeiion of he signal x(). where he signal x() has a finie duraion T. In k= s( ) = x( kt ) oher words, x() = for [ T /2, T /2]. Le X(f) be is Fourier ransform. Using he replicaion propery of he dela funcion, we can wrie: This means ha s( ) = x( ) δ ( kt ) k= S( f ) = X ( f ) F δ ( kt ) = fx ( f ) δ ( f kf ) k= k= 9
Using now he sampling propery, we can re-express he above as: S( f ) = f X ( kf ) δ ( f kf ) k= Example: Fourier ransform of a periodic rain of recangular pulses. kt Here, s( ) = A Π where τ < T. In our case, he funcion k= τ x() is: x( ) = AΠ τ wih a Fourier ransform X ( f ) Aτ sinc( fτ ) So: τ ( ) S( f ) = Af sinc nf τ δ ( f nf ) n= =. So, we have found an alernae way o compue he coefficiens of he Fourier series of a periodic waveform. In he above example, he coefficiens c n of he developmen are: ( τ ) sinc( ) c = Af τ sinc nf = Ad nd n τ where d is he duy cycle d = fτ =. T Fourier ransform of he uni sep funcion and of he signum funcion: The signum funcion sgn() is a funcion ha is relaed o he uni sep funcion. I is defined as: > sgn( ) = = <
I is eviden ha u() = ½sgn() + ½. The signum funcion has zero ne area. I can be seen also ha sgn() is he limi of he following funcion: b >. We have F [ z( ) ] [ ] b F sgn( ) = lim Z( f ) = = b b e > z( ) = = b e < 2 jπ f j4π f + ( 2π f ) 2. So, signum and he uni sep funcion, we ge:. And from he relaion beween he F [ u( ) ] = δ ( f ) j2π f +. 2 By dualiy, we also obain: F [ sgn( f )] = jπ The uni sep funcion ransform allows us o compue he Fourier ransform of he inegral of a signal. x( λ) dλ = x( λ) u( λ) dλ = x( ) u( ). X ( f ) So, F x( λ ) d λ = + X () ( f ) j2π f 2 δ Signals can be classified according o heir specral occupancy. A lowpass (or baseband) signal is a signal wih high componens a low frequencies and small componens a high frequencies. On he oher hand, if he specrum is significanly differen from zero only in a band of frequencies all differen from zero, he signal is call bandpass.
The widh of his band is called he bandwidh. If he raio of he bandwidh o he value of he cener of he band is small, he signal is said o be a narrow bandpass signal. Bandlimied Signals and he Sampling Theorem A signal x() wih Fourier ransform X(f) is said o be bandlimied if X(f) = for f > W. The frequency W is he bandwidh of he signal. Bandlimied signals have he propery o be uniquely represened by a sequence of heir values obained by uniformly sampling he signal. So, o a signal x(), we can associae a sequence x (n) = x(nt s ). T s is called he sampling period. Is inverse f s is he sampling frequency. The Sampling Theorem: Given a bandlimied signal x() wih specrum X(f ) = for f > W. The signal can be recovered from is samples x(nt s ) aken a a rae f s = /T s wih f s 2W. nt s x( ) = x( nts )sinc n= Ts Proof: We have seen ha x( ) δ ( nt ) = x( nt ) δ ( nt ). So, if we muliply s s s he signal by he impulse rain δ ( nt ), we obain he sequence n= of values x (n) = x(nt s ). Le us call he obained signal x s ().. The Fourier x ( ) = x( ) δ ( nt ) = x( nt ) δ ( nt ) s s s s n= n= ransform of his signal is obained by he following convoluion: s 2
X s ( f ) = X ( f ) F δ ( nts ) = X ( f ) fs δ ( f nfs ). n= n= Using he replicaion propery of he dela funcion, we obain immediaely: X ( f ) = f X ( f nf ). We see ha he specrum of s s s n= he signal x s () is a repeiion of he specrum of he signal x(). X(f) W W f X s (f) f s /2 f s /2 f s W W f s f The above figures show he relaionship beween specra. I is clear ha if f s /2 > W, The specrum of x s () and he one of x() will coincide 3
for he range of frequencies beween f s /2 and f s /2 (wihin he scale facor f s ). This means ha we can recover x() by compuing he inverse Fourier ransform of he specrum X s (f) muliplied by a "recangular" filer wih a ransfer funcion f Π. So, we have: f s f s fs fs x( ) = F X ( f ) Π The resul is finally (show i): nt s x( ) = x( nts )sinc n= Ts (q.e.d) In he above proof, we had o have f s > 2W. If his does no occur, we have he phenomenon of aliasing. Aliasing is a disorion ha canno be cured (in general). I is due o he superposiion of he differen shifed specra. We can observe aliasing when we wach wesern movies. The wagon wheels seem o roae in reverse. This is due o he sampling rae (number of images/second) which is oo small. So, sampling is he process of generaing a sequence (discree ime signal) from a coninuous ime one. The obained sequence can be analyzed in he frequency domain. Le us consider x (n) = x(nt s ). Is Fourier ransform is defined o be: = n= X ( ω) x ( n) e jnω We can remark ha his specrum is periodic (in he frequency domain), wih a period equal o 2π. In he definiion of he specrum of 4
a sequence, we can also observe ha he frequencies are now measured in radians and no in radians per second as for he coninuous ime signal. This is due o he fac ha, in a sequence, he "ime" variable n is an ineger indicaing jus he posiion of he sample and no he ime posiion measured in seconds. You should consul he lab manual in order o have he correspondence beween he specrum of he sequence and he one of he original signal. If he sequence exiss for a finie ime, i.e. for N samples, hen he sum is finie. = N X ( ω) x ( n) e n= jnω We can also compue he specrum of he sequence over a finie 2π k number of discree frequencies ω k =, k =,, N. N N N jnωk = ωk = = n= n= X ( k) X ( ) x ( n) e x ( n) e 2π j nk N The above relaion defines he Discree Fourier Transform (DFT). This ransform can be compued very efficienly using an algorihm called he Fas Fourier Transform (FFT). Linear Time Invarian Sysems Signals are processed by sysems. By he word sysem, we undersand a mapping from a signal se (inpu signals) o anoher signal se (oupu signals). 5
x() Sysem y() The above figure shows graphically he relaionship ha exiss beween he inpu signal and he oupu one. In he mapping, we undersand ha he whole signal x() is ransformed ino he whole signal y(). You can encouner he noaion: y() = H[x()]. This noaion can be misleading. I can also mean ha he value of he signal y a he ime is funcionally relaed o only he value of he signal x a. When we wan o indicae he funcional relaionship beween values, we will use he following noaion: [ ] y( ) = H, x( λ), λ, This means ha he value of he oupu y a he ime depends on all he values of he inpu signal a imes λ beween and 2 and also on he sae of he sysem a. Memoriless sysem: If y( ) H [ x( )] =, i.e. he oupu a ime depends only on he inpu 2 a he same ime, he sysem is said o be memoriless. Causal and anicausal sysem: If y( ) H [ x( λ), λ ] =, i.e. if he oupu depends only on he pas and on he presen (bu no on he fuure), he sysem is said o be causal. If y( ) H [ x( λ), λ ] = >, i.e. he sysem oupu depends only on he fuure, he sysem is called anicausal. 6
Sable sysem: If a bounded inpu ( x( ) M, ) produces a bounded oupu, we say ha he sysem is BIBO sable. Linear sysem: A sysem is linear if i saisfies he condiion of superposiion: If y () is he oupu corresponding o x () and y 2 () is he oupu corresponding o x 2 (), hen a y ()+a 2 y 2 () corresponds o a x ()+a 2 x 2 (). Time Invarian sysem: A sysem is ime invarian if i is no affeced by a shif of he ime origin. In oher words, is properies remain he same as ime goes by. One consequence is ha if x() produces y(), hen x( τ) will produce y( τ). Many sysems of ineres are linear and ime invarian (LTI). Among such sysems, we find mos of he filers used o selec signals in communicaion sysems. Linear Time Invarian sysems: LTI sysems are sysems ha can be compleely described by a single funcion: he Impulse Response. If he inpu of an LTI sysem is a Dirac impulse, he corresponding oupu is a funcion h(). We have seen ha an signal x() can be seen a linear combinaion of shifed dela funcions. x( λ) δ ( λ) dλ = x( ) 7
So, since he sysem is ime invarian, hen he oupu corresponding o δ( λ) is h( λ). The sysem is also linear, so he oupu corresponding o x(λ)δ( λ) is x(λ)h( λ). Finally, he oupu corresponding o x() is he sum of such values: y( ) = x( λ) h( λ) dλ The above relaion is a convoluion. I is easy o show ha he convoluion is a commuaive operaion. This means ha we can wrie also: y( ) = h( λ) x( λ) dλ The funcion h() is he impulse response. I describes compleely he LTI sysem and allows us he compue he oupu for any given inpu. We can es he sabiliy of he sysem by esing is impulse response. A necessary and sufficien condiion for a sysem o be BIBO sable is h( ) d < The above condiion also implies ha he Fourier ransform of a BIBO sable sysem exiss. I is called he ransfer funcion H(f) = F[h()]. If he inpu and oupu signals possess Fourier ransforms, we can wrie: Y ( f ) = H ( f ) X ( f ) Causaliy also imposes resricions on h(). If he sysem is causal, he oupu from he convoluion inegral should no depend on values of he inpu a imes λ coming afer he ime. This implies ha h( λ) = for λ >. So, we see ha in order o have causaliy, we 8
mus have h() = for <. So, when a sysem is causal, he inpu oupu relaion becomes: y( ) = x( λ) h( λ) dλ = h( λ) x( λ) dλ Response of an LTI sysem o a phasor and o a sinewave: j( ) Le he inpu of he LTI sysem be he phasor Ae ω + θ. The oupu will be: j( ω ( λ ) + θ ) j( ω+ θ ) jωλ y( ) = h( ) Ae d = Ae h( ) e d λ λ λ λ Replacing ω = 2π f, we recognize he Fourier ransform of he impulse response, he ransfer funcion H(f ). So j( ) y( ) = H ( f ) Ae ω + θ If we inroduce he modulus H(f ) and argumen ϕ(f ) of he ransfer funcion: y( ) = H ( f ) e Ae jϕ ( f ) j( ω + θ ) From he above resul, we can conclude wo imporan facs. s ) he response of a phasor of frequency f, is also a phasor. The oupu phasor is proporional o he inpu one. The consan of proporionaliy is he ransfer funcion. Since an LTI sysem is a linear operaor, we can say ha he phasors are he "eigenfuncions" of LTI sysems while he ransfer funcion is he "eigenvalue". 2 nd ) he oupu phasor is equal o he inpu phasor scaled by he modulus of he ransfer funcion and phase shifed by is argumen. Now, if he inpu is a sinewave x() = Acos(ω + θ), we can wrie: A A x( ) e e 2 2 j( ω+ θ ) j( ω+ θ ) = +, he oupu becomes: 9
A A y( ) H ( f) e H ( f) e 2 2 j( ω+ θ ) j( ω+ θ ) = + giving A A y( ) H ( f) e e H ( f) e e 2 2 jϕ ( f ) j( ω+ θ ) jϕ ( f ) j( ω+ θ ) = +. If he impulse response is real, hen H(f ) = H( f ) and ϕ(f ) = ϕ( f ). This implies: y( ) = A H ( f) e + e 2 2 j( ω + θ + ϕ ( f )) j( ω + θ + ϕ ( f )) ( ω θ ϕ ) y( ) = A H ( f ) cos + + ( f ) The oupu is a sinewave a he same frequency, scaled by he modulus of he ransfer funcion and phase shifed by he argumen of he ransfer funcion a ha frequency. Example: Consider he following RC circui: The ransfer funcion is equal o: (his is a simple volage divider) j2π fc H ( f ) = = = j2 frc f R + + π + j j2π fc f where f = 3 2π RC is he 3 db cu-off frequency. Assume we inpu a sinewave a he frequency 3 f, x( ) Acos( 2π f ) =. 3 3 2
2 π H ( f 3) = = and ϕ( f3) = an =. So: + 2 4 A π y( ) = cos 2π f3 2 4 When he LTI sysem is used o modify he specrum of a signal, i is called a filer. We can classify filers according o heir ampliude response. Le H(f) be he ransfer funcion. If H(f) for f > W, he filer is called Lowpass. If H(f) for f < W, he filer is called Highpass. If H(f) for < f < f and f > f 2, he filer is called Bandpass. If H(f) = consan for all frequencies, he filer is an Allpass filer. Bandpass Signals: Bandpass signals form an imporan class of signals. This is due o he fac ha pracically all mehods for ransporing informaion use modulaion sysems ha ransform he baseband informaion ino bandpass signals. In his secion, we are concerned wih real bandpass signals. Le x() IR be a bandpass signal. Due o he Hermiian symmery (X(f) = X * ( f)), he informaion in he posiive frequency is enough o characerize compleely he signal. A real bandpass signal is characerized by: X ( f ) = for < f < f and f > f2 ( f < f 2 ). The bandwidh of he bandpass signal is defined as being he difference B = f 2 f. If B << f, he signal is a narrow bandpass signal. 2
Bandpass signals are compleely described by heir low frequency envelop. In order o describe quie simply bandpass signals, we have o inroduce some mahemaical ools. The Hilber Transform In his secion of he course, we are going o inroduce a ool ha allows us o ransform a real signal, wih a wo sided specrum, ino a complex signal wih he same specrum, bu only for posiive frequencies. To begin, consider he real sinewave x() = cosω and he phasor x + () = exp(jω) = cosω + jsinω. The respecive specra are: X ( f ) f f f f 2 2 = δ ( ) + δ ( + ) and X ( f ) δ ( f f ) + =. We can remark ha he specrum of he phasor is he same (wihin a scale facor of 2) as he one of he sinewave for he posiive frequencies while i is zero for negaive frequencies. Furhermore, he phasor is equal o he sum of he sinewave and he same signal phase shifed by 9. We can generalize his relaionship o mos signals. In order o phase shif signals by 9, we inroduce a ransform called he Hilber Transform. The Hilber ransform of a signal x() is he signal x() equal o he original signal wih all frequencies phase shifed by 9. The operaion of shifing he phase of a signal by a consan value is a linear ime invarian operaion. This means ha he signal x () is 22
obained from x() by a filering operaion. In fac, he filer is an allpass one. This means ha: xˆ( ) = h( )* x( ) or Xˆ ( f ) = H ( f ) X ( f ) where H ( f ) = and [ H f ] arg ( ) π f > = 2 π f 2 < So, we can wrie π j sgn f 2 H ( f ) = e = jsgn f. We have already compued he Fourier ransform of he signum funcion. Using he dualiy propery, we obain: giving h( ) = π x( λ) xˆ( ) = dλ π λ In general i is easier o compue he Hilber ransform in he frequency domain since i amouns o shifing he frequencies by 9. cosω = sinω, sinω = cosω. If we apply he Hilber ransform o a signal x() ha is iself he Hilber ransform of a signal x(), we phase shif x() by 8. This means ha we simply inver he signal. xˆ( ) = x( ) The above relaion provides he inversion formula: xˆ ( λ) x( ) = dλ π λ 23
The following propery is imporan in he analysis of bandpass signals. Theorem: Given a baseband signal x() wih X(f) = for f W and a highpass signal y() wih Y(f) = for f < W (non-overlapping specra), hen x()y() = x()y(). Example: if y( ) = x( )cosω wih X(f) = for f ω, hen yˆ( ) = x( )sinω. The analyic signal using he analogy of he sinusoid and he phasor, we can define a signal having a specrum ha exiss only for posiive frequencies. I is he analyic signal associaed wih x(): x+ ( ) = x( ) + jxˆ ( ) We obain, in he frequency domain: [ ] X ( ) ( ) ˆ + f = X f + jx ( f ) = X ( f ) + sgn f 2 X ( f ) f > so, X + ( f ) = X () f = f < We can also define anoher analyic signal, bu ha exiss only for negaive frequencies. x ( ) = x( ) jxˆ ( ) 24
2 X ( f ) f < X ( f ) = X () f = f > We can remark ha x( ) Re [ x ( ) ] Re [ x ( ) ] = =. So, i is simple o + exrac he original signal from he analyic one. We have also x+ ( ) + x ( ) e x( ) = which is analogous o cosω = 2 jω + e 2 jω Using analyic signals, we can now give imporan properies of bandpass signals. Consider a real bandpass signal x(), such ha X ( f ) = for < f < f and f > f2 ( f < f 2 ), and consider a frequency f beween f and f 2, i.e. f f f 2, hen we can express he signal as: x( ) = a( )cos ω b( )sinω where a() and b() are baseband signals bandlimied o [ f f f f ] max,. The above represenaion is called he 2 quadraure represenaion. We can also represen he signal as ( ω ϕ ) x( ) = r( )cos + ( ) where r() and ϕ() are also baseband signals. This represenaion is called a modulus (ampliude) and phase (argumen) represenaion. Proof: Consider he analyic signal associaed wih x(). x + () = x() + jx(). Is specrum is X + (f) = 2X(f) for posiive frequencies and zero for negaive ones, i.e. X + (f) = 2X(f)u(f). If we shif is specrum down o dc by f, we obain a bandlimied signal 25
j m ( ) = x ( ) e ω x M f X f f X f f u f f + ( ) x = ( ) 2 ( ) ( ) + + = + + This signal is baseband and is in fac bandlimied o [ f f f f ] max,. Is specrum has no paricular symmery (in 2 general). So, he signal is complex. This means ha we can wrie: m ( ) ( ) ( ) x = a + jb where he wo signals are real and have he same bandwidh as m x (). The signal m x () is called he complex envelop of x(). We can recover he signal x() by shifing i back o f. x m e ω ( ) ( ) j + = + = a( )cos ω b( )sinω x ( )( ω ω ) jω x( ) = Re m ( ) e = Re a( ) + jb( ) cos + jsin We can also express he complex envelop in modulus and phase: (q.e.d.) ( ) = r e ϕ giving: x( ) r( )cos [ ω ϕ( ) ] m ( ) ( ) j x = + along wih 2 2 b( ) r( ) = a ( ) + b ( ), ϕ( ) = an a( ) a( ) = r( )cos ω, b( ) = r( )sinω In a bandpass signal, he informaion is conained in he complex envelop. In many cases, i is easier o process he envelop of he signal insead of processing direcly he bandpass signal. 26
Filering a bandpass signal: Consider a real narrow bandpass signal x() having a bandwidh W cenered around a frequency f and consider a real bandpass filer (wih impulse response h())wih a bandwidh B ha covers compleely he signal x(). The ransfer funcion H(f) is of course he Fourier ransform of h(). We define he equivalen lowpass filer h lp () as he lowpass filer having as ransfer funcion H lp (f) he posiive frequency half of H(f) ranslaed down o zero by f. So: H f H f f u f f ( ) lp = ( + ) ( ) + If we call x() he inpu of he bandpass filer and y() he oupu, we have: y( ) = h( ) x( ) or Y ( f ) = H ( f ) X ( f ). Inroducing he complex envelops: x m e ω ( ) ( ) j + = x and y y ( ) m ( ) e jω + =. In he frequency domain, his becomes: X + ( f ) = M x ( f f) and Y+ ( f ) = M y ( f f). Since Y ( f ) = H ( f ) X ( f ), hen Y ( f ) = H ( f ) X ( f ). This implies + + ha M ( f f) = H ( f ) M ( f f), his relaion is valid for posiive y x frequencies, so we can wrie, wihou affecing he previous relaion: M ( f f ) u( f ) = H ( f ) M ( f f ) u( f ). If we make he change of y x variable f' =f f, we obain: M ( f ') u( f ' + f ) = H ( f ' + f ) u( f ' + f ) M ( f ') or y M ( f ) = H ( f ) M ( f ) y lp x x 27
So, bandpass filering a bandpass signal amouns o lowpass filering is complex envelop by he equivalen lowpass filer. Example: Consider he following parallel RLC circui: The impedance of he circui is: Z( ω) = = + jcω + + R jlω QT = RCω. jqt R 2 2 ω ω ωω where ω = ; LC If Q T >, his impedance can be approximaed quie closely by: Z( ω) = + R ω ω j α, ω >, α = from zero only in he viciniy of ω. 2RC and i is essenially differen If he inpu is he curren flowing hrough he circui and he oupu is he volage, we have a narrow bandpass filer. Le he curren x() be: 28
x( ) Acos cos = ωm ω wih ωm ω <<. The signal is already in quadraure form wih a( ) = Acosω m, b( ) =. So, he complex envelop is m ( ) = Acosω. The equivalen lowpass filer has is: Z lp ( f ) = + x m R so he complex envelop of he oupu is he 2π f j α sinewave m x () filered by Z lp (f). So, So, ( ω ) my ( ) = Zlp ( fm) Acos m + Arg Zlp ( fm) AR m ( ) cos an y And finally 2π f m = 2 ωm 2π f α m + α AR 2π f = y = 2 m 2π f α m + α Group delay and phase delay: jω m y( ) Re m ( ) e cos ω an cosω Consider a very narrow bandpass signal cenered on a frequency f and having a bandwidh W (W << f ). This signal is o be filered by a filer having a ransfer funcion H(f). The signal is: [ ω θ ] x( ) = r( )cos + ( ) Because of he narrowness of he bandwidh of he signal, we can make he following approximaions for he ransfer funcion: [ ϕ ] H ( f ) = A( f )exp j ( f ) and around f, we can assume ha he ampliude response is consan, and ha he phase response can be approximaed by is firs order Taylor series. 29
A( f ) A and ϕ f ϕ + k f f ( ) ( ) for f around f. k dϕ =. df The complex envelop of he signal is: m ( ) r( )exp [ jθ ( ) ] equivalen lowpass filer ransfer funcion is: x f = f = and he ( ϕ ) H ( lp f ) = H ( f + f ) ( ) exp u f + f = A j + kf. So, he complex envelop of he oupu is given by: ( ϕ + ) ϕ M ( f ) A e M ( f ) A e M ( f ) e = =. Using he ime delay j kf j jkf y x x jϕ heorem: my ( ) = A e mx + k. Finally, he oupu of he filer is: 2π k j j k j j k θ + ϕ ω ϕ 2π jω y( ) = Re A e mx + e = Re A e r + e e 2π 2π So, we obain: k k y( ) = A r + cos ω + θ + + ϕ 2π 2π. Inroducing he ϕ( ω) "phase delay" τ p = and he "group delay" τ g ω finally ge: ( τ ) g ω τ p θ ( τ g ) y( ) = A r cos ( ) + dϕ =, we d ω ω= ω We can remark ha he carrier and he complex envelop are no delayed by he same amoun (unless he phase response is a linear funcion of he frequency). 3
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