1 I. Phasors (complex envelope) representation for sinusoidal signal narrow band signal II. Complex Representation of Linear Modulated Signals & Bandpass System Band Pass Systems, Phasors and Complex Representation of Systems KEY LEARNING OBJECTIVES Phasors and Complex Representation are useful for analyzing baseband component of a signal eliminates high frequency carrier components

2 x(t) is a narrowband signal (aka bandpass signal) if X(f) ≠ 0 in some small neighborhood of f 0, a high frequency X(f) ≡ 0 for | f – f 0 | ≥ W where W < f 0 f 0 is usually referred to as center frequency, but need not be center frequency or in signal bandwidth at all X(f) 2W -f 0 -W -f 0 - f 0 +Wf 0 -W f 0 f 0 +W I. Phasors for monochromatic & narrow band signals h(t) is a Bandpass System,, that passes signals with frequency components in the neighborhood of some frequency, f 0 H(f) = 1 for | f – f 0 | ≤ W otherwise H(f) ≈ 0 bandpass system h(t) passes a bandpass signal x(t) X(f) H(f)

3 output determined by multiplying X & frequency response of system computed at input frequency, f 0 input & output frequencies are same  output phasor gives output signal Consider LTI system driven by input x(t) H(f) X(f) Y(f)  determine the phasor for sinusoida1 signal and narrowband signal capture phase and magnitude of base band signal ignore effects of the carrier

4 z(t) = Aexp(j(2πf 0 t + θ)) = Acos(2πf 0 t + θ) + jAsin(2πf 0 t + θ) = x(t) + jx q (t) (i) define a signal z(t) as a vector rotating with angular frequency 2πf 0 1. determination of phasor, X for sinusoidal input signal x(t) x(t) = Acos(2πf 0 t + θ) x q (t) = Asin(2πf 0 t + θ) quadrature component shifted 90 o from x(t) (ii) obtain phasor X from z(t) by eliminating 2πf 0 rotation - rotate z(t) at an angular frequency = 2πf 0 in opposite direction - equivalent to multiplying z(t) by exp(2πf 0 t) X = z(t) exp(-j2πf 0 t ) = Aexp(j(2πf 0 t + θ))exp(-j2πf 0 t ) = Aexp(jθ) 2πf02πf0 Aexp(jθ) R I x q (t) x(t)

5 1a. determine Frequency Domain equivalent of z(t) and X Z(f) = [cos(θ)δ(f–f 0 ) + jsin(θ)δ(f–f 0 )] x(t) = Acos(2πf 0 t + θ) = Acos(θ)cos(2πf 0 t) + Asin(θ)sin(2πf 0 t) X(f) = cos(θ)[δ(f–f 0 ) + δ(f+f 0 )] sin(θ)[δ(f+f 0 ) - δ(f-f 0 )] - j (1) determine X(f) = F[x(t)], delete negative frequencies & multiply by 2 X = Aexp(jθ) (ii) then shift Z(f) by f 0  (i) obtain Z(f), using either or two methods z(t) = Aexp(j(2πf 0 t + θ)) = Aexp(jθ)exp(j2πf 0 t ) Z(f) = Aexp(jθ)δ(f – f 0 ) since F[exp(j2παt)] = {δ(f-α)}  (2) determine Z(f) = F[z(t)]

6 z(t) is known as the analytic signal or pre-envelope of x(t) 2. determine phasor for a narrowband signal, x(t) Z(f) = 2u -1 (f)X(f) based on definition of z(t) in sinusoid case: z(t) = x(t) + jx q (t) find Z(f) by deleting negative frequencies of X(f) & multiply result by 2 find z(t) using IFT  find signal whose Fourier transform = u -1 (f) we know that F[u -1 (t)] = by duality  = u -1 (f) by convolutionz(t) =  then z(t) = let

7 phase shift x(t) byfor positive frequencies phase shift x(t) byfor negative frequencies Hilbert Transform of x(t) is given by pre-envelope for two types of signals (ii) narrowband case z(t) = x(t) + j z(t)= x(t) + jx q (t) (i) sinusoid case x(t) = Acos(2πf 0 t+θ) x q (t)= Asin(2πf 0 t+θ)

8 determine phasor, x l (t) of bandpass signal x(t) x l (t) = low pass representation of x(t) determined by shifting spectrum of z(t) left by f 0 X l (f) = Z(f + f 0 ) = 2u -1 (f + f 0 )X(f + f 0 ) x l (t) = z(t)exp(-j2πf 0 t) x l (t) is a low pass signal X l (f) ≡ 0 for all | f | ≥ W phasor for band pass signal X(f) f 0 f Z(f) 2A f f X l (f) 2A A

9 x l (t) = x c (t) + jx s (t) Generally x l (t) is complex signal with real (in phase) & imaginary (quadrature) components z(t) = x l (t)exp(j2πf 0 t) = [x c (t) + jx s (t)]exp(j2πf 0 t) = x c (t)cos(2πf 0 t) - x s (t)sin(2πf 0 t) + j[x c (t)sin(2πf 0 t)+x s (t)cos(2πf 0 t)] z(t) =  rewrite in terms of quadrature & in-phase components equate real & imaginary parts of z(t) and x l (t) = Im{z(t)} = x c (t)sin(2πf 0 t)+x s (t)cos(2πf 0 t) x(t) = Re{z(t)} = x c (t)cos(2πf 0 t) - x s (t)sin(2πf 0 t) )( ˆ tx bandpass to lowpass transform describes relationship of x(t) & in terms of x c (t) & x s (t) )( ˆ tx

10 x l (t) R I Θ(t) V(t) monochromatic phasor has constant amplitude & phase bandpass signal’s phase & envelope vary slowly with time  vector representation moves on a curve in the complex plane V(t) & Θ(t) are slowly time varying x l (t) = V(t)exp( jΘ(t) )then = define envelope of x l (t) as V(t) = Θ(t) = define phase of x l (t) as Define x l (t) in terms of phase & envelope

11 II. Complex Representation of Linear Modulated Signals & Bandpass System s(t) = s I (t)cos(2πf c t) - s Q (t)sin(2πf c t) canonical representation of any bandpass signal, s(t) has 2 components s I (t) = in-phase component of s(t) s Q (t) = quadrature component of s(t) properties of s I (t) & s Q (t) are real valued functions are orthogonal to each other are uniquely defined in terms of the baseband signal m(t) two components can be used to synthesize modulated signal s(t)

12 circuit used to synthesize s(t) from s I (t) & s Q (t)  s(t) cos(2  f c t) sin(2  f c t) 90 o oscillator s I (t) s Q (t) s I (t) LPF s(t) 2cos(2  f c t) -2sin(2  f c t) oscillator 90 o s Q (t) LPF circuits used to analyze s I (t) & s Q (t) based on s(t),

13 1. Complex Envelope of a Band-Pass Signal s(t) is given as s̃̃(t) preserves information content of s(t), except for f c (t) s̃̃(t) = s I (t) + js Q (t) s(t) = Re{s̃̃(t)e (2πf c t) } = s I (t)cos(2πf c t) - s Q (t)sin(2πf c t) then, s̃̃(t)e (2πf c t) = [s I (t) + js Q (t)] [cos(2πf c t) + jsin(2πf c t)] = s I (t)cos(2πf c t) - s Q (t)sin(2πf c t) + j[s I (t)sin(2πf c t)+s Q (t)cos(2πf c t)] real imag

14 system is narrowband if bandwidth W << f c, the system’s center frequency input x(t) is modulated by carrier, f c output = y(t) h(t) x(t)y(t) 2. Consider a narrowband linear band-pass system x̃̃(t)2ỹ(t) h̃̃(t) use equivalent complex baseband model to simplify analysis impulse response given by h̃̃(t) = h I (t) + jh Q (t) canonical representation of system’s impulse response given by: h(t) = h I (t)cos(2πf c t) - h Q (t)sin(2πf c t)

Passband Analysis of LTI System y(t) = [x I ( )cos(2πf c )-x Q (t)sin(2πf c )]· [h I (t- )cos(2πf c t- )-h Q (t- )sin(2πf c t- )]d y(t) = = x I (t) h I (t - ) cos(2πf c t)cos(2πf c t- ) d x I (t)h Q (t- )cos(2πf c t)sin(2πf c t- ) d - + x Q (t) h Q (t - ) sin(2πf c t)sin(2πf c t- ) d x Q (t)h I (t- )cos(2πf c t- )sin(2πf c t) d -

16 Passband Analysis of LTI System (continued) y(t) = x I (t) h I (t - ) ½[ cos( ) + cos(4πf c t- ) ] d x I (t)h Q (t- )½[ sin(4πf c t ) + sin( ) ] d - + x Q (t) h Q (t - ) ½[ cos( ) - cos(4πf c t- ) ] d x Q (t)h I (t- )½[ sin(4πf c t ) - sin( ) ] d -

17 complex envelopes are related by complex convolution 2.2 Equivalent Complex Baseband Model ỹ (t) = y I (t) + jy Q (t)  is the complex envelope of y(t) complex input & output are complex envelopes of bandpass systems input & output x̃̃(t) = x I (t) + jx Q (t)  is the complex envelope of x(t) = [x I (t) + jx Q (t)] [h I (t-λ) + jh Q (t-λ)]dλ ỹ(t) = = = h I (t-λ)x I (t) - h Q (t-λ)x Q (t) + j[x Q (t)h I (t-λ) + h Q (t-λ)x I (t)]dλ

18 Equivalent Notation for complex baseband model ( ‘  ’ = convolution) ỹ(t) = ½ (x̃̃(t)  h̃̃(t)) = ½(h̃̃(t)  x̃̃(t)) ½ factor added to maintain equivalence between real & complex models f c is omitted from complex baseband model  simplifies analysis without loss of information x(t) = Re{x̃̃(t)exp(2πf c t)} y(t) = Re{ỹ(t)exp(2πf c t)} Passband signals are readily determined from ỹ(t) and x̃̃(t) Impulse response of band-pass system given by h(t) = Re{h̃̃(t)exp(2πf c t)} = Re{ (h I (t) + jh Q (t)) (cos(2πf c t) + jsin(2πf c t) ) } = h I (t)cos(2πf c t) - h Q (t) sin(2πf c t)

19 Appendix: More on Complex Envelope - viewed as an extension of phasor for a real harmonic signal x(t) x(t) =  x cos(2  f 0 t +  x ) t  R assume  x  0 and phase is 0   x < 2 , then: (i) exp( j(2  f 0 t+  x )) = cos(2  f 0 t +  x ) + jsin(2  f 0 t +  x ) = Re [  x exp(j(2  f 0 t +  x) )] t  R = Re [  x exp(j  x ) exp(j2  f 0 t )] t  R (ii) x(t) = Re[  x ( cos(2  f 0 t +  x ) + jsin(2  f 0 t +  x ) )] t  R phasor representing phase & magnitude of x(t) = complex envelope:  x exp(j  x ) =  x cos(  x ) + j  x sin(  x )  x = magnitude  x = argument (phase of x(t))

20 ii. suppress negative frequencies & multiply by 2 iii. shift left by f 0 to obtain frequency signal =  x exp(j  x )  (f 0 ) f  R iv. take Inverse Fourier Transform i. Take Fourier Transform of x(t) X(f) = F[  x cos(2  f 0 t+  x )] =  x exp(j  x )  (f-f 0 ) +  x exp(-j  x )  (f+f 0 ) derive complex envelope for any real continuous signal, x(t) assume x(t) = Re [x e (t) exp(j2  f 0 t )] t  R where x e (t)=  x exp(j  x ), x̃̃ e (f) =  x exp(j  x )  (f-f 0 ) f  R x̃̃ p (f) = x e (t) =  x exp(j  x ) F -1 [x̃̃ e (f) ]

21 x(t) =  cos(2  f 1 t +  x ) t  R e.g. Pure Harmonic signal given by if f 1 = f 0  complex envelope = phasor if |f 1 -f 0 | << f 0  x e varies slowly compared to exp(2j  f 0 t) where  x  0 0   x < 2  i. FT yields X(f) = ½  exp(j  x )  (f-f 1 ) + ½  exp(-j  x )  (f+f 1 ) ii. iii. x e (t) =  exp(j  )exp(2j  (f 1 -f 0 ))t t  R iv =  exp(j  )  (f-f 1 ) x̃̃ p (f) =  exp(j  )  (f-f 1 +f 0 ) x̃̃ e (f)

22 If x(t) = real, continuous function, & F(x) has no delta function at f = 0 pre-envelope (aka analytical) of x is complex valued signal x p with complex-envelope of x with respect to frequency f 0 is signal x e x̃̃ e (f) = x̃̃ p (f+f 0 ) = 2X(f+f 0 ) 1(f+f 0 ) f  R F[x̃̃ p ] = = 2X(f)1 (f) f  R x̃̃ p (f) x e (t) = F -1 [ x̃̃ e (f) ]

23 Complex Envelope for let x(t) = real, band-pass, band-limited signal f c = center frequency & W = bandwidth where W < f c, are positive real numbers (W << f c  x(t) is narrowband) f  R X(f) = 0 for| f | < f c -W and | f | > f c +W 0 W fcfc 0-f c W X(f) x p = analytical fcfc 0 )( ˆ fx p x e = complex envelope with respect to f 0 contains only low frequencies f 0  R+  x e is not uniquely defined 0