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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

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x(t) is a narrowband signal (aka bandpass signal) if

• X(f) ≠ 0 in some small neighborhood of f0 , a high frequency

• X(f) ≡ 0 for | f – f0 | ≥ W where W < f0

• f0 is usually referred to as center frequency, but need not be center frequency or in signal bandwidth at all

X(f)2W

-f0 -W -f0 - f0 +W f0 -W f0 f0 +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, f0

• H(f) = 1 for | f – f0 | ≤ W otherwise H(f) ≈ 0

• bandpass system h(t) passes a bandpass signal x(t)

X(f) X(f)H(f)

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• output determined by multiplying X & frequency response of system computed at input frequency, f0

• 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

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z(t) = Aexp(j(2πf0t + θ))

= Acos(2πf0t + θ) + jAsin(2πf0t + θ)

= x(t) + jxq(t)

(i) define a signal z(t) as a vector rotating with angular frequency 2πf0

1. determination of phasor, X for sinusoidal input signal x(t)

x(t) = Acos(2πf0 t + θ)

xq(t) = Asin(2πf0 t + θ)

• quadrature component shifted 90o from x(t)

(ii) obtain phasor X from z(t) by eliminating 2πf0 rotation

- rotate z(t) at an angular frequency = 2πf0 in opposite direction

- equivalent to multiplying z(t) by exp(2πf0t)

X = z(t) exp(-j2πf0t ) = Aexp(j(2πf0t + θ))exp(-j2πf0t )

= Aexp(jθ)

2πf0

Aexp(jθ)

R

I

xq(t)

x(t)

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1a. determine Frequency Domain equivalent of z(t) and X

Z(f) = [cos(θ)δ(f–f0 ) + jsin(θ)δ(f–f0 )] 2

2A

x(t) = Acos(2πf0t + θ) = Acos(θ)cos(2πf0t) + Asin(θ)sin(2πf0t)

X(f) = cos(θ)[δ(f–f0 ) + δ(f+f0)] 2

Asin(θ)[δ(f+f0) - δ(f-f0)] 2

A- j

(1) determine X(f) = F[x(t)], delete negative frequencies & multiply by 2

X = Aexp(jθ) (ii) then shift Z(f) by f0

(i) obtain Z(f), using either or two methods

z(t) = Aexp(j(2πf0t + θ)) = Aexp(jθ)exp(j2πf0t )

Z(f) = Aexp(jθ)δ(f – f0 ) since F[exp(j2παt)] = {δ(f-α)}

(2) determine Z(f) = F[z(t)]

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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) + jxq(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)] = fjf

2

1)(

2

1

by duality = u-1(f)

t

jt

2)(

2

1F

by convolution )()( txt

jt

z(t) =

)(ˆ)( txjtx then z(t) = let )(1

)(ˆ txt

tx

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2

phase shift x(t) by for positive frequencies

2

phase shift x(t) by for negative frequencies

Hilbert Transform of x(t) is given by )(1

)(ˆ txt

tx

pre-envelope for two types of signals

(ii) narrowband case

)(ˆ txz(t) = x(t) + j

z(t) = x(t) + jxq(t)(i) sinusoid case

x(t) = Acos(2πf0 t+θ)

xq(t)= Asin(2πf0 t+θ)

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determine phasor, xl(t) of bandpass signal x(t)

• xl(t) = low pass representation of x(t)

• determined by shifting spectrum of z(t) left by f0

Xl(f) = Z(f + f0) = 2u-1(f + f0)X(f + f0)

xl(t) = z(t)exp(-j2πf0t)

xl(t) is a low pass signal

• Xl(f) ≡ 0 for all | f | ≥ W

• phasor for band pass signal

X(f)

f0

f0

f0

f

Z(f)2A

f

f

Xl(f) 2A

A

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xl(t) = xc(t) + jxs(t)

Generally xl(t) is complex signal with real (in phase) & imaginary (quadrature) components

z(t) = xl(t)exp(j2πf0t)

= [xc(t) + jxs(t)]exp(j2πf0t)

= xc(t)cos(2πf0t) - xs(t)sin(2πf0t) + j[xc(t)sin(2πf0t)+xs(t)cos(2πf0t)]

z(t) = )(ˆ)( txjtx rewrite in terms of quadrature & in-phase components

equate real & imaginary parts of z(t) and xl(t)

= Im{z(t)} = xc(t)sin(2πf0t)+xs(t)cos(2πf0t)

x(t) = Re{z(t)} = xc(t)cos(2πf0t) - xs(t)sin(2πf0t)

)(ˆ tx

bandpass to lowpass transform describes relationship of x(t) & in terms

of xc(t) & xs(t)

)(ˆ tx

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xl(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

xl(t) = V(t)exp( jΘ(t) )then

= )()( 22 txtx sc

)(

)(1tan

tx

tx

c

s

define envelope of xl(t) as V(t) = )()( 22 txtx sc

Θ(t) =

)(

)(1tan

tx

tx

c

sdefine phase of xl(t) as

Define xl(t) in terms of phase & envelope

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II. Complex Representation of Linear Modulated Signals & Bandpass System

s(t) = sI(t)cos(2πfct) - sQ(t)sin(2πfct)

canonical representation of any bandpass signal, s(t) has 2 components

• sI(t) = in-phase component of s(t)

• sQ(t) = quadrature component of s(t)

properties of sI(t) & sQ(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)

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circuit used to synthesize s(t) from sI(t) & sQ(t)

s(t)

cos(2fct)

sin(2fct)90o

oscillator

sI(t)

sQ(t)

sI(t)LPF

s(t)

2cos(2fct)

-2sin(2fct)

oscillator

90o

sQ(t)LPF

circuits used to analyze sI(t) & sQ(t) based on s(t),

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1. Complex Envelope of a Band-Pass Signal s(t) is given as

s33 (t) preserves information content of s(t), except for fc(t)

s33 (t) = sI(t) + jsQ(t)

s(t) = Re{s33 (t)e(2πfct)}

= sI(t)cos(2πfct) - sQ(t)sin(2πfct)

then,

s33 (t)e(2πfct) = [sI(t) + jsQ(t)] [cos(2πfct) + jsin(2πfct)]

= sI(t)cos(2πfct) - sQ(t)sin(2πfct) + j[sI(t)sin(2πfct)+sQ(t)cos(2πfct)]

real imag

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• system is narrowband if bandwidth W << fc , the system’s center frequency

• input x(t) is modulated by carrier, fc

• output = y(t)

h(t)x(t) y(t)

2. Consider a narrowband linear band-pass system

x33 (t) 2ỹ(t)h33 (t)

use equivalent complex baseband model to simplify analysis

• impulse response given by

h33 (t) = hI(t) + jhQ(t)

canonical representation of system’s impulse response given by:

h(t) = hI(t)cos(2πfct) - hQ(t)sin(2πfct)

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2.1 Passband Analysis of LTI System

y(t) =

dthx )()(

[xI()cos(2πfc )-xQ(t)sin(2πfc )]·

[hI(t-)cos(2πfct-)-hQ(t-)sin(2πfct-)]dy(t) =

= xI(t) hI(t-) cos(2πfct)cos(2πfct-) d

xI(t)hQ(t-)cos(2πfct)sin(2πfct-) d

-

+ xQ(t) hQ(t-) sin(2πfct)sin(2πfct-) d

xQ(t)hI(t-)cos(2πfct-)sin(2πfct) d -

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Passband Analysis of LTI System (continued)

y(t) = xI(t) hI(t-) ½[ cos() + cos(4πfc t-) ] d

xI(t)hQ(t-)½[ sin(4πfc t) + sin() ] d

-

+ xQ(t) hQ(t-) ½[ cos() - cos(4πfc t-) ] d

xQ(t)hI(t-)½[ sin(4πfc t) - sin() ] d

-

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• complex envelopes are related by complex convolution

2.2 Equivalent Complex Baseband Model

ỹ(t) = yI(t) + jyQ(t) is the complex envelope of y(t)

• complex input & output are complex envelopes of bandpass systems input & output

x33 (t) = xI(t) + jxQ(t) is the complex envelope of x(t)

= [xI(t) + jxQ(t)] [hI(t-λ) + jhQ(t-λ)]dλ

ỹ(t) =

dthx )(~

)(~2

1=

dhtx )(~

)(~2

1

= hI(t-λ)xI(t) - hQ(t-λ)xQ(t) + j[xQ(t)hI(t-λ) + hQ(t-λ)xI(t)]dλ

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Equivalent Notation for complex baseband model ( ‘’ = convolution)

ỹ(t) = ½ (x33 (t) h33 (t)) = ½(h33 (t) x33 (t))

• ½ factor added to maintain equivalence between real & complex models

• fc is omitted from complex baseband model simplifies analysis without loss of information

x(t) = Re{x33 (t)exp(2πfct)}

y(t) = Re{ỹ(t)exp(2πfct)}

Passband signals are readily determined from ỹ(t) and x33 (t)

Impulse response of band-pass system given by

h(t) = Re{h33 (t)exp(2πfct)}

= Re{ (hI(t) + jhQ(t)) (cos(2πfct) + jsin(2πfct) ) }

= hI(t)cos(2πfct) - hQ(t) sin(2πfct)

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Appendix: More on Complex Envelope - viewed as an extension of phasor for a real harmonic signal x(t)

x(t) = x cos(2f0t + x) t R

• assume x 0 and phase is 0 x < 2, then:

(i) exp( j(2f0t+x )) = cos(2f0t +x) + jsin(2f0t +x)

= Re [x exp(j(2f0t + x))] t R

= Re [x exp(jx) exp(j2f0t )] t R

(ii) x(t) = Re[x ( cos(2f0t +x) + jsin(2f0t +x) )] t R

• phasor representing phase & magnitude of x(t) = complex envelope:

x exp(jx) = x cos(x) + jx sin(x)

x = magnitude

x = argument (phase of x(t))

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ii. suppress negative frequencies & multiply by 2

iii. shift left by f0 to obtain frequency signal

= x exp(jx)(f0) f R

iv. take Inverse Fourier Transform

i. Take Fourier Transform of x(t)

X(f) = F[x cos(2f0t+ x)] = x exp(jx)(f-f0) + x exp(-jx)(f+f0)2

1

2

1

derive complex envelope for any real continuous signal, x(t)

assume x(t) = Re [xe(t) exp(j2f0t )] t R

where xe(t)= x exp(jx),

x33 e(f)

= x exp(jx)(f-f0) f R x33 p(f)

= xe(t) = x exp(jx) F-1[x33 e(f) ]

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x(t) = cos(2f1t + x) t Re.g. Pure Harmonic signal given by

• if f1 = f0 complex envelope = phasor

• if |f1-f0| << f0 xe varies slowly compared to exp(2jf0t)

where x 0

0 x < 2

i. FT yields X(f) = ½ exp(jx)(f-f1) + ½ exp(-jx)(f+f1)

ii.

iii.

xe(t) = exp(j)exp(2j(f1-f0))t t Riv

= exp(j)(f-f1) x33 p(f)

= exp(j)(f-f1+f0) x33 e(f)

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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 xp with

complex-envelope of x with respect to frequency f0 is signal xe

x33 e(f) = x33 p(f+f0) = 2X(f+f0) 1(f+f0) f R

F[x33 p] = = 2X(f)1(f) f R x33 p(f)

xe(t) = F-1[ x33 e(f) ]

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Complex Envelope for let x(t) = real, band-pass, band-limited signal

• fc = center frequency & W = bandwidth

• where W < fc, are positive real numbers (W << fc x(t) is narrowband)f R• X(f) = 0 for | f | < fc-W and | f | > fc+W

0

W

fc0-fc

WX(f)

xp = analytical

fc0

)(ˆ fx p

xe = complex envelope with respect to f0 • contains only low frequencies• f0 R+ xe is not uniquely defined 0

)(ˆ fxe