I. Phasors (complex envelope) representation for

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Band Pass Systems, Phasors and Complex
Representation of Systems
KEY LEARNING OBJECTIVES

• I. Phasors (complex envelope) representation for
• sinusoidal signal
• narrow band signal

II. Complex Representation of Linear Modulated
Signals Bandpass System

• Phasors and Complex Representation are useful for
  analyzing
• baseband component of a signal
• eliminates high frequency carrier components

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I. Phasors for monochromatic narrow band signals

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

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

• x(t) Acos(2pf0 t ?)
• xq(t) Asin(2pf0 t ?)
• quadrature component shifted 90o from x(t)

(i) define a signal z(t) as a vector rotating
with angular frequency 2pf0
z(t) Aexp(j(2pf0t ?))
Acos(2pf0t ?) jAsin(2pf0t ?) x(t)
jxq(t)
(ii) obtain phasor X from z(t) by eliminating
2pf0 rotation - rotate z(t) at an angular
frequency 2pf0 in opposite direction -
equivalent to multiplying z(t) by exp(2pf0t)
X z(t) exp(-j2pf0t ) Aexp(j(2pf0t
?))exp(-j2pf0t ) Aexp(j?)
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1a. determine Frequency Domain equivalent of
z(t) and X
(i) obtain Z(f), using either or two methods
(1) determine X(f) Fx(t), delete negative
frequencies multiply by 2
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z(t) is known as the analytic signal or
pre-envelope of x(t)

• find z(t) using IFT ? find signal whose Fourier
  transform u-1(f)

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pre-envelope for two types of signals
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Xl(f) Z(f f0) 2u-1(f f0)X(f f0) xl(t)
z(t)exp(-j2pf0t)

• xl(t) is a low pass signal
• Xl(f) 0 for all f W
• phasor for band pass signal

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Generally xl(t) is complex signal with real (in
phase) imaginary (quadrature) components
xl(t) xc(t) jxs(t)
bandpass to lowpass transform describes
relationship of x(t) in terms of xc(t)
xs(t)
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Define xl(t) in terms of phase envelope
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II. Complex Representation of Linear Modulated
Signals Bandpass System
canonical representation of any bandpass signal,
s(t) has 2 components
s(t) sI(t)cos(2pfct) - sQ(t)sin(2pfct)

• 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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2. Consider a narrowband linear band-pass
system

• system is narrowband if bandwidth W ltlt fc ,
  the systems center
• frequency
• input x(t) is modulated by carrier, fc
• output y(t)

canonical representation of systems impulse
response given by
h(t) hI(t)cos(2pfct) - hQ(t)sin(2pfct)
use equivalent complex baseband model to
simplify analysis

• impulse response given by

h(t) hI(t) jhQ(t)
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2.1 Passband Analysis of LTI System
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Passband Analysis of LTI System (continued)
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2.2 Equivalent Complex Baseband Model

• complex envelopes are related by complex
  convolution

xI(t) jxQ(t) hI(t-?) jhQ(t-?)d?
hI(t-?)xI(t) - hQ(t-?)xQ(t) jxQ(t)hI(t-?)
hQ(t-?)xI(t)d?
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Equivalent Notation for complex baseband model (
? convolution)
?(t) ½ (x(t)? h(t)) ½(h(t) ? x(t))

• ½ factor added to maintain equivalence between
  real complex models
• fc is omitted from complex baseband model ?
  simplifies analysis
• without loss of information

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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(2?f0t ?x) t ? R

• assume ?x ? 0 and phase is 0 ? ?x lt 2?, then

(i) exp( j(2?f0t?x )) cos(2?f0t ?x)
jsin(2?f0t ?x)
(ii) x(t) Re?x ( cos(2?f0t ?x) jsin(2?f0t
?x) ) t ? R
Re ?x exp(j(2?f0t ?x))
t ? R
Re ?x exp(j?x) exp(j2?f0t )
t ? R
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i. Take Fourier Transform of x(t)
ii. suppress negative frequencies multiply by 2
iii. shift left by f0 to obtain frequency signal
?x exp(j?x)?(f0)
xe(f)
f ? R
iv. take Inverse Fourier Transform
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where ?x ? 0 0 ? ?x lt 2?
i. FT yields X(f) ½ ? exp(j?x)?(f-f1) ½ ?
exp(-j?x)?(ff1)
ii.
iii.
iv
xe(t) ? exp(j?)exp(2j?(f1-f0))t t ? R

• if f1 f0 ? complex envelope phasor
• if f1-f0 ltlt f0 ? xe varies slowly compared to
  exp(2j?f0t)

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xe(t) F-1 xe(f)
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xp analytical