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