Unformatted text preview: actice the propagation delay is unknown and time varying, and the
transmitter and receiver phase can drift with respect to each other. EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 8 Consider the case where the transmitter has a phase of −π /2, so that the
modulated signal is
m(t) cos(ωct − π /2) = m(t) sin(ωct).
The spectrum of the transmitted signal is now
j
j
F [m(t) sin(ωct)] = M (j (ω + ωc)) − M (j (ω − ωc)).
2
2
If we demodulate with a cosine, the result is shown in the next plot: EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 9 j
M ( j(! + !c))
2
−!c 0 !c !
j
− M ( j(! − !c))
2 ∗1 2! !"(# − #c) !"(# + #c)
−!c 0 −!c ! j
− M ( j(! − 2!c))
4
!c
2!c ! = j
M ( j(! + 2!c))
4
−2!c !c 0 The baseband signal we want cancels!
EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 10 One solution is to demodulate with a complex exponential
ej ωct ⇔ 2πδ (ω − ωc)
j
M ( j(! + !c))
2
−!c !c j
!
− M ( j(! − !c))
2 0 ∗1 2! 2!"(# − #c)
−!c !c 0 ! = j
M ( j!)
2
−!c
0 Lowpass Filter j
− M ( j(! − 2!c))
2
!c
2!c
! EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 11 This also works for an arbitrary phase shift.
If the input has a phase shift of −θ, the spectrum of the carrier is
F [cos(ωct − θ)] = F [cos(ωct) cos θ + sin(ωct) sin θ]
= π cos θ [δ (ω + ωc) + δ (ω − ωc)] +π sin θ [j δ (ω + ωc) − j δ (ω − ωc)]
jθ
= π e δ (ω + ωc) + e−j θ δ (ω − ωc) and the spectrum of the modulated signal is
F [m(t) cos(ωct − θ)] = ej θ
e−j θ
M (j (ω + ωc)) +
M (j (ω − ωc))
2
2 Demodulating with a complex exponential can be plotted as: EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 12 e j!
M ( j(" + "c))
2
−!c !c e− j!
!
M ( j(" − "c))
2 0 ∗1 2! 2!"(# − #c)
−!c ! =
Lowpass Filter e j!
M ( j")
2
−!c The lowpass ﬁlter extracts
ej θ
2 m(t). !c 0 e− j!
M ( j(" − 2"c))
2
!c
2!c
! 0 ej θ
2 M (j ω ) corresponding to the complex signal EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 13 The system now looks like
m(t ) × Propagation m(t ) cos(!ct − ") cos(!ct − ")
Transmitter × j!
2 H (j ω ) m(t )e
ω e j!ct
Receiver This is a quadrature receiver, common in radar, sonar, ultrasound, and MRI
systems. Often m(t) is a simple pulse, and the interesting information is in
θ, such as doppler shift for weather radar.
Also common in communications systems. You can buy a digital chip that
implements a quadrature receiver for your cell phone.
Cost is that the receiver has to be implemented for complex signals. This is
done the way you do it, by keeping track of two real signals, the real part
and the imaginary part (the I and Q channels).
EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 14 Commercial AM Modulation
The DSBSC systems we’ve been looking at have a problem with
s...
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This note was uploaded on 09/28/2013 for the course EE 108b taught by Professor Mukamel during the Fall '13 term at Singapore Stanford Partnership.
 Fall '13
 Mukamel
 Signal Processing

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