Transcript Linearity
EE345S Real-Time Digital Signal Processing Lab Spring 2006
Analog Sinusoidal Modulation
Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin
Lecture 9
Single-Carrier Modulation Methods
•
Analog communication
Transmit/receive analog waveforms Amplitude Modulation (AM) Freq. Modulation (FM) Phase Modulation (PM) Quadrature Amplitude Mod.
Pulse Amplitude Modulation
Digital communication
Same but treat transmission and reception as digitized Amplitude Shift Keying (ASK) Freq. Shift Keying (FSK) Phase Shift Keying (PSK) QAM PAM
m
(
t
)
Signal Processing Carrier Circuits TRANSMITTER Transmission Medium
s
(
t
)
CHANNEL
r
(
t
)
Carrier Circuits Signal Processing RECEIVER
9 - 2
m
(
t
)
Radio Frequency (RF) Modem
• • •
Message signal: stream of bits Digital sinusoidal modulation in digital signaling Analog sinusoidal modulation in carrier circuits for upconversion to RF Error Correction Digital Signaling D/A Converter
m
[
k
]
Signal Processing Carrier Circuits TRANSMITTER Transmission Medium
s
(
t
)
CHANNEL
r
(
t
)
Carrier Circuits Signal Processing RECEIVER
9 - 3 ˆ [
k
]
Modulation
• •
Modulation: some characteristic of a carrier signal is varied in accordance with a modulating signal For amplitude, frequency, and phase modulation, modulated signals can be expressed as
s
(
t
)
A
(
t
) cos( 2
f c t
(
t
))
A
(
t
) is real-valued amplitude function
f c
is carrier frequency (
t
) is real-valued phase function 9 - 4
Review
Amplitude Modulation by Cosine
• • •
Multiplication in time: convolution in Fourier domain (let
0 = 2
f
0 ):
y
f t
0
t Y
1 2
F
0
0
Sifting property of Dirac delta functional
x
t
d
x
x
t
t
0
t
0
t
d
x
t
t
0
Fourier transform property for modulation by a cosine
Y
1 2
F
0
1 2
F
0
9 - 5
Review
Amplitude Modulation by Cosine
•
Example:
y
(
t
) =
f
(
t
) cos( 0
t
) Assume
f
(
t
) is an ideal lowpass signal with bandwidth 1 Assume 1 << 0
lower sidebands
F
( ) 1
½F
0
Y
( )
½ ½F
0 • • 1 0 1 0 1 0 0 + 1 0
Y
( ) is real-valued if
F
( ) is real-valued 0 1 0 0 + 1
Demodulation: modulation then lowpass filtering Similar derivation for modulation with
sin( 0 t) 9 - 6
Review
Amplitude Modulation by Sine
• • •
Multiplication in time is convolution in Fourier domain
Y y
f
0
t
1 2
F
j
0
0
Sifting property of the Dirac delta functional
x
x
t
t
0
x
t
t
0
d
t
x
d
x
t
t
0
Fourier transform property for modulation by a sine
Y
j
2
F
0
2
j F
0
9 - 7
Review
Amplitude Modulation by Sine
•
Example:
y
(
t
) =
f
(
t
) sin( 0
t
) Assume
f
(
t
) is an ideal lowpass signal with bandwidth 1 Assume 1 << 0
lower sidebands
F
( ) 1
Y
( )
j ½ -j ½F
0 0 1 0 0 + 1 1 0 1
j ½F
0 0 1 0 0 + 1
-j ½
•
Y
( ) is imaginary-valued if
F
( ) is real-valued
Demodulation: modulation then lowpass filtering
9 - 8
Amplitude Modulated (AM) Radio
• •
Double sideband large carrier (DSC-LC)
Carrier wave varied about mean value linearly with baseband message signal
m
(
t
)
s
(
t
)
A c A c
1 cos(
k m a
2 (
t f
)
c
cos( ) 2
k a f m c
(
t t
) ) cos( 2
f c t
)
k a
is the amplitude sensitivity,
k a
Modulation factor is =
k a A m
> 0 where
A m
amplitude of
m
(
t
) is maximum
Envelope of s(t) has about same shape as m(t) if
|
k a m
(
t
) | < 1 for all
t f c
>>
W
where
W
is bandwidth of
m
(
t
) 9 - 9
Amplitude Modulation
• • •
Disadvantages
– Redundant bandwidth is used – Carrier consumes most of the transmitted power
Advantage
– Simple detectors (e.g. AM radio receivers for cars)
Receiver uses a simple envelope detector
– Diode (with forward resistance
R f
) in series – Parallel connection of capacitor
C
and load resistor
R l R s
+
v s
(
t
) –
R f C
9 - 10
R l
Amplitude Modulation (con’t)
• • •
Let R
s
be source resistance Charging time constant (R
f
+ R
s
) C must be short when compared to 1/ f
c
, so (R
f
+R
s
) C << 1/ f
c
Discharging time constant R
l C
– Long enough so that capacitor discharges slowly through load resistor
R l
between positive peaks of carrier wave – Not so long that capacitor voltage will not discharge at max rate of change of modulating wave 1/
f c
<<
R l C
<< 1/
W
9 - 11
Other Amplitude Modulation Types
• • •
Double sideband suppressed carrier (DSB-SC)
s
(
t
)
A c m
(
t
) cos( 2
f c t
)
Double sideband variable carrier (DSB-VC)
s
(
t
)
A c m
(
t
) cos( 2
f c t
) cos( 2
f c t
)
Single sideband (SSB): Remove either lower sideband or upper sideband by
– Extremely sharp bandpass or highpass filter, or – Phase shifters using a Hilbert transformer 9 - 12
Quadrature Amplitude Modulation
• •
Allows DSB-SC signals to occupy same channel bandwidth provided that the two message signals are from independent sources
s
(
t
)
A c m
1 (
t A
(
t
) ) cos( cos( 2 2
f c t f c t
)
(
t A
))
c m
2 (
t
) sin( 2
f c t
)
A
(
t
)
A c m
1 2 (
t
)
m
2 2 (
t
) (
t
) arctan
m
2
m
1 ( (
t
)
t
)
Two message signals m
1
(t) and m
2
(t) are sent
A c m 1
(
t
) is in-phase component of
s
(
t
)
A c m 2
(
t
) is quadrature component of
s
(
t
) 9 - 13
Frequency Modulated (FM) Radio
• • •
Message signal: analog audio signal Transmitter
– Signal processing: lowpass filter to reject above 15 kHz – Carrier circuits: sinusoidal modulatation from baseband to FM station frequency (often in two modulation steps)
Receiver
– Carrier circuits: sinusoidal demodulation from FM station frequency to baseband (often in two demodulation steps) – Signal processing: lowpass filter to reject above 15 kHz
m
(
t
)
Signal Processing Carrier Circuits TRANSMITTER Transmission Medium
s
(
t
)
CHANNEL
r
(
t
)
Carrier Circuits Signal Processing RECEIVER
9 - 14
m
(
t
)
Frequency Modulation
• • •
Non-linear, time-varying, has memory, non-causal
s
(
t
)
A c
cos
i
(
t
)
A c
cos 2
f c t
2
k f
t m
(
t
)
dt
0
For single tone message m(t) = A
m
i
(
t
) 2
f c t
f f m
sin( 2
f m t
cos(2
) where
f
f m
t)
k f A m f i
(
t
) 1 2
d dt
i
(
t
)
f c
f
cos( 2
f m t
) Instantaneous frequency
Modulation index is
=
f / f
m
<< 1 => Narrowband FM (looks like double-sideband AM) >> 1 => Broadband FM 9 - 15
Carson's Rule
• •
Bandwidth of FM for single-tone message at f
m
– Narrowband:
B T
2
f m
– Wideband:
B T
2
f
Carson’s rule for single-tone FM:
B T
2
f m
( 1
)
f f m
Peak freq. deviation (
F) Peak message freq. (W) Deviation ratio (D) Bandwidth B
T
= 2 f
m
(1+
) Station Spacing FM Radio 75 kHz 15 kHz 5 180 kHz 200 kHz TV Audio 25 kHz 15 kHz 1.66
80 kHz 6 MHz
•
For a general message signal, f
m
= W
9 - 16
Summary
•
General form of modulation:
Modulation
DSB-LC DSB-SC DSB-VC SSB
A c
A(t)
A c
[ 1
k a m
(
t
)]
A c m
(
t
)
A c m
(
t
)
m
2 (
t
) [
m
(
t
)
h
(
t
)] 2
(t)
arctan
0
0
0
m
(
t
)
m
(
t h
(
t
) )
s
(
t
)
A
(
t
) cos( 2
f c t
(
t
))
Carrier
Yes No Yes No
Type
Amp.
Amp.
Amp.
Amp.
Use
AM Radio Marine Radios QAM
A c m
1 2 (
t
)
m
2 2 (
t
) arctan
m m
1 2 ( (
t
)
t
)
No Hybrid Satellite comm.
Phase Frequency
A c A c
0 2
k f
k p m
(
t
)
t
0
m
(
t
)
dt
No No Angle Angle FM Radio
h(t)
is the impulse response of a bandpass filter or phase shifter to effect a cancellation of one pair of redundant sidebands.
9 - 17
Optional
Angle Modulation
• • • •
Angle modulation general form
s
(
t
)
A c
cos(
i
(
t
))
A c
is constant carrier amplitude
i
(
t
) is instantaneous angle of modulation in radians
Average frequency in rad/s over interval
t
f
t
(
t
)
i
(
t
2
t
)
t
1
d
Instantaneous frequency in rad/s
f i
(
t
) lim
t
0
f
t
(
t
) 2
dt
Instantaneous angle
i
(
t
) 2
f i
Phase modulation
i
(
t
) 2
f c t
k
i
(
t p
(
t
(
t
)
dt
) )
m
(
t
) Frequency modulation
i
(
t
) 2
f c t
2
k f
0
t m
(
t
)
dt
9 - 18