Transcript Linearity - The University of Texas at Austin

EE445S Real-Time Digital Signal Processing Lab
Fall 2011
Analog Sinusoidal Modulation
Prof. Brian L. Evans
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
Lecture 19
Outline
• Introduction
• Amplitude sinusoidal modulation
Double-sideband large carrier
Quadrature amplitude modulation
Other amplitude modulation types
• Frequency modulation
• Conclusion
19 - 2
Example: Radio Frequency Modem
• Message signal: stream of bits
• Digital sinusoidal modulation in digital signaling
• Analog sinusoidal modulation in carrier circuits
for upconversion to radio frequencies (RF)
Error
Correction
m[k ]
Signal
Processing
Digital
Signaling
Carrier
Circuits
TRANSMITTER
D/A
Converter
Transmission
Medium
s(t)
CHANNEL
Carrier
Circuits
r(t)
Signal
Processing mˆ [ k ]
RECEIVER
19 - 3
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
y(t )  f (t ) cos( 2  f c t   (t ))
f(t) is real-valued amplitude function
fc is carrier frequency
(t) is real-valued phase function
• See Modulation handout (Appendix I)
19 - 4
Amplitude Modulation
• By cosine
y t   f t  cosc t 
1
Y   
F   
2
    c      c 
• Fourier property
Y   
1
1
F   c   F   c 
2
2
• Spectrum F() is
Shifted left by c and
scaled by ½ and
Shifted right by c and
scaled by ½
• By sine
y t   f t sin c t 
1
Y   
F   
2
j    c      c 
• Fourier property
Y   
j
j
F   c   F   c 
2
2
• Spectrum F() is
Shifted left by c and
scaled by j/2 and
Shifted right by c and
19 - 5
scaled by –j/2
Amplitude Modulation by Cosine
• Example: y(t) = f(t) cos(c t)
Assume f(t) is an ideal lowpass signal with bandwidth 1
Assume 1 << c
lower sidebands
F()
½F  c
1
Y()
½F  c
½
-1
0
1

-c - 1
c
-c + 1
0
c - 1
c
c + 1

Y() is real-valued if F() is real-valued
• Demodulation: modulation then lowpass filtering
• Similar derivation for modulation with sin(c t)
19 - 6
Amplitude Modulation by Sine
• Example: y(t) = f(t) sin(c t)
Assume f(t) is an ideal lowpass signal with bandwidth 1
Assume 1 << c
lower sidebands
F()
j ½F  c
1
Y()
j
-1
0
1

-j ½F  c
½
c - 1
-c - 1
c
c
c + 1
-c + 1
-j
½
Y() is imaginary-valued if F() is real-valued
• Demodulation: modulation then lowpass filtering
19 - 7

Amplitude Modulated (AM) Radio
• Double sideband large carrier (DSC-LC)
Carrier wave varied about mean value linearly with
baseband message signal m(t)
y (t )  Ac 1  k a m(t )  cos( 2  f c t )
 Ac cos( 2  f c t )  Ac k a m(t ) cos( 2  f c t )
ka is the amplitude sensitivity, ka > 0
Modulation factor is  = ka Am where Am is maximum
amplitude of m(t)
• Envelope of s(t) has about same shape as m(t) if
| ka m(t) | < 1 for all t
fc >> W where W is bandwidth of m(t)
19 - 8
Amplitude Modulated (AM) Radio
• 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
Rs
resistance Rf ) in series
Parallel connection of
+
capacitor C and load vs(t)
–
resistor Rl
Rf
C
Rl
19 - 9
Amplitude Modulated (AM) Radio
• Let Rs be source resistance
• Charging time constant (Rf + Rs) C must be short
when compared to 1/ fc, so (Rf +Rs) C << 1/ fc
• Discharging time constant Rl C
Long enough so that capacitor discharges slowly through load
resistor Rl between positive peaks of carrier wave
Not so long that capacitor voltage will not discharge at max
rate of change of modulating wave 1/fc << Rl C << 1/W
19 - 10
Quadrature Amplitude Modulation
• Allows DSB-SC signals to occupy same channel
bandwidth provided that the two message signals
are from independent sources
y(t )  Ac m1 (t ) cos(2  f c t )  Ac m2 (t ) sin( 2  f c t )
 A(t ) cos(2  f c t   (t ))
 m2 (t ) 
2
2

A(t )  Ac m1 (t )  m2 (t )
 (t )  arctan  
 m1 (t ) 
• Two message signals m1(t) and m2(t) are sent
Ac m1(t) is in-phase component of s(t)
Ac m2(t) is quadrature component of s(t)
19 - 11
Other Amplitude Modulation Types
• Double sideband suppressed carrier (DSB-SC)
s(t )  Ac m(t ) cos(2  f c t )
• Double sideband variable carrier (DSB-VC)
s(t )  Ac m(t ) cos(2  f c t )   cos(2  f c t )
• Single sideband (SSB) removes either lower
sideband or upper sideband by
Extremely sharp bandpass or highpass filter, or
Phase shifters using Hilbert transformer (slides 15-7 to 15-10)
19 - 12
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
Carrier
Circuits
r(t)
Signal
Processing mˆ (t )
RECEIVER
19 - 13
Frequency Modulation
• Non-linear, time-varying, has memory, non-causal
t


s (t )  Ac cos i (t )   Ac cos 2  f c t  2  k f  m(t ) dt 
0


• For single tone message m(t) = Am cos(2  fm t)
f
 i (t )  2  f c t 
sin( 2  f m t ) where f  k f Am
fm
1 d
Instantaneous
f i (t ) 
 i (t )  f c  f cos( 2  f m t )
frequency
2 dt
• Modulation index is  = f / fm
 << 1 => Narrowband FM (looks like double-sideband AM)
 >> 1 => Broadband FM
19 - 14
Carson's Rule
• Bandwidth of FM for single-tone message at fm
Narrowband:
Wideband:
BT  2 f m
BT  2f
• Carson’s rule for single-tone FM: BT  2 f m (1   )
FM Radio
f
fm

Peak freq. deviation (F)
Peak message freq. (W)
Deviation ratio (D)
Bandwidth BT = 2 fm (1+ )
Station Spacing
75 kHz
15 kHz
5
180 kHz
200 kHz
TV Audio
25 kHz
15 kHz
1.66
80 kHz
6 MHz
• For message signal of bandwidth W, let fm = W19 - 15
Conclusion
• Amplitude modulation
Digital and analog versions may be used in same system
Analog amplitude modulation is one method for upconversion
• Double sideband amplitude modulation
Transmission bandwidth is twice message bandwidth (wasteful)
• Quadrature amplitude modulation
Uses cosine and sine to modulate two different message signals
and subtracts resulting waveforms
Two messages in same transmission bandwidth (efficient)
19 - 16