1 carson’s rule for a sinusoidal signal angle-modulated signal by a sinusoidal message total...
Post on 28-Dec-2015
213 Views
Preview:
TRANSCRIPT
1
Carson’s rule for a Sinusoidal Signal Angle-modulated signal by a sinusoidal message
Total power,
Power up to Nth harmonic,
Find N such that satisfies
N
nn
cN JJ
AP
1
220
2
)](2)([2
nmcnc
mcc
tnffJA
tftfAtu
)(2cos)(
)2sin2cos()(
2)](2)([
2
2
1
220
2c
nn
cT
AorJJ
AP
98(%) T
N
P
P
2
Carson’s rule for a Sinusoidal Signal values
From above table, N such that satisfies is
Bandwidth,
N 0 1 2 3 4 5 6 7 8 9 10 11
1 58.6 97.3 99.9
5 3.2 24.6 25.1 51.7 82.3 95.9 99.4
10 6.1 6.4 19.4 20.1 29.7 40.7 40.8 50.1 70.3 87.4 96.0 99.0
(%)T
N
P
P
1N
98(%) T
N
P
P
mm fNfW )1(22
3
Angle Modulators One method for directly generating an FM signal is to desig
n an oscillator whose frequency changes with the input voltage => voltage-controlled oscillator (VCO)
What is VCO? When the input voltage is zero, the oscillator generates a sinusoid with
frequency fc When the input voltage changes, this frequency changes accordingly.
An approach to designing VCO is to use a varactor diode. A varactor diode is a capacitor whose capacitance changes with the app
lied voltage. So if this capacitor is used in the tuned circuit of the oscillator and the
message signal is applied to it, the frequency of the tuned circuit and the oscillator changes according to the message signal.
4
Angle Modulators Let us assume that the capacitance of the varactor diode show
n in below is given by
When m(t) = 0, the frequency of the tuned circuit is given by
In general, for nonzero m(t), we have
Assuming that
)()( 00 tmkCtC
002
1
CLfc
)(1
1
)(1
1
2
1
)(
1)(
0
0
0
000000 tmC
kf
tmC
kCLtmkCLtf ci
1)(0
0 tmC
k
Varactor-diode:
5
Angle Modulators And using the approximations
Hence,
Another approach to designing VCO is to use a reactance tube.
In the reactance-tube implementation, an inductor whose inductance
varies with the applied voltage is employed
PM signals can be generated by relation between FM and PM
signals (that is, PM is serial connection of differentiator and FM)
211
11
1and
21
1
1
)(
21)(
0
0 tmC
kftf ci
6
Angle Modulators Another approach for generating an angle-modulated signal is to gen
erate a narrowband angle-modulated signal and then change it to a wideband signal. => Indirect method
Design of Indirect method Due to the similarity with conventional AM signals, the narrowband angle-mod
ulated signals are easily generated by modifing conventional AM signal
: Conventional AM signal
: Narrowband angle-modulated signal
)2cos()](1[ tftamA cc
Narrowband angle modulator
tftAtfA cccc 2sin)(2cos
7
Angle Modulators Next, we use the narrowband angle-modulated signal to gen
erate a wideband angle-modulated signal.
The narrowband angle-modulated signal enters a frequency multiplier which multiplies the instantaneous frequency of the input by some constant n.
This is usually done by applying the input signal to a nonlinear element and then passing its output through a bandpass filter tuned to the desired central frequency.
Indirect Angle modulator
8
Angle Modulators If the narrowband modulated signal is represented by
the output of the frequency multiplier is given by
This is a wideband angle-modulated signal. However, the carrier frequency of this signal, nfc, may not be the desired carrier frequency.
In the last stage, the modulator performs an up/down conversion to shift the modulated signal to the desired center frequency.
This stage consists of a mixer and a bandpass filter. If the frequency of the local oscillator of the mixer is fLo and we are us
ing a down converter, the final wideband angle-modulated signal is given by
Since we can freely choose n and fLo, we can generate any modulation index at any desired carrier frequency using this method.
)(2cos)( ttfAtu ccn
)(2cos)( tntnfAty cc
)()(2cos)( tntfnfAtu LOcc
top related