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