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

2 Carson’s rule for a Sinusoidal Signal values From above table, N such that satisfies is Bandwidth, N

3 Angle Modulators One method for directly generating an FM signal is to design 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 f c  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 applied 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 shown 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 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)

6 Angle Modulators Another approach for generating an angle-modulated signal is to generate 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- modulated signals are easily generated by modifing conventional AM signal : Conventional AM signal : Narrowband angle-modulated signal Narrowband angle modulator

7 Angle Modulators  Next, we use the narrowband angle-modulated signal to generate 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, nf c, 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 f Lo and we are using a down converter, the final wideband angle-modulated signal is given by Since we can freely choose n and f Lo, we can generate any modulation index at any desired carrier frequency using this method.