Description of FM

Since is not a linear function of , it is, in general, not possible to express in terms of .

Test Tone

The factor

is called the modulation index. The function is periodic with period . Thus, it may be expanded into a Fourier Series:

where

which is the order Bessel function of the first kind. ( is called the ``argument''.) Thus

and

Carrier Power

This can also be determined for the case of sinusoidal modulation by using Parseval's Theorem. The power spectral density of is

By Parseval's Theorem,

FM Bandwidth

For arbitrary , the bandwidth is virtually impossible to determine exactly. For sinusoidal modulation, the bandwidth is theoretically infinite, but for practical purposes, is used. is determined by the value of k for which

so that

Table 3.2 reveals . Thus,

This rule of thumb is called Carsons's Rule in honor of John Carson, a radio engineer who first observed this in the 1940's. For arbitrary , experimental results have shown that the same rule of thumb works well:

where

FM Generation

1. VCO
   A Voltage Controled Oscillator (VCO) is an oscillator whose output frequency is proportional to the input voltage. It really is a frequency modulator. Typically, these devices are not able to produce large frequency deviations.
2. Indirect Modulation
   Narrow Band Frequency Modulation (NBFM) is FM where . In this case

   

   Since , so that

   

   and

   

   This is almost like AM-TC where

   

   Note the differences between NBFM and AM-TC.

   The bandwidth occupied by NBFM modulated carrier is seen to be 2W. Carson's rule also gives this result for small D (or )!

   Block diagram in Figure 3.26 (page 177).

   em Wide Band Frequency Modulation (WBFM) is FM where . WBFM is usually generated by using a frequency multiplier to perform narrow band to wideband conversion. A frequency multiplier multiplies the angle of a sinusoid by N. (Compare this operation to that of a mixer which performs a linear translation of the carrier frequency but does not effect the instantaneous phase or frequency deviation).

   Demodulation (FM Discriminator)

   There are four basic types of FM discriminators:

   1. Ideal FM Discriminator
      The ideal FM discriminator consists of a differentiator followed by an envelope detector as shown in Figure 3.34 (page 190). In this case let so that

      

      Thus,

      

      if which is almost always the case. Removing the DC component yields

      

      If the amplitude of the carrier is perturbed by noise or other interference so that

      

      then the discriminator output is

      

      The carrier amplitude variations cause distortion at the output. This distortion can be removed by passing through a limiter prior to the differentiator as shown in Figure 3.35 (page 191). The output of the limiter is square wave given by

      

      The limiter output must then be converted to a sinusoid by passing it through a band-pass filter with center frequency and sufficient bandwidth to pass the varying fundamental. The result is a constant amplitude sinusoid which is differentiated and passed through the envelope detector to produce the desired signal.

   2. Time-Delay/Phase-Shift Demodulator
      The time derivative of is defined by the following limit:

      

      For small

      

      so that the differentiation process may be approximated by time delay and subtraction operation. This is illustrated in Figure 3.36 (page 192).

      A Phase shift demodulator uses a linear phase shift network (or delay line) to produce the delayed verion of the modulated carrier as shown in Figure . The phase-shift network has a group delay and a carrier delay designed so that

      

      Let the output of the bandpass filter be

       

      The output of the phase shift network is thus

       

      The low pass filter output of the product of () and (

      Now if is small and that the change is is very small during seconds,

      

      Since the phase-shift network produces a quadrature version of the modulated carrier, this demodulator is often called a quadrature detector.

   3. Zero Crossing Detector
      All the message information is contained in the frequency of the carrier not in the amplitude (that is why the limiter is able to remove all amplitude information from the carrier without destroying any of the information content of the carrier). The frequency of the carrier can be measured by observing the zero crossings of the carrier.

      The demodulator is outlined in Figure . The square-wave FM signal from the hard limiter triggers a monostable pulse generator which produces a short pulse of duration at each positive (or negative) zero crossing of the FM square wave. The monostable output is a pulse train with period

      

      During any interval of T seconds where

      

      there are pulses. Continually integrating v(t) over this T second interval yields

      

   4. PLL Discriminator (See Section 3.3 (pages 199 -- 213) of the text.)

     
   Figure: Phase-Shift (Delay-Line) FM Demodulator.
   

     
   Figure: Zero-Crossing FM Demodulator.

   Interference in Angle Modulation

   Let the modulated carrier be given by

   

   where

   

   The interference signal is a sinusoid given by

   

   so that the received signal is

   

   Clearly, for both FM and PM, the carrier and interference powers are

   

   so that the carrier-to-interference ratio is

   

   FM Interference

   The signal-to-interference ratio is derived for the ideal FM discriminator. The received signal is expressed as

   

   where

   

   After passing through the hard-limiter, the received signal is expressed as

   

   Using

   

   for small x, it is seen that for

   

   The outputs of the differentiator and envelope detector are

   

   The first term represents the signal the power of which is

   

   The second term represents a combination of the signal and the interference at the discriminator output. The interference power is determined by observing the power of the interference signal in the presence of an unmodulated carrer. Setting , it is seen that and interference component of is

   

   which is a sinusoid with a frequency dependent amplitude. The interfernece power is

   

   and the signal-to-interference ratio is

   

   PM Interference

   The PM demodulator is effectively an ideal FM demodulator followed by an integrator. Thus

   

   The signal power is

   

   The interference power is determined by calculating the power of the demodulator output when the input is an unmodulated carrer. With

   

   The signal-to-interference ratio is