# Communication Systems/Frequency Modulation

## Frequency Modulation

If we make the frequency of our carrier wave a function of time, we can get a generalized function that looks like this:

$s_{FM} = A\cos (2 \pi [f_c + ks(t)]t + \phi)$

We still have a carrier wave, but now we have the value ks(t) that we add to that carrier wave, to send our data.

As an important result, ks(t) must be less than the carrier frequency always, to avoid ambiguity and distortion.

### Deriving the FM Equation

Recall that a general sinusoid is of the form:

$e_c = \sin \left( {\omega _c t + \theta } \right)$

Frequency modulation involves deviating a carrier frequency by some amount. If a sine wave is used to deviate the carrier, the expression for the frequency at any instant would be:

$\omega _i = \omega _c + \Delta \omega \sin \left( {\omega _m t} \right)$
where:
$\omega _i =$instantaneous frequency
$\omega _c =$carrier frequency
$\Delta \omega =$carrier deviation
$\omega _m =$ modulation frequency

This expression describes a signal varying sinusoidally about some average frequency. However, we cannot simply substitute this expression into the general equation for a sinusoid to get the FM equation. This is because the sine operator acts on angles, not frequency. Therefore, we must define the instantaneous frequency in terms of angles.

It should be noted that the modulation signal amplitude governs the amount of carrier deviation while the modulation signal frequency governs the rate of carrier deviation.

The term $\omega$ is an angular velocity (radians per second) and is related to frequency and angle by the following relationship:

$\omega {\rm{ = 2}}\pi {\rm{f = }}\frac{{d\theta }}{{dt}}$

To find the angle, we must integrate $\omega$ with respect to time:
$\int {\omega dt = \theta }$

We can now find the instantaneous angle associated with the instantaneous frequency:
$\begin{array}{l} \theta = \int {\omega _i dt = \int {\left( {\omega _c + \Delta \omega \sin \left( {\omega _m t} \right)} \right)} } dt = \omega _c t - \frac{{\Delta \omega }}{{\omega _m }}\cos \left( {\omega _m t} \right) = \omega _c t - \frac{{\Delta f}}{{f_m }}\cos \left( {\omega _m t} \right) \\ \end{array}$

This angle can now be substituted into the general carrier signal to define FM:
$e_{fm} = \sin \left( {\omega _c t - \frac{{\Delta f}}{{f_m }}\cos \left( {\omega _m t} \right)} \right)$

The FM modulation index is defined as the ratio of the carrier deviation to modulation frequency:
$m_{fm} = \frac{{\Delta f}}{{f_m }}$

Consequently, the FM equation is often written as:
$e_{fm} = \sin \left( {\omega _c t - m_{fm} \cos \left( {\omega _m t} \right)} \right)$

### Bessel's Functions

This is a very complex expression and it is not readily apparent what the sidebands of this signal are like. The solution to this problem requires a knowledge of Bessel's functions of the first kind and order p. In open form, it resembles:

$J_p \left( x \right) = \sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k \left( {\frac{x}{2}} \right)^{2k + p} }}{{k!\left( {k + p} \right)!}}}$
where:
$J_p \left( x \right) =$ Magnitude of the frequency component
$p =$ Side frequency number (not to be confused with sidebands)
$x =$ Modulation index

As a point of interest, Bessel's functions are a solution to the following equation:
$x^2\frac{{d^2 y}}{{dx^2 }} + x\frac{{dy}}{{dx}} + \left( {x^2 - p^2 } \right) = 0$

Bessel's functions occur in the theory of cylindrical and spherical waves, much like sine waves occur in the theory of plane waves.

It turns out that FM generates an infinite number of side frequencies (in both the upper and lower sidebands). Each side frequency is an integer multiple of the modulation signal frequency. The amplitude of higher order side frequencies decreases rapidly and can generally be ignored.

The amplitude of the carrier signal is also a function of the modulation index and under some conditions, its amplitude can actually go to zero. This does not mean that the signal disappears, but rather that all of the broadcast energy is redistributed to the side frequencies.

A plot of the carrier and first five side frequency amplitudes as a function of modulation index resembles:

The Bessel coefficients have several interesting properties including:

$J_0^2 + 2\left( {J_1^2 + J_2^2 + J_3^2 + \cdots } \right) = 1$

One very useful interpretation of this is: $J_0$ represents the voltage amplitude of the carrier, $J_1$ represents the amplitude of the 1st side frequency, $J_2$ the 2nd side frequency etc. Note that the sum of the squares (power) remains constant.

### FM Bandwidth

FM generates upper and lower sidebands, each of which contain an infinite number of side frequencies. However, the FM bandwidth is not infinite because the amplitude of the higher order side frequencies decreases rapidly. Carson's Rule is often used to calculate the bandwidth, since it contains more than 90% of the FM signal.

Carson's Rule
$B_{fm} \approx 2\left( {m_{fm} + 1} \right)f_m = 2\left( {\Delta f + f_m } \right))$

In commercial broadcast applications, the maximum modulation index ($m_{fm}$) = 5, the maximum, carrier deviation ($\Delta f$) = 75 kHz, and maximum modulation frequency ($f_m$) = 15 kHz. The total broadcast spectrum according to Carson's rule is 180 kHz, but an additional 20 kHz guard band is used to separate adjacent radio stations. Therefore, each FM radio station is allocated 200 kHz.

### Noise

In AM systems, noise easily distorts the transmitted signal however, in FM systems any added noise must create a frequency deviation in order to be perceptible.

The maximum frequency deviation due to random noise occurs when the noise is at right angles to the resultant signal. In the worst case the signal frequency has been deviated by:

$\delta = \theta f_m$

This shows that the deviation due to noise increases as the modulation frequency increases. Since noise power is the square of the noise voltage, the signal to noise ratio can significantly degrade.

To prevent this, the amplitude of the modulation signal is increased to keep the S/N ratio constant over the entire broadcast band. This is called pre-emphasis.

### Pre & De-emphasis

Increasing the amplitude of high frequency baseband signals in the FM modulator (transmitter) must be compensated for in the FM demodulator (receiver) otherwise the signal would sound quite tinny (too much treble).

The standard curves resemble:

In commercial FM broadcast, the emphasis circuits consist of a simple RC network with a time constant of 75 $\mu$Sec and a corner frequency of 2125 Hz.

The magnitude of the pre-emphasis response is defined by:

## FM Transmission Power

The equation for the transmitted power in a sinusoid is a fundamental equation. Remember it.

Since the value of the amplitude of the sine wave in FM does not change, the transmitted power is a constant. As a general rule, for a sinusoid with a constant amplitude, the transmitted power can be found as follows:

$P(t) = \frac{A^2}{2R_L}$

Where A is the amplitude of the sine wave, and RL is the resistance of the load. In a normalized system, we set RL to 1.

The Bessel coefficients can be used to determine the power in the carrier and any side frequency:

$P_C = P_T \left( {J_0^2 + 2\left( {J_1^2 + J_2^2 + J_3^2 + \cdots } \right)} \right)$
$P_C$ is the power in the unmodulated carrier.
$P_T$ is the total power and is by definition equal to the unmodulated carrier power.

As the modulation index varies, the individual Bessel coefficients change and power is redistributed from the carrier to the side frequencies.

## FM Transmitters

FM Transmitters can be easily implemented using a VCO (see why we discussed Voltage Controlled Oscillators, in the first section?), because a VCO converts an input voltage (our input signal) to a frequency (our modulated output).

Signal ----->|VCO|-----> FM Signal


Any angle modulation receiver needs to have several components:

1. A limiter, to remove abnormal amplitude values
2. bandpass filter, to separate the out-of-band noise.
3. A Discriminator, to change a frequency back to a voltage
4. A lowpass filter, to remove noise added by the discriminator.

A discriminator is essentially a differentiator in line with an envelope detector:

FM ---->|Differentiator|---->|Envelope Filter|----> Signal


Also, you can add in a blocking capacitor to remove any DC components of the signal, if needed.