Calculate the approximate bit rate and signal levels for a 4

Calculate the approximate bit rate and signal level(s) for a 4.2 MHz bandwidth system
with a signal to noise ratio of 170.

Solution

FM Theory

Recall that a general sinusoid is of the form:

Frequency modulation involves deviating a carrier frequency by some amount. If a sine wave was used to frequency modulate a carrier, the mathematical expression would be:

where

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

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

The term w is an angular velocity and it is related to frequency and angle by the following relationship:

To find the angle, we must integrate w with respect to time, we obtain:

We can now find the instantaneous angle associated with an instantaneous frequency:

This angle can now be substituted into the general carrier signal to define FM:

The FM modulation index is defined as the ratio of carrier deviation to modulation frequency:

As a result, the FM equation is generally written as:

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 knowledge of Bessel’s functions of the first kind and order p. In open form, it resembles:

where

As a point of interest, Bessel’s functions are a solution to the following equation:

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. Each frequency is an integer multiple of the modulation signal. It should be noted that the amplitude of the higher order sided frequencies drops off quickly.

It is also interesting to note that the amplitude of the carrier signal is also a function of the modulation index. Under some conditions, the amplitude of the carrier frequency 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.