Signal Bandwidth

For this month we wanted to explore how much bandwidth is taken up by a Ham Band FM signal as compared to SSB and CW. As stated in last months RMVHF+ Newsletter, most handy toys come out of the factory with the deviation set to some place between 3 and 5 kHz. We assume out highest modulating frequency to be 3 kHz as the human audio voice range is considered between 300 and 3000 Hz. There are several acceptable ways of calculating Occupied Bandwidth and who is to say who is correct; however, when filing for a government controlled frequency on which FM is to be used their suggestion is to use twice the deviation plus twice the highest modulating frequency (2D + 2F). Applying this to a ham transceiver running ‘belly band’ (3 kHz), we’d have 6 + 6 kHz or an occupied bandwidth of 12 kHz. The ARRL on the other hand suggests two times one half the total frequency deviation plus the maximum audio frequency. So, for ‘belly band’ their method (2 X 6/2 + 3) indicates an occupied bandwidth of 9 kHz. So who is right? Throughout many years in engineering and research I have always found that when in doubt go back to the basics. In this case the basics involve mathematics, in a scientific hobby or occupation you can’t live without it (mathematics). Let’s first look at the equation for the instantaneous voltage (e) of an FM signal;

e = A sin(wct + d/fm sin wmt)

Right in the middle of the equation is the expression (d/fm) “the modulation index” that will determine the width of an FM signal. The d or sigma is the maximum frequency deviation divided by the fm the highest modulating frequency. For “belly band” the modulation index is found by dividing the total deviation by the modulating frequency, (6/3) or a modulation index of 2. Great what do we do with this? Well there is a chart used in FM systems engineering known as the Bessel Functions. This is used to determine the amplitude and the number of harmonics (or sidebands) contained in an FM signal of various modulation indices. We have a modulation index of 2 in our example. Going to the chart of Bessel Functions we find that a modulation index of 2 will produce four significant sidebands. Now this is only on one side of the deviation swing. The signal swings an equal amount on the other side of center or resting frequency so we have a total of eight significant sidebands or a multiple of four times the frequency deviation (3 kHz) which amounts to an occupied bandwidth of 12 kHz. This is equal to the expression used by the government of (OBW = 2D + 2F). Now as we can see if we were to increase our deviation to let’s say 5 kHz our OBW goes out to 16 kHz. Still not as bad as the old 15 kHz deviation systems where the OBW was 36 kHz. Sure it sounds good but we don’t need fidelity in a communications system.

Now how about an SSB signal? Properly modulated (mic gain set so that the fluctuations on the ALC meter are within the suggested ALC range) the signal should be no more than 3 kHz wide. Many SSB rigs have a 2.4 or 2.8 kHz filter system so that some of the highs are attenuated because they are not needed in a communications system, the voice power is contained in the lower frequencies anyway.

A CW signal requires much less bandwidth than any other communication mode. We find the bandwidth of a CW signal by the equation, BW = B X K. B is the keying speed in baud, K is a factor relating to the keying envelope. The higher the sending speed the shorter the rise and fall time of the signal elements (dots and dashes) and the higher the harmonic content of the signal. Baud rate is determined by using the standard word “PARIS”. It is used because it contains an even 50 signal elements, sent in one minute (60 seconds) the baud rate is 0.83. Most amateur radio contacts have the their keying such that rise and fall times are on the order of 5 milliseconds or a K factor of 4.8. First calculate baud rate (WPM X 0.83), this times the K factor (4.8). At a comfortable sending and copying speed of 10-WPM the Occupied Bandwidth of the CW signal will be right around 40 Hz. At 20 WPM we still only occupy 80 Hz of spectrum.

I trust that this column has somehow imparted some knowledge and usefulness to the readership; however, without participation or feed back it is difficult to discern the worth of the effort. I have attempted to bring to light the usefulness and necessity of mathematics in our hobby through the request of the readership to solve some common problems yet participation has been faint. Perhaps a different approach to gain participation is in order, although this has been an effort to stimulate the “thinker” thing. An attempt at philosophy may garner some responses. So, let’s see how folks feel about the use of amplifiers and filters. In VHF and above, we find that preamps are very much a necessity for “weak signal work”. We know that our present-day technology, low noise, Gallium-Arsenide Field Effect Transistors (GAsFET’s) have a very broad range over which they exhibit gain (amplify). Our Electromagnetic Spectrum is becoming more and more crowded with a proliferation of varied devices some legal, some not so legal, unlicensed and dirty. In this day and age we have to filter our spectrum of interest in order to use our Radio Frequency Spectrum to any advantage and consequence. Now, I’d like to hear from the readership as to just what your philosophy is as to use of filter/preamp or preamp/filter combinations and reasons why you feel one is better than the other.

From: The Rocky Mountain VHF+ Newsletter THINK TANK December 1998

Dave W6OAL –

Olde Antenna Laboratory 41541 Dublin Drive Parker, CO 80138