Bandwidth versus Keying Speed

Introduction

A popular article on this reflector says that it is a "misconception" that the "bandwidth" of a CW transmitter is a function of the keying speed. On the other hand, The ARRL Handbook says that the keying speed is a factor in determining the bandwidth "occupied" by a CW signal. What's going on here?

Although there are a variety of potential pitfalls when discussing this topic, the most obvious one is the meaning of the word "bandwidth" itself. Many definitions of bandwidth exist and it is important that the precise definition or type of signal bandwidth be made clear early on.

Fourier Analysis of Signals

I will focus here on the "power" bandwidth of a CW transmitter when keyed by a long string of dits. According to Couch [1], the power bandwidth is f₂ - f₁, where f₁ < f < f₂ defines the frequency band in which 99% of the total average power resides. Note that this definition is similar to the FCC's definition of "occupied" bandwidth:

Occupied bandwidth. The frequency bandwidth such that, below its lower and above its upper frequency limits, the mean powers radiated are each equal to 0.5 percent of the total mean power radiated by a given emission [2].

In studying the specific waveforms presented here, I have used classical Fourier analysis as described in practically all of the communication systems and signal analysis textbooks written over the past 50 years or more.

Let's start with a very good keying wave shape, a sinusoidal-shaped pulse with identical rise and fall times of 5 milliseconds as shown in Figure 1. Now let's assume we form a periodic signal using these shaped pulses with a duty cycle of 50% to represent a long string of dits. Since we want to maintain the same rise/fall characteristics (shapes and times) regardless of speed, we will only vary the time duration of the constant amplitude portion of each pulse (designated as tc) to increase or decrease the number of dits sent per second. I have chosen two specific speeds here, 2.4 words per minute (wpm) and 30 wpm. These speeds were convenient choices mathematically and are also considered to be reasonably representative of the speed range employed by many hams. The value of tc is 0.490 seconds and 0.030 seconds for 2.4 wpm and 30 wpm, respectively. (These values of tc were found by first determining that the number of dits per second is 1 and 12.5 for 2.4 wpm and 30 wpm, respectively.)

Once the exact keying waveform has been decided, the detailed Fourier analysis can begin. If a periodic pulse signal is assumed for the keying waveform, one can calculate its Fourier series and the resulting average power (on a 1-ohm basis) for each discrete frequency component. Once the frequency characteristics of the keying waveform (which is the modulating signal) are known, the frequency characteristics of the radiated CW signal (which, mathematically, is binary ASK) can be found via trigonometric identities because the output signal is the product of the modulating signal and the high frequency carrier.

It turns out that the 99.1% power bandwidth of the 2.4-wpm CW/ASK signal is 34 Hz for the 5-ms sinusoidal-shaped keying waveform. (That is, 99.1% of the total power in the 2.4-wpm CW signal is contained by the carrier and the first 17 sideband pairs.) However, the 99.1% power bandwidth of the 30-wpm CW signal is 150 Hz. (99.1% of the total power in the 30-wpm CW signal is contained by the carrier and the first 6 sideband pairs.) Even though both signals have exactly the same rise and fall characteristics, Fourier analysis indicates that the 99.1% power bandwidth of the 30-wpm CW signal is over four times as large as that of the 2.4-wpm signal!

Figure 1. Sinusoidal keying waveform with symmetrical rise and fall times

Now suppose our keying waveform is changed from the very good one described above to one having zero rise and fall times. "Square-wave" keying is quite a bit easier to examine mathematically than sinusoidal keying and it will yield a "worst-case" value of power bandwidth for a given speed. As before, the speed of the dits is set so that we are sending at a rate of either 2.4 wpm or 30 wpm. In the case of square-wave keying, the 99.1% power bandwidth for our CW signal is 42 Hz and 525 Hz when sending dits at 2.4 wpm and 30 wpm, respectively. Once again we see that the power bandwidth increases significantly in going from 2.4 to 30 wpm. In fact, since both the rise and fall times of this keying waveform are zero, the power bandwidth ratio will be exactly equal to the speed ratio (12.5 to 1 in this specific example). Of course, we also expected the power bandwidth to increase significantly in going from sinusoidal keying to square-wave keying at the same speed. At 2.4 wpm, the 99.1% power bandwidth ratio is 1.24 for the two keying waveforms, but at 30 wpm the ratio is a whopping 3.50. The results are summarized in Table 1 below.

Table 1. 99.1% power bandwidth for CW/ASK transmitter for periodic signaling

I have purposely avoided cluttering this article with the mathematical details used to obtain the values shown in Table 1. If anyone is interested, I will be glad to post the details in a follow-up article or via email. Although I have carefully checked all of my calculations, in some cases by both time-domain and frequency-domain approaches, there is always the possibility of some computational errors. However, these answers appear to be quite consistent with my expectations and with such sources as The ARRL Handbook.

Summary

The concept of "bandwidth" commonly used in electrical engineering (and certainly used by the FCC in its definition of occupied bandwidth) is a "time averaged" quantity. As shown in Table 1, 99.1% of the total mean (average) power of the 30-wpm signal resides in a wider bandwidth as compared to the 2.4-wpm signal. A crucial point to be made here is that the concept of occupied bandwidth does not say that the strength of the individual key clicks generated by a poorly designed transmitter is reduced when the sending speed is decreased! The information provided by the occupied bandwidth is exactly that described in its definition. In closing, however, I do want to point out that if I have to be subjected to key clicks when operating CW, I would much prefer that the offending transmitter send at a speed of 1 word per 20 minutes rather than 20 words per minute!

References

[1] Leon W. Couch, Digital and Analog Communication Systems, 7th ed., Pearson

Prentice Hall, 2007.

[2] FCC Rules and Regulations, 47 CFR 2.202

http://www.access.gpo.gov/nara/cfr/waisidx_06/47cfr2_06.html