Although it is usually simplest to describe CW as turning on and off a carrier frequency, that isn't really a very good description from a signal point of view. Actually CW is 100% modulated AM. When the signal is turned off, its amplitude is exactly zero, and when it is turned on its amplitude is the carrier signal amplitude. So that means we can analyze a CW signal using the same techniques that are used to analyze AM signals.

Our first thought might be to evaluate a CW signal as if it were an AM signal modulated by a square wave with an amplitude of 1. That is the same as switching the carrier instantaneously on and off. As with any AM signal, we can write the resulting signal as a product of a carrier frequency and a modulating signal, in this case a square wave. But a square wave can also be represented in terms of sine functions as is shown in most basic signal processing books. In fact, a square wave is composed of an infinite series of sine waves all added together. The formula is the sum from zero to infinity of all of the odd harmonics of the modulating frequency, with a decreasing amplitude as the harmonic order increases. Now that may sound complicated, but what it means is that if we instantaneously turn on and off a carrier wave to generate CW, we end up with a signal that has an enormous bandwidth. If you listen to a signal actually generated that way, what we hear are terrible key-clicks, that represent all of the higher frequency components of the modulating square wave.

Surely that doesn't sound so good! We want a narrow band signal, but we find that simply turning on and off a carrier generates a wideband signal instead. In fact, the normal technical definition of bandwidth is the frequency range where the signal is above 30 dB of its peak value. Since the coefficients are equal to 4/Πk where k is an odd integer, the peak value is when k=1 (the fundamental modulating frequency) and the coefficients decrease with increasing values of k, the drop in magnitude of the harmonics is 30 dB < 20 log(k) or log(k) > 1.5, which implies k > 31. Recall in another section we found that the time interval for a CW element was about 1.2/W (W in words per minute) or the frequency of the symbols was about (0.8333 W). Considering that the wavelength of a modulating square wave would be equivalent to a dit and a space element, that means that the equivalent modulating square wave would have frequency of about (0.4167 W), so that the 31st harmonic would be (12.92 W) on each side of the carrier for a total bandwidth of about (26 W) Hz. That means that at 5 WPM, the bandwidth would be about 130 Hz and at 20 WPM the bandwidth would be over 500 Hz! Clearly that isn't acceptable and our experience says that it isn't usually true in practice, either.

The problem is that real electronic equipment, like radios, don't generate square waves precisely. Instead the signal rises over some time period controlled by the electronic components used. But the lesson is simple: Don't use square waves to modulate a CW signal!