Why is short wave radio transmission like preparing a signal for spectral analysis? If you have always wondered what the answer to this question is, this short section on multiplying signals will make you happy.
Multiplication of signal values for each point in time is the basic operation by which the amplitude of signals is modulated. The operation is known as amplitude modulation, because the instantaneous amplitude of the carrier signal varies with the amplitude of the modulation signal. The modulating signal is added to its own negative peak amplitude in order to ensure that all modulation values are in the positive range.
A volume control on an amplifier does just this: one signal is provided by a radio, CD player or other familiar signal source, the other signal is provided by manual movements of the control knob by the user.
Amplitude modulated long, medium and short wave radio signals are produced by multiplying a radio frequency signal (the carrier wave) with an audio frequency signal superimposed on a constant positive signal (DC component) of at least its own amplitude, the modulation signal. The instantaneous amplitudes of the audio frequency wave is raised before the multiplication operation, so that all peaks in the modulation signal are either positive or negative.
Figure 24: Sine wave (high frequency).
Figure 25: Sine wave (low frequency).
Figure 12: Sine wave (raised to DC).
Figure 13: Sinusoid amplitude modulation.
In signal processing, this operation is used to preprocess the signal before applying the Fourier transformation: the raised cosine window is a time segment of the signal, amplitude modulated by a cosine (actually by a raised inverted cosine). Technically, segment is not `simply' a chunk of the signal, but a window, which is an amplitude modulation of the signal by a value of 0 for all signals outside the window time segment, and by 1 for all values within the window time segment.
If you think this is pedantic, you're right. But it can still be important. The reason the raised cosine (or some variant of it) is used is to permit approximation to the the `ideal' mathematical function: if the signal were suddenly cut off at the edge of the window, the sharp edge would imply, falsely, the presence of very high frequencies (cf. the sawtooth and square wave illustrations above). Informally, if the amplitude tapers off to zero, there is no need to consider a signal of infinite length, as required by the mathematical function, since properties outside the window are relevant.
Figure 14: Cosine (low frequency).
Figure 16: Signal amplitude modulated by raised cosine (raised cosine window).
