Single Sideband (SSB) Modulation 1
I found that Haykin and Moher's Introduction to Analog & Digital Communications gave a quick overview of the development of SSB for a general signal. I decided to expand upon it and give my own explanation for how it all works.
First, consider the motivation. Wikipedia shows Double Sideband Suppressed Carrier (DSB-SC) is simply the result of multiplying a carrier signal by a message signal. In this way, the amplitude of the carrier is varied in proportion to the message.
From the perspective of a transmitter, it is relatively simple to construct. All that is needed is something that can multiply two signals together (e.g. a mixer), a local oscillator, and an information source. However, the transmitter wastes a lot of power and bandwidth transmitting both sidebands when only one is necessary to properly transmit information. The reason that both sidebands go out is that real signals have equal amounts of positive and negative frequencies. When a cosine carrier, which can be expressed as a summation of two complex exponentials, multiplies a message signal, the result is that the positive and negative frequencies are copied and shifted to the right as well as copied and shifted to the left.
This is where SSB comes in. SSB only transmits one sideband as the name suggests. Wikipedia makes a good point in the Single Sideband (SSB) article that SSB is really a special case of Quadrature Amplitude Modulation (QAM) where only one signal is sent out instead of the usual two. But other than that, the Wikipedia article requires a lot of clicking on links to figure out what all the words mean.
Intuitively, all SSB has to do is take a DSB-SC wave and chop off one sideband that's translated up near a carrier frequency f_c. Theoretically, this would require an infinitely sharp transition band, meaning the filter can suddenly stop passing frequencies. This is difficult and limits the use of SSB in practice to situations where there is an energy gap in the message signal.
I wanted one resource where all this information could be found. Lectures, in general, are easier to follow than concise textbooks, but they take time and it's difficult to pinpoint information unless someone has already done it for you or you got around to identifying key points yourself. However, I find that the Indian Institute of Technology video lectures on YouTube to be of very good quality. Some of the insights here are due to the lecture on SSB from IIT Delhi. Note that some of their videos, particularly for the communication systems lectures have horrible quality (very distorted, difficult to hear even with the volume up, but it is audible).
With the motivation explained and some of the resources used pointed out, there's not much else left to do but to get technical and do some math. SSB is very easy to understand in the frequency domain, but it's not as easy to understand in the time domain.
Consider a DSB-SC signal spectrum first. The goal is to lop off one sideband (in this case, first, the upper sideband) represented by two triangles: the right angle triangle with side length W (from f_c up to f_c+W) and the right angle triangle with side length W (from -f_c down to -f_c-W). This is LSSB (lower sideband).
See how it looks when an ideal low-pass filter does just that. How can this be written mathematically then? The low-pass filter is lopping off the upper sideband leaving only the lower sideband behind. So LSSB can be represented, in the frequency domain, as the product of the DSB-SC wave's spectrum and the ideal low-pass filter's spectrum. A rectangular function can also be written in terms of two sgn (signum) functions.
Wikipedia defines the signum function as being -1 below x=0, as being +1 above x=0, and as being 0 at x=0. This alternate representation of the rectangular function works because one signum is 1 for a portion of the frequency axis while the other signum is -1. The net result is a sort of "destructive interference" and they cancel out. This explains the stopband region (i.e. outside of -f_c to f_c). How about f=-f_c and f=f_c? At those points, one signum is 1 and the other is 0, so the result is 0.5. Within -f_c to f_c, both signums are 1 and so the result of the expression is 1. More next time.