## 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.

1. > "the Wikipedia article requires a lot of clicking on links..."

Not nearly as many as this multi-part blog requires. And the Wiki links go straight to the relevant information instead of adf.ly promotionals.

1. I agree that there are a lot of ad supported links I have posted, but I did it because I did spend a lot of time piecing all this information together.

However, more importantly, I feel like I've fleshed out the information more than Wikipedia does and all in one place. On Wikipedia's SSB article you have to hop from link to link.

I understand your comment, but also I hope you can appreciate that I want to be able to cash in for the insights I provide to the world. I don't make much. So far I've made 3 cents with about 26 clicks.

2. Thank you for posting that comment and for your honesty. But I disagree with several things. I don't see how you can claim that your information is "all in one place". By that definition, Wikipedia is also just "one place".

Respectfully, I believe you are deluding yourself because you spent a lot of time on your derivation, and you want more to show for it than just your own enlightenment. (I know the feeling!) But frankly, you did it the hard way. What you call "fleshing out" is actually obfuscation. It's well hidden, but the key fact underlying your derivation is still the concept of "analytic signal" (of which complex baseband is one case.) You derive it in an obtuse way, and you don't even realize it.

The blog is about you, of course, and you can do what you want here. People understand that. But I suggest you swallow your pride and admit that you took the long route around the mountain. I've done it many times myself, and sometimes that's what it takes to really appreciate a more elegant derivation. No doubt many of the concise concepts we see in our textbooks are the product of someone learning something "the hard way" and then distilling the essence of their discovery and discarding the clutter. You haven't reached that point yet.

I do think some people will appreciate your work. But it needs to be represented differently. For instance, the key difference between it and Wikipedia is not frequency domain vs time domain. It is RF (aka bandpass, aka narrowband) vs baseband. They both rely on frequency domain arguments and inverse Fourier transforms. The reader should also be forewarned that the most interesting thing about your derivation is that it demonstrates why it is more desireable to do the "hard stuff" at baseband. It demonstrates the long way around the mountain, and that is somewhat interesting, perhaps interesting enough for a section of its own in the Wikipedia article. That's not for me to say. You can only try and see what happens.

And I assume you're joking about the money. If you're actually serious, then it is certainly not appropriate to direct Wikipedia readers here. The blog is about you, but Wikipedia is above that.

2. Regarding Wikipedia, my point is more that many different people write the articles on Wikipedia and there isn't a consistent or reliable means to know ahead of time if the links will be approachable for people of different math backgrounds. In contrast, this place has just one author (me) and the links I post are ultimately part of one article (that was split in 3 because of the blog's limitations; I would have made it one long post).

I intentionally wrote it this way. In fact I got a lot of feedback from people asking if I can explain the complex baseband representation of signals and I am happy to do that (that's for the next post).

When I first went through the derivation I followed a different route -- one that used a special case and kept generalizing. For example, start with a tone modulated SSB wave, then represent any periodic message by its Fourier series, then let the "period" of an aperiodic signal be represented as by a Fourier transform. That is one way to do this derivation.

I find it just makes sense to keep the translation from the idea of SSB to the graphical representation in both frequency and time domain as simple as possible. This, to me, means use the minimum number of different concepts that can cause a reader to feel that the material is out of reach.

My intent was not to demonstrate the usefulness of the math done in baseband world, but it was to show another derivation which people (especially students struggling with the concept of SSB).

3. Thanks. I now understand your "in one place" concept. It wasn't clear, because you also seemed to be complaining about lots of links, which I think is a great strength of Wikipedia... the ability to stay "on message" without getting distracted by tangential issues. And I agree with you that the Achilles heel of Wikipedia is the inability to achieve notational and perhaps ideological consistency as one hops from one article to another, or even internal consistency within one article. I very much dislike the Wikipedia's Fourier Transform article, for instance. There we see the dominance of mathmeticians pushing their own agenda, oblivious to the disharmony with many related articles, in my humble opinion.

In the case of SSB, I think Wikipedia's treatment is reasonably consistent. But there is no guarantee of stability. So I like that aspect of what you are doing here. But I would still recommend a more conventional approach. Start with the elegant baseband derivation. Then explain that it can also be done at RF, but it involves larger formulas. And explain your reason for doing it anyway. E.g. some people who gave up on the baseband derivation might be more inclined to persevere through the RF derivation (really?). Or it serves as a double-check, for those who think the baseband derivation is too much like magic. My reason, as you know, is that it helps one appreciate the elegance of baseband.

While editorial consistency is your strength, the associated weakness is that you probably can't expect much editorial "help" either. Even the best textbook writers rely extensively on the help of others to improve the presentation or provide different perspectives. Keep an open mind, and best of luck.

4. Thanks for your comment. I agree with you 100% regarding Wikipedia's weaknesses. I find Wikipedia's frequent use of links analogous to a person who keeps splintering off a conversation into hundreds of side stories. I like simple.wikipedia.org's idea, but there are far too few articles on there to say it is a good alternative to the en.wikipedia.org "main" site.

The other thing that I don't to deal with here is petty people hijacking articles and justifying amendments to articles with "WP:SOMEOBSCURERULE." I don't find Wikipedia's SSB article to be all that consistent. Some parts call LSSB simply SSB and other parts specifically derive USSB while calling it USSB.

I welcome alternate (valid!) explanations to many matters related to science and technology. I'm sure you're very familiar with how seeing a problem from many different angles helps the material soak into your brain. The "conventional" approach is explained with varying degrees of success in B. P. Lathi's book and Haykin and Moher. Many foundational signals and systems books also explain it, but authors often "leave it up to the reader as an exercise." I prefer to flesh everything out as I have no real space constraints.

There are several reasons people gave for giving up on baseband derivations of the material. The common theme seems to be that the people sending in questions are not comfortable with taking the information in the negative frequencies and "dumping their content" into the positive frequencies. Sure, it can be explained mathematically, but I think a few graphics could help. I've been a bit busy with other things, so sorry, but it may take me a few more weeks to get it up.

I agree that I don't have Linus' law on my side with a niche blog. I will also try my best to keep an open mind. Thanks for the well wishes.