The colonial British writer Rudyard Kipling (same guy who wrote The Jungle Book) wrote a book in 1902 called Just So Stories that is now considered a classic of children’s literature. It is a very good book by all accounts and consists of “origin stories” which are fanciful stories about how various animals came to be the way they are. Here’s an illustration from the story of how the Rhino got his Skin:
This is a wonderful way to tell folk tales about animals, but we also often tell this kind of “just so” story (without perhaps realizing it) about the way we solve technical problems: things are the way they are because well, they are Just So! This note is about one example of a Just So story in comms design: the idea of signal constellations. Let’s start from the basics.
Coding and modulation are important building blocks in communication systems. See e.g. this block diagram:
from Massey, James L. “Coding and modulation in digital communication.” Zurich Seminar on Digital Communications, 1974. Vol. 2. No. 1. 1974. There is a version of this diagram in every text including ours.
In modern comms systems, the input U to the coding block is always in the form of a sequence of bits (“all comms is digital”) representing digitized information from a message source. In electronic wired and wireless comms systems, the output s(t) of the modulation block is an analog (typically passband) voltage waveform.
The output X of the coding block – which is of course the input of the modulation block – is an intermediary between the digital information bits and the analog modulated waveform. This intermediate representation is entirely in the control of the comms engineer. Traditionally, it is specified in the form of a signal constellation e.g. BPSK, 16-QAM and so on.
Why Constellations?
These signal constellations are presented without explanation or justification as obvious and essential features of comms systems. But they impose an extremely strong a-priori restriction on the modulated waveforms; specifically, they require that (with some simplification) every sample of the modulated analog waveform s(t) are restricted to a finite – and typically very small – set of allowed values represented by the points in the signal constellation.
So how did the concept of signaling constellation become so prevalent? And what are their benefits and drawbacks? Are there alternatives i.e. is it possible to design comms systems that do not use signaling constellations in their traditional form? These are big questions that touch on many subtopics in comms design. Below are a few notes, comments and links. We may revisit this at the end of the term, time permitting.
- In the comms literature, the questions we raised above falls under the technical problem of “coded modulation”. The coding and modulation blocks taken together perform the channel encoding function in the standard Shannon comms model.
- The purpose of the channel encoder is to produce a set of output waveforms that are as dissimilar to each other as possible, to allow the decoder at the receiver can tell them apart with maximum efficiency. In other words, we want the possible outputs of the channel encoder to be “spaced out” far apart from each other. (To make this concept precise we need a physically meaningful concept of “distance” between waveforms. This is something we will talk about at length.)
- There is no theoretical justification for splitting the channel encoding function in this way. We still choose this design for practical convenience, but this will in general incur a performance penalty. The idea behind coded modulation is to minimize this performance penalty by co-designing the two blocks to work well together. This is a somewhat strange dance (what does it mean to “co-design” two functional blocks together, while keep them “separate”?).
- The main practical reason for maintaining a separation between the coding and modulation blocks is that we do not really know how to design a “nice” set of analog real- or complex-valued waveforms. “Nice” means spaced out far apart from each other (in terms of waveform distance – see above), but also able to be stored, listed and sorted in a compact form.
- On the other hand, “we” have invented extremely powerful methods for designing “nice” set of symbol sequences from a finite alphabet (like a signaling constellation). These methods come from the mathematics of finite fields, which is also the basis for many tools of modern cryptography.
- There is a lot more to say about this, and hopefully we will return to this, but briefly, comms engineers now know how to make this “split” architecture work well enough to basically reach the Shannon limit for many waveform channels.
hawkee 