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I’m looking for the simplest, precise definition of a Symbol

e.g.

  • The symbols found within a signal
  • The signal has a N symbols per second

I'm following this tutorial. The fellow defines the term symbol informally as "different states" and I do grasp the concept intuitively, however I am seeking a more formal definition.

Here are various definitions I have found:

  • My definition 1: "A distinct, observable state of a property of a physical medium ... which persists for a fixed unit of time in a communication channel"

  • My definition 2: "A distinct state of a quantity of a waveform ... which persists for a fixed unit of time in a communication channel"

  • Khan Academy definition: “A symbol can be broadly defined as the current state of some observable signal, which persists for a fixed period of time.” 4m 15s on this video

  • Signal Processing StackExchange Definition: "A symbol is a symbolic representation of a baseband signal in digital communication."

  • Electrical Engineering StackExchange Definition 1: "A symbol is any distinct state of the communication channel."

  • Electrical Engineering StackExchange Definition 2: "A symbol is an information entity"

  • Wikipedia Definition 1 (under Symbol Rate): “A symbol may be described as either a pulse in digital baseband transmission or a tone in passband transmission using modems. A symbol is a waveform, a state or a significant condition of the communication channel that persists, for a fixed period of time”

  • Wikipedia Definition 2 (under Symbol (disambiguation): “Symbol (data), the smallest amount of data transmitted at a time in digital communications”

Out of interest, is there a mathematical/formal definition of a symbol? Perhaps that would help me.

I'm not certain if this is the appropriate StackExchange to post the question if it's not should I try one of the following? :

  1. Mathematics (under Information Theory)
  2. Electrical Engineering
  3. Network Engineering

Thank you for reading

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    $\begingroup$ In many cases the "exact" definition depends on the context in which the word is used. None of the definitions you list are wrong, but they are not the same thing either. They just term the term in a slightly different context. $\endgroup$
    – Hilmar
    Sep 14 at 11:16
  • $\begingroup$ Welcome to SE.SP! Good question. And nice list of potential answers, too. As @Hilmar says, I doubt there is a single definition. Context helps. “When I use a word,” Humpty Dumpty said in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.” $\endgroup$
    – Peter K.
    Sep 14 at 13:17
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    $\begingroup$ @PeterK. Please do complete the story to include Alice asking whether one could make words mean anything that one pleases, and Humpty Dumpty responding that the question is which is to be the master. $\endgroup$ Sep 18 at 2:47
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In the words of that great communications theorist, William Shakespear, "There are more things in heaven and earth, Horatio, than are dream'd of in your philosophy".

The reason it's hard to pin down an exact definition of "symbol" is because it's a really handy concept that can be stretched to aid us in doing a lot of useful math, but an all-inclusive definition gets tediously vague, while the nice concise definition that I might use for my problem over here may not fit with the nice concise definition that you need to use for your problem over there.

Worse, the definition may reasonably change even within a problem. For example, MSK modulation can be defined as frequency-shift modulation where a symbol is one bit-period long, the transmitted frequency can take on one of two values where the frequency shift is exactly $\frac{1}{2}$ the bit rate, and (this is crucial) the phase remains continuous from one bit to the next (no phase jumps).

After you grind through about a page of math, you can show that "really", MSK isn't frequency shift keying, it's "really" quadrature phase shift keying, with pulses that are two bit periods long, and are half a period of a sinusoid (i.e., one bump).

There's other examples. You can think of OFDM as being a chunk of spectrum that's been subdivided into a whole bunch of teeny sub-chunks, with each sub-chunk modulated one way or another (often QPSK). In that case you're sending $N$ channels, with some number of $m < N$ symbols ($m < N$ because OFDM often has "quiet channels"), then putting them together into a word later. Or, you can think of each frame of OFDM as being one giant symbol that's encoding an unreasonably large number of bits. Or, you can think of OFDM completely in the time domain, as being $m$ possible symbols all added together and modulated onto a wave.

If I remember correctly, the CDMA cell phone standards have one layer where they define 64 different possible "symbols", defined as 64 different possible Hadamard codes.

All the above was just to soften you up -- now that you're reeling, here's a general-to-the-point-of-uselessness definition of a symbol:

A symbol is a known pattern that is modulated onto the transmit signal, that can be usefully distinguished from other symbols at the receiver.

Slightly more useful:

Usually, symbols are all of the same duration, and are sent in such a way that they are orthogonal or nearly orthogonal to one another (look up "intersymbol interference" (ISI) for counterexamples -- ISI is usually unintentional, but sometimes it's profitable to let some creep in intentionally, as in GMSK).

Usually, symbols are emitted on a known schedule (i.e., always starting at the same time, or, in the case of OQPSK, staggered but on an even interval, and the "odd numbered" symbols are guaranteed to be orthogonal to the "even numbered" symbols).

The most important definition for a symbol is whatever the heck the author meant when they wrote the text you're reading. If you're lucky, they tell you. If you're kinda lucky, you can infer what they meant by reading on a bit and scratching your head (usually there's examples, either intentional or not). If you're not lucky at all, then you need to find help and ask.

Personally, I consider it good style to say what I mean by a term when I'm using it, unless I feel it's super-obvious from the surrounding text. (I.e., if I assume you know what PSK is, then I might talk about "symbols" without defining them in context). But you can't always count on that, particularly in papers where you get a fixed amount of column space and are thus motivated to make your writing very concise.

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  • $\begingroup$ Hi Tim, thank you for your detailed response , I will print it out and stick it on my fridge! Your general & slightly-more-useful definitions are well-phrased and I shall be plagarizing them! Haha yes “whatever the heck the author meant when they wrote the text you're reading” sounds right! I guess you could say its a “fuzzy concept”. I was surprised trying to read some papers that they didn’t define their terms but you make a good point about publications having a paper-real-estate scarity! $\endgroup$ Sep 19 at 11:10
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The best definition (in the sense of being specific, clear and useful) of a symbol is the following, in my opinion.

In a digital communication system bits are transmitted using an analog (continuous-time) signal. The process of converting from bits to this analog signal is called "modulation". It consists of these steps:

  1. Select a Nyquist pulse $p(t)$ and a pulse interval $T_p$.
  2. Group the bits to be transmitted in sets of $k$ elements. There will be $M=2^k$ different bit combinations.
  3. Assign each of the $M$ bit combinations to a real number $c_i \in \mathcal{C}, i=1,\ldots,M$. These are the symbols. The set $\mathcal{C}$ is called the "constellation".
  4. Generate the signal $$\sum_l a_l p(t-lT_p)$$ to transmit the bits. During the $l$-th time interval, the pulse $a_l p(t-lT_p)$ is transmitted; $a_l \in \mathcal{C}$ is a symbol, and its purpose is to set the pulse amplitude in order to convey a specific combination of $k$ bits.

In summary, then, a symbol is a real number that belongs to a set called a constellation, selected as a pulse amplitude in order to convey a specific group of $k$ bits to the receiver.

Notes:

  1. Symbols can also be complex.
  2. The pulse $p(t)$ determines the bandwidht required, and the pulse interval determines the bit rate $R_b = k / T_p$.
  3. The choice of symbols determine the energy required to transmit; the energy of $ap(t)$ is $a^2 E_p$.
  4. The choice of symbols also determines the error rate of the system.
  5. The requirement that the pulse $p(t)$ is Nyquist ensures that the symbols $a_l$ (and therefore the transmitted bits) can be recovered at the receiver, for example using a matched filter.
  6. The modulation process described above is called "linear modulation". Other modulation schemes exist, such as FSK, which are not linear; they still use symbols in the same sense used here. Other modulations where "symbol" can take a (related, but) different meaning are MSK and OFDM.

Example:

Let $k=2$, $T_p=1$ and $\mathcal{C}=\lbrace -3, -1, 1, 3 \rbrace$. This scheme is called 4-PAM. Let the bits be associated to symbols as follows: $00 \rightarrow -3$, $01 \rightarrow -1$, $11 \rightarrow 1$, $10 \rightarrow 3$. If the bits to transmit are $00011101$, then the transmitted symbols are $a_1=-3$, $a_2=-1$, $a_3=1$, $a_4=-1$. The transmitted signal is $$-3p(t-T_p)-p(t-2T_p)+p(t-3T_p)-p(t-4T_p)$$ If the energy of $p(t)$ is $E_p$, then the energy required to transmit this signal is $E_p(9+1+1+1)=12E_p$. There are $k=2$ bits transmitted per symbol, and $T_p=1$ symbols per second, so the bit rate is $R_b=k/T_p=2\text{ bps}$.

By the way: my definition of "symbol" is heavily influenced by the book "A foundation in digital communication", by A. Lapidoth, which is the book I'd take to a desert island if I could choose only one. It is available for free here. Symbols are discussed in chapter 10, "Mapping bits to waveforms".

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  • $\begingroup$ "Assign each of the M bit combinations to a real number $c_i$" -- better wording would be "integer" rather than "real number" -- a real number implies continuity. Not all signals are impressed on Nyquist pulses (i.e., GMSK), or on the same Nyquist pulse (i.e. OFDM, with each frame viewed as a collection of a whole lot of signals). $\endgroup$
    – TimWescott
    Sep 14 at 14:50
  • $\begingroup$ @TimWescott Thanks for the comments. The symbols are in fact real numbers. In note six I mentions FSK. I will add a note on OFDM, where each subcarrier is still linearly-modulated. $\endgroup$
    – MBaz
    Sep 14 at 15:00
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    $\begingroup$ in more complex schemes M doesn't need to be a power of 2 because it doesn't need to exactly represent a certain number of bits. $\endgroup$
    – user253751
    Sep 14 at 16:41
  • $\begingroup$ @user253751 That is correct, but I thought that was too much detail to add to an already long answer. $\endgroup$
    – MBaz
    Sep 14 at 17:29
  • $\begingroup$ Hi Mbaz, thank you for your detailed, formally defined response, it will take some work to decode, but now I have a useful future reference on hand! Also your lexical definition is helpful: “a symbol is a real number that belongs to a set called a constellation, selected as a pulse amplitude in order to convey a specific group of 𝑘 bits to the receiver.” $\endgroup$ Sep 19 at 11:12
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I do like @TimWescott answer a lot. less cultural than Shakespeare, the symbol wikipedia page asserts that:

"Symbols are a means of complex communication that often can have multiple levels of meaning"

and this goes well within our signal processing/information theory/data compression/communication community. The etymology for symbol is often associated with the word token (something serving as a visible/tangible representation of a fact, quality, feeling). The Greek prefix sym- additionally denotes the notion of togetherness: a symbol is unlikely to be alone, it ought to fit well with other symbols of similar nature, so that they could be assigned some meaning, when considered together.

In our digital signal processing context, I consider that the signals at hand are usually discrete, and appear with some coherent (eg spatial/temporal) ordering from discrete alphabets (bits, substrings). A set of symbols does not need to provide a perfect coverage of all combinations (not a paving nor a segmentation).

So I am trying this:

Symbols (in DSP) are coherent arrangements (potentially overlapping or not complete) of elementary chunks of information that altogether make sense in a given context

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