Confusion about collection of bits and symbols

Question

I'm taking an intro communications module and I'm quite confused on what the term "symbol" really means?

Consider this setup:

My understanding is as folllows

1. After the information source, we have a stream of bits coming in 0 1 0 1 0 0 0 1 1.....
2. The source encoder takes these bits and groups them into sets of N bits.
3. The channel encoder takes these sets of N bits and based on what coding scheme we use (gray coding) for example, it will multiply it by a certain coefficient of a orthogonal basis function (carrier signal) depending on what exact gray code it is.
4. The digital modulator now receives a coefficient of it's carrier/basis function and just mulitplies it by the fixed carrier/basis function (either sin or cos of something) and sends it onto the channel.

My question is - what point in this transmitter do we call "the symbol"? Is it the group of N bits after the source encoder? Or is it the coefficient of the orhtogonal basis function after the channel encoder?

Answer 1

In digital communications, information is transmitted in quantized form. We are constrained to transmit items that belong to a finite, discrete set. The items are called "symbols", and the set is called a "constellation". The symbols may be transmitted by changing a signal's frequency, phase and/or amplitude.

As an example, pulse amplitude modulation (PAM) is a technique where the symbols are pulse amplitudes, and the transmitted signal is a sequence of pulses. Let's say that you are using the 4-PAM constellation $\mathcal{C} = \lbrace -3, -1, 1, 3 \rbrace$, and the pulse shape is $h(t)$. Then, you group your bits 2 by 2 and assign them symbols as follows:

bits | symbol | pulse
-----------------------
 00      -3     -3h(t)
 01      -1     - h(t)
 11       1       h(t)
 10       3      3h(t)

Recall that a PAM signal can be written as $$s(t) = \sum_k c_k h(t-kT_p),$$ where $T_p$ is the pulse interval (the pulse rate is $R_p = 1/T_P$) and $c_k \in \mathcal{C}$ are the symbols. Then, transmission of the bit sequence $00111001$ would be done with the signal $$h(t) = -3h(t) + h(t-T_p) + 3h(t-2T_p) - h(t-3T_p).$$ Note that the information (bits) is conveyed by the "symbols" or pulse amplitudes (hence, PAM = pulse amplitude modulation). The task of the receiver is to recover the transmitted symbols from the received signal.

(Note that there's no unique or "official" way to map bits to symbols in 4-PAM or any other constellation; the mapping above is just an example. The only rule is to use Gray encoding, in order to minimize the bit error rate. And, or course, the mapping must be known at the receiver!)