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Suppose I have an analog signal I want to transmit. After sampling and quantization, I can map the samples to bits. So now I have a bit stream that represents the original analog signal.

My questions are

  1. Why do we choose to map bits to symbols before transmitting? And
  2. what exactly are these symbols?

One can understand a bit by saying '0' is 0 Volts and '1' is for example 5 Volts. So, 3) How exactly do the symbols modify the analog signal that is transmitted?

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What you did by saying a $0$ bit is mapped to a voltage of zero Volts, and a $1$ bit is mapped to $5$ Volts is exactly what it means to map bits to symbols. Your symbols are 'zero Volts' and '$5$ Volts'. A bit is just a unit of information, and you have to decide which type of analog signal you choose to transmit the source information. You always have to map information bits to some analog signal in order to transmit it over an analog channel.

There are many ways to choose appropriate symbols. Your suggested mapping from bits to symbols requires a baseband channel that can also transmit DC, because your transmitted signal will have a DC value of about $2.5$ Volts, assuming that $0$s and $1$s are equally likely. There are baseband channels that are AC-coupled, i.e., you can use them to transmit very low frequencies, but not DC. In that case you could choose a positive voltage for transmitting a $1$, and a negative voltage of equal magnitude for transmitting a $0$ ("binary antipodal signaling"). There are many standard ways to encode bits for transmitting them over a baseband channel. Here you can start reading up on these so-called line codes.

Note that you can also map more than one bit to a symbol. E.g., you could choose voltages $-3$, $-1$, $1$, and $3$ to encode all possible two-bit sequences. This makes your symbol rate (baud rate) half the bit rate, which requires less channel bandwidth. There's a trade-off between bandwidth and noise immunity. For a fixed transmit power, the symbols are now closer to each other than before, so it is more likely to confuse one symbol for another when there's additive noise on the channel.

There are also channels that have a bandpass characteristic (such as a radio channel). In that case you need to use carrier modulation, which shifts the spectrum of the transmitted signal to the desired frequency band. You can use two orthogonal carriers at the same frequency, such as $\sin(2\pi f_c t)$ and $\cos(2\pi f_ct)$ to transmit two symbols at the same time (quadrature modulation). In that case, symbols are represented by complex numbers (because you have two symbols at a time) and you get a two-dimensional symbol constellation.

It is common to use constellations with many more than just two symbols, e.g., $256$-QAM, which uses one symbol to encode $8$ bits.

All the above gives a very basic and simplistic view of how encoding and modulation works. If you want to dig a bit deeper you should get a good textbook on digital communications.

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