Telecommunications

Bit, Symbol and Valence

A symbol contains multiple bits. The number of bits per symbol is denoted as n.

The number of possible signal states is denoted by M - called valence in French.

It is calculated from the valence (number of possible states) as:

M = 2^n

The symbol rate is also called modulation rate. It is the inverse of the symbol time:

Symbol rate (in baud)

R = \frac{1}{T_s}

The bitrate is the inverse of the bit time:

Bitrate (in bits/s)

R_b = \frac{1}{T_b}

We can convert between symbol rate and bitrate using valence:

Bitrate from symbol rate and valence (in bits/s)

R_b = R \cdot \log_2(M) = R \cdot n

The Shannon-Hartley theorem allows us to determine channel capacity:

C = B \cdot \log_2\left(1 + \frac{S}{N}\right)

Caption:

C: capacity in bits/s

B: bandwidth in Hz

S/N: signal-to-noise ratio

(1 + S/N): the maximum valence

Noise limits the valence, which in turn limits the channel capacity.

So we have:

R_b \le C

Caption:

R_b: the source bitrate

C: the channel capacity

Nyquist’s theorem defines the minimum required bandwidth for a channel (to avoid intersymbol interference - ISI):

B \ge R

Caption:

B: bandwidth

R: symbol rate

Since channel bandwidth is limited, symbol time is limited too — so the transmission speed is limited.

The eye diagram is used to detect intersymbol interference (ISI).

To plot it, we take a period (either T or 2T) and overlay the signal on that period.

We can measure the horizontal distance, also called the “eye opening”.

The constellation diagram measures the distance between symbols generated by the IQ modem.

Gray coding helps reduce errors by ensuring only one bit changes for each π/2 phase shift.