What is the limit capacity of the channel?

2.2. 1 Some basic concepts about channels

Of course, in order to communicate between computers, there must be a circuit for transmitting electromagnetic wave signals. The circuits mentioned here also include wireless circuits. But in many cases, we often use the term "channel". Channels and circuits are different. A channel is usually used to represent a medium that transmits information in a specific direction. Therefore, the communication circuit includes at least one transmitting channel and/or one receiving channel. A channel can be regarded as a logical component of a circuit.

From the perspective of information interaction between the two parties, there are three basic ways:

One-way communication is also called simplex communication, that is, there can only be communication in one direction, and there can be no interaction in the opposite direction. Radio broadcasting or cable broadcasting and TV broadcasting all belong to this type.

Two-way alternating communication is also called half-duplex communication, that is, both parties can send information, but they can't send it at the same time (and certainly can't receive it at the same time). This way of communication is that one party sends the other party to receive it, and then the other way around after a period of time.

Two-way simultaneous communication is also called full duplex communication, which means that both parties can send and receive information at the same time.

One-way communication requires only one channel, while two-way alternating communication or two-way simultaneous communication requires two channels (one in each direction). Obviously, two-way simultaneous communication has the highest transmission efficiency. However, it should be pointed out that although the telecommunications bureau provides a two-way simultaneous communication channel for telephone users, the effective telephone conversation is generally alternating communication between the two parties. When the two sides have a quarrel, they often communicate in two directions at the same time.

What I want to remind readers here is that sometimes people often use the word "simplex" to mean "two-way alternate communication". As people often say, "simplex radio" is not only used for one-way communication. Because of this, ITU-T does not use confusing terms "simplex", "half duplex" and "full duplex" as formal terms.

From the form of signals generated by the communication sender, these signals can be divided into the following two categories:

Analog signals are continuous signals, such as voice signals and current radio and television signals.

Digital signals are discrete signals, such as signals composed of binary codes "1" and "0" used in computer communication.

Similar to this classification of signals, channels can also be divided into two categories: analog channels for transmitting analog signals and digital channels for transmitting digital signals. However, it should be noted that digital signals can be transmitted on analog channels after digital-to-analog conversion, and analog signals can also be transmitted on digital channels after digital-to-analog conversion.

Signals transmitted on the channel can also be divided into baseband signals and broadband signals. Simply put, the so-called baseband signal means that the digital signal 1 or 0 is directly represented by two different voltages and then sent to the line for transmission. Broadband signal is a frequency division multiplexed analog signal formed by modulating baseband signal. After the baseband signal is modulated, its spectrum shifts to a higher frequency. Because the frequency spectrum of each subgrade signal is moved to a different frequency band, it will not interfere with each other after merging. In this way, multiple digital signals can be transmitted in one circuit at the same time, thus improving the utilization rate of the line.

In the early days of communication network development, all communication channels were analog channels. However, due to the rapid development of digital technology, digital channels can provide higher quality communication services, so the analog channels established in the past are being replaced by new digital channels. At present, the communication channels used in computer communication are basically digital channels on trunk lines, but a large number of subscriber lines currently used are basically traditional analog channels. The coexistence of analog channel and digital channel also makes the content of physical layer more complicated.

With the above basic concepts about the channel, let's discuss the ultimate capacity of the channel. This is the highest symbol transmission rate and the highest information transmission rate on the channel.

2.2.2 Maximum symbol transmission rate on the channel

Any practical channel is not ideal. This is because the bandwidth of the channel is limited (that is, the bandwidth through which the signal can pass is limited), and various distortions will occur when transmitting the signal; All kinds of interference will also enter the channel in different ways. This makes the symbol transmission rate on the channel have an upper limit. As early as 1924, Naquist derived the formula of the highest symbol transmission rate when the channel has ideal low-pass rectangular characteristics. This is the Nyquist criterion:

Maximum symbol transmission rate of ideal low communication channel = 2 W baud (2-l)

Where w is the bandwidth of the ideal low communication channel in Hz;

Porter is porter, which is the unit of symbol transmission rate, and 1 porter is the transmission of 1 symbol per second.

Formula (2- 1) is the famous Nyquist criterion. Another expression of Nyquist criterion is that the highest symbol transmission rate per hertz in an ideal low-bandwidth communication channel is 2 symbols per second. If the transmission rate of symbols exceeds the value given by Nyquist criterion, the symbols will interfere with each other, so that the receiver cannot correctly judge whether the symbol sent by the sender is 1 or 0.

Here we want to emphasize the following two points:

The channel with ideal low-pass characteristics mentioned above is an ideal channel, which is of course very different from the actually used channel. Therefore, the highest symbol rate that the actual channel can transmit is obviously lower than the upper limit given by Nyquist criterion.

Potter and bit are two different concepts.

Porter is the rate unit of symbol transmission, indicating how many symbols are transmitted per second. The symbol transmission rate is also called modulation rate, waveform rate or symbol rate.

Bit is the unit of information, which is completely different from the transmission rate of the symbol "Potter".

However, the information transmission rate "bits per second" and the symbol transmission rate "baud" are related in quantity. If 1 symbol only carries 1 bit information, then "bits per second" and "Potter" are equal in value. However, if 1 symbol carries n bits of information, the information transmission rate corresponding to the symbol transmission rate of m baud is m× Nb/s, for example, there is an ideal low communication channel with a bandwidth of 3 kHz, and its highest symbol transmission rate is m baud. If 1 symbol can carry 3 bits of information, the highest information transmission rate is18000 b/s.

For the channel with ideal bandpass rectangular characteristics (bandwidth w), Nyquist criterion becomes:

Maximum symbol transmission rate of ideal band communication channel = W baud (2-2)

That is, the highest symbol transmission rate of a band communication channel with a bandwidth per Hz is 1 symbol per second.

2.2.3 Limit the information transmission rate of the channel.

In 1948, Shannon derived the limit information transmission rate of the channel with limited bandwidth and Gaussian white noise interference by using the theory of information theory. When transmitting at this rate, no errors will occur. If expressed by a formula, the limit information transmission rate c of the channel can be expressed as

C= W log 2( 1+SNR) b/s (2-3)

Where w is the bandwidth of the channel (Hz);

S is the average power of the signal transmitted in the channel;

N is the Gaussian noise power in the channel.

Formula (2-3) is the famous Shannon formula. Shannon formula shows that the greater the bandwidth of the channel or the greater the signal-to-noise ratio in the channel, the higher the limit transmission rate of information. But more importantly, Shannon formula points out that as long as the information transmission rate is lower than the limit information transmission rate of the channel, some way can be found to realize error-free transmission. But Shannon didn't tell us the specific implementation method. This should be discovered by experts who study communication.

It can be seen from Shannon's formula that if there is no upper limit for the channel bandwidth w or the signal-to-noise ratio S/N (which is certainly impossible for an actual channel), then there is no upper limit for the channel's ultimate information transmission rate C..

Since the publication of Shannon formula, various new signal processing and modulation methods have appeared constantly, all in order to approach the transmission rate limit given by Shannon formula as much as possible. The information transmission rate that can be achieved on the actual channel is much lower than Shannon's limit transmission rate. This is because in the actual channel, the signal will suffer other damages, such as various pulse interference and distortion in transmission. These factors are not considered in the derivation of Shannon formula.

Because the transmission rate of symbols is limited by Nyquist criterion, in order to improve the transmission rate of information, each symbol must carry as much bit information as possible. This requires the use of multi-system (also known as multi-system) modulation methods. For example, when the binary system 16 is adopted, a symbol can carry 4 bits of information. The frequency band of a standard telephone conversation is 300-3400 Hz, that is, the bandwidth is 3 100 Hz. In this frequency band close to the ideal channel, that is, the middle section, its bandwidth is about 2400 Hz. If the symbol transmission rate is 2400 baud (equivalent to the symbol transmission rate of bandwidth per hertz 1 baud), the information transmission rate can reach 9600 b/s. In fact, to achieve such information transmission rate, a high signal-to-noise ratio is needed. The reader can easily calculate the required minimum signal-to-noise ratio from Equation (2-3). However, it should be noted that the required signal-to-noise ratio of the actual channel is much higher than this minimum value.

For a standard telephone channel with a bandwidth of 3. 1 kHz, if the signal-to-noise ratio S/N = 2500, it can be known from Shannon formula that no matter what advanced coding technology is adopted, the information transmission rate should not exceed the limit value calculated by formula (2-3), that is, about 35 KB/s. Compared with this limit value, the current coding technology level is not far behind.