Table of contents
2. Maximum transmission rate limit for digital signal transmission without distortion
3. Minimum sampling frequency limit for analog signal restoration without distortion
1. Bit rate and baud rate
Bit rate is usually also called information transmission rate, which refers to the number of effective bits (bits) of binary code transmitted per unit time, represented by Rb (note that it is a lowercase b here). Bit rate is used to describe the transmission rate of digital data that can be directly transmitted on digital lines of a computer network. Commonly used units are bit/s (b/s or bps, bits per second), kbit/s (kb/s or kbps, kilobits per second) or Mbit/s (Mb/s or Mbps, megabits per second). But note that k and M here are 1000 and 1 000 000 respectively, not 1024 and 1048576 when it comes to computer memory capacity.
Baud rate refers to the modulation rate of the digital signal to the carrier after a digital signal is modulated, that is, the number of times the carrier parameters (such as frequency, phase, etc.) change per unit time, and the unit is B. If it is a non-modulated binary digital signal, the baud rate and the bit rate are numerically equal, so the baud rate is usually based on the modulated digital signal transmission rate .
The unit of baud rate is Baud, represented by RB (note the capital B here). A digital pulse, or directly a binary bit, is called a symbol, and the baud rate can also be understood as the number of symbols transmitted per unit time. The code elements here can be binary or multi-ary. For example, the ASCII code of the binary letter A is 01000001, which can be represented by 7 pulses (the highest bit is the check digit and is not included in the code element), so A can be considered to be composed of 7 code elements. If 2400 code elements are transmitted within 1 second, the baud rate is 2400B.
For digital data, the relationship between bit rate and baud rate is related to the base system used by the data. Because each code element or symbol usually contains a certain number of bits of information, the bit rate (Rb) and the baud rate (RB) have the following relationship:
Rb=RB×log2M(b/s)
M in the formula is the base number of the symbol, such as binary, octal, and hexadecimal. For example, if the code element rate is 1200B and octal (M=8) is used, the corresponding information rate can be calculated Rb=1200×log28=3600(b/s); if binary (M=2) is used, the information rate The rate is Rb=1200×log22=1200(b/s). It can be seen that the binary baud rate and bit rate are numerically equal.
In digital communications using digital modulation technology, the relationship between bit rate and baud rate is also related to the modulation technology used.
The value of the bit rate of two-phase modulation (a single modulation state corresponds to 1 binary bit) is equal to the value of the baud rate; the value of the bit rate of four-phase modulation (a single modulation state corresponds to 2 binary bits) is the value of the baud rate 2 times; the bit rate value of eight-phase modulation (a single modulation state corresponds to 3 binary bits) is 3 times the baud rate value; and so on.
"Bandwidth" refers to the maximum amount of information transmitted per second in the channel, which is the maximum data transmission rate of a channel, and the unit is also "bits/second" (b/s or bps). Bandwidth is the channel transmission rate in an ideal state (without any interference, without any attenuation), and the "data transmission rate" introduced above refers to the current actual data transmission rate. Normally, the "data transmission rate" is always less than Or equal to the "bandwidth" value (in fact, it is generally impossible to be equal to the bandwidth, because the transmission process cannot be completely free of interference, and the line performance generally cannot fully reach the ideal state).
2. Maximum transmission rate limit for digital signal transmission without distortion
Any channel in the physical layer has a limited signal frequency (or channel bandwidth) it can bear, just like any river can only bear a certain peak water flow. In the channel, signals that exceed the limited frequency range will be directly filtered out and cannot pass through the channel, just like any water flow in a river that exceeds its flow rate limit will eventually not form the actual flow rate.
In terms of calculating the maximum transmission rate in digital channels, there are two very famous criteria or theorems, namely the Nyquist criterion (referred to as Nyquist criterion) and Shannon's formula.
Since the frequency band (here refers to the frequency bandwidth of the signal) of the digital signal (here specifically refers to the original digital signal that has not been modulated, which is the digital baseband signal mentioned above) can be very wide (from DC to infinitely high frequencies), However, its main energy is concentrated in the low frequency band, and the cable transmission channel in data communication only allows relatively low frequency components to pass through the ideal low-pass channel, and the high frequency components will be filtered out, which causes distortion of the output waveform. The distorted output waveform becomes rounded at the top and widened at the bottom (called waveform tailing), causing the waveform of one symbol to broaden to the position of other symbols, affecting other symbols. This effect is called inter-symbol interference.
An ideal "low-pass channel" means that as long as the signal frequency does not exceed a certain upper limit, it can pass through this channel without distortion; while all high-frequency components whose frequencies exceed this upper limit cannot pass through this channel. In contrast, there is an ideal "bandpass rectangular channel", which only allows signal frequency components between the upper and lower limits to pass without distortion, and other frequency components cannot pass.
To ensure that the digital signal transmitted in the channel is not distorted, the data transmission rate in the channel must be limited to a certain range, which is related to the channel type. In 1924, the famous American physicist Nyquist proved through many experiments that in order to ensure the distortion-free transmission of digital signals, the actual highest code element transmission rate under an ideal low-pass channel (channel in baseband transmission) (That is, the highest baud rate without inter-symbol interference) must be within the calculation result of the following formula:
Maximum symbol transmission rate (MaxRB)=2W
In the formula, W is the bandwidth of the ideal low-pass channel, in Hertz (Hz), and MaxRB is the highest symbol transmission rate, in Raud (Baud). This is the famous Nyquist first criterion, which states that in an ideal low-pass channel (that is, the channel in baseband transmission introduced in Section 4.2.4), in order to ensure the undistorted transmission of digital signals, the channel bandwidth and The relationship between the maximum symbol transmission rate.
The above Nyquist first criterion describes the relationship between the maximum data transmission rate and the channel bandwidth of a limited-bandwidth, noise-free channel. Another famous American physicist, Claude Elwood Shannon, also made an authoritative discovery in the field of data communications, which is the " Shannon formula " . This formula describes the relationship between the maximum transmission rate of a limited channel bandwidth, a channel with random thermal noise (closer to the real environment), the channel bandwidth, and the signal-to-noise ratio (the ratio of the average signal power to the noise power). Shannon's formula states that when transmitting data signals on a channel with random thermal noise
When signaling, the relationship between the maximum data transmission rate Rmax, channel bandwidth B and signal-to-noise ratio S/N is as follows:
Rmax=B×log2(1+S/N)
In the formula, the unit of Rmax is bps, the unit of channel bandwidth B is Hz, and the signal-to-noise ratio S/N is usually expressed in dB (decibel). The 10×log10 (S/N) formula can be used to calculate the decibel number of the signal-to-noise ratio. If S/N=1000 is known, according to the formula, it can be quickly concluded that the signal-to-noise ratio expressed in decibels is 30dB. If the channel bandwidth B=3000Hz is known, then Rmax≈30kbps. It means that for a communication channel with a bandwidth of only 3000Hz, when the signal-to-noise ratio is 30dB, no matter the data is expressed in binary or more discrete level values, the data transmission rate cannot exceed 30kbps.
3. Minimum sampling frequency limit for analog signal restoration without distortion
Only digital signals can be received in computer equipment. If analog signals are transmitted in the channel ( telephone lines connected to remote networks through the public switched telephone network transmit analog signals ), analog/digital processing must be performed at the receiving end. Conversion to generate a digital signal; similarly, if a digital signal in a computer device is restored to an analog line for transmission, digital/analog conversion is required to restore it to the original analog signal. The most critical thing here is to convert the analog signal into a digital signal, because during this conversion process, it is necessary to ensure that the data is not distorted.
Because analog signals change continuously with time, while digital signals are discrete, during the analog-to-digital conversion process, it is necessary to discretely extract certain signal samples from the analog signals to form digital signal samples so that they can be without distortion. It accurately represents the original data itself, and also enables the extracted digital signal samples to be restored to the original analog signals when needed. This involves a question of "sampling frequency" (simply put, how often the signal wave is extracted).
To ensure signal recovery without distortion, what is the minimum sampling rate required? The law that the sampling process should follow is called the sampling theorem, also known as the sampling theorem and the sampling theorem. The sampling theorem explains the relationship between sampling frequency and signal spectrum, and is the basic basis for discretization of continuous signals.
As long as the time interval of these sampling points Δt ≤ 1/2F , the original signal f(t) can be completely restored based on each sampling value . Simply put, during the conversion process of analog / digital signals, when the sampling frequency fs , max is greater than or equal to 2 times the highest frequency fmax in the signal (fs , max ≥ 2fmax) , the digital signal after sampling can be complete The information in the original signal can be perfectly retained, and the original continuous signal can be completely reconstructed from the sampled samples . Otherwise, the original continuous signal can be completely reconstructed from the sampled samples.
The frequencies behind the sampled signal will be aliased. In general practical applications, the sampling frequency is guaranteed to be 5 to 10 times the highest frequency of the signal.
It can be known from the above theorem that if the highest frequency fH of the signal is known, it can be concluded that the lowest sampling frequency to ensure complete reconstruction of the signal is 2fH. On the contrary, if the sampling frequency is known, the highest signal frequency allowed to guarantee complete reconstruction of the signal can be derived. But it should be noted that there is a condition here, that is, the sampled signal must be band limited, that is, with a limited bandwidth. The component in the signal higher than a given frequency must be 0, or at least very close to 0. Only in this way can the influence of these frequency components be ignored when reconstructing the signal. If the sampled signal is not band-limited, the frequencies of the sampled signals will overlap, that is, components higher than half the sampling frequency will be reconstructed into signals lower than half the sampling frequency, ultimately resulting in the signal not being fully recovered.