Wireless Coding and Modulation
Coding and modulation provide a means to map digital information to the underlying signal so that a receiver can retrieve the information from the signal using appropriate decoder and demodulator. As coding and modulation directly affect the achievable capacity and data rate of the communication system, new coding and modulation techniques are constantly proposed and implemented to keep up with the demand for mobile data. This chapter will cover the basic theories and terminologies of coding and modulation in digital wireless communications.
2.1 Frequency, Wavelength, Amplitude, and Phase
Signal waveforms are the fundamental carriers of all types of data that we send over a communication system. It is therefore important to understand the basic properties of such waveforms. Figure 2.1 shows the waves that are created when a rock is thrown into the water. Similar waves are also created in wireless networking for communication purposes, but these are called electromagnetic waves and special electronics circuits are used to generate and receive them.
Copyright By PowCoder代写 加微信 powcoder
In the simplest form, a wave is mathematically represented by a sine wave 𝐴 sin(2𝜋𝑓𝑡 + 𝜃), where A = Amplitude, f = Frequency, q = Phase, and Period T = 1/f. The small t is the current time, which allows us to obtain the value of the wave at any time using this formula.
Figure 2.2 illustrates the frequency, amplitude, and phase of a sine wave. Amplitude is the height of the wave, measured from zero to the maximum value, either up or down. Note that the sine wave is cyclic, i.e., it keeps repeating the pattern. One complete pattern is called a cycle.
Frequency is measured in cycles/sec or Hertz or simply Hz. For example, if a wave completes 1 cycle per sec, like the wave shown in Figure 2.2(a), then it has a frequency of 1 Hz. On the other hand, the wave in Figure 2.2(c) has a frequency of 2 Hz.
Phase is the amount of shift from a given reference point. For example, if we consider zero amplitude as our reference point, then the waves that start at zero while gaining their amplitudes, like most of the cases in Figure 2.2, then their phase is zero. The maximum phase shift is 360°, i.e., if the wave is shifted by 360°, then its phase is back to zero again. For example, in Figure 2.2(d), phase is shifted by 45°. Usually, the phase is measured in radians, where 360°=2π radian. Therefore, a 45° phase in radian would be π/4.
Figure 2.1 Waves in the water.
Figure 2.2 Frequency, amplitude, and phase of a sine wave 𝐴 sin(2𝜋𝑓𝑡 + 𝜃). 2.1.1 2D Representation of Phase and Amplitude
The phase can be represented on a 2D graph. A sine wave can be decomposed into its sine and cosine parts. For example, a sine wave with a phase of 45°, can be written as the summation of two parts:
The first part is called the In-phase component I, and the second part the Quadrature
componentQ.Inthiscase,wehaveI=! andQ= !.IandQarethenplottedina2Dgraph √#√# !
as shown in Figure 2.3. In this figure, the x-axis is sine and the y-axis cosine. With I=√# and Q = ! , we get a single point in the graph, which has a length (amplitude) of 1 (=0𝐼# + 𝑄#), and
an angle (phase) of 45°.
Figure 2.3 Phase and amplitude representation of sin(2πft+ π/4) in I-Q graph.
2.1.2 Wavelength
Waves propagate through space and cover distances over time. The distance occupied by one cycle is called the wavelength of the wave, and it is represented by l. This is the distance between two points of corresponding phase in two consecutive cycles as shown in Figure 2.4.
In the air or space, all electromagnetic waves, irrespective of their frequencies, travel at the speed of light, which is a universal constant of 300 m/μs. Given that it takes T sec for the wave to complete a cycle (T is called the period of the wave), and that T=1/f, we have
l =cT=c/f (2.2)
Now we see that the wavelength is inversely proportional to its frequency.
Equation (2.2) is a universal formula that can be used to derive the wavelength for any types of communication medium. For example, for acoustic communications, which use sound waves to transmit data, the parameter c in Equation (2.2) should represent the speed of sound, which is only 343 m/s in dry air at 20o Celsius. Table 2.1 lists the wavelengths for some of the popular electromagnetic frequencies.
Table 2.1 Wavelengths of popular electromagnetic frequencies
Wireless Technology IoT Bluetooth/WiFi
Advanced Wi (3G/4G/5G)
mmWave and Terahertz for 5G and beyond
Example 2.1
Frequency 915 MHz 2.4 GHz 700 MHz 5 GHz 60 GHz 1.7 GHz 2.1 GHz 28 GHz 73 GHz 140 GHz 1 THz 10 THz
Wavelenth 32.7 cm 12.5 cm 42.8 cm 6 cm
5 mm 17.6 cm 14.2 cm 1 cm
4 mm 2.1 mm 0.3 mm 0.03 mm
What is the wavelength of a 2.5 GHz electromagnetic signal propagating through air?
Example 2.2
What is the frequency of a signal with 5 mm wavelength?
Wavelength = λ = 5 mm Frequency = f = c/λ
= (3 x 108 m/s) / (5 x 10-3 m) = (300 x 109) / 5 = 60 GHz
Figure 2.4 Wavelength of a sinusoidal signal.
2.2 Time and Frequency Domains
So far, we have seen how to represent waves in time domain. It turns out, that every wave can be represented in both time and frequency domains. Given the time domain representation, we can convert it to its frequency domain representation, and vice versa.
Using three different sine waves, Figure 2.5 illustrates the conversion from time domain representation (left hand side) to frequency domain (right hand side). The top sine wave has a frequency 1 (f) and amplitude A. In the frequency domain it is therefore just a pulse at frequency f (x-axis) having a height A (y-axis). The second sine wave has three times the frequency as the original one, but one third its amplitude. Therefore, in the frequency domain, its pulse is located at 3f and has a height of A/2.
The third sine wave is actually a combination of the first and the second waves. One can actually just add the wave values at each time instant to derive the third one. In the frequency domain, it therefore has pulses at two frequencies. The pulse at frequency f has a height of A and the pulse at 3f has A/2.
The transformation of a wave from time domain to frequency domain is called Fourier transform and from frequency domain to time domain is called inverse Fourier transform. There are fast algorithms to do this, such as Fast Fourier Transform (FFT) and Inverse FFT (IFFT). Most mathematical packages, such as MATLAB has library functions for FFT and IFFT. In recent years, general purpose programming languages, such as Python, are also offering library functions for FFT and IFFT.
Figure 2.5 Time domain to frequency domain conversion.
2.3 Electromagnetic Spectrum
Wireless communications use the airwaves, which are basically electromagnetic waves that can propagate through the air or even in the vacuum. Any electricity or current flow will generate these electromagnetic waves. Therefore, many things we use generate or utilize some forms of electromagnetic waves. TV, power supply, remote control, microwave oven, wireless router, etc. all use or generate electromagnetic waves of different frequencies. Even the light is basically electromagnetic waves as we use electricity to generate light.
Electromagnetic waves can have a frequency of just 10 Hz, or 300 THz! The spectrum is all of the ‘usable’ frequency ranges. It is a natural resource and like most natural resources, it is limited. Spectrum use is therefore highly regulated by government authorities, such as the FCC in the US or ACMA in Australia.
A large portion of the spectrum is reserved for various government use, such as radar, military communications, atmospheric research and so on. The rest of the spectrum is often licensed to
competing network operators, which give the operators exclusive rights to specific parts of the spectrum. For example, different TV channels or radio stations license different frequencies. Interestingly, part of the spectrum is also allocated for use without having to license it. Such spectrum is called licensed-exempt and sometimes referred to as ‘free’ spectrum. The spectrum used by Wi-Fi, such as the 2.4 GHz band, is a good example of such license-exempt spectrum. Table 2.2 lists some of the currently available license-exempt bands.
Table 2.2 Examples of license-exempt spectrum and their use
License-exempt Spectrum 433 MHz
5.2/5.3/5.8 GHz
Example use Keyless Entry
Amateur Radio, IoT (e.g., LoRaWAN) WiFi, Microwave Oven
WiFi, Cordless Phone
It is important to note that although manufacturers of any product can use license-exempt frequencies for free, they are subject to certain rules, such as power limitation for transmitting those frequencies. For example, the maximum transmit power of Wi-Fi products are often limited to about 100 mW depending on the region of operation.
Given the diverse needs of spectrum, spectrum allocation authorities must follow a set of principles when allocating spectrum. These principles include maximizing the spectrum utilization, adapt to market needs by promoting new technologies that may require some specific spectrum, promote market competition by strategically making certain spectrum license-exempt, ensure fairness in licensing spectrum among competing operators, as well as allocating spectrum to satisfy core national interests such as public safety, health, defense, scientific experiments and so on.
Certain services use specific bands of frequencies. For example, the basic Wi-Fi services use a frequency band of 2.4 GHz, which contains frequency in the range of 2.4 GHz to 2.5 GHz. Figure 2.6 shows the spectrum uses for different services. As we can see, historically wireless communications mostly use the spectrum between 100 kHz to 6 GHz. To meet the exponential demand for mobile data, the industry is now exploring spectrum beyond 6 GHz. For example, 60 GHz is now used in some Wi-Fi standards and the fifth generation (5G) mobile networks are targeting spectrum beyond 60 GHz.
Figure 2.6 Spectrum allocation for different services. Wireless communication mostly uses 100 kHz to 6 GHz
2.4 Decibels
When waves travel, they lose power. We say that the power is attenuated. The question is: what would be a practical unit to measure power attenuation that is universal in all wireless communication systems?
Power loss for electromagnetic waves can be many orders of magnitude. For example, Wi-Fi chipsets can decode signals as weak as pico Watts, which allows them to offer reasonable communication coverage and range around the house or office buildings. Now imagine a signal that was transmitted by the Wi-Fi access point at full power of 100 mW but was received at a distant laptop with only 1 pW of power. The loss is one trillion folds!
Because the power loss can be many orders of magnitude, the attenuation is measured in
logarithmic units. After the inventor , power attenuation was originally measured
as Bel, where Bel = log10( P / P ) with P representing the transmitted powered and P the in out in out
attenuated power.
Bel was found to be too large for most practical systems. Later, a new quantity called decibel, written as dB, was introduced to measure power loss, where
dB = 10log10( P / P ) in out
Example 2.3
What is the attenuation in dB if the power is reduced by half (50% loss)?
Attenuation in dB = 10 log10 (2) = 3 dB © 2019
Example 2.4
Compute the loss in dB if the received power of a 100mW transmitted signal is only 1 mW
Power is reduced by a factor of 100. Attenuation = 10log10(100)= 20 dB
The concept of decibel is also used to measure the absolute signal power, i.e., decibel can be used to measure the strength of a transmitted or received signal. In that case, it is a measure of power in reference to 1 mW and the unit is dBm. In other words, dBm is obtained as:
dBm = 10log10 (power in milliwatt)
Example 2.5
Convert 1 Watt to dBm.
Wehave 10log10(1W/1mW)=10log10(1000mW/1mW)=30dBm Example 2.6
Express 1 mW in units of dBm
10 log10 (1) = 10 ́ 0 = 0 dBm (ZERO dBm does not mean there is no power!)
Now we have a way to convert various electrical measurements, which are measured in Watts, into “networking measurements”, which is in dB and dBm. Another important networking measurement is noise, which is often produced at the receiver due to the movement of electrons in the electronic circuits. The presence of such noise can make it difficult to decode data from the received signal if the signal-to-noise (SNR) is too low. Because decibel (dB) is basically a method to measure a ratio, it is also used to measure the SNR in wireless communications.
Example 2.7
With 100 μW of noise, what would be the SNR in dB if the received signal strength is 1 mW?
P = 1 mW (received signal strength), P = 100 μW signal noise
SNR=10log10(1000/100)=10log10(10)=10dB Example 2.8
Received signal strength is measured at 10 mW. What is the noise power if SNR = 10 dB?
SNR=10dB=10log (10mW/P ) 10 noise
P =1mW noise
Now we can see that decibel is a versatile method to measure three different wireless communication phenomena, path loss, signal strength, and SNR. We have also seen that transmission or received power is measured in dBm, while path loss or antenna gain is measured in dB. It should be noted that dB can be added or subtracted to dBm, which would produce dBm again. For example, for a transmission power of 20dBm and a path loss of 25dB, the received power can be calculated as 20dBm-25dB = -5dBm. Similarly, if there was a 5dBi antenna gain for the previous example, the received power would be calculated as 20dBm+5dBi-25dB=0dBm.
It does not, however, make any sense to add dBm with dBm. For example, given P1=20dBm and P2=20dBm, it would be incorrect to say that P1+P2=20dBm+20dBm = 40dBm. The correct way to add powers would be to add them in linear scale and then convert the final value back to dBm. For the previous example, we have P1=100mW and P2=100mW, so P1+P2=100mW+100mW=200mW, which is only 23dBm!
2.5 Coding Terminology
The following terminology is often used to explain the coding of digital data on the carrier signal.
Symbol is the smallest element of a signal with a given amplitude, frequency, and phase that can be detected. Shorter symbol duration means that more signal elements carrying bits can be transmitted per second, and vice versa.
Baud rate refers to the number of symbols that can be transmitted per second. It is the inverse of symbol duration and hence sometimes referred to as the symbol rate. It is also called the modulation rate because this is how fast the property of the signal, i.e., its amplitude, frequency, or phase, can be changed or modulated.
Data rate, measured in bits per second, is the number of bits that can be transmitted per second. For example, for a binary signal, only 1 bit is transmitted for a given signal status, i.e., only 1 bit is carried over a baud and hence baud rate and data rate are equivalent. However, an M-ary signal has M distinct symbols and hence can carry log# 𝑀 bits per baud or symbol. As we will see shortly, most modern wireless modulation techniques transmit multiple bits per symbol.
2.6 Modulation
Carrier waves are usually represented in sine waves. Data can be sent over a sine carrier by modulating one or more properties of the wave. As we have learned earlier in this chapter, there are three main properties of a wave, amplitude, frequency, and phase, that we can modulate.
Figure 2.7 shows how 0’s and 1’s can be transmitted over a sine carrier by modulating one of these three properties. When amplitude is modulated, it is called Amplitude Shift Keying (ASK). Similarly, we have Frequency Shift Keying (FSK) and Phase Shift Keying (PSK). In such modulations, the value of amplitude, frequency, and phase remains constant during a fixed period called bit (or symbol) interval. The receiver observes the signal value during this bit interval to demodulate the signal, i.e., extract the bit or bits transmitted during that interval. Note that the bit interval is essentially the inverse of the baud rate.
In Figure 2.7, we used only two different values of the amplitude, frequency, or phase to represent 0’s and 1’s. Therefore, we can send only 1 bit per different value of the signal, i.e., 1 bit per baud. In practical communication systems, usually more than 1 bit is transmitted per baud. For example, if we can modulate the amplitude in a way so we have four different values of the amplitude, then we need 2 bits to represent each amplitude, enabling us to transmit two bits per baud.
Figure 2.7 Modulation of a sine wave.
For PSK, the phase values can be absolute, or the difference in phase with respect to the previous phase. In Figure 2.8, the top graph shows that when there is no change in phase from the previous bit-interval to the next, it is treated as a 0 and when there is a change it is a 1. This is called differential BPSK. With differential, the receiver does not have to compare the phase against some pre-established value, but rather observe the change only, which is easier to implement.
The top graph in Figure 2.8 also shows that the phase is shifted by 180o and there is only one value to change. The corresponding 2D (I-Q) graph shows that the two dots are 180o apart. The bottom graph shows that the phase can switch to any of the four different values, which is called Quadrature Phase Shift Keying (QPSK). In QPSK, 2 bits can be sent per baud. Here in the I-Q graph, there are four dots and they are separated by 90 degrees.
Figure 2.8 Differential BPSK and QPSK
To push the data rate even higher, we can combine amplitude and phase modulations together, which is called Quadrature Amplitude and Phase Modulation (QAM).
Note that in QPSK, the amplitude was kept constant. However, we could vary amplitude and get more than 2 bits per baud. A constellation diagram is often used to visually represent a QAM. Using constellation diagrams, Figure 2.9 shows three examples of amplitude and phase combinations to achieve different levels of QAMs. In the left most graph, we have constant amplitude, but 2 different phases. In total we have 1×2 = 2 combinations, so we get 1 bit per baud or 1 bit per symbol.
In the middle graph, we have four different phases, but just 1 amplitude. This is actually QPSK, but we could call it 4-QAM. It has a total of 1×4 = 4 combinations, so we have 2 bits per symbol. In the third graph (16-QAM), we have 3 different amplitudes; there are 4 different phases for each of the smallest and the largest amplitudes while the medium amplitude has 8 different phases. Thus, from 3 different amplitudes and 12 different phases, we use a total of 4+4+8 = 16 combinations of amplitudes and phases, which allows us to transmit 4 bits per symbol.
It is clear that we can increase the bit rates by going for higher QAMs. Table 2.3 lists the use of different QAMs in practical wireless networks, which shows that latest wireless standards employ as high as 1024 QAMs. As hardware and signal processing technology improves, we can expect even higher QAMs in the future.
Figure 2.9 Constellation diagrams illustrating 2, 4, and 16 QAMs. Table 2.3 QAMs used in practical wireless technologies
Wireless Technology 4G
WiFi 802.11n
WiFi 802.11ac WiFi 802.11ax
2.8 Channel Capacity
Supported QAM Technique 256-QAM
16-QAM, 64-QAM 256-QAM
The capacity of a channel basically refers to the maximum data rate or the number of bits that can be reliably transmitted over the channel. There are two basic theorems that explain the capacity, one by Nyquist and th
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com