Research on Digital Demodulation Technology of Analog Modulation Signal

Research on Digital Demodulation Technology of Analog Modulation Signal

Software-Defined-Radio (SDR) is a current communicaTIon research hotspot and the development direcTIon. Digital modulaTIon and demodulaTIon are important content of SDR.
The SDR demodulation generally uses digital correlative demodulation. Digital correlative demodulation is the same with analog correlative demodulation, but its calculation is complicated. Discrete Fourier Transform (DFT) is used for digital signal analysis and processing. The paper is about a DFT-based algorithm for AM (DSB, SSB and VSB with strong carrier) signal demodulation. The main idea is to filter the digitalized AM signal received with BPF, then do DFT on samples in every one (or several) carrier circle to get the amplitude of AM signal and get rid of the zero frequency current. Compared with digital orthogonal demodulation framework, it is simple and easier to be realized for it doesn't need to resume local carrier and filter with LPF in two branches. Simulation of this demodulation scheme indicates that noise -resisting property has been improved obviously. It is hopeful for this demodulation scheme to be applied to the design of AM signal digital receiver.
The paper also discussed the SSB, VSB signal modulation theory and simulation of the formation of SSB, VSB signal scheme.

KEYWORDS: software radio, digital demodulation, discrete fourier transform, Matlab

Table of Contents ................................................................................................ III
ABSTRACT ……………………………………………………………………………… IV
Chapter 1 Introduction .............................................................................. 1
1.1 Software Radio Technology ........................................................................ 1
1.1.1 Basic principles ........................................................................ 1
1.1.2 System structure ........................................................................ 2
1.2 Practical application of software radio ............................................................ 3
1.3 Modulation and demodulation technology in software radio ................................................ 5
1.4 The content of the project design ........................................................................ 6
Chapter 2 Digital Modulation Demodulation of Analog Modulation Signal .......................................... 8
2.1 Two demodulation schemes ..................................................................... 8
2.1.1 Digital quadrature demodulation ............................................................ 8
2.1.2 Digital demodulation based on DFT ………………………………………… 10
2.2 Comparison of plans ........................................................................ 11
Chapter 3 Structure of Digital Demodulation System Based on DFT ……………………………… 12
3.1 Part A / D .............................................................................. 12
3.2 Part of the band-pass filter ………………………………………………………… 13
3.3 DFT calculation part ........................................................................ 14
3.3.1 Selection of DFT formula ............................................................ 14
3.3.2 Discussion on the principle of DFT processing digital signal .................................... 15
3.4 Signal recovery part .................................................................. 16
Chapter 4 Realization and Simulation of Digital Demodulation of Analog Modulation Signal .................................... 18
4.1 Introduction to Matlab and Communication Simulation ...................................................... 18
4.2 Realization of analog modulated signal ...................................................... 19
4.2.1 AM signal modulation algorithm and implementation .......................................... 19
4.2.2 DSB signal modulation algorithm and implementation .......................................... 20
4.2.3 SSB signal modulation algorithm and implementation ................................................ 22
4.2.4 VSB signal modulation algorithm and implementation ................................................ 24
4.3 Digital demodulation method and simulation of analog modulation signal .................................... 26
4.3.1 AM signal demodulation method and simulation ................................................ 26
4.3.2 DSB signal demodulation method and simulation ................................................ 27
4.3.3 SSB signal demodulation method and simulation .......................................... 29
4.3.4 VSB signal demodulation method and simulation ................................................ 32
Chapter 5 Summary and Prospects .................................................................. 33
5.1 Advantages of plan design and areas to be improved .......................................... 33
5.2 Summary .............................................................................. 34
5.3 Gain and Experience ........................................................................ 35
Conclusion ....................................................................................... 36 References ................................................ .......................................... 37
Appendix ................................................................................................... 38

Abstract Software radio (SDR) is a research hotspot and development direction of current communication technology. Digital modulation and demodulation are an important part of SDR. SDR demodulation generally uses digital coherent demodulation. The digital coherent demodulation method is the same as the analog coherent demodulation method in principle, but the calculation amount is relatively large. Discrete Fourier Transform (DFT) is a commonly used and effective method in digital signal analysis and processing. This article proposes a digital demodulation algorithm based on discrete Fourier transform (DFT) for analog modulation signals (AM, DSB, SSB and VSB inserted into a strong carrier). The method is to band-pass filter the sampled digital signal. Perform discrete Fourier transform (DFT) according to the sampled value in each (or several) carrier period, find the amplitude of the carrier, and then remove the DC component. Using Matlab language programming, the demodulation method of the aforementioned analog modulation signal based on DFT operation was simulated, and the results show that the anti-interference performance of the demodulation scheme has been significantly improved. Compared with the digital quadrature demodulation structure, local carrier recovery is omitted, two low-pass filtering is simple and easy to implement, and it is expected to be applied in the design of digital receivers using AM signal mode.
The modulation principle of SSB and VSB signals and the methods of SSB and VSB signals generated in the simulation are also discussed.

Keywords: software radio, digital demodulation, discrete Fourier transform, Matlab

ABSTRACT

Chapter One Introduction
1.1 Software Radio Technology Software radio, as the name implies, uses modern software to manipulate and control the traditional "pure hardware circuit" wireless communication. The important value of software radio technology is that the traditional hardware radio communication equipment is only used as the basic platform of radio communication, and many communication functions are implemented by software, breaking the historical pattern that the realization of device communication functions only depends on hardware development. . The emergence of software radio technology is the third revolution in the communication field after fixed communication to mobile communication, analog communication to digital communication [1].
1.1.1 Basic principles of software radio The basic idea of ​​software radio is to convert broadband analog-to-digital converter (A / D, Analog / Digital, analog / digital) and digital-to-analog converter (D / A, Digital / Analog, digital / analog) ) As close to the antenna as possible, establish a universal, open hardware platform with the "AD-DSP (Digital Signal Process, Digital Signal Processing)-DA" model, on this platform as much as possible to use software technology to achieve various radio stations Function module [2]. Such as the use of broadband ADC (Analog Digital Convert, analog-to-digital converter) through programming to achieve the selection of various communication frequency bands, such as HF (High Frequency, high frequency), VHF (Very High Frequency, UHF), UHF (Ultra High Frequency (UHF), SHF (Super High Frequency), etc., through the software programming to complete the transmission signal sampling, quantization, coding, decoding operation processing and conversion, in order to achieve the radio frequency transceiver function; through software programming Realize the selection of different channel modulation methods, such as amplitude modulation, frequency modulation, single sideband, data, frequency hopping and spread spectrum, etc., through software programming to achieve different security structures, network protocols and control terminal functions. Software radio is a software-based, computationally intensive form of operation.
From the perspective of the technical realization of software radio, the decisive step is to apply the wideband antenna or multiband antenna to the A / D and D / A converters as close as possible to the radio frequency end, and perform A / D conversion of the entire mid-band. The processing is implemented with programmable digital devices, especially software. It can be seen that such an architecture has great versatility. It has great potential to solve the problems mentioned above, and can be used to realize a multi-band, multi-user and multi-system universal wireless communication system. To realize the above system, the antenna, high-speed A / D converter, high-speed digital signal processor and general-purpose CPU (Central Processing Unit) are all very demanding.
The above requirements were almost unachievable in the past (even some requirements are now). However, we can refer to the experience in the field of personal computers. In the early days when the concept of personal microcomputers was proposed, the computer industry also competed with different machines. There was no standard at all. Since the microelectronics technology at that time was still very backward, most people thought that it was unrealistic for an individual to own a computer. In just over a decade, the development of microelectronics technology has made personal microcomputers the most popular industry today, and those companies and countries that did not seize the opportunity in the early stages of development have fallen far behind. Now the competition in the field of microcomputers has shifted the focus to the competition of software. The personal communication system of the next century will most likely be a universal hardware platform with amazing processing capabilities and standard RF interfaces, relying on different software to provide exceptionally rich functions and services, that is to say, the communication field will experience similar personal computers The changes we experienced in the 1980s and 1990s are now the critical moment for this change.
The background of the concept of software radio is [3]:
1. The hardware technology level is improving rapidly, and the performance of A / D / A, DSP and CPU is getting better and better.
2. New communication systems and standards are constantly being proposed. The survival time of communication products is shortened and the development costs are increased. It is difficult for traditional communication systems to adapt.
3. Various communication systems coexist, and the requirements for interconnection among various systems are also becoming stronger. This is particularly prominent in military communications, and it is also the main reason for the first development of software radio in the military field.
4. The wireless frequency band is becoming more and more crowded, and the frequency band utilization rate and anti-interference ability of the communication system are constantly increasing. Along with the development direction of the current communication system, it is difficult to re-plan the frequency band, and adopting a new anti-jamming method requires major changes to the system structure, which is very expensive.
1.1.2 The system structure of software radio can be seen from Figure 1.1, the so-called software radio, its key ideas and the main difference from the traditional structure are:
1. Move A / D and D / A closer to the RF (Radio Frequency) end. Move from baseband to intermediate frequency. Sampling the entire system band.
2. Use high-speed DSP / CPU instead of traditional dedicated digital circuits and low-speed DSP / CPU to do a series of processing after A / D. The above two points are just structural differences.

Figure 1.1 Schematic diagram of the software radio system

With the development of microelectronic technology, the performance of various digital devices continues to improve, and the existing digital radio will continue to develop, which will also make A / D / A approach the RF end step by step. So will software radio be just a further development of digital radio? The answer is no. We believe that the further development of software radio and digital radio is conceptually different. This is mainly because the move of A / D / A to the RF end only provides the essential conditions for the realization of software radio, and the really critical step is to use general programmable devices with strong programmable capabilities (DSP, CPU, etc.) instead. Dedicated digital circuit. The resulting series of benefits is the real purpose of software radio.
The ultimate goal of software radio is to free the communication system from the constraints of the hardware system structure. Under the circumstance that the system structure is relatively universal and stable, various functions are realized through software, which makes the improvement and upgrade of the system very convenient and low cost, and different systems can be interconnected and compatible. The further development of digital radio cannot do this [4]. It can only lead to more reliance on hardware and system architecture. However, at present, software radio appears more in the form of a concept and conjecture, and the specific definition and architecture are inconclusive. It can be said that in addition to the two key ideas mentioned above being generally accepted, the contents of other aspects are being debated and discussed. What this article discusses is the digital modulation and demodulation technology in software radio, no further discussion will be done in other aspects.
1.2 The practical application of software radio As an emerging technology, software radio currently has the following application areas:
1. Cellular mobile communication system In a cellular mobile communication system, the base station and the mobile terminal use a software radio structure, the hardware is simple, and the functions are defined by the software. The radio frequency band, channel access mode and channel modulation are programmable. In this system, the transmission of software radio is different from other systems. It divides the available transmission channels, detects the propagation path, performs modulation for the channel, electronically controls the transmission beam to point in the correct direction, selects the appropriate power, and then transmits. The same is true for reception. It can divide the energy distribution of the current channel and adjacent channels, identify the mode of the input transmission signal, adaptively cancel interference, estimate the dynamic characteristics of the required signal multipath, and perform coherent combination on the multipath required signal. With adaptive equalization, the channel modulation is grid-decoded, and then FEC (Forward Error Correct, Forward Error Correction) decoding corrects the remaining errors to reduce the bit error rate as much as possible. In addition, software radio can increase value-added services through many software tools. These software tools can help analyze the radio environment, define the required additional content, and test the model of value-added services developed by software in a wireless environment, and finally open the value-added services through software or hardware.
2. Smart antennas Smart antennas were originally used in the fields of radar, sonar, and military communications. Due to price and other factors, they have not been popularized in other communication fields. In recent years, digital signal processing technology has developed rapidly, the processing power of digital signal processing chips has been continuously improved, and chip prices have been accepted. At the same time, the use of digital technology can form an antenna beam in the baseband, replacing the analog circuit, improving the reliability and flexibility of the antenna system. In China's TD-SCDMA (Time Division-Synchronous Code Division Multiple Access, Time Division Synchronous Code Division Multiple Access) scheme, the base station uses smart antenna technology and digital signal processing technology to identify the direction of user signal arrival and form the antenna main beam.
After introducing the SDMA (Spase Division Multiple Access, space division multiple access) mode, different spatial channels are provided according to different spatial propagation directions of user signals. The digital method is used to weight the processing of the received signals of the array elements to form a wireless beam, and the main beam is directed to the direction of the user signal. The direction of the interference signal forms zero defects or lower power gain in the antenna direction to achieve the purpose of suppressing interference.
The advantage of using smart wireless is that the result of wireless beamforming is equivalent to increasing the gain of the antenna; after the antenna beamforming, it can greatly reduce multipath interference; the direction of signal arrival provides user terminal position information for user positioning ; Replace high-power amplifiers with multiple low-power amplifiers, reducing base station costs and improving equipment reliability.
3. Multi-frequency multi-mode mobile phone In the ACTS FIRST (Acoustic Control and Telemetry System) project in Europe, software radio technology is used to design multi-frequency / multi-mode (compatible with GSM (Global System of Mobile System)), CDMA (Code Division Multiple Access) and most existing analog systems) programmable mobile phones. It can automatically detect the received signal and access different networks, and can meet the requirements of different connection times. Software radio technology can use different software to realize various functions of different radio equipment, can change the channel access mode or modulation mode arbitrarily, and use different software to adapt to different standards, form a multi-mode mobile phone and multi-function base station, with a high degree of flexibility.
With its appearance, the development of communications has undergone three changes from fixed to mobile, from analog to digital, and from hardware to software. Software radio technology is becoming more and more widely used in the field of mobile communications. During the transition from the second generation mobile communication system to the third generation mobile communication system, software radio technology will play an important role.
4. Satellite communication In today's communication field, satellite communication is one of the most important communication methods. However, due to the wide variety of equipment in the satellite communication system and the complicated equipment management and maintenance work, the satellite communication system has a long replacement cycle and cannot be well adapted to the pace of modern high-tech development. The software radio can solve the problems of satellite communication systems with its software-defined functions and open modular structure. Therefore, it is very meaningful to study satellite communication systems with software radio characteristics.
In satellite communication systems, system functions mainly refer to multiple access methods, network structure, networking protocols, and communication services; while device functions refer to interface standards, modem methods, channel coding methods, source coding methods, information rates, and multiplexing By way of waiting. The idea of ​​software radio technology is to adopt advanced technical means, so that the above functions can be defined by software. Through the friendly man-machine interface, people can change the function of the communication system in real time without changing the hardware equipment, so that the system can adapt to various application environments, so it has strong applicability and flexibility.
Considering the characteristics of satellite communication frequency bandwidth, high information rate and wide range of changes, at the current computer technology level, if the device functions are all implemented by software, due to the characteristics of the software's operation instructions one by one, even if multiple processors are used to coordinate It is also impossible to realize real-time processing at high information rates due to calculations, which limits its use in satellite communications.
The application prospect of software radio technology in the field of commercial communication is very broad. At present, software radio technology has been applied in 800MHz commercial cellular radio frequency bands, satellite communication and other fields. As a strong structural framework, it helps us provide advanced, Economic wireless business. Software radio also has some shortcomings, such as the difficulty in designing wide-band, low-loss antennas and radio frequency converters; it is difficult to estimate the demand for processing power in practice and the configuration of reprogrammable DSP / CPU processing power; it is difficult to guarantee internal processing The data rate of the controller interface. At present, there are no open structural standards for key components of the software radio structure. The DSP function library cannot mix and match real-time software from different software vendors like the mixing and matching VME (Versamodel Eurocard, a traditional telecommunications equipment bus) board. However, with the rapid development of modern communication technology, most of these shortcomings can be avoided, and at the same time overcoming these obstacles, it can further reduce costs and enable software radio stations to be put on the market as soon as possible.
1.3 Modulation and demodulation technology in software radio Modulation and demodulation technology has been continuously developed and improved in recent decades. In general, it can be divided into two categories: single-tone modulation and multi-tone modulation. The single tone modulation method uses input data to modulate different components of a single carrier (such as amplitude, frequency, phase, etc.) at a certain time, so it is also called single carrier modulation. Multi-tone modulation usually divides the original channel into multiple orthogonal sub-channels at equal intervals, and each sub-channel uses a different carrier for modulation. Therefore, multi-tone modulation is also called multi-carrier or multi-channel parallel modulation, sometimes also called OFDM (Orthogonal Frequency Division Multiplexing, Orthogonal Frequency Division Multiplexing).
Because the single carrier modulation technology is relatively mature, the current data communication system mostly uses this modulation method. However, since Weinstein, Ebert and others proposed to use DFT for frequency division multiplexing in multitone modulation systems in 1971, multitone modulation technology has received more and more attention [5]. Compared with single-tone modulation, it has the following characteristics: the maximum transmission rate obtained by using a multi-tone modulation scheme and a single-tone modulation scheme using decision feedback equalization is approximately equal. However, for channels with distortion, fading, or non-white noise, multi-tone modulation can achieve higher transmission rates; because multi-tone modulation has the characteristics of multi-channel parallelism, its modulation signal does not require any special at the receiving end. The processing can obtain the signal-to-noise ratio or signal-to-interference ratio equivalent to that obtained by the single-tone modulation and demodulation system at the receiving end; in order to obtain better transmission performance, you can use equalization in a multi-tone modulation system Technology, because the channel characteristics in each narrow-band sub-channel are approximately linear and the impulse response tailing is less, the equalization of multi-tone modulation is much simpler than that of single-tone modulation; phase jitter Will cause the signal to rotate in space, which seriously affects the decision: in a multi-tone modulation system, the distortion caused by phase jitter is evenly distributed in each sub-channel, so that its impact is greatly reduced; at the same transmission rate In the case of multi-tone modulation system, the longer symbol period makes the impact of pulse interference on it much weaker than that of single-tone modulation Impact; In a single tone modulation system, it is more sensitive to single frequency interference, while in a multitone modulation system, each sub-channel can transmit different numbers of bits according to their respective signal-to-noise ratio, and can close channels with severe interference, which can both Make full use of the frequency band, and can overcome a variety of interference.
The modulation and demodulation of signals in software radio is one of the key issues of research. On the common hardware platform, using different software algorithms to achieve different modulation and demodulation is the core idea of ​​software radio.
In software radio systems, both modulation and demodulation are implemented by programs (also called fully digital modulation and demodulation). To write modulation and demodulation software for various types of modulated signals, the key is to determine the signal processing algorithm. You can use FPGA (Field Programmable Gate Array, field programmable logic device) to achieve the required modulation and demodulation algorithm, its calculation speed is faster than DSP, but the flexibility and control functions are poor, and it needs to be used in conjunction with DSP or single chip microcomputer. The latest technology is to use DFT (Discrete Fourier Transform, Discrete Fourier Transform) to implement a digital modulation and demodulation algorithm. This is a method that does not require a local carrier. This article will focus on the introduction.
One way to establish modem algorithms and procedures is to softwareize the working principle of analog circuits. For example, the method of coherent demodulation of AM (Amplitude Modulation) signal, or the establishment of carrier synchronization, multiplier, low-pass filtering and other software modules is feasible, but it is very computationally intensive. In fact, according to the characteristics of software radio, a modulation and demodulation algorithm that differs from the working principle of the modulation and demodulation circuit can be established. What this question proposes is a different modulation and demodulation algorithm-a digital demodulation algorithm for analog modulation signals based on DFT.
In summary, the modulation and demodulation of signals in software radio is one of the key issues of research.
1.4 Project design content and research purpose and significance Digital modulation refers to the use of software to generate a sample sequence of modulated signals, and then through D / A conversion to obtain an analog modulated signal, digital demodulation refers to the A / A D conversion, and then demodulate the signal through data processing. Digital modulation and demodulation is an important content in SDR (Software-Defined Radio, software radio technology). SDR mainly relies on software to complete various functions of the receiving system, such as modulation and demodulation, smart antenna, signal identification, etc. Its advantage is that it can greatly simplify the hardware of the product, greatly improve the reliability, and facilitate production and maintenance. You can update the software To achieve product function upgrades, etc. SDR is an important research field and development direction of current communication technology.
The title of this graduation project is: Specific Requirements for the Research of Digital Demodulation Technology for Analog Modulation Signals:
1. To study the basic concept of SDR, focusing on digital demodulation technology.
2. Design an AM signal digital demodulation algorithm based on discrete Fourier transform
3. Program the AM signal with MATLAB language and realize digital demodulation.
4. Study the digital modulation and demodulation methods of DSB, SSB and VSB signals.
The research content of this subject involves the digital demodulation method in software radio. The author introduces the existing digital signal demodulation method based on discrete short-time Fourier transform (DSTFT) into the widely used demodulation in communication systems and carries out To improve, a digital demodulation method for AM signals based on discrete Fourier transform (DFT) is proposed. The author also used MATLAB language programming to simulate the demodulation system to verify its feasibility and anti-interference performance (signal-to-noise ratio). The significance of the DFT-based AM signal digital demodulation method lies in two aspects. First of all, this demodulation method does not require the local carrier extraction and two low-pass like the existing digital orthogonal demodulation method to demodulate the AM signal. The digital filter greatly reduces the calculation amount of the demodulation algorithm; secondly, compared with the traditional AM signal demodulation method, the performance of the demodulator is improved to a certain extent under the same input signal-to-noise ratio.
The article is arranged as follows: we will talk about two commonly used digital modulation demodulation schemes in the second chapter: digital quadrature demodulation and DFT-based digital demodulation, and compare the schemes, and finally choose an obvious Solution-digital demodulation based on DFT. The third chapter discusses the system structure of the digital demodulation scheme based on DFT, and introduces the functions and theoretical knowledge of each part in detail. Starting from the fourth chapter, the modulation algorithm and demodulation algorithm of analog modulation signal are discussed, and the simulation is realized with Matlab. Finally, it summarizes the deficiencies in the design process and the areas to be improved, analyzes the prospects of the design, and prospects the next step.

Chapter 2 Digital Demodulation Scheme of Analog Modulation Signal
2.1 Two kinds of demodulation schemes Software radio has become a new subject of modern communication technology research. Its basic idea is to implement different communication functions by installing different software on a common hardware platform. Software radio has an open modular structure, which is mainly composed of broadband A / D & D / A, programmable DSP module, narrowband A / D & D / A, user terminal, etc. [6]. During reception, the signal from the antenna undergoes RF processing and conversion, is digitized by wideband A / D, and then realizes various required signal processing through a programmable DSP module, and sends the processed data to a multi-function user terminal; similarly, Data can also be transmitted through the antenna through a similar process. In addition, using online and offline software, software radio can also realize the analysis and management of the communication environment, as well as business and performance upgrades. One of the main features of software radio is complete programmability, that is, RF frequency band and bandwidth, channel access method, transmission rate, interface type, service type, encryption method, etc. can be changed by software programming.
The open modular structure of software radio provides a good software and hardware platform for the realization of modulation and demodulation, but at the same time it also puts forward higher requirements for modulation and demodulation, that is, the modulation and demodulation method used can be adapted to different Bandwidth and different transmission rates. For this reason, how to design a modulation and demodulation technology to meet the needs of software radio will be the main issue to be discussed in this article. Various technologies of contemporary wireless communication are developing rapidly, and there are many modulation methods of various communication systems, including AM, FM (Frequency Modulation, frequency modulation), DSB (Double Side Band, double side band modulation signal), FSK (Frequency Shift Key, frequency Shift keying), PSK (Phasic Shift Key, phase shift keying), etc., its multiple access method is sometimes divided into multiple access, frequency division multiple access and code division multiple access. The differences in the modulation methods, multiple access methods, communication protocols, etc. of each communication system cause the differences between the receivers of each communication system. A receiver can only meet certain specific needs, but cannot meet various needs, so it increases the reception. The versatility of the machine is very meaningful. Among them, the receiving and processing of signals is the key to achieving universality. In order to achieve versatility, this article uses two commonly used methods of digital demodulation to process the signal.
2.1.1 Digital quadrature demodulation The following first introduces the digital quadrature demodulation scheme. Digital quadrature demodulation schemes are widely used in software radio receivers [5]. For AM signals, the baseband demodulation algorithm is. The data extraction of the output of the LPF (Low Pass Filter) is because the sampling rate required by the baseband signals I (In-phase) and Q (Quadrature) is much lower than the sampling rate of the modulation signal. This demodulation scheme eliminates the complex carrier synchronization process by using the square sum square root operation that can be implemented in the software, which not only reduces the amount of calculation, but also avoids the demodulation error (phase synchronization error and The relatively small frequency synchronization error does not affect the demodulation effect). Because it is still coherent demodulation, this demodulation scheme has good anti-interference performance.
Traditional quadrature demodulation circuits use analog devices. The inherent errors introduced by a series of analog devices reduce the performance of the quadrature demodulator, such as gain balance, quadrature balance, DC offset, impedance matching, and LO leakage etc. Nowadays, the traditional analog demodulation mechanism is being gradually replaced by digital demodulation, thereby improving the stability of the system and the flexibility of signal analysis.
Figure 2.1 Digital quadrature demodulation scheme

The quadrature modulation and demodulation algorithm can use almost the same hardware circuit, and only by performing different software processing on the amplitude and phase can the demodulation of different modulation signals be met to meet the design requirements of software radio. In practice, the structure of FPGA and DSP working together is generally adopted. FPGA mainly completes the work that needs to be completed at high rates such as down-conversion and filter extraction, which is the hardware general part of the demodulation algorithm. DSP is responsible for the amplitude and amplitude of different modulated signals. Phase is the final software demodulation algorithm. In summary, the orthogonal demodulation algorithm has certain theoretical and practical application value.
Figure 2.1 is a basic model of a digital quadrature demodulation circuit. This is a demodulator through digitization. Different types of modulated signals require different baseband demodulation algorithms. For AM signals, the baseband demodulated signal needs to complete sampling data sampling (lower the sampling frequency of the output signal of the low-pass filter), calculate the carrier amplitude, etc. The input analog intermediate frequency signal is first subjected to AD conversion to achieve digital sampling. The data stream is divided into two channels through a digital multiplier to multiply the cos component and sin component generated by the local digital oscillator to realize the movement of the input signal in the frequency domain, namely The carrier frequency is zero, and then enters the digital low-pass filter and extracts according to the signal bandwidth to obtain two baseband signals of the in-phase component I and the quadrature component Q, thereby realizing the down conversion of the intermediate frequency signal and the acquisition of the two orthogonal baseband signals . The above process can be explained in the form of a mathematical expression, and the signal after AD conversion is expressed in an orthogonal form:
(2.1)
Among them are the in-phase component and the quadrature component of the signal, respectively, the carrier frequency of the input intermediate frequency signal, and the digital oscillator signal (2.2)
Multiply to achieve down conversion:

(2.3)
After filtering the second harmonic component through the digital low-pass filter, the expected two baseband signals can be obtained:
(2.4)
According to different signal processing requirements, the output result is further processed by FPGA plus DSP. Due to the application of digital local oscillator, digital mixing and digital filter, the stability of the circuit is well guaranteed. By changing the frequency and phase of the digital local oscillator and the passband characteristics of the digital filter, the input can be obtained conveniently and flexibly The amplitude and phase characteristics of the intermediate frequency signal, and has a good consistency.
Orthogonal demodulation technology is widely used in radar, sonar, communications and other fields. The fully digital quadrature demodulation method based on direct intermediate frequency sampling technology has been widely researched and applied in recent years.
2.1.2 DFT-based digital demodulation literature [2] proposes an amplitude modulation signal (AM) digital demodulation algorithm based on the discrete Fourier transform (DFT) algorithm in software radio, by sampling the digitized AM signal After the band-pass filter, the discrete Fourier transform (DFT) is performed according to the sampling value in each (or several) carrier period to find the amplitude of the carrier, and the DC quantity is removed.与数字化正交解调结构相比，省去了本地载波恢复，两路低通滤波，简单而易于实现，该解调方案仿真结果表明抗干扰性能也有所改善，可望在采用AM信号方式的数字化接收机的设计中得到应用。
基于DFT的AM信号数字化解调框图如图2.2所示。与图2.1相比较，去掉了复杂的载波恢复，不需要进行两路低通滤波，解调过程实现起来容易得多。
AM信号是使载波信号的包络输入调制信号呈线性对应关系，在接收方只要能够将载波的包络值提取出来即可恢复原来的调制信号。

图2.2 基于DFT的AM信号数字化解调框图

在该算法中，首先对采样后的数字化AM信号进行带通滤波，取出带外信号，并对噪声有一定的抑制效果，然后进行DFT解调恢复；设采样频率AM信号的载波频率的m倍，即每个载波周期采样m个点，m是大于或等于3的整数，每取得m格采样数据（记为）进行一次DFT，求出载波幅值：
（2.5）

（2.6）

（2.7）
由DFT的性质可知，序列就是AM信号包络的采样值，只要求出并去掉其中的直流成分便可正确恢复原信号。
2.2 方案比较前面所述的数字化正交解调方法存在如下缺点:
首先要进行本地载波恢复，且本地载波和信号载波之间的频率偏差超出一定范围时，会导致信号超出数字信道而发生失真；
其次计算量大，因为对每一个采样值要分两路乘法和阶数较高的低通滤波：与本地载波的相乘之后要进行两路低通滤波器来提取同相分量和正交分量；
最后是由于这种解调方案只是对传统的相干解调方法的数字化实现，其抗干扰性能没有得到改进。其本振、混频、低通滤波均采用模拟技术实现，数字化在I、Q基带信号生成之后进行。由于模拟器件的一致性及稳定性等因素，两路正交通道间幅度一致性及相位正交性难以做得很好；此外，基带采样还容易受零漂、1/f噪声的影响。这些将导致系统性能的下降。而本文后面所提出的基于DFT的AM信号数字化解调方法则省去了载波恢复，码元确定时，是一种简单实用的算法。
与数字化正交解调结构相比，基于DFT的数字化解调方案计算量大大降低，对采样数据基本上只做加减运算，每8个采样点才做一次平方、开方运算；采用较低的采样频率也可以正常解调；省去了本地载波恢复，两路低通滤波，简单而易于实现，该解调方案仿真结果表明抗干扰性能也有所改善，可望在采用AM信号方式的数字化接收机的设计中得到应用。

第三章基于DFT的数字化解调系统结构如第二章所述，基于DFT的数字化解调系统结构分为四个主要的部分，A/D，带通滤波器，DFT运算以及信号恢复部分。
3.1 A/D部分随着通信技术的迅速发展以及计算机的广泛应用，利用数字系统处理模拟信号的情况变得更加普遍。数字电子计算机所处理和传送的都是不连续的数字信号，而实际中遇到的大都是连续变化的模拟量，模拟量经传感器转换成电信号的模拟量后，需经模/数转换变成数字信号才可输入到数字系统中进行处理和控制，因而作为把模拟电量转换成数字量输出的接口电路A/D转换器是现实世界中模拟信号向数字信号的桥梁。
采样是模拟信号数字化的第一个步骤，研究的重点是确定合适的采样频率，使得既要能够从采样信号(采样序列)中无失真地恢复原模拟信号，同时又尽量降低采样频率，减少编码数据速率，有利于数据的存储、处理和传输.从”在采样信号的频谱中要完整地保留原模拟信号的频谱”的要求出发，提出了两种采样方式，即低通信号采样和带通信号采样。
1.低通信号采样:如果被采样信号是(或者看成是)低通信号，则只要选取采样频率( 是模拟信号的截止频率)，则采样信号的频谱中就完整地保留了原模拟信号的频谱，只要让采样信号通过一个理想低通滤波器就能够无失真地恢复原模拟信号。如果，则称为过采样.
2.带通信号采样(欠采样):如果被采样信号是带通信号(中心频率大于带宽)，则只要按照下面的条件
(3.1)

来选择采样频率。
其中， 是带通信号的下限频率， 是带通信号的上限频率，n是的整数部分，也可记为表示带通信号的带宽，则采样信的频谱中就完整地保留了原带通信号的频谱，只要让采样信号通过一个理想带通滤波器就能够无失真地恢复原带通信号。对于窄带信号欠采样的采样频率只需要稍大于带宽的2倍就行了。各种已调波信号实际上都可以看作是带通信号，是否可以按照带通信号采样的情况来确定一个很低的采样频率，从而取得很低的数据速率呢?作者认为答案是否定的。因为在软件无线电中，对已调波信号进行采样的目的不是要从采样信号去恢复原来的信号，而是要用采样信号来进行解调(通过欠采样直接恢复基带信号只是特例，几乎没有实际意义)，所以对采样速率的选择必须考虑到解调过程(包括位同步等操作)的要求，而不能盲目地套用带通信号采样的公式。
现代应用中经常要求对模拟信号采样，将其转换为数字信号，然后对其进行计算处理，最后再重建为模拟信号。而基于DFT的模拟调制信号数字化解调的方案采用MATLAB作为仿真工具：取AM信号为多音信号，频率为：1000Hz，2000Hz，3000Hz。载波频率为：10000Hz，根据采样定理，我们取采样频率至少为20000Hz。
具体信号为：
（3.2）
载波信号：
（3.3）
AM信号为：
（3.4）
3.2 带通滤波器部分数字滤波器是具有一定传输选择特性的数字信号处理装置，其输入、输出均为数字信号，实质上是一个由有限精度算法实现的线性时不变离散系统。它的基本工作原理是利用离散系统特性对系统输入信号进行加工和变换，改变输入序列的频谱或信号波形，让有用频率的信号分量通过，抑制无用的信号分量输出。
数字滤波器和模拟滤波器有着相同的滤波概念，根据其频率响应特性可分为低通、高通、带通、带阻等类型，与模拟滤波器相比，数字滤波器除了具有数字信号处理的固有优点外，还有滤波精度高(与系统字长有关)、稳定性好(仅运行在0与l两个电平状态)、灵活性强等优点。数字滤波器按单位脉冲响应的性质可分为无限长单位脉冲响应滤波器和有限长单位脉冲响应滤波器两种。
基于MATLAB的信号处理工具箱为数字滤波器设计带来了全新的实现手段，设计快捷方便，仿真波形直观。上述三种设计方案均可实现设计指标，但以直接原型变换法最为简便。实际应用中，数字滤波器也可以对连续时间信号进行处理，但需要先对连续信号进行A/D变换，经数字滤波后，再经D/A转换得到所需要的连续信号。
MATLAB提供了多种FIR数字滤波器的设计方法。选用ParksMcClellan最优滤波器设计是在与其他类型的滤波器进行仿真比较后决定的。作者用窗函数法中的fir1函数进行设计，滤波后的波形延迟比较大，而且在稳定区内的波形也有所削弱。用基于最小二乘约束设计方法的fircls函数进行设计，仿真结果表明码元稳定区的波形幅度有所减少，而采用cremez函数设计出来的滤波器是非线性相位的，升余弦函数则主要是低通滤波。所以选用了ParksMcClellan设计算法。将上述带通滤波器应用于AM信号的数字化解调仿真系统，取得了比较满意的结果。
3.3 DFT运算部分傅立叶变换在通信与控制系统的理论研究和实际应用之中，采用频率域（频域）的分析方法比经典的时间域（时域）方法有许多突出的优点。当今，傅里叶分析方法已成为信号分析与系统设计不可缺少的重要工具。20世纪70年代，出现的各种二值正交函数（沃尔什函数），它对通信、数字信号处理等技术领域的研究提供了多种途径和手段。使人们认识到傅里叶分析不是信息科学与技术领域中唯一的变换域方法。
DFT开辟了频域离散化得到了，使数字信号处理可以在频域采用数字运算的方法进行，这样就大大增加了数字信号处理的灵活性。更重要的是DFT有多种快速算法。统称为快速傅立叶变换（FFT，Fast Fourier Transform）。从而使信号的实时处理和设备的简化得以实现。因此，时域离散系统的研究与应用在许多方面取代了传统的连续时间系统。所以说，DFT不仅在理论上有重要的意义，而且在各种信号的处理中亦起着核心的作用。
3.3.1 DFT公式的选择目前关于DFT存在着两套公式，在大多数著作和文献中给出的公式为:
（3.5）

（3.6）

在少数著作中给出的公式为:
（3.7）

（3.8）

这两套公式是成对的，即每套公式中正变换成立，则逆变换也一定成立，反之亦然。它们的差别仅仅在于系数I/N的位置不同，前者将之放在DFT中，而后者将之放在IDFT中。那么它们有什么区别呢?应当如何选择呢?实际上，公式(3.5)采用的是零延拓原理，是对有限序列采用傅立叶变换了分析而得到的，它代表FT谱的”抽样值”，反映有限长序列的总体量，与序列的长度有关；而公式(3.7)是时限信号x(t) ( )周期延拓后的傅立叶系数近似值，它反映的是信号的平均性质，与序列的长度无关。在具体选用时，应根据实际情况来选择使用哪种公式。
在AM信号的数字化解调算法中，采用的DFT公式是(3.7)，好处在于:该公式的计算结果(载波的幅值)与有限序列的长度无关，即其值不会随着DFT点数的增多而大幅度变化，这样概念上就比较直观、正确，同时可以避免由于采样点数太多而发生数据溢出(求和可以分段进行，对每段的和先除以系数再相加，避免总的和数发生溢出)。当然如果确信不会发生求和数据发生溢出，采用公式(3.5)也是可以的。
3.3.2 DFT处理数字信号原理讨论首先需要了解DFT处理数字信号的过程和原理对幅值A ,频率为、初相为的正弦波按采样频率f进行均匀采样，每周期采样点数，则得到时域离散周期序列，其主值序列为:
（3.9）

其中n=0,1，…N-1
根据定义对x(n)进行离散傅立叶变换(这里选取公式3.7)，为保证频谱分析的准确性，取变换区间长度为N，则有:
（3.10）
其中，
根据该计算式可得到频域的离散序列X(0),X(1) …. X(N-1)当时域波形为正弦函数时，它所对应的傅立叶变换是一对冲激函数，即只有基波分量X（1）不为零.将带入公式(3.7)，并根据欧拉公式作数学推导，求解X(1):
（3.11）
（3.11）即是基波分量的数学表达式，首先可以看出频域基波分量的幅度只与时域波形的峰值存在线性关系，而与采样点数不存在关系。其次，还可以发现初相位对基波的幅度没有影响，这一点很有意义。
为了分析相频特性，将X(1)分解成实部与虚部:

（3.12）
可以解得正弦信号的幅值为

（3.13）

由此可知DFT运算完全可以提取出正弦信号的幅度信息。
由（3.13）式可以看出，解出的正弦信号的幅度值与（2.5）式得出的结果是一样的。由此可知，基于DFT的数字化解调系统方案是可行的。
3.4 信号恢复部分在软件无线电、雷达等系统中，通常需要对带通信号进行数字化，为了降低后续数字信号处理的数据量，可以采用均匀欠采样和非均匀采样技术。关于均匀采样技术，很多文献都有论述对于非均匀采样，由于信号的采样间隔不均匀，传统的均匀采样定理不再适用。如何选取采样参数及如何根据非均匀采样序列重建带通信号是这种处理方法的一个基本问题。文献用多维线性系统理论研究了带限信号的m阶采样和重建问题。文献讨论了带通信号的M阶非均匀采样理论，它采用M个相互错开的均匀采样序列以2/m倍的信号带宽为采样频率对信号进行采样。文献[5]中采样频率为带通信号非均匀采样的最小采样频率（奈奎斯特速率），当m为奇数时，要求信号下截止频率为信号带宽的整数倍；当m为偶数时，对信号的频带位置没有要求。本文将带通信号非均匀采样的采样频率范围进行了拓宽，研究了频谱混叠和信号重建。
对于带限信号，只要满足采样定理，就可以用时域上的采样信号完整的重建出来。对于时限信号，可以通过频域上的采样样本X（k/T）完整的表示出来，这里T为信号的时域长度； 对于时限信号，可以对时域上有限个采样样本或者频域上有限个采样样本完整的表示出来。也就是说，可以通过对时域上有限个样本进行变换，得到频域上的表示，并且不会丢失任何信息。 因此，我们可以对时域信号进行加窗，到达近似时限信号。直接将信号截断，相当于矩形窗。只有采用了加窗之后，才能实现时限，这样才能在频域上的谱线表示（采样表示）出来。因此，才能使用DFT/FFT进行计算。
本文对低通和带通信号的采样及重建进行了理论分析，指出当用最低采样频率2B(2倍信号带宽)进行采样时，如果信号的边缘频率分量(即信号的最高及最低频率分量)为冲激函数，则大多数条件下不能精确重建原信号，而如果边缘频率分量为有限值，尽管此时信号频谱发生混叠，仍然能够精确重建原信号．结论不仅适用于带通信号，也同样适用于低通信号．
实际应用中，一方面，不论带通信号还是低通信号，如果已知待采样信号的边缘频率分量不含冲激函数，采样频率可以选择2B，此时的频谱虽然发生了混叠，仍然能够重建原信号;另一方面，如果预先无法得知待采样信号的边缘频率分量是否含有冲激函数，选择的采样频率最好大于2B，这样就不会引起频谱混叠，且当边缘频率为冲激函数时，也能精确重建原信号．
带通信号广泛应用于通信、雷达、声纳等领域，在这些领域中常常需要对信号进行数字化处理。传统的数字化方法是对信号进行均匀采样，均匀采样理论已很成熟。另一种方法是对信号进行高阶周期性非均匀采样，由于信号的采样间隔不均匀，传统的采样定理不再适用，如何根据非均匀采样序列重建带通信号是这种信号处理的一个基本问题。许多文献都对这个问题进行了探讨，它们都是从消除信号的频谱迁移项之间频谱混叠出发讨论信号的重建问题。文献用多维线性系统理论讨论了带限信号的广义采样问题，本文将文献中的广义采样定理从带限信号扩展到带通信号，讨论了非均匀采样时带通信号的重建问题;将带通信号重建像函数计算变成了一个线性方程组求解问题，利用克拉默法则(Cramer's rule)，通过求解线性方程组得出重建像函数;最后给出了计算机仿真实例。

第四章模拟调制信号数字化解调实现与仿真
4.1 MATLAB简介与通信仿真
MATLAB语言是一种广泛应用于工程计算及数值分析领域的新型高级语言，自1984年由美国MathWorks公司推向市场以来，历经十多年的发展与竞争，现已成为国际公认的最优秀的工程应用开发环境。MATLAB功能强大、简单易学、编程效率高，深受广大科技工作者的欢迎。
在欧美各高等院校，MATLAB已经成为线性代数、自动控制理论、数字信号处理、时间序列分析、动态系统仿真、图像处理等课程的基本教学工具，成为大学生、硕士生以及博士生必须掌握的基本技能。
MATLAB特点:
1.数值计算和符号计算功能
MATLAB的数值计算功能包括：矩阵运算、多项式和有理分式运算、数据统计分析、数值积分、优化处理等。符号计算将得到问题的解析解。
2.MATLAB语言
MATLAB除了命令行的交互式操作以外，还可以程序方式工作。使用MATLAB可以很容易地实现C或FORTRAN语言的几乎全部功能，包括Windows图形用户界面的设计。
3.图形功能
MATLAB提供了两个层次的图形命令：一种是对图形句柄进行的低级图形命令，另一种是建立在低级图形命令之上的高级图形命令。利用MATLAB的高级图形命令可以轻而易举地绘制二维、三维乃至四维图形，并可进行图形和坐标的标识、视角和光照设计、色彩精细控制等等。
4.应用工具箱基本部分和各种可选的工具箱。基本部分中有数百个内部函数。其工具箱分为两大类：功能性工具箱和学科性工具箱。功能性工具箱主要用来扩充其符号计算功能、可视建模仿真功能及文字处理功能等。学科性工具箱专业性比较强，如控制系统工具箱、信号处理工具箱、神经网络工具箱、最优化工具箱、金融工具箱等，用户可以直接利用这些工具箱进行相关领域的科学研究。
MATLAB与通信仿真一般来说，通信电路与系统仿真过程可以分为五个步骤：
1.系统建模：根据要分析的通信电路与系统，建立相应的数学模型。
2.仿真算法：找到合适的仿真算法。 MATLAB已经被确认为准确、可靠的科学计算标准软件。
3.仿真语言：应用仿真语言编写计算程序。MATLAB语言有非常突出的优点，是通信电路与系统仿真首选的仿真语言。
4.仿真计算：根据初步的仿真结果对该数学模型进行验证。
5.系统仿真：进行系统仿真，并认真地分析仿真的结果。
仿真算法、仿真语言和仿真程序构成了数字仿真软件。数学模型的正确性、仿真算法的可行性、仿真程序的准确性和可靠性，最后编制成一个成熟的仿真软件。
通信电路与系统仿真在教学实践中应用越来越普遍。对于改进教学效果、给学生提供形象化的信息、激发学生的学习兴趣、提高学生的自学能力、加强学生对授课内容的理解等无疑是十分有益的。有利于对学生分析问题的能力和解决问题的能力的培养。
4.2 模拟调制信号的实现软件无线电具有灵活性，可扩展性等主要特点，这主要是因为软件无线电的所有功能都是由软件来实现（定义）的，通过软件的增加，修改或者升级就可以实现新的功能。可以说，功能的软件化是软件无线电的最大优势之一。在所有的软件中，数字信号处理软件占据着重要的位子，如：调制，解调，编码，译码，信号识别，同步提取等都可以采用信号处理算法来实现。
4.2.1 AM信号调制算法与实现
AM波是怎样的波?前面已经简单提到，用期望信号去调制一个等幅信号的振幅的过程叫调幅，调制后的波就叫调幅波(AM波)，这个被调制的信号叫载波。
设正弦型载波为：
（4.1）
式中：载波角频率为；载波的初相位为；载波振幅为。幅度调制信号的一般表达式为（4.2）
其中A0=20，A1=5，A2=5，A3=5；
fc=10000Hzï¼›fm1=1000Hzï¼›fm2=2000 Hzï¼›fm3=3000 Hzï¼›
最后得到AM信号的时域表达式：

（4.3）

图4.1 AM信号时域及频域图

从图4.1可以看出，AM波的振幅变化与音频信号一致，其波形是上下对称的。在图4.1中，可以看出包含了3个部分，第1部分是原来的载波，频率是fc，振幅还是A0，第2部分为比载波高一个音频频的波(fc+fm1,fc+fm2,fc+fm3)，第3部分为比载波低一个音频频率的波(fc-fm1,fc-fm2,fc-fm3)，这二个部分分别被称为上侧边带和下侧边带(USB，LSB)，从式（4.2）可以知道这二个边带波各含有一个音频信号fm1，fm2，fm3，AM波中的这3个成分除了一个原载波是等幅波外，从式（4.2）中推断，上下边带这二个波成分其实也是等幅波(假定音频fm1，fm2，fm3此时为一固定频率比如1KHZ，2KHZ，3KHZ的正弦波)，在调幅指数m为1时振幅分别为载波振幅的一半(A/2)，频率分别为fc+fm1,fc+fm2,fc+fm3和fc-fm1,fc-fm2,fc-fm3，也就是说音频信号fm1，fm2，fm3分别包含在这二个等幅波中。
4.2.2 DSB信号调制算法与实现
DSB信号是一种与AM信号差不多形式的信号，与AM信号相比，只是其中不含有直流分量，如图4.3所示，其中是理想带通滤波器。
其时域表达式为：
(4.4)

调制信号为：
（4.5)
图4.2 抑制载波双边带调制（DSB-SC）信号

DSB信号产生原理方框图如图4.3所示，其中详细介绍了DSB信号产生的过程。
图4.3 DSB信号产生原理方框图

从图4.2可以看出，DSB-SC波的振幅变化与音频信号一致，其波形是上下对称的。在图4.2中，可以看出包含了2个部分，第1部分为比载波高一个音频频的波(fc+fm1,fc+fm2,fc+fm3)，第2部分为比载波低一个音频频率(fc-fm1,fc-fm2,fc-fm3) 的波，这二个部分分别被称为上侧边带和下侧边带(USB，LSB)，从式（4.2）可以知道这二个边带波各含有一个音频信号fm1，fm2，fm3。
4.2.3 SSB信号调制算法与实现采用下边带调制时的单边带信号时域表达式为

（4.6）

采用上边带调制时的单边带信号时域表达式为

（4.7）
图4.4 SSB信号产生原理方框图

产生上边带信号时：

（4.8）
产生下边带信号时：

（4.9）

SSB信号通过MATLAB信号仿真，产生如图4.5,4.6所示的截图：图上方为时域部分,图下方为频域部分。其产生原理即由图4.4所示的原理方框图。
图4.5 抑制载波单边带调制（SSB）信号（上边带）及频谱

图4.6 抑制载波单边带调制（SSB）信号（下边带）及频谱
4.2.4 VSB信号调制算法与实现残留边带(VSB)调制是一种幅度调制法，它是在双边带(DSB)调制的基础上，通过设计适当的输出滤波器，使信号一个边带的频谱成分原则上保留，另一个边带频谱成分只保留小部分(残留)。所以说，残留边带调制是介于单边带调制与抑制载波双边带调制之间的一种调制方式该调制方法由于其传输带宽介于DSB和单边带(SSB)之间，既比双边带调制节省频谱，又比单边带易于解调．残留边带调制的另一优点是便于实现，对发射机功放的峰均比要求比较低，因此它在广播、电视技术等许多领域得到了广泛的应用，如美国ATSC数字电视地面传输采用的就是残留边带调制方。
对于具有低频即直流分量的调制信号，用滤波法实现单边带调制时所需要的过渡带是无限陡的理想滤波器，在残留边带调制中已不再需要，这就避免了实现上的困难。其代价是传输频带增宽了一些。
残旁边带常被运用在电视信号的传输上， 因为VSB信号无SSB调变信号的低频响应差的缺点，且无DSB-SC调变信号波的频宽，更无AM调变信号消耗大功率的缺点。因此，对于视频基带讯号，既可节省边带的频宽，又可简化接收电路的成本， 故使用VSB调变就显得十分重要。 讨论残边带调变作为前提时，让我们來考虑双边带与载波在一起的情形。 假设输入的基频信号为（4.10）
载波信号为（4.11）
DSB信号可表示为

（4.12）
图4.7 残留边带调制(VSB)的滤波法形成用滤波法实现残留边带调制的原理如图4.6所示。图中为残留边带滤波器，残留部分上边带时滤波器的传递函数如图4.7所示。由滤波法可知，残留边带信号的频谱为（4.13）
其时域表达式为（4.14）
由图示的滤波器函数，可以知道，VSB信号的时域表达式为：
（4.15）
图4.8 残留部

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