Research on π / 4QPSK modulation technology based on digital generation of baseband waveform

Software-Defined Radio (Software-Defined Radio) is a new architecture for wireless communication proposed in recent years. Its basic concept is to use hardware as the basic platform for wireless communication, and to implement as many wireless and personal communication functions as possible with software, breaking the historical pattern that the realization of device communication functions only depends on hardware development. SDR mainly relies on software to complete various functions of the receiving system, such as smart antennas, signal identification, modulation and demodulation, etc. Its advantages are 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. Digital modulation and demodulation is an important part of software radio technology (SDR). The π / 4QPSK signal has the advantage that the frequency spectrum is more concentrated than the general QPSK signal, which is more conducive to achieving bit synchronization. The basic requirement of digital modulation is to generate a modulated signal waveform with good performance and a small amount of calculation.
This subject is based on the research of π / 4QPSK signal modulation technology based on the digital generation of baseband waveforms. First introduces the software radio and modulation and demodulation technology, then proposes a digital modulation algorithm, the baseband waveform is modulated into π / 4QPSK signal, and the MATLAB language is used to simulate the typical π / 4QPSK signal, and finally the various digital modulation Technology for comparative analysis.

Keywords: software radio, π / 4QPSK signal, digital modulation, simulation

Summary
ABSRACT
Chapter 1 Introduction .............................................................................. 1
1.1 Development of Software Radio ........................................................................ 1
1.2 Principles of Software Radio ........................................................................ 1
1.2.1 Basic Concepts of Software Radio ......................................................... 1
1.2.2 Basic structure of software radio ......................................................... 2
1.3 The practical application and development prospects of software radio ................................................ 3
1.4 Contents of Course Design .............................................................................. 4
Chapter 2 Modulation and Demodulation Technology in Software Radio .......................................... 5
2.1 Modulation technology in software radio ............................................................ 6
2.1.1 Introduction of Common Digital Modulation Technology ......................................................... 6
2.1.2 The realization principle of modulation technology ............................................................ 7
2.2 Demodulation technology in software radio ............................................................ 8
2.2.1 The realization principle of digital quadrature demodulation .............................. 8
2.2.2 Digital demodulation technology based on DFT .............................. 9
Chapter 3 Research on π / 4QPSK signal modulation algorithm .......................................... 10
3.1 Principle of QPSK signal and π / 4QPSK signal ...................................................... 10
3.1.1 QPSK signal ........................................................................... 10
3.1.2 π / 4QPSK signal ........................................................................... 11
3.2 Research on the realization of π / 4QPSK signal circuit …………………………………… 12
3.2.1 Analysis of π / 4QPSK signal circuit block diagram ……………………………… 12
3.2.2 Analysis of π / 4QPSK signal modulation principle ...................................................... 13
3.2.3 Waveform Realization of π / 4QPSK Signal Circuit ................................................ 14
3.3 Implementation of digital modulation algorithm for π / 4QPSK signal .................................... 15
3.3.1 Block diagram of digital modulation of π / 4QPSK signal .................................... 15
3.3.2 Analysis of π / 4QPSK signal digital modulation algorithm .................................... 16
Chapter IV Simulation of Digital Modulation of π / 4QPSK Signals .......................................... 27
4.1 Introduction to MATLAB .............................................................................. 27
4.2 Analysis of MATLAB Implementation of Digital Modulation of π / 4QPSK Signal .................................... 29
4.2.1 Implementation block diagram of digital modulation MATLAB .......................................... 29
4.2.2 MATLAB code analysis .................................................................. 30
4.3 Typical π / 4QPSK signal waveform ................................................... 32
4.4 π / 4QPSK signal performance study .................................................................. 33
4.5 Other digital modulation methods and their comparison ................................................ 34
Chapter V Summary of Graduation Design ........................................................................ 36
5.1 What needs to be improved in the design .................................................................. 36
5.2 Gain and Experience .................................................................................... 36
Conclusion ................................................................................................ 38
References ....................................................................................... 39
Appendix ................................................................................................... 40

Summary

ABSTRACT
SDR (Software-Defined Radio) is a new wireless communicaTIon design arising in recent years. Its basic concept is to make hardware as the basic wireless communicaTIonal platform, while maximally using software to implement wireless and personal communicaTIonal funcTIons, which breaks functional realization of the communicational equipment relying only on hardware development. SDR depends mainly on software to complete systemic functions of the receiver, such as smart antenna, signal identification, modulation and demodulation. The advantage of SDR is greatly simplifying the hardware products, largely enhancing reliability, ease of production and maintenance, being able to upgrade the products` function through updating software. Digital modulation, demodulation is one of the important parts in SDR. π / 4QPSK signal has a more focused spectrum compared with the general QPSK signal, more favorable to achieve synchronization . The basic requirement of digital modulation is producing modulati on signal waveform with better performance, less calculation.
This paper is the modulation technical research of digital generating π / 4QPSK signal based on the base-band waveform. Firstly, the paper introduces SDR and modulation and demodulation, then raises a digital modulation algorithm, and modulates base-band waveform into π / 4QPSK , uses MATLAB to simulate out typical π / 4QPSK signal, finally, compares with various modulation techniques.

KEY WORDS: Software-Defined Radio (SDR), π / 4QPSK signal, digital modulation, simulation
Chapter One Introduction
1.1 Development of Software Radio Software Radio (SDR) is a new architecture for wireless communication proposed in recent years.
In May 1992, Jeo Mitola of MILTRE clearly put forward the concept of soft radio for the first time. Its basic idea is to build an open, standardized, and modular universal hardware platform, using hardware as the basis of wireless communication. Platform, and as many wireless and personal communication functions as possible, such as working frequency band, modulation and demodulation type, data format, encryption mode, communication protocol, etc., are implemented by software [1]. In this way, the development of new wireless communication systems and new products will gradually be transferred to software, and the output value of the wireless communication industry will be more and more reflected in software. This is another major breakthrough in the field of wireless communication after analog to digital and fixed to mobile. Therefore, some people call software radio "supercomputer".
The main features of software radio can be summarized as follows:
1. Has great flexibility. Software radio can easily add new functions by adding software modules. It can communicate with any other radio station and can be used as a radio frequency relay for other radio stations. You can change the software module or update the module through wireless loading. In order to reduce expenses, you can choose the software module you choose according to the strength of the required function.
2. Has a strong openness. Because the software radio adopts a standardized and modular structure, its hardware can be updated or expanded with the development of devices and technologies, and the software can also be continuously upgraded as needed. Software radio can not only communicate with the new system radio, but also compatible with the old system radio. In this way, not only the service life of the old system radio is extended, but also the software radio itself has a long life cycle.
Once the new concept of software radio is proposed, it has received widespread attention in the radio field worldwide. Due to the flexibility and openness of software radio, it will not only be applied in military and civilian wireless communications, but will also be promoted in other fields such as electronic warfare, radar, and information appliances. This will greatly Promote the rapid development of software radio technology and related industries (integrated circuits).
1.2 Principle of Software Radio
1.2.1 The basic concept of software radio The basic idea of ​​software radio is to bring the wideband analog-to-digital converter (A / D) and digital-to-analog converter as close to the antenna as possible, and to establish a universal AD-DSP-DA model. , An open hardware platform, on this platform as much as possible to use software technology to achieve various functional modules of the radio. For example, a wideband ADC can be used to realize the selection of various communication frequency bands through programming, such as HF, VHF, UHF, SHF, etc., through software programming to complete the transmission signal sampling, quantization, encoding, decoding, arithmetic processing and conversion to achieve the radio frequency Transceiver function; realize the selection of different channel modulation methods through software programming, such as amplitude modulation, frequency modulation, single sideband, data, frequency hopping and spread spectrum, etc., realize different security structures, network protocols and control terminal functions through software programming, 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 for solving the problems mentioned above, and can be used to realize a multi-band, multi-user and multi-system general wireless communication system. To realize the above system, the antenna, high-speed A / D converter and high-speed digital signal processing Both the CPU and the general-purpose CPU are very demanding. These 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.
1.2.2 Basic structure of software radio A typical software radio system includes antennas, multi-band radio frequency converters, chips containing A / D and D / A converters, and on-chip general-purpose processors and memory, which can effectively implement radio stations Functions and required interface functions [2]. Its functional structure block diagram is shown in Figure 1.1:
Figure 1.1 The basic structure of the software radio. The key ideas and the main differences from the traditional structure are: 1) A / D and D / A are closer to the RF end, and the entire system band is sampled from the baseband to the intermediate frequency band; 2) The high-speed DSP / CPU replaces the traditional dedicated digital circuit and low-speed DSP / CPU to do a series of processing after A / D. The move of A / D and D / A to the RF side only provides the necessary conditions for the realization of software radio. The key step is to use general programmable devices (DSP, CPU, etc.) to replace the dedicated digital circuits, which brings The series of benefits is the real purpose of software radio.
1.3 The practical application and development prospects of software radio Software radio develops rapidly and has a very wide range of applications in the field of communication. It is mainly summarized as follows:
1. Application in personal mobile communication Personal mobile communication has developed from the first generation FDMA analog cellular mobile communication to the second generation cellular mobile communication (GSM and CDMA), and is currently developing to the third generation WCDMA mobile communication (3G) system. The goal of personal mobile communications in the future is that anyone can communicate with any other person (voice, data, images, etc.) at any time and any place. Due to the increasing communication demand, on the one hand, the life cycle of communication products is short, and development costs are rising; on the other hand, the coexistence of new and old system communications, the interconnection between various communication systems becomes more complicated and difficult, so it is necessary Seeking a new personal mobile communication architecture that not only meets the needs of the new generation of mobile communications, but also is compatible with the old system and has more expansion capabilities has become the direction of people's efforts. The software radio just provides a technical way to solve this problem, and it has become the research hotspot of the third generation mobile communication system.
2. The term software radio used in military communications was originally a new concept proposed by the US military in order to solve the interoperability, interconnection and interoperability problems encountered by the multinational forces of the Gulf War in joint operations. Because of the previous military communication equipment, whether it is a working frequency band, an information transmission format or a communication system, the three armed forces are independent and incompatible with each other, resulting in rapid communication and mutual transmission of information between the military services during joint operations. The result is a nominal The joint operations of the United States, in fact, are only the simple participation of various services, and they cannot form a "joint" in the true sense. Software radio can solve the above-mentioned problems, and it is the real "union" of various services.
3. Application of satellite communication Satellite communication is one of the most important communication methods in the contemporary era. However, due to the wide variety of satellite communication system equipment and the complicated equipment management and maintenance work, the satellite communication system has a long replacement cycle and cannot be well adapted. The pace of modern high-tech development. At the same time, 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, even if multiple processors are used Collaborative computing also cannot achieve real-time processing at high information rates, which limits its use in satellite communications. The software radio with its software-defined functions and open modular structure can solve the problems of satellite communication systems.
4. Application in digital TV system
In the 1990s, an epoch-making digital revolution was launched in the field of broadcasting and television. The third-generation TV marked by high-definition television (HDTV) achieved ideal audio-visual effects and became the development direction of a new generation of digital TV. The encoding rate of HDTV source is up to 25MHz. In order to enable HDTV to transmit and broadcast on the existing analog TV channel (with a bandwidth of 6MHz to 8MHz), the HDTV must be channel-encoded to compress the transmission bandwidth. The so-called channel coding is to choose the appropriate modulation method, modulate the 25Mbps video data to the radio frequency, and keep its bandwidth in the range of 6MHz ~ 8MHz. Source coding now has a unified standard, using MPEG-2, while channel coding has no unified standard internationally, it will coexist with various systems. To complete source coding and multiple

The systematic channel coding is more convenient to implement by software radio.
SDR has a bright future, and it represents the future of some wireless technologies. Like any other technology, with the continuous improvement of devices and application systems, SDR technology is also constantly developing. Although the current application scale of SDR is relatively small, its definition and status in the industry are gradually forming. Cellular phone base stations and military radios are just the beginning of SDR applications. With the improvement of chip integration and the advancement of software technology, SDR will be more widely adopted.
1.4 Project design content Digital modulation refers to the use of software to generate a sampling sequence of modulated signals, and then through D / A conversion to obtain an analog modulated signal, digital demodulation refers to the A / D conversion of the modulated signal, and then through the data Process to demodulate the signal. Digital modulation and demodulation is an important part of software radio technology (SDR). SDR mainly relies on software to complete various functions of the receiving system, such as smart antennas, signal identification, modulation and demodulation, etc. Its advantages are that it can greatly simplify the hardware of the product, greatly improve the reliability, and facilitate production and maintenance. To achieve product function upgrades, etc. The π / 4QPSK signal has the advantage that the spectrum is more concentrated than the general QPSK signal, which is more advantageous for achieving bit synchronization. The basic requirement of digital modulation is to generate a modulated signal waveform with good performance and a small amount of calculation. This topic requires:
(1) To study the basic content of SDR, focusing on digital modulation and demodulation technology.
(2) Design a π / 4QPSK signal digital modulation algorithm based on baseband pulse digital shaping.
(3) Programming with MATLAB language produces a typical π / 4QPSK signal.
(4) Study the performance of the modulated signal and compare it with other digital modulation methods.

Chapter 2 Modulation and Demodulation Techniques in Software Radio 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.
Software modulation and demodulation algorithm is the focus of software radio research. 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 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. FPGA (Field Programmable Gate Array) can be used to implement the required modulation and demodulation algorithm. The calculation speed is faster than DSP, but the flexibility and control function are poor, so it needs to be used in conjunction with DSP or microcontroller. The latest technology is to use DFT to realize the digital modulation and demodulation algorithm, which is a method that does not require a local carrier. This article will focus on the introduction.
Modulation and demodulation technology has been continuously developed and perfected in recent decades. In general, it can be divided into two categories: single-tone modulation and multi-tone modulation [3]. 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. But since Weinstein, Ebert and others proposed to use DFT for frequency division multiplexing in multi-tone modulation systems in 1971, multi-tone modulation technology has received more and more attention. 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 is at the receiving end of the single-tone modulation system 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.
It can be seen from the above characteristics that multi-tone modulation can obtain higher transmission performance under the condition of channel distortion or interference, and can also perform optimal rate allocation for each sub-channel according to different conditions of the channel, which can be applied to the rate Variable information transmission. Therefore, we will use multi-tone modulation technology to achieve modulation and demodulation in software radio.
2.1 Modulation technology in software radio
2.1.1 Introduction of common digital modulation technology In a digital transmission system, the transmission object is usually binary digital information, which may come from various digital codes of computers, networks or other digital devices. It may also be a pulse-coded signal from a digital telephone terminal. The basic consideration in designing a digital transmission system is to choose a limited set of discrete waveforms to represent digital information. These discrete waveforms can be unmodulated signals of different levels, or they can be in the form of modulated signals. The basic modulation methods of digital signals are as follows:
(1) 2ASK signal modulation technology:
Keying the amplitude of the carrier with a binary code is called amplitude shift keying (ASK). During a symbol duration, the ASK signal is either a "sign" or a "empty number". which is:
(2) 2FSK signal modulation technology:
The binary code is used to key the carrier frequency, which is called Frequency Shift Keying (FSK), and its expression can be expressed as:

(3) 2PSK signal modulation technology:
Carry out keying on the carrier amplitude with a bipolar non-return-to-zero code sequence to form phase shift keying (PSK). Its expression is:

Digital modulation technology has the advantages of strong anti-interference ability, easy encryption, and low voice gap noise. With the development of digital communications, stricter requirements have been imposed on the frequency band occupancy and utilization. For example, the United States requires that at least 99% of the entire signal spectrum be included in the occupied frequency band, that is, the out-of-band radiated power must not exceed 1%; the transmission bit rate must be equal to or greater than the specified frequency bandwidth. This means that the digitally modulated modulated signal must be band-limited, and the narrower the band-limited range, the better. For example, the channel spacing of mobile communications is only 25 kHz, and in order to make full use of frequency resources, it is advancing toward 12.5 kHz. At the same time, the nonlinearity of the channel transmission characteristics is only

The so-called amplitude modulation and phase modulation, when the amplitude of the input signal changes, will be converted into the phase change of the output signal, thereby generating new out-of-band components again, causing spectrum regeneration. Therefore, the traditional digital modulation method must be improved to meet the needs of development. Common modulation schemes currently under study include modulation schemes such as coherent phase shifting (CPSK), quadrature phase shift keying (QPSK), and Gaussian minimum phase shift keying (GMSK). When designing a digital system, it is very important to choose which digital modulation method. However, the choice of digital modulation method is often the result of comprehensive consideration of factors such as frequency band utilization, bit error rate, signal-to-noise ratio, and complexity of device implementation. It must be compared according to specific use conditions to make a judgment.
2.1.2 The realization principle of modulation technology With the rapid development of contemporary communication, the changes in communication system are also changing with each passing day: some old communication methods are either improved or eliminated, and new communication methods suitable for the contemporary communication system are constantly emerging and improving . The modulation methods commonly used a few days ago have been introduced in section 2.1.1. If according to the conventional method, each kind of signal requires a hardware circuit or even a template, then if a communication machine generates several or more than ten kinds of communication signals, its circuit will be very complicated, volume and weight It will be big. It is very difficult to add a new communication method.
The various modulation signals in the software radio are supported by a general-purpose digital signal processing platform and are generated using various software. Each jump-to algorithm is made in the form of a software template. To generate a certain modulation signal, you only need to call the corresponding module [1]. It is implemented by software for various modulations. Therefore, in software radio, the software of the modulation module can be continuously updated to adapt to the constantly developing modulation system, which has considerable flexibility and openness. Various modulations of software radio can be implemented based on digital signal processing technology.
In contemporary communication, there are many types of communication signals. In theory, various communication signals can be implemented by orthogonal modulation, as shown in Figure 2.1.

Figure 2.1 Implementation block diagram of quadrature modulation Any radio signal can be expressed as (2.1)
Digitize the formula (2.1) to obtain (2.2)
In order to facilitate information modulation, for digital modulation systems, equation (2.1) is usually orthogonally decomposed:
(2.3)
In the formula

The modulation method is to first calculate according to the modulation method, and then multiply and sum the two orthogonal local oscillators respectively to obtain the modulation signal.
2.2 Demodulation technology in software radio
2.2.1 The realization principle of digital quadrature demodulation In the software radio system, modulation and demodulation are implemented by programs (also called full 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. FPGA (Field Programmable Logic Device) can be used to implement 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.
One way to establish modem algorithms and procedures is to softwareize the working principle of analog circuits. For example, it is necessary to demodulate AM signals coherently, or establish carrier synchronization! Multiplier! Low-pass filtering and other software modules are feasible, but they are computationally expensive. 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.
Figure 2.2 shows the digital quadrature demodulation scheme widely used in SDR receivers [4]. This is a versatile demodulation model. For different modulation signals, only the corresponding baseband demodulation algorithm needs to be designed. For AM signals, the baseband demodulation algorithm is. Data extraction is performed on the output of the LPF because the sampling rate required for the baseband signals I and Q is much lower than the sampling rate for the modulated signal. This demodulation scheme utilizes the square sum square root operation that can be implemented in the software to avoid the complex carrier synchronization process, 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.
Figure 2.2 Digital quadrature demodulation scheme
2.2.2 DFT-based digital demodulation technology literature [4] The calculation amount of the demodulation method is still relatively large, because each sampled value must be multiplied in two ways, and after a higher order and lower Pass filter. The AM signal demodulation algorithm based on DFT (Discrete Fourier Transform), the main point is to sample the low and intermediate frequency AM signals over the entire period (for example, the sampling frequency is 8 times the carrier frequency), and the sampling points within each carrier period Dx for x1 ～ x8), calculate the amplitude of the carrier
A (n):

(2.4)

(2.5)

(2.6)
Obviously, after removing the DC component, the A (n) sequence is the required demodulation output. Compared with the general quadrature demodulation algorithm, since the low-pass filtering and data extraction process is omitted, the sampled data is basically only added and subtracted, and the square and square root operations are only performed every 8 sampling points, and the calculation amount Greatly reduced, creating the conditions for the use of "IF sampling-DSP demodulation" program. Using a lower sampling frequency (such as sampling 4 points per carrier cycle) can also demodulate normally. Of course, a higher sampling frequency is beneficial to suppress noise.
Digital demodulation is an important content in software radio. Proceeding from the characteristics of SDR, a small amount of calculation is proposed for various types of modulated signals. A good performance demodulation algorithm has obvious significance for the improvement and promotion of SDR technology. The demodulation technology of AM and QDPSK signals based on the DFT budget eliminates the need for filters and data extraction, reduces the amount of calculation, and is beneficial to the use of the "IF sampling-DSP demodulation" scheme. This demodulation algorithm can also be extended to MQAM,
In demodulation of modulation signals such as π / 4QPSK.
Chapter 3 Research on π / 4QPSK signal modulation algorithm
3.1 QPSK signal and π / 4QPSK signal principle
3.1.1 QPSK signal Four-phase phase shift keying modulation QPSK, its essence is to use the relative change of carrier oscillation phase between the front and back symbols to transfer information, so QPSK signal can be regarded as the synthesis of two carrier orthogonal 2PSK signals. Because the phase of the previous symbol signal replaces the phase of the extracted fundamental frequency, the uncertainty of the standard oscillation phase is overcome, and its frequency band utilization is also doubled compared to the binary PSK signal. However, when the two signals change simultaneously, the phase of the QPSK signal will change abruptly by 180 °. Since the instantaneous frequency is a differential of the phase, the sudden change of the phase is equivalent to the instantaneous frequency tending to infinity. When the signal with a sudden phase change of 180 ° passes through a band-pass filter with limited bandwidth, the output waveform changes. That is, at the moment when the phase changes by 180 °, the envelope of the modulated wave drops to zero, thus causing the envelope to fluctuate is too big. After the non-linear period of the waveform of this modulated signal, due to the limiting amplification, the envelope that originally fell to zero at the symbol conversion point will rise again. This is equivalent to reverting to a constant envelope, that is, reverting to the original unrestricted frequency band. In other words, the side lobes of the power spectrum filtered by the band-pass filter will emerge again, or broadened frequency bands [6].
QPSK modulation uses four different phases of the carrier to characterize digital information. Since each carrier phase represents two bits of information, each quaternary symbol is also called a two-bit symbol. We represent the previous information bit constituting the two-bit symbol as a, and the latter information bit as b. The two information bits ab in the double-bit symbol are generally arranged according to the Gray code, and its relationship with the carrier phase is shown in Table 3.1 and Table 3.2. Its vector relationship is shown in Figure 3.1, where (a) is the vector diagram of the QPSK signal in mode A, and (b) is the vector diagram of the QPSK signal in mode B:

(A) (b)
Figure 3.1 The vector diagram of QPSK modulation can also use Gray code to represent the QPSK signal, as shown in the following table 3.1 logical coding table (1) and table 3.2 logical coding table (2), with 0 ° and 45 ° carrier phase as the reference phase respectively . The left carrier phase is 0 °, 90 °, 180 °, and 270 °, and the representative information is 00, 01, 11, 10. The carrier phase on the right is 45 °, 135 °, 225 °, and 315 °, and the representative information is 11, 01, 00, and 10, respectively.
Table 3.1 QPSK logical coding table in A mode
Relative displacement of symbols before and after a channel b symbol
ab △ θ = θ = θ

0 0 0

0 1 90

1 0 180

1 1 270
Table 3.2 QPSK logical coding table in mode B
Relative displacement of symbols before and after a channel b symbol
ab △ θ = θ = θ

0 0 225

0 1 135

1 0 270

1 1 45
When using QPSK signals, there is often a "phase blur" phenomenon. For the "phase blur" of the local coherent carrier recovered by the customer service receiver, it is the same as the method of using binary codes to form relative phase shift keying (DPSK). Metaphase modulation can also use the relative changes of the carrier phases of the two symbols before and after to convey information, that is, to form four-phase differential phase modulation (QDPSK).
3.1.2 π / 4QPSK signal The standard of the American digital mobile phone is formulated by the American Telecommunications Industry Association (TIA). It uses the 800MHz frequency band, the channel spacing is 30kHz, and the digital rate is 48.6bit / s. The digital modulation method uses the π / 4QPSK signal that I want to study. Digital mobile phones in Japan also use this method, but the channel spacing is 25kHz, the digital rate is 42kbit / s, and the frequency bands allocated are 800MHz and 1.5GHz.
由于虽然QPSK调制具有比2PSK调制频带利用率提高一倍的优点，但是QPSK调制的载波有4种相位变化，即0°、90°、180°、270°或45°、135°、225°、315°。当载波相位突变，特别是出现180°突变时，载波包络为零，使载波信号功率谱扩展，从而造成信号带限失真，也就是上一节介绍的展宽频带。
为改进QPSK调制信号的频谱特性，把QPSK调制的A、B两种方式的矢量图合二为一，并且使载波相位只能从一种模式（A或B）向另一种模式（B或A）跳变，其中，”●”表示QPSK调制A方式的矢量图，”○”表示QPSK调制B方式的矢量图，从而构成π/4QPSK调制的矢量图，如图3.2所示。矢量图中的箭头表示载波相位的跳变路径，显然，相位变化只有±45°和±135°4种状态,不存在180°相位跳变，因此较QPSK调制具有更好的频谱特性【6】。

图3.2 π/4ＱＰＳＫ调制的矢量图π/4QPSK虽然不是恒定包络，但包络线的变化很小。它要求高频放大器线性工作范围比一般QPSK所需要的线性工作范围要小。π/4QPSK的设计就是以能使用具有一定程度线性的甲乙类高频放大器为出发点的。
3.2 π/4QPSK信号电路实现研究
3.2.1 π/4QPSK信号电路框图分析π／4QPSK调制的系统框图如图3.3示，在QPSK调制系统的基础上，增加了一个映射逻辑电路。输入的数据流经串／并电路后，变换成I 和Q 双bit符号，输至映射逻辑电路【10】。映射逻辑电路的功能为：
(3.1)
(3.2)
图3.3 π／4QPSK调制的系统框图其中，Δθ 是输入双bit符号{ a , a }所对应的相移值，相移值的大小符合表3.1所示规律。需要说明的是，与图3.3对应的相移值是QPSK调制的B方式； I 和Q 分别为双bit符号a 与a 经映射逻辑变换后输出的同相和正交支路双bit符号；I 和Q 分别为双bit符号经映射逻辑变换后输出的同相和正交支路双bit符号。a 和a 有(0，0)，(0，1)，(1，0)，(1，1)4种组合，经映射逻辑变换后，输出有8种取值： 如图3.2所示。在映射逻辑输出的数据流中，第k个同相正交双bit符号I 、Q 的合成相位值用表示，第k－1个同相正交双bit符号I 、Q 的合成相位值用θ 表示。
3.2.2 π/4QPSK信号调制原理分析为了求得I 、Q 的合成相位，将I 、Q 的8种取值逐一代入（3.1）式和（3.2）式：
（1）(I ，Q ) =( 1,0)时, θ =0°

（2）（I ，Q ) =( 1/ ,1/ )时, θ =45°

（3）(I ，Q )=( 0,1)时, θ =90°

……
同理，(I ，Q )=

æ—¶,
均有

如果令（3.3）
则有

有上可得：

于是得到π/4QPSK调制系统输出信号数学表达式为（3.4）
由此可知：π/4QPSK调制系统输出信号的相位为， 随输入数据流变化的跳变关系，就是π/4QPSK调制系统输出信号相位随输入数据流变化的跳变关系。再由式(3.4)可知，π/4QPSK调制系统输出信号相位由决定，根据( 3.3 )式对的定义，很容易求出输入数据流变化时，π/4QPSK调制系统输出信号的相位变化关系。
3.2.3 π/4QPSK信号电路实现波形根据3.2.1的原理分析，对于任一组输入数据流”00110110010110… …”，都可以有相应的输出，输出波形如图所示，显然，相位变化有±45°和±135°4种状态,不存在180°相位跳变，因此较QPSK调制具有更好的频谱特性。电路实现的π/4QPSK信号波形如下图3.4所示。
图3.4 π/4QPSK信号电路实现波形
3.3 π/4QPSK信号数字化调制算法
3.3.1 π/4QPSK信号数字化调制框图在本课题设计中，要求设计一个基于基带波形数字成形的π/4QPSK信号数字化调制算法，也即是使用数字化调制技术把基带波形调制成π/4QPSK波形。传统π/4QPSK信号调制算法中往往要使用数字滤波器，当阶数较高时，一方面不容易计算量非常大，另外又产生很多高频分量。例如使用一个32阶的IIR数字滤波器完成16点滤波，计算一点要用到65次乘法和加法运算，计算16个点时就要用到65*16次乘法和加法运算，带来了很大的麻烦。而使用数字化调制实现，则可以避免出现以上所提到的问题，并且它还对过渡区实现余弦函数平滑过渡，避免了相位改变时的剧烈跳变，大大减少了高频分量。数字化调制方案如下图3.5所示：

图3.5 π/4QPSK信号调制方案对于任意一组数据流，通过串/并转化成两组码元，分别为偶数流和奇数流，然后通过映射逻辑电路得到波形生成的I 和Q ，在此过程中不需要经过低通滤波器进行，而是直接进行函数的调用，最后对I 和Q 进行采样，就得到π/4QPSK信号。
3.3.2 π/4QPSK信号数字化调制算法分析根据（3.2）中关于π/4QPSK信号调制原理的分析，可知IQ 不仅与输入数据（Δθ）有关，而且与I 和Q 有关。利用星座中前一时刻的绝对相位θ和有当前时刻输入IQ 决定相位偏移量Δθ，找到由当前绝对相位θ+Δθ决定的输出I 和Q 。相位偏移量Δθ可以转化成相应的编号偏移量ΔN【11】。Δθ和ΔN的对应关系如下表3.3：
表3.3 Δθ和ΔN的对应关系
IQ
00 01 10 11
Δθ （- ）

偏移量编号5 3 7 1

又由图（3.5）中π/4QPSK信号数字化调制方案，可知如何求出经过映射逻辑电路的I 和Q 是本课题设计的一个关键。
首先设前一码元的矢量号为S ，对应的I 、Q 分量为I 、Q ；本码元的矢量号为S ，对应的I 、Q 分量为I 、Q 。前一码元和本码元的分析如下：（其中1、3、5、7分别代表11、01、00、10的矢量号）
若本码元为00，则S =mod（S +5，N ）；
若本码元为01，则S =mod（S +3，N ）；
若本码元为10，则S =mod（S +7，N ）；
若本码元为11，则S =mod（S +1，N ）；
其中为前一码元最后一个采样点的归一化相位（整数， ），因为信号的每个载波周期要进行16点采样，所以这里的N 是小于等于16的。
有了本码元就可以知道相位偏移量Δθ，也就是找到由当前绝对相位θ+Δθ决定的输出I 和Q ，可先列出绝对相位和当前输出IQ 之间的关系，如表3.4：
表3.4 绝对相位和当前输出IQ 之间的关系
N 0 1 2 3 4 5 6 7
θ 0

Ï€
I
1 0.707 0 -0.707 -1 -0.707 0 0.707
Q
0 0.707 1 0.707 0 -0.707 -1 -0.707

可见，只要有了相位偏移量，就可以得到所要求的当前时刻的输出IQ ，然后对波形成形的I ，Q 进行采样，最后把所得采样后信号相加，就得出所需的经过数字化调制的π/4QPSK信号。
为了简单易行，首先不妨先假设输入一组数据流”00110110… …”，来进行数字化调制。按照数字化调制结构框图（3.5）可得，这组数据首先经过串/并变换，于是可以得到奇、偶数组{ a ,a }，即{ a ,a }={ 00；11；01；10… …}，其中每两个数据为一组码元，然后然这组码元再通过映射逻辑电路，经过数据的调用，可以得到波形生成的I 、Q 。
在设计中，设每个码元包含10个载波周期（即f =10R ），其中码元波形的前面3个载波周期为过渡区，后面7个载波周期为稳定区。每个载波周期采样16点，因此可求的过渡区共有16 3=48个采样点，稳定区有160-48=112个采样点。
由于前三个载波周期是过渡区，因此首先研究过渡区的情况，在保持码元稳定区为余弦函数的前提下，在信号相邻码元之间的过渡区采用余弦函数进行过渡，直到下一个码元的稳定区。这样一来,在信号相邻码元之间的过渡区内最大相位差的绝对值趋近于零，从而避免了相位改变时的剧烈跳变，可以大大抑制谐波分量【12】。由此可见, 平滑相位π/ 4QPSK 调制与普通π/ 4QPSK调制的区别主要体现在引入了平滑函数。
由3.1.2节中的分析，π/ 4QPSK信号的相位突变有± 和± 四种情况，因此分以下四种进行讨论：
（1）码元相位跳变为设载波峰值为1，前一码元的矢量相位为0，则本码元矢量相位为，即从（1，0） 跳到（0.707，0.707），过渡区采用余弦函数过渡，信号矢量变化过程如图：

图3.6 信号矢量变化过程一个码元过渡区共有采样48个采样点，对于过渡区内的第i个采样点，I 和Q 值可以得到分别为：

， i＝1、2、……48 （3.5）

， i＝1、2、……48 （3.6）

在求I 和Q 值的过程中要不断的调用余弦函数c（i）：

（3.7）

可先列出该函数调用表3.5如下：
表3.5 余弦函数调用表

i 1 2 3 4 5 6 7 8
c（i） 0.99786 0.99144 0.98079 0.96593 0.94693 0.92388 0.89687 0.86603
i 9 10 11 12 13 14 15 16
c（i） 0.83147 0.79335 0.75184 0.70711 0.65935 0.60876 0.55557 0.5
i 17 18 19 20 21 22 23 24
c（i） 0.44229 0.38268 0.32144 0.25882 0.19509 0.13053 0.065403 6.1232e-017
i 25 26 27 28 29 30 31 32
c（i） -0.06540 -0.13053 -0.19509 -0.25882 -0.32144 -0.38268 -0.44229 -0.5
i 33 34 35 36 37 38 39 40
c（i） -0.55557 -0.60876 -0.65935 -0.70711 -0.75184 -0.79335 -0.83147 -0.86603
i 41 42 43 44 45 46 47 48
c（i） -0.89687 -0.92388 -0.94693 -0.96593 -0.98079 -0.99144 -0.99786 -1
同时也可以列出余弦函数的调用图，如下图3.7：
图3.7 余弦函数的调用图有了调用表之后，就可以求出当相位偏移量为时，经过余弦函数过渡的I 和Q 值。因此可以根据（3.5）和（3.6）式列出波形生成后的I 和Q ，如下表3.6：
表3.6 输出I 和Q 的调用表
i 1 2 3 4 5 6 7 8

0.99969 0.99875 0.99719 0.99501 0.99223 0.98885 0.98489 0.98037

0.00075687 0.0030242 0.0067924 0.012045 0.01876 0.026909 0.036455 0.04736
i 9 10 11 12 13 14 15 16

0.97531 0.96973 0.96364 0.95709 0.95009 0.94268 0.93489 0.92675

0.059575 0.07305 0.087725 0.10354 0.12042 0.1383 0.15711 0.17675
i 17 18 19 20 21 22 23 24

0.9183 0.90956 0.90059 0.89142 0.88208 0.87262 0.86308 0.8535

0.19715 0.21822 0.23987 0.26201 0.28454 0.30736 0.33038 0.3535
i 25 26 27 28 29 30 31 32

0.84392 0.83438 0.82492 0.81558 0.80641 0.79744 0.7887 0.78025

0.37662 0.39964 0.42246 0.44499 0.46713 0.48878 0.50985 0.53025
i 33 34 35 36 37 38 39 40

0.77211 0.76432 0.75691 0.74991 0.74336 0.73727 0.73169 0.72663

0.54989 0.5687 0.58658 0.60346 0.61928 0.63395 0.64742 0.65964
i 41 42 43 44 45 46 47 48

0.72211 0.71815 0.71477 0.71199 0.70981 0.70825 0.70731 0.707

0.67054 0.68009 0.68824 0.69495 0.70021 0.70398 0.70624 0.707
根据数字化调制框图（3.5），下面需要考虑的是如何把得到的来自两支路的IQ 经过同相和正交，再把所得的数值相加得到π/4QPSK信号。π/4QPSK调制信号计算公式如下：

，i=1、2、… …16 （3.8）

普通的周期载波调制算法完成一点的载波调制就要用两次乘法和一次加法运算，而在数字化调制过程中则是采用数据的调用来实现，不需要做乘法运算。对每个正弦、余弦波周期中取16个点：0， ， ，… …， 进行载波调制。
在上述过程中又要调用余弦函数bc(i)：

，i=1、2、… …16 （3.9）

列出函数调用表3.8以及函数调用图3.7如下：

表3.7 余弦函数调用表
i 1 2 3 4 5 6 7
bc(i) 0.38268 0.70711 0.92388 1 0.92388 0.70711 0.38268
i 8 9 10 11 12 13 14
bc(i) 1.2246e-016 -0.38268 -0.70711 -0.92388 -0.92388 -0.70711 -0.38268
i 15 16
bc(i) -2.4493e-016 0.38268

图3.8 余弦函数调用图然后再得出正弦函数调用表3.8以及函数调用图3.8如下

表3.8 正弦函数调用表
i 1 2 3 4 5 6 7
d 0.92388 0.70711 0.38268 6.1232e-017 -0.38268 -0.70711 -0.92388
i 8 9 10 11 12 13 14
d -1 -0.92388 -0.70711 -0.38268 -1.837e-016 0.38268 0.70711
i 15 16
d 1 0.92388

图3.9 正弦函数调用图由式（3.8）可知载波调制不需要做乘法运算。即是第1采样点的输出为0.38268 +0.92388 ，第2个采样点的输出为0.70711 +0.70711 ，… …，第48个采样点的输出为0.38268 +0.92388 ，如此循环。在过渡区内共有三个载波周期，共需采样16×3=48个点，可一次根据式（3.12）求出48个采样值，从而得到所需的过渡区的π/4QPSK信号的波形。
另外也对（3.9）式做如下处理，分为求出合矢量的幅值和相位偏移量来求所需的过渡区的π/4QPSK信号的波形。首先得到合矢量的幅值为：
（3.10）
相位偏移量为： ，归一化相位偏移量为：

于是采样点i的值为： ，在算法中需要用BC表，数据要经插值运算才能得到，比如。归一化相位以32为模，比如。码元相位跳变为的合向量幅值为： …… 。码元相位跳变为的归一化相位偏移量为： …… ，那么输出的就是π/4QPSK信号各个采样点的值。
（2）码元相位跳变为设载波峰值为1，前一码元的矢量相位为0，则本码元适量相位为。可得相位跳变为时，即从（1，0）跳到（-0.707，0.707），过渡区采用余弦函数过渡，信号矢量变化过程如图3.3所示：

图3.9 信号矢量变化过程对于过渡区第i个采样点，I 和Q 值分别为：

， i＝1、2、……48 （3.11）

， i＝1 、2、……48 （3.12）

在求和过程中同样要不断调用余弦函数c（i），也可以列出波形成形后出和，如表3.9：
表3.9 输出和的调用表
i 1 2 3 4 5 6 7 8

0.99817 0.9927 0.9836 0.97092 0.9547 0.93503 0.91198 0.88565

0.00075687 0.0030242 0.0067924 0.012045 0.01876 0.026909 0.036455 0.04736
i 9 10 11 12 13 14 15 16

0.85616 0.82363 0.7882 0.75002 0.70925 0.66608 0.62068 0.57325

0.059575 0.07305 0.087725 0.10354 0.12042 0.1383 0.15711 0.17675
i 17 18 19 20 21 22 23 24

0.52399 0.47312 0.42085 0.3674 0.31301 0.2579 0.20232 0.1465

0.19715 0.21822 0.23987 0.26201 0.28454 0.30736 0.33038 0.3535
i 25 26 27 28 29 30 31 32
I
0.090678 0.035096 -0.02001 -0.074402 -0.12785 -0.18012 -0.23099 -0.28025

0.37662 0.39964 0.42246 0.44499 0.46713 0.48878 0.50985 0.53025
i 33 34 35 36 37 38 39 40

-0.32768 -0.37308 -0.41625 -0.45702 -0.4952 -0.53063 -0.56316 -0.59265

0.54989 0.5687 0.58658 0.60346 0.61928 0.63395 0.64742 0.65964
i 41 42 43 44 45 46 47 48

-0.61898 -0.64203 -0.6617 -0.67792 -0.6906 -0.6997 -0.70517 -0.707

0.67054 0.68009 0.68824 0.69495 0.70021 0.70398 0.70624 0.707
然后用同样的方法产生所需要的π/4QPSK信号。
（3）码元相位跳变为-
设载波峰值为1，前一码元的矢量相位为0，则本码元矢量相位为- ，即从（1， 0）跳到（0.707，-0.707），过渡区采用余弦函数过渡，信号矢量变化过程如图3.10：

图3.10 信号矢量变化过程一个码元要采样48个点，对于过渡区内的第i个采样点，I 和Q 值分别为：

， i＝1、2、……48 （3.13）

， i＝1、2、……48 （3.14）

在求所需要的π/4QPSK信号时，同样要调用上面提到的两个函数。
（4）码元相位跳变为-3
设载波峰值为1，前一码元的矢量相位为0，则本码元矢量相位为-3 ，即从（1，0）跳到（-0.707，-0.707），过渡区采用余弦函数过渡，信号矢量变化过程如图：

3.3 信号矢量变化过程一个码元的过渡区共要采样48个点，对于过渡区内的第i个采样点，I 和Q 值分别为：

， i＝1、2、……48 （3.15）

， i＝1、2、……48 （3.16）

在求所需要的π/4QPSK信号时，同样要调用上面提到的两个函数。
再来考虑稳定区的情况，不妨先讨论初相θ =0时，可以列出下相位偏移量、IQ 与输入数据的关系，如图3.10所示，观察表达式的特征。
表3.10 相位偏移量、IQ 与数据之间的对应关系数据Δθ θ =θ +Δθ
（I ，Q ）

00

（-0.707，-0.707）
11

（0，-1）
01

（0.707，0.707）
10
¬¬0 （1，0）
… … … … … … … …

通过上表可以求出：
当码元为00时，

，其中i=1、2、3…16； （3.17）

当码元为11时，

， 其中i=1、2、3…16； （3.18）

当码元为01时，

，其中i=1、2、3…16； （3.19）

当码元为10时，

，其中i=1、2、3…16。 （3.20）

通过计算可以得到，（3.5）～(3.8)均是余弦函数，对于稳定区内的波形，可知应该是余弦波。有了输入码元以及初始相位就可以很容易的求出波形成形后的I 和Q ，然后就是对周期载波进行16点采样，由于码元稳定区的I 和Q 是恒定的，因此求输出π/4QPSK信号的波形相对于过渡区要容易得多。稳定区共有7个周期载波，则共需要采样112个点，求出一个周期的16个采样值就可以了。
需要调用表（3.7）和表（3.8），再结合式3.12，先拿式（3.5）来加以说明如何得到采样值的，由于初相位为0 ，当输入码元为00时，可求出本码元的绝对相位是，通过表（3.4）可以求出当前波形成形I 和Q 为（-0.707，-0.707），可以得到式（3.17）

不断的调用就可以得到所求的采样值，不需要做乘法运算。即是第1个点输出为（-0.707）×0.38268+（-0.707）×0.92388，第2个采样点的输出为（-0.707）×0.70711+（-0.707）×0.70711，… …，第16个采样点的输出为（-0.707）×0.38268+（-0.707）×0.92388，然后由于是周期载波，其他6个周期的采样值与第一个周期的是相同的。也就得到了稳定区的信号输出。
再仔细观察（3.17）式可以得到：
由表（3.3）可知，码元00的偏移量编号为5，则可想到上式可表示成
=bc(N +i),其中i=1,2,… …112, N 是码元过渡区结束时采样点的相位偏移量量，bc(i)函数上面已经介绍过。
以上分析都是以初相位是0时的情况，当前一码元的矢量相位不是0时，上述算法仍然正确。此处假设前码元相位为0，只是为了叙述更加直观方便。在实际实现过程中初相位可以是八种情况中的任意一种。
上面分别介绍了求码元稳定区调制算法和过渡区的调制算法。数字化调制生成π/4QPSK信号就有许多的优点，码元经过映射逻辑后不需要通过低通滤波器来进行波形成形，而是根据输入数据流与相位偏移量的关系得到矢量号，得到I 和Q ，这样就大大减少由滤波器引起的计算量。得到了I 和Q 后，该调制算法通过调用函数来实现周期载波的采样。另外，该算法还解决了在码元过渡区相位跳变的问题，使用余弦函数实现平滑过渡，这样得到的信号的高频分量很少，频谱性能更好。

第四章π/4QPSK信号数字化调制的仿真
4.1 MATLAB简介
MATLAB是矩阵实验室（Matrix Laboratory）之意，除具备卓越的数值计算能力外，它还提供了专业水平的符号计算，文字处理，可视化建模仿真和实时控制等功能。它的基本数据单位是矩阵，它的指令表达式与数学,工程中常用的形式十分相似,故用MATLAB来解算问题要比用C,FORTRAN等语言完全相同的事情简捷得多. 当前流行的MATLAB 5.3/Simulink 3.0包括拥有数百个内部函数的主包和三十几种工具包(Toolbox).工具包又可以分为功能性工具包和学科工具包.功能工具包用来扩充MATLAB的符号计算,可视化建模仿真,文字处理及实时控制等功能.学科工具包是专业性比较强的工具包,控制工具包,信号处理工具包,通信工具包等都属于此类. 开放性使MATLAB广受用户欢迎.除内部函数外,所有MATLAB主包文件和各种工具包都是可读可修改的文件,用户通过对源程序的修改或加入自己编写程序构造新的专用工具包.
在70年代中期,Cleve Moler博士和其同事在美国国家科学基金的资助下开发了调用EISPACK和LINPACK的FORTRAN子程序库.EISPACK是特征值求解的FOETRAN程序库,LINPACK是解线性方程的程序库.在当时,这两个程序库代表矩阵运算的最高水平. 到70年代后期,身为美国New Mexico大学计算机系系主任的Cleve Moler,在给学生讲授线性代数课程时,想教学生使用EISPACK和LINPACK程序库,但他发现学生用FORTRAN编写接口程序很费时间,于是他开始自己动手,利用业余时间为学生编写EISPACK和LINPACK的接口程序.Cleve Moler给这个接口程序取名为MATLAB,该名为矩阵(matrix)和实验室(laboratory)两个英文单词的前三个字母的组合.在以后的数年里,MATLAB在多所大学里作为教学辅助软件使用,并作为面向大众的免费软件广为流传. 1983年春天,Cleve Moler到Standford大学讲学,MATLAB深深地吸引了工程师John Little John Little敏锐地觉察到MATLAB在工程领域的广阔前景.同年,他和Cleve Moler Steve Bangert一起,用C语言开发了第二代专业版.这一代的MATLAB语言同时具备了数值计算和数据图示化的功能. 1984年,Cleve Moler和John Little成立了Math Works公司,正式把MATLAB推向市场,并继续进行MATLAB的研究和开发. 在当今30多个数学类科技应用软件中,就软件数学处理的原始内核而言,可分为两大类.一类是数值计算型软件,如MATLAB Xmath Gauss等,这类软件长于数值计算,对处理大批数据效率高;另一类是数学分析型软件Mathematica Maple等,这类软件以符号计算见长,能给出解析解和任意精确解,其缺点是处理大量数据时效率较低.MathWorks公司顺应多功能需求之潮流,在其卓越数值计算和图示能力的基础上,又率先在专业水平上开拓了其符号计算,文字处理,可视化建模和实时控制能力,开发了适合多学科,多部门要求的新一代科技应用软件MATLAB.经过多年的国际竞争,MATLAB以经占据了数值软件市场的主导地位. 在MATLAB进入市场前，国际上的许多软件包都是直接以FORTRANC语言等编程语言开发的。这种软件的缺点是使用面窄，接口简陋，程序结构不开放以及没有标准的基库，很难适应各学科的最新发展，因而很难推广。MATLAB的出现，为各国科学家开发学科软件提供了新的基础。在MATLAB问世不久的80年代中期，原先控制领域里的一些软件包纷纷被淘汰或在MATLAB上重建。MathWorks公司1993年推出了MATLAB 4.0版，1995年推出4.2C版（for win3.X）1997年推出5.0版。1999年推出5.3版。MATLAB 5.X较MATLAB 4.X无论是界面还是内容都有长足的进展，其帮助信息采用超文本格式和PDF格式，在Netscape 3.0或IE 4.0及以上版本，Acrobat Reader中可以方便地浏览。 时至今日，经过MathWorks公司的不断完善，MATLAB已经发展成为适合多学科，多种工作平台的功能强大的大型软件。在国外，MATLAB已经经受了多年考验。在欧美等高校，MATLAB已经成为线性代数，自动控制理论，数理统计，数字信号处理，时间序列分析，动态系统仿真等高级课程的基本教学工具；成为攻读学位的大学生，硕士生，博士生必须掌握的基本技能。在设计研究单位和工业部门，MATLAB被广泛用于科学研究和解决各种具体问题。在国内，特别是工程界，MATLAB一定会盛行起来。可以说，无论你从事工程方面的哪个学科，都能在MATLAB里找到合适的功能。 介绍一下MATLAB的主要特点：
1. 语言简洁紧凑，使用方便灵活，库函数极其丰富。MATLAB程序书写形式自由，利用起丰富的库函数避开繁杂的子程序编程任务，压缩了一切不必要的编程工作。由于库函数都由本领域的专家编写，用户不必担心函数的可靠性。可以说，用MATLAB进行科技开发是站在专家的肩膀上。
2. 运算符丰富。由于MATLAB是用C语言编写的，MATLAB提供了和C语言几乎一样多的运算符，灵活使用MATLAB的运算符将使程序变得极为简短。
3. MATLAB既具有结构

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