Friday, July 17, 2009
Itroduction to Z-Transform
The z-transform is useful for the manipulation of discrete data sequences and has acquired a new significance in the formulation and analysis of discrete-time systems. It is used extensively today in the areas of applied mathematics, digital signal processing, control theory, population science, economics. These discrete models are solved with difference equations in a manner that is analogous to solving continuous models with differential equations. The role played by the z-transform in the solution of difference equations corresponds to that played by the Laplace transforms in the solution of differential equations.
Reasons for converting Analog Signal to Digital
There are many reasons why we process these analog signals in the digital world (or domain) but these can all be reduced to two primary reasons, which are :
Cost - DSP systems are almost always cheaper than analog.
Functionality - DSP systems can perform many operations that are impossible in the analog world.
Cost - DSP systems are almost always cheaper than analog.
Functionality - DSP systems can perform many operations that are impossible in the analog world.
Applications of DSP
DSP technology is nowadays commonplace in such devices as mobile phones, multimedia computers, video recorders, CD players, hard disc drive controllers and modems, and will soon replace analog circuitry in TV sets and telephones. An important application of DSP is in signal compression and decompression. Signal compression is used in digital cellular phones to allow a greater number of calls to be handled simultaneously within each local "cell". DSP signal compression technology allows people not only to talk to one another but also to see one another on their computer screens, using small video cameras mounted on the computer monitors, with only a conventional telephone line linking them together. In audio CD systems, DSP technology is used to perform complex error detection and correction on the raw data as it is read from the CD.
Although some of the mathematical theory underlying DSP techniques, such as Fourier and Hilbert Transforms, digital filter design and signal compression, can be fairly complex, the numerical operations required actually to implement these techniques are very simple, consisting mainly of operations that could be done on a cheap four-function calculator. The architecture of a DSP chip is designed to carry out such operations incredibly fast, processing hundreds of millions of samples every second, to provide real-time performance: that is, the ability to process a signal "live" as it is sampled and then output the processed signal, for example to a loudspeaker or video display. All of the practical examples of DSP applications mentioned earlier, such as hard disc drives and mobile phones, demand real-time operation.
The major electronics manufacturers have invested heavily in DSP technology. Because they now find application in mass-market products, DSP chips account for a substantial proportion of the world market for electronic devices. Sales amount to billions of dollars annually, and seem likely to continue to increase rapidly.
DSP technology is nowadays commonplace in such devices as mobile phones, multimedia computers, video recorders, CD players, hard disc drive controllers and modems, and will soon replace analog circuitry in TV sets and telephones. An important application of DSP is in signal compression and decompression. Signal compression is used in digital cellular phones to allow a greater number of calls to be handled simultaneously within each local "cell". DSP signal compression technology allows people not only to talk to one another but also to see one another on their computer screens, using small video cameras mounted on the computer monitors, with only a conventional telephone line linking them together. In audio CD systems, DSP technology is used to perform complex error detection and correction on the raw data as it is read from the CD.
Although some of the mathematical theory underlying DSP techniques, such as Fourier and Hilbert Transforms, digital filter design and signal compression, can be fairly complex, the numerical operations required actually to implement these techniques are very simple, consisting mainly of operations that could be done on a cheap four-function calculator. The architecture of a DSP chip is designed to carry out such operations incredibly fast, processing hundreds of millions of samples every second, to provide real-time performance: that is, the ability to process a signal "live" as it is sampled and then output the processed signal, for example to a loudspeaker or video display. All of the practical examples of DSP applications mentioned earlier, such as hard disc drives and mobile phones, demand real-time operation.
The major electronics manufacturers have invested heavily in DSP technology. Because they now find application in mass-market products, DSP chips account for a substantial proportion of the world market for electronic devices. Sales amount to billions of dollars annually, and seem likely to continue to increase rapidly.
DSP technology is nowadays commonplace in such devices as mobile phones, multimedia computers, video recorders, CD players, hard disc drive controllers and modems, and will soon replace analog circuitry in TV sets and telephones. An important application of DSP is in signal compression and decompression. Signal compression is used in digital cellular phones to allow a greater number of calls to be handled simultaneously within each local "cell". DSP signal compression technology allows people not only to talk to one another but also to see one another on their computer screens, using small video cameras mounted on the computer monitors, with only a conventional telephone line linking them together. In audio CD systems, DSP technology is used to perform complex error detection and correction on the raw data as it is read from the CD.
Although some of the mathematical theory underlying DSP techniques, such as Fourier and Hilbert Transforms, digital filter design and signal compression, can be fairly complex, the numerical operations required actually to implement these techniques are very simple, consisting mainly of operations that could be done on a cheap four-function calculator. The architecture of a DSP chip is designed to carry out such operations incredibly fast, processing hundreds of millions of samples every second, to provide real-time performance: that is, the ability to process a signal "live" as it is sampled and then output the processed signal, for example to a loudspeaker or video display. All of the practical examples of DSP applications mentioned earlier, such as hard disc drives and mobile phones, demand real-time operation.
The major electronics manufacturers have invested heavily in DSP technology. Because they now find application in mass-market products, DSP chips account for a substantial proportion of the world market for electronic devices. Sales amount to billions of dollars annually, and seem likely to continue to increase rapidly.
Although some of the mathematical theory underlying DSP techniques, such as Fourier and Hilbert Transforms, digital filter design and signal compression, can be fairly complex, the numerical operations required actually to implement these techniques are very simple, consisting mainly of operations that could be done on a cheap four-function calculator. The architecture of a DSP chip is designed to carry out such operations incredibly fast, processing hundreds of millions of samples every second, to provide real-time performance: that is, the ability to process a signal "live" as it is sampled and then output the processed signal, for example to a loudspeaker or video display. All of the practical examples of DSP applications mentioned earlier, such as hard disc drives and mobile phones, demand real-time operation.
The major electronics manufacturers have invested heavily in DSP technology. Because they now find application in mass-market products, DSP chips account for a substantial proportion of the world market for electronic devices. Sales amount to billions of dollars annually, and seem likely to continue to increase rapidly.
DSP technology is nowadays commonplace in such devices as mobile phones, multimedia computers, video recorders, CD players, hard disc drive controllers and modems, and will soon replace analog circuitry in TV sets and telephones. An important application of DSP is in signal compression and decompression. Signal compression is used in digital cellular phones to allow a greater number of calls to be handled simultaneously within each local "cell". DSP signal compression technology allows people not only to talk to one another but also to see one another on their computer screens, using small video cameras mounted on the computer monitors, with only a conventional telephone line linking them together. In audio CD systems, DSP technology is used to perform complex error detection and correction on the raw data as it is read from the CD.
Although some of the mathematical theory underlying DSP techniques, such as Fourier and Hilbert Transforms, digital filter design and signal compression, can be fairly complex, the numerical operations required actually to implement these techniques are very simple, consisting mainly of operations that could be done on a cheap four-function calculator. The architecture of a DSP chip is designed to carry out such operations incredibly fast, processing hundreds of millions of samples every second, to provide real-time performance: that is, the ability to process a signal "live" as it is sampled and then output the processed signal, for example to a loudspeaker or video display. All of the practical examples of DSP applications mentioned earlier, such as hard disc drives and mobile phones, demand real-time operation.
The major electronics manufacturers have invested heavily in DSP technology. Because they now find application in mass-market products, DSP chips account for a substantial proportion of the world market for electronic devices. Sales amount to billions of dollars annually, and seem likely to continue to increase rapidly.
DSP technology is nowadays commonplace in such devices as mobile phones, multimedia computers, video recorders, CD players, hard disc drive controllers and modems, and will soon replace analog circuitry in TV sets and telephones. An important application of DSP is in signal compression and decompression. Signal compression is used in digital cellular phones to allow a greater number of calls to be handled simultaneously within each local "cell". DSP signal compression technology allows people not only to talk to one another but also to see one another on their computer screens, using small video cameras mounted on the computer monitors, with only a conventional telephone line linking them together. In audio CD systems, DSP technology is used to perform complex error detection and correction on the raw data as it is read from the CD.
Although some of the mathematical theory underlying DSP techniques, such as Fourier and Hilbert Transforms, digital filter design and signal compression, can be fairly complex, the numerical operations required actually to implement these techniques are very simple, consisting mainly of operations that could be done on a cheap four-function calculator. The architecture of a DSP chip is designed to carry out such operations incredibly fast, processing hundreds of millions of samples every second, to provide real-time performance: that is, the ability to process a signal "live" as it is sampled and then output the processed signal, for example to a loudspeaker or video display. All of the practical examples of DSP applications mentioned earlier, such as hard disc drives and mobile phones, demand real-time operation.
The major electronics manufacturers have invested heavily in DSP technology. Because they now find application in mass-market products, DSP chips account for a substantial proportion of the world market for electronic devices. Sales amount to billions of dollars annually, and seem likely to continue to increase rapidly.
Analog and Digital Signals
In order to be able to process real world signals they need to be converted to a format that the computers can understand (digital) this process is called analog to digital conversion and the reverse process is called digital to analog conversion. Although there is no definitive DSP system, the following diagram shows the configuration of a typical DSP system.
An analog signal must be converted into digital form before DSP techniques can be applied. An analog electrical voltage signal, for example, can be digitised using an electronic circuit called an analog-to-digital converter or ADC. This generates a digital output as a stream of binary numbers whose values represent the electrical voltage input to the device at each sampling instant.
Analog and digital signals
Analog signals are continuous where digital signals are discrete. Anolog signals are continuously varying where digital signals are based on 0's and 1's (or as often said------- on's and off's). As an analogy, consider a light switch that is either on or off (digital) and a dimmer switch (analog) that allows you to vary the light in different degrees of brightness. As another analogy, consider a clock in which the second hand smoothly circles the clock face (analog) versus another clock in which the second hand jumps as each second passes (digital). Digital computers work with a series of 0's and 1's to represent letters, symbols, and numbers. In addition, numbers are represented by using the binary code (where only 0's and 1's are used). Number Binary equivalent
1----------------------------- -----------------12----------------------------- ----------------103----------------------------- ----------------114----------------------------- ---------------1005----------------------------- ---------------1016----------------------------- ---------------1107----------------------------- ---------------1118----------------------------- --------------1000and so on. So each number (that we are accustomed to, such as 5) is represented by 0's and 1's. Morse code uses dits (or dots) and dashes. Digital signals are similar to Morse code. The signal is either a dit or a dash for Morse code and it is either a 0 or 1 for digital. A series of these dits and dashes might represent SOS to a navy radio man, and a series of 0's and 1's might represent the question mark to a computer.When an e-mail is sent that says "Hello Joe", Hello Joe doesn't mysteriously appear on Joe's computer. What is sent through the phone line is a series of 0's and 1's and Joe's computer "interprets" these into the words Hello Joe. If you type the letter A into your computer, it converts this A into 01000001. This 01000001 goes to Joe's computer and his computer interprets it as A. Each 0 or 1 is "bit" and the series of eight 0's and 1's is a byte. Well, that is about as simple as it gets and about as simple as I can state it.
1----------------------------- -----------------12----------------------------- ----------------103----------------------------- ----------------114----------------------------- ---------------1005----------------------------- ---------------1016----------------------------- ---------------1107----------------------------- ---------------1118----------------------------- --------------1000and so on. So each number (that we are accustomed to, such as 5) is represented by 0's and 1's. Morse code uses dits (or dots) and dashes. Digital signals are similar to Morse code. The signal is either a dit or a dash for Morse code and it is either a 0 or 1 for digital. A series of these dits and dashes might represent SOS to a navy radio man, and a series of 0's and 1's might represent the question mark to a computer.When an e-mail is sent that says "Hello Joe", Hello Joe doesn't mysteriously appear on Joe's computer. What is sent through the phone line is a series of 0's and 1's and Joe's computer "interprets" these into the words Hello Joe. If you type the letter A into your computer, it converts this A into 01000001. This 01000001 goes to Joe's computer and his computer interprets it as A. Each 0 or 1 is "bit" and the series of eight 0's and 1's is a byte. Well, that is about as simple as it gets and about as simple as I can state it.
Introduction to Digital Signal Processing
DSP, or Digital Signal Processing, as the term suggests, is the processing of signals by digital means. A signal in this context can mean a number of different things. Historically the origins of signal processing are in electrical engineering, and a signal here means an electrical signal carried by a wire or telephone line, or perhaps by a radio wave. More generally, however, a signal is a stream of information representing anything from stock prices to data from a remote-sensing satellite. The term "digital" comes from "digit", meaning a number (you count with your fingers - your digits), so "digital" literally means numerical; the French word for digital is numerique. A digital signal consists of a stream of numbers, usually (but not necessarily) in binary form. The processing of a digital signal is done by performing numerical calculations.
Digital Signal Processing (DSP) is the processing of real world (analog) signals with microprocessors. It is essentially the real-time execution of mathematical algorithms and is one of the fastest growing fields of technology and computer science in the world, with typically a growth rate in excess of 30%. The growth of the computer industry has affected every corner of daily life and everyone is aware of this effect. In today's western world almost everyone uses DSPs in their everyday life, but unlike users of PCs almost no one knows that they are using DSPs. Digital Signal Processors (DSPs) are special purpose microprocessors and they are used in every form of electronic product, from mobile phones, modems and CD players to the automotive industry; medical imaging systems to the electronic battlefield and from dishwashers to satellites.
Digital Signal Processing (DSP) is the processing of real world (analog) signals with microprocessors. It is essentially the real-time execution of mathematical algorithms and is one of the fastest growing fields of technology and computer science in the world, with typically a growth rate in excess of 30%. The growth of the computer industry has affected every corner of daily life and everyone is aware of this effect. In today's western world almost everyone uses DSPs in their everyday life, but unlike users of PCs almost no one knows that they are using DSPs. Digital Signal Processors (DSPs) are special purpose microprocessors and they are used in every form of electronic product, from mobile phones, modems and CD players to the automotive industry; medical imaging systems to the electronic battlefield and from dishwashers to satellites.
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