Signal and System

Signal and System1.Mathematical Representation of SignalsAnything that bears information can be considered a signal. Signals may describe a wide variety of physical phenomena. For example, speech, music, interest rates, the speed of an automobile are signals. Although signals can be represented in many ways, in all cases the information in a signal is contained in a pattern of variations of some form. For example, consider the human vocal mechanism, which produces speech by creating fluctuations in acoustic pressure.The signal represents acoustic pressure variations as a function of time for the spoken words “should we chase.” The top line of the figure corresponds to the word “should,” the second line to the word “we,” and the last two lines to the wo rd “chase.”Signals are presented mathematically as functions of one or more independent variables. For example, a speech signal can be represented mathematically by acoustic pressure as a function of time, and a picture can be presented by brightness as a function of two spatial variables. For convenience, we will generally refer to independent variable as time, although it may not in fact represent time in specific time in specific applications. For example in geophysics, signals representing variations with depth of physical quantities such as density, porosity, and electrical resistivity are used to study the structure of the earth.Two basic types of signals will be considered: continuous-time signal and discrete-time signal. In the case of continuous-time signals the independent variable is continuous, and thus these signals are defined for a continuum of values of the independent variable. On the other hand, discrete-time signals are defined only at discrete times, foe these signals, the independent variable takes on only a discrete set of values. A speech signals as a function of time and atmospheric pressure as a function of altitude are example of continuous-time signals. The weekly Dow-Jones stock market index is an example of a discrete-time signal. Other examples of discrete-time signals can be found in demographic studies in which various attributes, such as average budget, crime rate, or pounds of fish caught, are tabulated against such discrete variable as family size, total population, or type of fishing vessel, respectively.To distinguish between continuous-time signals and discrete-time signals, we will use the symbol t to denote the continuous-time independent variable and n to denote the discrete-time independent. In addition, for continuous-time signals we will enclose the independent variable in parentheses (.), whereas for discrete-time signals we will use brackets [.] to enclose the independent variable. We will also have frequent occasions when it will be useful to represent signals graphically. It is important to note that the discrete-time signal x[n] is defined only for integer values of the independent variable. For further emphasis we will on occasion refer to x[n] as discrete-time sequence.A discrete-time signal x[n] may represent a phenomenon for which theindependent variable is inherently discrete. Signal such as demographic data are example of this .On the other hand, a very important class of discrete-time signalsarises from the sampling of continuous-time signals. In this case, the discrete-time signal x[n] represent successive samples of an underlying phenomenon for which the independent variable is continuous. No matter what source of the data,however, the signal x[n] is defined only for integer values of n. It makes no matter sense to refer to the 7/2th sample of a digital speech signal than it does to refer to the average budget of family with 5/2 family members.2. Mathematical Representation of SystemsPhysical systems in the broadest sense are an interconnection of components, devices, or subsystems. In contexts ranging from signal processing and communications to electromechanical motors, automotive vehicles, and chemical-processing plants, a system can be viewed as a process in which input signals are transformed by the system or cause the system to respond in some way, resulting in other signals as outputs, For example, a high fidelity system takes a recorded audio signal and generates a reproduction of that signal. If the hi-fi system has tone controls, we can change the tonal quality of the reproduced signal. Similarly, the circuit can be viewed as a system with input voltage x (t) and output voltage v (t), while the automobile can be thought of as a system with input equal to the force f (t) and output equal to the velocity v(t) of the vehicle An image-enhancement system transforms an input image into an output image that has some desired properties, such as improved contrast.A continuous-time system is a system in which continuous-time input signals are applied and result in continuous-time output signals. Such as system will be represented pictorially as in FIG. 3-4(a), where x(t) is the input and y(t) is the output. Alternatively, we will often represent the input-output relation of a continuous-time system by the notationX (t)→y(t)Similarly, a discrete-time system-that is, a system that transforms discrete-time inputs into discrete-time outputs-will be depicted as in FIG.3-4(b) and will sometimes be represented symbolically asX (n)→y(n)(t)Fig.3-4 (a) Continuous-time system; (b) Discrete-time system;Many real systems are built as interconnections of several subsystems. Be viewing such a system as an interconnection of its components, we can use our understanding of the components systems and of how they are interconnected in order to analyze the operation and behavior of the overall system. In addition, by describing a system in terms of an interconnection of simpler subsystems, we may in fact be able to define useful ways in which to synthesize complex systems out of simpler, basic building blocks.While one can construct a variety of system interconnections, there are several basic ones that are frequently encountered. A series or cascade information of two systems is illustrated in Fug.3-5(a). Diagrams such as this are referred to as block diagrams. Here, the output of System 1 is the input to System 2 and the overall system transforms an input by processing it first by System 1 and then by System 2 . An example of a series interconnection is a radio receiver followed by an amplifier. Similarly, one can define a series interconnection of three or more systems.(a)(b)(c)Fig.3-5 Interconnection of two systems (a) series (cascade) interconnection;(b) Parallel interconnection; (c) series-parallel interconnectionA parallel interconnection of two systems is illustrated in Fig.3-5(b).Here, the same i nput signal is applied to systems 1to 2.the symbol ”⊕” in the figure denotes addition, so that the output of the parallel interconnection is the sum of the outputs of Systems 1 and 2. An example of a parallel interconnection is a simple audio system with several microphones feeding into a signal amplifier and speaker system. In addition to the simple parallel interconnection in Fig.3-8(b), we can define parallel interconnections more than two systems, and we can combine both cascade and parallel interconnection to obtain more complicated interconnections. An example of such an interconnection is given in Fig.3-5(c).Another important type of system interconnection is a feedback interconnection, an example of which is illustrated in Fig.3-6. Here, the output of system 1 is the input to system 2, while the output of system 2 is fed back and added to the external input to produce the actual input to system 1. Feedback systems arise in a wide variety of applications. For example, a cruise control system on an automobile senses sense thevehicle’s velocity and adjusts the fuel flow in order to keep the speed at the desired level.Fig.3-6 Feedback interconnection3. Fourier Transforms and Frequency-Domain DescriptionSignals encountered in practice are mostly continuous-time signals and can be denoted as x (t), where t is a continuum. Although some signals such as stock markets, savings account and inventory are inherently discrete time, most discrete-time signals are obtained from continuous-time signals by sampling and can be denoted as x[n]:=x(n T), where T is sampling period and n is the time index and can assume only integers. Both x (t) and x[n] are functions of time and are called the time-domain description. In signal analysis, we study frequency contents of signals. In order to do so, we must develop a different but equivalent description, called the frequency-domain description. From the description, we can more easily determine the distribution of power in frequencies.In digital processing of a continuous-time signal x (t), the first step is to select a sampling period T and then to sample x (t) to yield x(n T).It is clear that the smaller T is, the closer x (n T) is to x(t). However, a smaller T also requires more computation. Thus an important task in DSP is to find the largest possible T so that all information (if not possible, all essential information) of x(t) is retained in x(x T). Without the frequency-domain description, it is not possible to find such a sampling period. Thus computing the frequency content of signals is a first step in digital signal processing.The frequency-domain description is developed from the Fourier transform. If the Fourier transform of a signal is defined, the transform is called the frequency spectrum of the signal that isFourier transform ←→frequency spectrumThe continuous-time Fourier transform is defined by the following pair of equations:Forward Continuous- Time Fourier TransformX(j w)= e -jwt dt (3-1)And Inverse Continuous-Time Fourier TransformX(t)=1/2e jwt dt (3-2)Equation (3-1) and (3-2) are referred to as the Fourier transform pair, with the function X(jw) referred to as Fourier Transform or Fourier integral of x(t) and eq.(3-2)as the inverse Fourier Transform equation. X(jw) is commonly referred to as the frequency-domain representation or the spectrum of the signal, as it provides us with the information needed for describing x(t) as a liner combination (specifically, an integral) of sinusoidal signals at different frequencies. Likewise, x(t) is the time-domain representation of the signal. We indicate this relationship between the two domains as 360毕业设计网友情提供Time-Domain Frequency-domainX (jw)The notation F signifies that it is possible to go back and forth uniquely between the time-domain and the frequency-domain.If we are given x(t) as a mathematical function, we can determine the corresponding spectrum function X(jw) by evaluating the integral in (3-1). In other words, (3-1) defines a mathematical operation for transforming x(t) into a new equivalent representation X(jw). It is common to say that we take the Fourier transform of x(t),meaning that we determine X(jw) so that we can use the frequency-domain representation of the signal.Similarly, given X(jw) as a mathematical function, we can determine the corresponding time function x(t) using (3-2) by evaluating an integral, Thus, (3-2) defines the inverse Fourier transform operation that goes from the frequency-domain to the time-domain.Armed with the powerful tool of Fourier transform, we will be able to (1) define a precise notion of bandwidth for a signal, (2) explain the inner workings of modern communication systems which are able to transmit many signals simultaneously by sharing the available bandwidth, and (3) define filtering operations that are needed to separate signals in such frequency-shared systems. There are many other applications of the Fourier transform, so it is safe to say that Fourier analysis provides the rigorous language needed to define and design modern engineering systems.4. The Sampling TheoremUnder certain conditions, a continuous-time signal can be completely represented by and recoverable from a sequence of its values, or samples, at points equally space in time. This somewhat surprising property follows from a basic result that is referred to as the sampling theorem. This theorem is extremely important and useful. It is exploited, for example, in moving pictures, which consist of a sequence of individual frames, each of which represent an instantaneous view (i.e., a sample in time) of a continuously changing scene. When these samples are viewed in sequence at a sufficiently fast rate, we perceive an accurate representation of the original continuously moving scene.Much of the important of the sampling theorem also lies in its role as a bridge between continuous-time signals and discrete-time signals. The fact that under certain conditions a continuous-time signal can be completely recovered from a sequence of its samples provides a mechanism for repressing a continuous-time signal by a discrete-time signal. In many contexts, processing discrete-time signals is moreflexible and is often preferable to processing continuous-time signals. This is due in large part to the dramatic development of digital technology over the past few decades, resulting in the availability of inexpensive, lightweight, and programmable, and easily reproducible discrete-time system. We exploit sampling to convert a continuous-time signal to a discrete-time signal, process the discrete-time signal using a discrete-time system, and then convert back to continuous-time signal.Sampling theorem can be stated as follows:Let x(t) be a band-limited signal with X(jw)=0 for |w|>wm. Then x(t) is uniquely determined by its samples x(nT),n=0,+1,-1 (i)Ws>2w mWhereW s=2/TGiven these samples, we can reconstruct x(t) by generating a periodic impulse train in which successive impulses have amplitudes that are successive sample values. This impulse train in which successive through an ideal low pass filter with gain T and cutoff frequency greater than w m and less than w s-2 w m. The resulting output signal will exactly equal x (t).The frequency 2w m, which, under the sampling theorem, must be exceeded by the sampling frequency, is commonly referred to as the Nyquist rate.In previous discussion, it was assumed that the sampling frequency was sufficiently high that the conditions of sampling theorem were met. With w s>2w m the spectrum of the sampled signal consists of scaled replications of the spectrum of x(t), and this forms the basis for the sampling theorem. When w s<2w m X(jw),the spectrum of x(t),is no longer replicated in X p(jw) and thus is no longer recoverable by lowpass filtering. The reconstructed signal will no longer be equal to x(t). This effect is referred to as aliasing.Sampling has a number of important applications. One particularly significant set of applications relates to using sampling to process continuous-time signal with discrete-time systems, by means of minicomputers, or any of a variety of devices specifically oriented toward discrete-time signal processing.5.DSP Processor FundamentalsIn the literature, the define of a digital signal processor takes many forms. In a strict sense, a DSP is any microprocessor that processes digitally represented signals.A DSP filter for example,takes one or more discrete inputs, x i[n],and produces one corresponding output,y[n] for n=….-1,0,1,2,…and i=1,…，N，where n is the nth input or output at time n, i is the ith coefficient and N is the length of the filter. In effect, the DSP implements the discrete-time system. As its name implies, it is assumed that there must be some form of an analog to digital converter(ADC).In general, DSP functions are mathematical operations on real-time signals and are repetitive and numerically intensive. Samples from real-time signals can number in the millions and hence a large memory bandwidth is needed.It is because of this very nature that DSP processors are created with an architecture unlike those of conventional microprocessors. Most DSP algorithms are not complicated and onlyrequire multiply and accumulate calculations. Most, if not all, DSP processors have circuitry built and hard wire to execute these calculations as fast as possible.Most DSP calculations are repetitive, require a large a memory bandwidth and numeric precision, all executed in real time. One might also argue that moder GPPs have clock speeds and cycles per instruction(CPI) that outperform DSP processors but GPPs have operations and program flexibility that are unnecessary for DSP. DSPs must execute their takes efficiently while keeping cost, power consumption, memory and development time low, especially in the age of mobile computing.Since many signal processing applications process millions of samples of data for every second of operation, the minimum sample period is usually more important than the computational latency of the processor. We define the period as the time between each sequential sample of the input data. The time difference between the input data and the result of its computation is known as the computational latency. Once the initial sample is calculated with certain latency, the subsequent results will however, be produced at the sample period rate. As the number of calculations increases, the relatively larger latency of the processor will be negligible compared to the sample rate.。