信号与系统双语ppt课件.ppt

1.4 The unit impulse and unit step function,1.4.1 The discrete-time unit impulse and unit step sequence,1 Unit impulse:,Throughout the book, we refer interchangeably as the unit impulse or unit sample.,2 Unit step: un,3 Relationship between the discrete-time unit impulse and unit step,(1) The unit impulse is the first difference of the step:,(2) The unit step is the running sum of the impulse:,By changing the variable of summation from m to k=n-m,(a) n0,(a) n0,4 The sampling property of unit impulse: Unit impulse can be used to sample the value of signal at n=0,More generally, if we consider a unit impulse at,Note :The sampling property of unit impulse is very important.,1.4.2 The continuous-time unit step and unit impulse function,1 Unit step:,2 Unit impulse:,has no duration but unit area, we adopt the graphical notation for it shown in figure before, where the arrow at t=0 indicates that the area of the pulse is concentrate at t=0 and the height of the arrow and the “1” next to the arrow are used to represent the area of the impulse.,More generally, a scaled impulse will have an area k.,The continuous-time unit step is the running integral of the unit impulse:,Unit impulse can be thought of as the first derivative of the continuous-time unit step:,3 relationship between continuous-time unit step and unit impulse:,Therefore,As with the discrete-time impulse, the continuous-time impulse has a very important sampling property.,We also have an analogous expression for an impulse concentrated at an arbitrary point,t0,that is:,4 Sampling property,1.5 Continuous-time and discrete-time system,Physical system in the broadest sense are an interconnection of components, devices, or subsystem. 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.,output,input,system,A continuous-time system is a system in which continuous-time input signals are applied and result in continuous-time output signals.,We often represent the input-output relation by the notion:,Discrete-time system:,Simple example of system:,(1),(2),Finally ,we obtain a differential equation describing the relationship between the input and output :,1.5.2 Interconnections of system,Many real systems are built as interconnections of several subsystems. By viewing system as an interconnection of its components, we can use our understanding of the component 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 subsystem, we may in fact be able to define useful ways in which to synthesize complex systems out of simpler, basic building blocks.,1 Series interconnection :,Here , the output of system 1 is the input of system 2.,2 Parallel interconnection :,Here , the same input signals is apply to system 1 and system 2.,addition,We can combine both series and parallel interconnections to obtain more complicated interconnections. An example is as follow:,3 Feedback interconnection :,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.,Ch1.6 Basic system properties,1. System with and without memory 2. Invertibility and inverse system3.Causality 4. Stability 5.Time-Invariance 6. linearity,1.6.1 System with and without memory,1. A system is said to be memoryless if its input for each value of the independent variable at a given time is depended only on the input at that time.,For example:,memoryless,Similarly ,a resistor is a memoryless system,Voltage,Current,One particular simple memoryless system is the identity system, whose output is identical to input. That is,Or,2. A system with memory if the current output is dependent on past value or the future value of the input and output.,An example of the discrete-time system with is an accumulator or summer:,And a second example is a delay:,A capacitor is an example of a continuous-time system with memory:,1.6.2 Invertibility and inverse system,A system is said to be invertible if distinct input lead to distinct output. If an system is invertible, then an inverse system exists.,The inverse system is,An example of invertibility continuous-time system is,Another example of invertibility system is the accumulator;,The inverse system is :,Examples of noninvertibility system are:,This system produces zero output for any input sequence.,This system we cannot determine the sign of input from the knowledge of the output,Ex: Moving-Average Systems (滑动平均系统) y(n)=x(n)+x(n-1)+x(n-2)/3 Is this system causal?,1.6.3 Causality,A system is causal if the output at any time depends only on value of the input at the present time and in the past.,All memoryless system are causal. Why ?,Series RC circuit driven from an ideal voltage source v1(t), producing output voltage v2(t).,Ex: Consider the RC circuit. Is this system causal or non-causal?,Stable system ：Bounded Input-Bounded Output.,Unstable system：Bounded input-Unbounded output,1.6.4 Stability,Ex 1: Moving-Average Systems (滑动平均系统). Show that the System is BIBO stable: y(n)=x(n)+x(n-1)+x(n-2)/3.,Solution：y(n)=x(n)+x(n-1)+x(n-2)/3 （Mx+Mx+Mx）/3=Mx y(n)Bounded,Solution：if x(n)Mx1，rn，y(n) diverge.,Ex 2: Unstable System. , r1,Dramatic photographs showing the collapse of the Tacoma Narrows suspension bridge on November 7, 1940. (a) Photograph showing the twisting motion of the bridges center span just before failure. (b) A few minutes after the first piece of concrete fell, this second photograph shows a 600-ft section of the bridge breaking out of the suspension span and turning upside down as it crashed in Puget Sound, Washington. Note the car in the top right-hand corner of the photograph.,An Unstable System,1.6.5 Time Invariance,A system is time invariant if the behavior and characteristic of the system is fixed over time.,For example, the RC circuit is time invariant if the resistance and capacitance value R and C are constant over time.,On the other hand, if the resistance and capacitance value R and C are changed or fluctuate over time, then it is not time invariant. Because we would expect the result of our experiment to depend on the time at which we run it.,Ex: Are these systems time-invariant?,time-invariant,time-variant,A system is time-invariant if the input,(1) y(t) = 3x(t)(2) y(t) = tf(t),Then the output,A linearity system is a system that possesses the important property of superposition.,(2) Homogeneity,(1) Additivity,1.6.6 Linearity,The two properties defining a linear system can be combined into a single statement:,Continuous-time :,Discrete-time :,a and b are complex constants.,Ex: Consider the systems described by the input-output relation. Are the systems linear?,Analysis method :,Check,Is the systems linear?,solution：,The system is nonlinear.,solution：,Is the systems linear?,The system is linear。,The system is linear。,Is the systems linear?,solution：,