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Lee, Varaiya. Structure and interpretation of signals and systems (Berkeley, 2000)(441s).pdf |
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Size 2.7Mb Date Dec 22, 2004 |
Preface
“The computer revolution is a revolution in the way we think and in the way we express what we think. The essence of this change is what might best be called procedural epistemology – the study of the structure of knowledge from an imperative point of view, as opposed to the more declarative point of view taken by classical mathematical subjects. Mathematics provides a framework for dealing precisely with notions of ‘what is.’ Computation provides a fram-work for dealing precisely with notions of ‘how to’.”...
Notes to Instructors
Assume we have a signal that contains frequencies in the range of about 100 to 300 Hz, and we have a channel that can pass frequencies from 700 to 1300 Hz. The task is to modulate the first signal so that it lies entirely within the channel passband, and then to demodulate to recover the original signal....
Discussion
The first few times we offered this course, automata appeared after frequency domain concepts. The new ordering, however, is far better. In particular, it introduces mathematical concepts gradually....
Functions as Values
Most texts call the expression x(t) a function. A better interpretation is that x(t) is an element in the range of the function x. The difficulty with the former interpretation becomes obvious when talking about systems. Many texts pay lip service to the notion that a system is a function by introducing a notation like y (t) = T (x(t)). This makes no distinction between the value of the function at t and the function y itself. Why does this matter? Consider our favorite type of system, an LTI system. We write y (t) = x(t) ∗ h(t) to indicate convolution. Under any reasonable interpretation of mathematics, this would seem to imply that y (t − τ ) = x(t − τ ) ∗ h(t − τ ). But it is not so! How is a student supposed to conclude that y (t − 2τ ) = x(t − τ ) ∗ h(t − τ )? This sort of sloppy notation could easily undermine the students’ confidence in mathematics. In our notation, a function is the element of a set of functions, just as its value for a given element in the domain is an element of its range. Convolution is a function whose domain is the cross product of two sets of functions. Continuous-time convolution, for example, is Convolution : [Reals → Reals] × [Reals → Reals]...
Example 1.7: When you enter your car the starting trace of events might be StartEngine, SeatbeltSignOn, BuckleSeatbelt, SeatbeltSignOff, · · · Thus event streams are functions of the form EventStream: Indices → EventSet . We will see in chapter 3 that the behavior of finite state machines is best described in terms of event traces, and that systems that operate on event streams are often best described as finite state machines....
1.2.1 Systems as functions
Consider a system S that transforms input signal x into output signal y . The system is a function, so y = S (x). Suppose x: D → R is a signal with domain D and range R. For example, x might be...
1.2.2 Telecommunications systems
We give some examples of systems that occur in or interact with the global telecommunications network. This network is unquestionably one of the most remarkable accomplishments of humankind. It is astonishingly complex, composed of hundreds of distinct corporations and linking billions of people. We often think of it in terms of its basic service, POTS, or plain-old telephone service. POTS is a voice service, but the telephone network is in fact a global, high-speed digital network that carries not just voice, but also video, images, and computer data, including much of the traffic in the Internet....
DTMF Even in POTS, not all of the information transported is voice. At a minimum, the telephone needs to be able to convey to the central office a telephone number in order to establish a connection. A telephone number is not a voice signal. It is intrinsically discrete. Since the system is designed to carry voice signals, one option is to convert the telephone number into a voice-like signal. A system is needed with the structure shown in figure 1.16. The block labeled “DTMF” is a system that transforms a sequence of numbers (coming from the keypad on the left) into a voice-like signal....
1.2.5 Feedback control system
Feedback control systems are composite systems where a plant, the nature of which we have little control over, is fed a control signal. A plant may be a mechanical device, such as the power train of a car, or a chemical process, or an aircraft with certain inertial and aerodynamic properties, for example. Sensors attached to the plant produce signals that are fed to the controller, which then generates the control signal. This arrangement, where the plant feeds the controller and the controller feeds the plant, is a complicated sort of composite system called a feedback control system. It has extremely interesting properties which we will explore in much more depth in subsequent chapters. In this chapter, we construct a model of a feedback control system using the syntax of block diagrams. Each system model consists of several interconnected components. We will identify the input and output signals of each component and how the components are interconnected, and we will argue on the basis of a common-sense physics how the overall system will behave. In later chapters we consider mathematical specifications of these component systems from which we can mathematically analyze the system behavior. Example 1.13: Consider a forced air heating system, which heats a room in a home or office to a desired temperature. Our first task is to identify the individual components of the heating system. These are • a furnace/blower unit (which we will simply call the heater) that heats air and blows the hot air through vents into a room, • a temperature sensor that measures the temperature in a room, and...
1.3. SUMMARY
(b) An image from a scanner stored in computer memory, (c) The height of points on the surface of the earth, (d) The location of the chairs in a room....
The exponential of a complex number, exp: Comps → Comps, is given by ∀ z ∈ Comps, exp(z ) =
n∞ z n
=0...
Thus, the imperative approach has the weakness that ensuring correctness is more difficult. Humans have developed a huge arsenal of techniques and skills for thoroughly understanding declarative definitions (thus lending confidence in their correctness), but we are only beginning to learn how to ensure correctness in imperative definitions....
2.3.3 Difference equations
Consider a class of systems given by functions S : DiscSignals → DiscSignals where DiscSignals is a set of discrete-time signals. Depending on the scenario, we could have DiscSignals = [Ints →...
The system of figure 2.7 is obtained from that of figure 2.6 by connecting the output signal z to the input signal w. As a result the new system has input signal x, output signal z , internal signals y and w, and it is described by the function S : X → Z , where ∀ x ∈ X, S (x) = S2 (S (x), S1 (x)). (2.17)...
States is the state space, Inputs is the input alphabet, Outputs is the output alphabet, initialState ∈ States is the initial state, and update : States × Inputs → States × Outputs is the update function. This five-tuple is called the sets and functions model of a state machine. Inputs and Outputs are the sets of possible input and output values. The set of input signals consists of all infinite sequences of input values, InputSignals = [Nats0 → Inputs]. The set of output signals consists of all infinite sequences of output values, OutputSignals = [Nats0 → Outputs]. Let x ∈ InputSignals be an input signal. A particular element in the signal can be written x(n) for any n ∈ Nats0 . We write the entire input as a sequence (x(0), x(1), · · · , x(n), · · ·). This sequence defines the function x in terms of elements x(n) ∈ Inputs, which represent particular input values. We reiterate that the index n in x(n) does not refer to time, but rather to the step number. This is an ordering constraint only: step n occurs after step n − 1 and before step n + 1. The state machine evolves (i.e. moves from one state to the next) in steps.1
1 Of course the steps could last a fixed duration of time, in which case there would be a simple relationship between step number and time. The relationship may be a mixed one, where some inputs are separated by a fixed amount of time and some are not....
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