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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]...
1.1.1 Audio signals
Our ears are sensitive to sound, which is physically just rapid variations in air pressure. Thus sound can be represented as a function Sound : Time → Pressure...
so each pixel value is an element of {0, 1, · · · , 255}3 . Such a pixel can be represented as three 8-bit words. A common method which saves memory is to use a colormap. Define the set ColorMapIndexes = {0, · · · , 255}, together with a Display function, Display: ColorMapIndexes → Intensity3 . (1.3)...
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....
Digital networks The first widely available service that gave direct access to the global digital telephone network was ISDN — integrated services digital network. The ISDN service required that a different line card be installed at the central office; it was therefore not as universally available as POTS. In fact, it took nearly 10 years in the U.S. for ISDN become widely installed after it was developed in the early 1980s. The configuration for ISDN is shown below the voiceband data modem in figure 1.14. It requires a special modem on the customer side as well as a special line card in the central office. ISDN typically provides two channels at rates of 64,000 bits per second plus a third control channel with a rate of 16,000 bits per second. One of the 64 kbps channels can be used for voice while simultaneously the other two channels are used for data. A more modern service is DSL — digital subscriber line. As shown at the lower right in figure 1.14, the configuration is similar to ISDN. Specialized modems and line cards are required. ADSL, asymmetric DSL, is a variant that provides an asymmetric bit rate, with a much higher rate in the direction from the central office to the customer than from the customer to the central office. This asymmetry recognizes the reality of most Internet applications, where relatively little data flows from the client, and torrents of data (including images and video) flow from the server. Modems are used for many other channels besides the voiceband channel. Digital transmission over radio, for example, requires that the bit sequence be transformed into a radio signal that conforms with regulatory constraints on that radio channel. Digital transmission over electrical cable requires transforming the bit sequence into a form that propagates well over such cable and that does not radiate too much radio-frequency interference. Digital transmission over optical fiber requires transforming the bit sequence into a light signal, usually with the intensity being modulated at very high rates. At each stage in the telephone network, therefore, a voice signal has a different physical form with properties that are well suited to the medium through which the signal propagates. For example, voice, which in the form of sound only travels well over the short distances, is converted to an electrical signal that carries well over copper wires for the few kilometers. Copper wires, however, are not as well matched for long distances as optical fiber. Most long distance communication channels today use optical fiber, although satellites still have certain advantages....
1.2.3 Audio storage and retrieval
We have seen how audio signals can be represented as sequences of numbers. Digital audio storage and retrieval is all about finding a physical and persistent representation for these numbers. These numbers can be converted into a single sequence of bits (binary digits) and then “printed” onto some physical medium from which they can later be read back. The transformation of sound into its persistent representation can be modeled as a system, as can the reverse or playback process. Example 1.12: In the case of compact discs (CDs), the physical medium is a layer of aluminum on a platter into which tiny pits are etched. In the playback device, a laser aimed at the platter uses an interference pattern to determine whether or not a pit exists at a particular point in the platter. These pits, thus, naturally represent binary digits, since they can have two states (present or not present). While a voiceband data modem converts bit sequences into voice-like signals, a musical recording studio does the reverse, creating a representation of the sound that is a bit
A phone conversation relayed by satellite has a much larger delay. Most satellites traditionally used in the telecommunications network are geosynchronous, meaning that they hover at the same point over the surface of the earth. To do that, they have to orbit at a height of 22,300 miles or 35,900 kilometers. It takes a radio signal about 120 ms to traverse that distance; since a signal has to go up and back, there is an end-to-end delay of at least 240 ms (not counting delays in the electronics). If you are using this channel for a telephone conversation, then the round-trip delay from when you say something to when you get a reaction is a minimum of 480 ms. This delay can be quite annoying, impeding your ability to converse until you got used to it. If you use Internet telephony, the delays are even larger, and they can be irregular depending upon how congested the Internet is when you call.
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