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Signals can be represented by discrete quantities instead of as a function of a continuous variable. These discrete time signals do notnecessarily have to take real number values. Many properties of continuous valued signals transfer almost directly to the discretedomain.

So far, we have treated what are known as analog signals and systems. Mathematically, analog signals are functions having continuous quantities as theirindependent variables, such as space and time. Discrete-time signals are functions defined on the integers; they are sequences. One ofthe fundamental results of signal theory will detail conditions under which an analog signal can be converted into a discrete-time one andretrieved without error . This result is important because discrete-time signals can be manipulated bysystems instantiated as computer programs. Subsequent modules describe how virtually all analog signal processing can beperformed with software.

As important as such results are, discrete-time signals are more general, encompassing signals derived fromanalog ones and signals that aren't. For example, the characters forming a text file form a sequence,which is also a discrete-time signal. We must deal with such symbolic valued signals and systems as well.

As with analog signals, we seek ways of decomposing real-valueddiscrete-time signals into simpler components. With this approach leading to a better understanding of signal structure,we can exploit that structure to represent information (create ways of representing information with signals) and to extractinformation (retrieve the information thus represented). For symbolic-valued signals, the approach is different: We develop acommon representation of all symbolic-valued signals so that we can embody the information they contain in a unified way. Froman information representation perspective, the most important issue becomes, for both real-valued and symbolic-valued signals,efficiency; What is the most parsimonious and compact way to represent information so that it can be extracted later.

Real- and complex-valued signals

A discrete-time signal is represented symbolically as s n , where n -1 0 1 . We usually draw discrete-time signals as stem plots toemphasize the fact they are functions defined only on the integers. We can delay a discrete-time signal by an integerjust as with analog ones. A delayed unit sample has the expression δ n m , and equals one when n m .

Discrete-time cosine signal

The discrete-time cosine signal is plotted as a stem plot. Can you find the formula for this signal?

Complex exponentials

The most important signal is, of course, the complex exponential sequence .

s n 2 f n

Sinusoids

Discrete-time sinusoids have the obvious form s n A 2 f n φ . As opposed to analog complex exponentials and sinusoids thatcan have their frequencies be any real value, frequencies of their discrete-time counterparts yield unique waveforms only when f lies in the interval 1 2 1 2 . This property can be easily understood by noting that addingan integer to the frequency of the discrete-time complex exponential has no effect on the signal's value.

2 f m n 2 f n 2 m n 2 f n
This derivation follows because the complex exponential evaluated at an integer multiple of 2 equals one.

Unit sample

The second-most important discrete-time signal is the unit sample , which is defined to be

δ n 1 n 0 0

Unit sample

The unit sample.

Examination of a discrete-time signal's plot, like that of the cosine signal shown in [link] , reveals that all signals consist of a sequence of delayed andscaled unit samples. Because the value of a sequence at each integer m is denoted by s m and the unit sample delayed to occur at m is written δ n m , we can decompose any signal as a sum of unit samples delayed to the appropriate location and scaled bythe signal value.

s n m s m δ n m
This kind of decomposition is unique to discrete-time signals, and will prove useful subsequently.

Discrete-time systems can act on discrete-time signals in wayssimilar to those found in analog signals and systems. Because of the role of software in discrete-time systems, many moredifferent systems can be envisioned and “constructed” with programs than can be with analog signals. In fact, a specialclass of analog signals can be converted into discrete-time signals, processed with software, and converted back into ananalog signal, all without the incursion of error. For such signals, systems can be easily produced in software, withequivalent analog realizations difficult, if not impossible, to design.

Symbolic-valued signals

Another interesting aspect of discrete-time signals is thattheir values do not need to be real numbers. We do have real-valued discrete-time signals like the sinusoid, but wealso have signals that denote the sequence of characters typed on the keyboard. Such characters certainly aren't realnumbers, and as a collection of possible signal values, they have little mathematical structure other than that they aremembers of a set. More formally, each element of the symbolic-valued signal s n takes on one of the values a 1 a K which comprise the alphabet A . This technical terminology does not mean we restrict symbols to being members of the Englishor Greek alphabet. They could represent keyboard characters, bytes (8-bit quantities), integers that convey dailytemperature. Whether controlled by software or not, discrete-time systems are ultimately constructed from digitalcircuits, which consist entirely of analog circuit elements. Furthermore, the transmission andreception of discrete-time signals, like e-mail, is accomplished with analog signals and systems. Understandinghow discrete-time and analog signals and systems intertwine is perhaps the main goal of this course.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
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can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
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