# Introductidn inspired by the linearity properties which are a

If you can show that a system has both properties, then you have proven that the system is linear. Likewise, if you can show that a system doesn't have one or both properties, you have proven that it isn't linear.

A third property, shift invarianceis not a strict requirement for linearity, but it is a mandatory property for most DSP techniques. When you see the term linear system used in DSP, you should assume it includes shift invariance unless you have reason to believe otherwise. These three properties form the mathematics of how linear system theory is defined and used.

Later in this chapter we will look at more intuitive ways of understanding linearity. For now, let's go through these formal mathematical properties. As illustrated in Fig. In mathematical terms, if an input signal of x[n] results in an output signal of y[n]an input of kx[n] results in an output of ky[n]for any input signal and constant, k. A simple resistor provides a good example of both homogenous and non-homogeneous systems.

If the input to the system is the voltage across the resistor, v tand the output from the system is the current through the resistor, i tthe system is homogeneous. Ohm's law guarantees this; if the voltage is increased or decreased, there will be a corresponding increase or decrease in the current. Now, consider another system where the input signal is the voltage across the resistor, v tbut the output signal is the power being dissipated in the resistor, p t.

Since power is proportional to the square of the voltage, if the input signal is increased by a factor of twothe output signal is increase by a factor of four. This system is not homogeneous and therefore cannot be linear. The property of additivity is illustrated in Fig. Consider a system where an input of x 1 [n] produces an output of y 1 [n]. Further suppose that a different input, x 2 [n]produces another output, y 2 [n].

In words, signals added at the input produce signals that are added at the output. The important point is that added signals pass through the system without interacting. As an example, think about a telephone conversation with your Aunt Edna and Uncle Bernie.

Aunt Edna begins a rather lengthy story about how well her radishes are doing this year. In the background, Uncle Bernie is yelling at the dog for having an accident in his favorite chair. The two voice signals are added and electronically transmitted through the telephone network. Since this system is additive, the sound you hear is the sum of the two voices as they would sound if transmitted individually. You hear Edna and Bernienot the creature, Ednabernie. A good example of a nonadditive circuit is the mixer stage in a radio transmitter.

Two signals are present: an audio signal that contains the voice or music, and a carrier wave that can propagate through space when applied to an antenna. The two signals are added and applied to a nonlinearity, such as a pn junction diode. This results in the signals merging to form a third signal, a modulated radio wave capable of carrying the information over great distances. As shown in Fig. Pay particular notice to how the mathematics of this shift is written, it will be used in upcoming chapters.

By adding a constant, sto the independent variable, nthe waveform can be advanced or retarded in the horizontal direction. Shift invariance is important because it means the characteristics of the system do not change with time or whatever the independent variable happens to be.

If a blip in the input causes a blop in the output, you can be assured that another blip will cause an identical blop. Most of the systems you encounter will be shift invariant.Linearity is the property of a mathematical relationship function that can be graphically represented as a straight line. Linearity is closely related to proportionality.

Examples in physics include the linear relationship of voltage and current in an electrical conductor Ohm's lawand the relationship of mass and weight. By contrast, more complicated relationships are nonlinear. Generalized for functions in more than one dimensionlinearity means the property of a function of being compatible with addition and scalingalso known as the superposition principle. The word linear comes from Latin linearis"pertaining to or resembling a line".

In mathematics, a linear map or linear function f x is a function that satisfies the two properties: [1]. These properties are known as the superposition principle. In this definition, x is not necessarily a real numberbut can in general be an element of any vector space. A more special definition of linear functionnot coinciding with the definition of linear map, is used in elementary mathematics see below.

The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operatorand other operators constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.

Linear algebra is the branch of mathematics concerned with the study of vectorsvector spaces also called 'linear spaces'linear transformations also called 'linear maps'and systems of linear equations. For a description of linear and nonlinear equations, see linear equation. In a different usage to the above definition, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a straight line.

Over the reals, a linear equation is one of the forms:. Note that this usage of the term linear is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. A Boolean function is linear if one of the following holds for the function's truth table :. Another way to express this is that each variable always makes a difference in the truth value of the operation or it never makes a difference.

NegationLogical biconditionalexclusive ortautologyand contradiction are linear functions. In physicslinearity is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation. In instrumentation, linearity means that a given change in an input variable gives the same change in the output of the measurement apparatus: this is highly desirable in scientific work.

In general, instruments are close to linear over a certain range, and most useful within that range. In contrast, human senses are highly nonlinear: for instance, the brain completely ignores incoming light unless it exceeds a certain absolute threshold number of photons. In electronicsthe linear operating region of a device, for example a transistoris where a dependent variable such as the transistor collector current is directly proportional to an independent variable such as the base current.

This ensures that an analog output is an accurate representation of an input, typically with higher amplitude amplified. A typical example of linear equipment is a high fidelity audio amplifierwhich must amplify a signal without changing its waveform.

Others are linear filterslinear regulatorsand linear amplifiers in general. In most scientific and technologicalas distinct from mathematical, applications, something may be described as linear if the characteristic is approximately but not exactly a straight line; and linearity may be valid only within a certain operating regionâ€”for example, a high-fidelity amplifier may distort a small signal, but sufficiently little to be acceptable acceptable but imperfect linearity ; and may distort very badly if the input exceeds a certain value, taking it away from the approximately linear part of the transfer function.If early Las Vegas odds are unavailable, offshore NFL playoff odds may be displayed in the interim.

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### 3.3 Linearity

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The moment a US Tier VIII Premium anything goes on sale, it's gonna get bought up hella quickly. Tier VIII is pretty much straight up the 'sweet spot' in this game, premiums at this tier make by far the most money, the experiance gain is nothing short of superb, and of course, since Tier VIII is generally the highest tier you'll get Premiums at, these ships become amazing Captain trainers. Not just that, Tier VIII just in general excels at ranked.

WG obviously knows about the Tier VIII hotspot, because they keep releasing Premium ships at this tier. What I don't understand is why it's taking so long for the US Navy, a faction that has been in the game since day one, to get its first Tier VIII Premium. Yukikaze is a possible Tier 8 now that Kagero is downtiered. BBs: With the Russian BB line coming (. But then, we thought that about Warspite. Also will be hilariously trollish if Novorossiysk is sold as a Russian premium before an Italian premium happens.

Probably another 30 Russian ships total next year.

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Russian carrier line and second Russian cruiser line. As well as some 1980s designed Russian tier 6 premium cruiser. Also, several sub-branches for "what if" scenarios for other ships, like "what if they decided to replace the triple turrets of the Imperator Nikolai I for single turrets with freakishly big guns. Hood will probably be in the game when the RN BB line get in.

Given how confused that battle was, it's possible that Yudachi could've done less than she's traditionally credited with or considerably more (as in, the torpedo kills attributed to other DDs might actually have been Yudachi).

She really didn't do much aside from not get sunk for a while. I don't really think she's premium ship material.