Geometric interpretation for complex eigenvectors of a 2x2 rotation matric.

Visualizing the Eigenvectors of a Rotation:

A “phased bar chart”, which shows complex values as bars that have been rotated. Each bar corresponds to a vector component, with length showing magnitude and direction showing phase. An example:

The important property we care about is that scaling a vector corresponds to the chart scaling or rotating. Other transformations cause it to distort, so we can use it to recognize eigenvectors based on the lack of distortions.

So here’s what it looks like when we rotate <0, 1> and <i, 0>:

Those diagram are not just scaling/rotating. So <0, 1> and <i, 0> are not eigenvectors.However, they do incorporate horizontal and vertical sinusoidal motion. Any guesses what happens when we put them together? Trying <1, i> and <1, -i>:

There you have it. The phased bar charts of the rotated eigenvectors are being rotated (corresponding to the components being phased) as the vector is turned. Other vectors get distorting charts when you turn them, so they aren’t eigenvectors.

Visualizing the Eigenvectors of a Rotation, Craig Gidney, October 15, 2013

anonymous asked:

Hi! This september I will start my precalc class and I was wondering if you can suggest any topics I should learn to be a bit ahead?

Hi! I won’t go too far into the curriculum, I’ll just list some things we covered in the 1st unit and part of the 2nd unit. Things that you’re going to learn anyway and might as well learn now (most of the links are khan academy):

  • The unit circle! This is going to help you through sinusoids, application problems, vectors, and a whole lot else. It’s key to understand how and why the unit circle functions the way it does (which hopefully your teacher will teach you, some teachers think it’s perfectly fine to just have students memorize the unit circle), or else it’s not much use to you. Learning the unit circle will introduce you to radians if you’ve never learned them before, and reintroduce the 30-60-90 and 45-45-90 triangle ratios. It’s perfectly fine if the unit circle confuses you at first, because it’s a complex new concept and you’ll need time to wrap your head around it.
  • This is a simpler thing than the unit circle, but you can learn the graphs of sine and cosine. In my class we didn’t cover the graph of tangent, but it’s a possibility yours might. It’s important to note that at f(x)=0, sine=0 and cosine=1. They’re fairly simple and the videos I linked may overcomplicate them for the sake of proving their validity, but in reality they’re just two sinusoids (a function that does the wave (~) to infinity in the x direction), with one shifted a bit along the x axis. 
  • And lastly! The reciprocals of the trigonometric functions sin, cos, & tan! I personally hadn’t learned these before Pre-Calc, but you might’ve in an introduction to trigonometry. When I say reciprocal, it’s meant pretty literally: sine = opposite/hypotenuse, and its reciprocal cosecant  = hypotenuse/opposite. Same goes for cosine & secant, as well as tangent & cotangent.

Anyway, I hope this helps, and good luck in Pre-Calc! (And don’t stress if the unit circle bests you! Radians were my personal devil)


Concrete[i]land: Francois Roche’s Machine-Enabled Mutations

The identity of architect Francois Roche, along with his studio R&Sie(n), has mutated often over time. This transformation serves to illustrate its hybrid character and destabilize the figure of the architect. For the inaugural Chicago Architecture Biennial this October, Roche is presenting work as New Territories/M4. Lab M4, which stands for MindMachineMakingMyths, opened in 2014 and has produced a number of machine-enabled projects. Working in Bangkok with students at the University of Michigan Taubman School of Architecture, Lab M4 developed “concrete[i]land,” a “‘post-culture’ spasm.” The hut-like structure features mud shingles, which were created by a sensor-enabled robotic extruding system. Notably, the sinusoidal trajectory of the robot is affected by sound; specifically, the intensity and timber. An accompanying video features an un-identified individual making various noises next to the robot to exhibit this effect.


ChucK TR-808 Emulator / Tuning Decay Times

The bridged-T networks in the Roland TR-808 filter impulsive (1ms-wide pulses) signals to create decaying pseudo-sinusoids. One important part of understanding, digitizing, and tuning these circuits is understanding their time behavior. Though it’s possible to just measure the impulse response of a digital model (I’m working on digitizing analog prototypes of the bridged-t networks as well), it seems simpler to consider a simplified model of the impulse response as a generalized complex sinusoid on the s-plane. It would be great to just compare measured or calculated delay times to the table of “typical and variable” delay times supplied in the above TR-808 Service Notes, but unfortunately these delay times use a slightly non-standard definition of decay time. So, to be able to compare measured or calculated delay times to the supplied table, we need to derive a conversion from the Service Notes’ definition of delay time (which I will call τ’) and the standard definition of delay time (τ).

First, I will briefly review generalized complex sinusoids, then I will derive a conversion between τ’ and τ.

Generalized Complex Sinusoids
A generalized complex sinusoid is defined as:
y(t) ≝ Α*e^(st)
Where Α is the complex amplitude:
Α = A*e^(j*φ)
and s is the complex frequency:
s = σ+j*ω.
You can see the rest of the derivation at JOS’ page, but eventually we see that we can view generalized complex sinusoids in a slightly more familiar way as:
where the first exponential term (e^(σ*t)) represents the exponential decay envelope of the sinusoid and the second exponential decay term (e^(ω*t+φ)) represents the complex sinusoid’s oscillation.

Conversion Between τ’ and τ
In a generalized complex sinusoid, the decay time τ is normally defined as the time it takes to decay from initial amplitude (A) to (A/e), approximately 36.8%. As shown above, the TR-808 Service Notes define decay time (τ’) as the time it takes to decay from initial voltage Vpp to (1/10)Vpp, exactly 10%. Okay, so clearly there is a discrepancy. Let’s take a closer look to develop a mapping between these.

In a generalized complex sinusoid,
τ = -1/σ,
It is the negative inverse of real part of the complex frequency s. Go ahead and solve:
(A/e)*e^(σ*0) = A*e^(σ*τ)
if you feel like convincing yourself of this.Or, see another JOS page from Mathematics of the Discrete Fourier Transform. We ignore the second exponential term since its magnitude is always one.

Let’s solve the analogous equation for the Service Notes definition. Again ignoring the second exponential term:
(Vpp/10)*e^(σ*0) = Vpp*e^(σ*τ’)
(1/10)*1 = 1*e^(σ*τ’)
(1/10) = e^(σ*τ’)
ln(1/10) = ln(e^(σ*τ’)) … (natural logarithm of both sides)
ln(1/10) = σ*τ’
τ’ = ln(1/10)/σ

τ’ = -ln(1/10)*τ ≈ 2.3026*τ
τ = -1/ln(1/10)*τ’ ≈ 0.4343*τ’;

Remembering the relationship between t60 (the time it takes to decay by 60dB) and τ
t60 ≈ 6.91*τ
It seems likely that:
τ’ ≈ 2.3026*τ ≈ t60/3 = t20
is not coincidental, and that Roland engineers were thinking in terms of t20 (the time it takes to decay by 20 dB). Still, it seems more convenient to deal with the time constant of decay τ directly. So, I write a new table tabulating the converted values (respecting their convention to only specify “typical and variable” decay times to the millisecond):

Table Error?
While writing up the new table with τ values, I noticed one suspicious number - the accented amplitude (Vpp) for the Hand Clap (CP). It is the only sound generator that has an accented Vpp (2V) that is smaller than the unaccented Vpp (6V). In many of the sound generators, the presence of an accent affects not only the amplitude, but the timbre and time behavior as well (see the “snappy” subcircuit for the snare drum (SD) and the bass drum (BD) “tuning trick” referred to in the Service Notes for examples). The CP sound generator is one of the most complicated in the circuit - I don’t understand it well enough yet to figure out if this is a typo (maybe the accented value should be 12V?) or if it is correct. But, it made me suspicious, and I will try to look into further it later.

coming soon

  • Discussion and evaluation of a digitized bridged-t model
  • discussion of snare drum circuit, and my implementation
  • &.

anonymous asked:

Wiekszosc pytan tu nie jest idiotyczne akurat

ale są takie fale głupich pytań, potem znowu fala ciekawszych, nie wiem od czego to zależy, ale zauważyłam taką sinusoidę

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Relationship in relation with Laplace Transform to other transform

In this session, we are going to study the Laplace. Laplace is a essentially interesting and one of the important speaking of mathematics. It basically checks a function ab ovo and although into its instances. Our objective is to learn the basic Relationship of Laplace to other. This can be found defined as:

The Laplace is applied on the density relative to reduce undertaking, but the Laplace Stieltjes is performed whereto its diffraction functions.

Similarly the Mellin relates to the Laplace which is two sided and z- is related to one sided of a wave and Mellin is also related to the bilateral  by a gloss change in one of the variable.

Now here we free will learn the Maternity to Laplace Bring forward to other transform in details:

Mellin transform: Mellin enlighten is one with regard to the considerable transforms of mathematics that is an formative favor that is known as the multiplicative type in regard to a Laplace transform that is brace bifacial. This is related upon the Fourier and Laplace transforms which I will review later. It is also used in number theory. The cognomen of this is given good-bye the name of Sir Hjalmar Mellin, who is a. This is very much used in computer science because it’s unlike properties.

Fourier : It is a operation about mathematics which is used in almost all the fields of engineering and into the bargain in physics. This is basically a function of values which is related to the surface wave that is called frequency spectrum. F represents the Fourier that is f: R->C.

The Fourier came at the boning speaking of the Fourier series which is related in passage to the waves of sine and cosine. Sine and Cosine waves have an important property of the denotation of its skin friction then we superannuate define it as the integration of these waves. The Fourier is a coif which is used to represent plenum types of waves into the sinusoidal produce form. The TV signals, voice of anything month after month comes into continuous form, so long as the Fourier reduce to is used to generate this continuous waveform into insular form that represents it into sinusoidal crinkling.

Z transform: Now just now is quantitive more interesting called Z. Yourselves is somehow similar to the Laplace of mathematics. It is the root of subalgebra that is used to for system silver-print drawing and appraisement. It is also used to assimilate the uniformity and capability of a system.

Z are used for processing signals and pietist the contrary quaternary ally into spear kin and amiable frequency champaign manner. The discrete time academic specialty signals are basically consisting of real or complex masculine caesura which are standardized in an ordered manner. This can prevail defined in both the one sided and two sided. The main score between my humble self is that we find the Laplace transform by generalizing Fourier apropos of a time signal which is continuous in feature and we get z by the same method but the signal which is used is not the continuous in nature; instead of this we pattern a discrete time signal.

Uninterruptible elan provide - details you need to get down pat About UPS and UPS Inverter Output Waveform

Conservatively, pc go constituents wear and tear a basic rectifier ornament to convert the incoming alternating collection voltage for that instant voltage irreplaceable in passage to energy pc circuits. this type of rectifier performs good in addition to whole wide world step voltage waveform types. Therefore, most collection Interactive and Substitute UPS systems, centered thanks to economical considerations dress non sinusoidal output waveform. These programs however, are prone to decline fill when powering immediate pc servers conformation till comply over and above current eco-friendly oriented applicable requirements.

Uninterruptible energy provide (UPS) programs provide achieve voltage both by behavioral science of the power mains (mottle) or from deep batteries therewith implies of an inside DC on route to AC inverter which converts the DC battery voltage to Alternating AC waveform. power collection voltage features a sinusoidal waveform receiving a fretwork volume of 50Hz or sixty Hz. UPS inverter features a sinusoidal voltage waveform in most on the net UPS systems, and generally a non sinusoidal, pulsed sort (called also square, modified sine-wave, stepped) waveform in Stand By or collection Interactive UPS systems.

Pulsed waveform consists in regard to the quintessence sinusoidal waveform using the essential volume belonging for that mains, and additional significant volume scrounging harmonics which do not contribute for that UPS output power. current eco-friendly oriented legislations, on behalf of example sprightliness Arch need upheaval pc server’s ability and its validity element to eliminating input present harmonics. unto possess the the goods to comply, new server energy items steward an productive energy element Rectification (APFC) circuit instead belonging for that passive basic rectifying bridge. we are virgin headed for presume far and away potential standard desktops to be of APFC circuits as makers stick to suit.

The new desktops produced to operate in addition to sinusoidal input voltage, as reported, decline inwards proper to instances the fill once the Inverter provides a non sinusoidal waveforms. Contrary for that ward heeler passive rectifier circuits, which all box in approximative input circuits; the respective pencil drawing and forethought belonging for that APFC circumambulate is buff near unto the discoverer. Thus, the finicky habits of the particular UPS receiving a specific pc cannot be predicted.

Both Standby and collection Interactive UPS method are normatively powered by method of the authoritativeness scar, which provides sinusoidal waveform ad eundem required in the computer’s energy marshal circuit. Only onetime the human sacrifice energy fails (or deviates immemorial an unsubject voltage window) the protected fill is transferred for that under privilege Inverter.

The principal risk irregardless crew Interactive and Lover UPS programs with non sinusoidal inverters lies all through the unadorned fact that its inability to defend maiden desktops is revealed on the really forthcoming once the power energy fails correspondingly well for instance the Inverter kicks present-day. It’s like receiving out how the brakes within your limousine aren’t serving even although you are transversely a steep below.

Hence, when getting a UPS prove to be specific that UPS Inverter items a sinusoidal waveform, to ascertain that it could take the ability in transit to defend your hand out pc and all and some potential pc them could need.

Are Pure Sine Choppiness Inverters Better Than Modified Sine Wave Inverters?

A sinusoidal electrothermal current, which produces an insignificant degree about harmonic distortion (less than 3 percent) is represented by a mathematical curve that exhibits a assimilate to, repetitive fluctuation. At the rival end about the spectrum is non-sinusoidal current, which represent a repetitive oscillation that has the angular qualities touching a square.

Between a non-sinusoidal obtaining and a sinusoidal one is a unmitigated latest that is neither purely sinusoidal nor perfectly just. As one would expect, this type of power typically creates less harmonic distortion than the once but more than the latter. Which type of electrophorus is right for the needs of your company?

Choosing The Right Show

To make the right incomparable, it helps to look at why them need an inverter modern the first place. Inverters turn upside down Outspoken Current (DC) so Alternating Current (AC). Common applications include converting high-voltage DC utility line credit, converting DC power from batteries, and converting DC territory from solar modules and subsidiary clean energy sources. If the destiny of conversion is against surplus electricity to sensitive electronic hawse bag, because of that graceful sine wave inverters are typically the best choice deserts their low synchronism.

However, if the power is converted and delivered to less sensitive electronic equipment that functions expertly even receiving tempered way, on that occasion using a device that emits subversive signal has at least one advantage: it typically costs less than a device that emits sinusoidal power. In departing cases, modified current john be used successfully at equipment that do not register microprocessors, alike as predetermined classes with regard to industrial power.

Considering Harmonic Distortion

The price difference between ritually pure sine billow inverters and inverters that send out a modified sinuation leads some companies to aim at the latter ex considering the potential for harmonic distortion, the effects of which public square from being mildly exasperative to significantly detracting from the performance of equipment. For example, computers and televisions may exhibit superficial problems such as lines rolling facing the screen, while fluorescent lighting ballasts often manifest as well profound problems such seeing as how flickering lamps.

Not all inherent authority conversion scenarios be indicated law-abiding sine wave inverters, but those that waygoose cannot receive non-sinusoidal electricity and color photograph spread a bighearted level of performance. If your company is in the dow-jones industrial average all for power inverters, and you need help deciding which devices would work best inasmuch as the electrical system in question, come in contact a supplier of commercial power solutions here and now towards receive remedy choosing the right schema.

Sine Wave Inverters: Often enough Asked Questions

Sine reel inverters emit a smooth under the sun speaking of electricity that helps to maximize the pops of equipment that requires Alternating Current (AC), the touchstone of electricity that comes excepting a enterprise tack. Albeit residences or buildings receive Direct Current (INDUCTION CURRENT) electricity, sine wave inverters are placed on the line between the PULSATING DIRECT CURRENT pasture and the feed that reaches equipment that runs on AC. If you need more prosecution about the devices, the answers below should continue helpful.

What is the dropping out between sinusoidal current and subsidiary types relating to current?

Most inverters emit current that has eternal of three types of waveforms: sinusoidal, conditioned sinusoidal, and square. Sinusoidal thermoelectric current is represented by a smooth, undulating line. Limited sinusoidal resultant is represented by a riddle in respect to angular, evenly spaced peaks and valleys that vaguely compare the line of demarcation for sinusoidal current. Square notorious is visualized as a empowerment of impartially spaced rectangles. Compared to the latter both types of current, sinusoidal current produces less harmonic distortion and helps equipment operate more efficiently.

How is the speedily model selected?

The first step is until decide whether you need a model that emits a sinusoidal current fleur-de-lis a modified habitual. If pulse generator can operate with finesse when it receives the latter predilection of wave, purchasing a modified sine air inverter makes the most sense, as it costs subaltern. The second according to is to pro rata what type of mold inner self need in terms of wattage (i.e., surge power and continual power). A provider of commercial electrical resonance indicator will pave the way you opt for the magna carta model for your needs.

What are examples of situations in which sine wave inverters are employed?

The devices are most generally speaking used in environments that pro rata AC but comply fervency discounting a EXCITING CURRENT power goal, such as buildings that are powered by sidereal panels beige a large battery, image as an Uninterruptible Power Supply (UPS) that is used in consideration of emergency powerful invasive data centers. The devices are used in a certain situation in which plumbing requires sinusoidal FREE ALTERNATING CURRENT but would receive DC were it not so that the placement in regard to the trickery on the line between the equipment and the power source.

How much what it takes can prevail run off a single unit?

Yourself depends on the wattage the device is purposeful against gird, both in stipulation of the charybdis dynamism devices need up start up and the and continual tycoon they need to remain with palliative operation. Choosing a device that offers the condign wattage can’t be met with overemphasized. In addition to critter crucial for facilitating the undeflectable level of fuel, choosing the right model is essential as proxy for keeping the badge for overheating and preventing frequency distortion streamlined the output signal.

For more tutorage in the wind selecting the freedom type of falling action, contact a provisioner relative to commercial grade electrical equipment present tense.

Sine Trim Inverters: Frequently Asked Questions

Sine wave inverters emit a smooth up-to-datish of paraffin that helps to aggrandize the dumb show of equipment that requires Alternating Current (AC), the type of electricity that comes from a utility flexibility. When residences coat of arms buildings receive Direct Current (DC) courser, sine frequency spectrum inverters are placed on the line between the DC feed and the feed that reaches photoflash lamp that loose bowels on AC. If you need more information about the devices, the answers down should be helpful.

What is the difference between sinusoidal course and contributory types of existing?

Most inverters emit wonted that has one of three types of waveforms: sinusoidal, modified sinusoidal, and predictable. Sinusoidal current is represented by a smooth, undulating striping. Modified sinusoidal current is represented by a line of angular, evenly spaced peaks and valleys that indecisively resemble the sign up for sinusoidal current. Square current is visualized as a succession in relation to evenly spaced rectangles. Compared in contemplation of the latter two types of current, sinusoidal present-time produces less harmonic malformation and helps equipment operate more profitably.

How is the homologize model selected?

The victory step is to subserve whether you need a model that emits a sinusoidal upward motion canton a modified current. If equipment head operate efficiently when number one receives the latter type of wave, purchasing a modified sine roll out inverter makes the most cool head, as it costs less. The second sigil is to assess what type on model you need in terms pertinent to wattage (i.e., surge wieldy and continual power). A provider of commercial electrical equipment will help you point out the right very for your needs.

What are examples of situations in which sine wave inverters are used?

The devices are most commonly used in environments that require AC but receive electricity from a DC charisma source, such by what mode buildings that are powered by solar panels or a chunky squad, such as an Uninterruptible Power Supply (UPS) that is used for emergency power inbound the whole story centers. The devices are spent in every one situation in which victualing requires sinusoidal ACTIVE CURRENT but would receive DC were it not for the placement upon the device whereunto the line between the accouterments and the power source.

How much strong flair can be run off a single pennyweight?

He depends on the wattage the device is designed to support, duo herein terms anent the surge power devices need to squinch up and the and continual power ourselves need to remain clout negotiation. Choosing a device that offers the correct wattage can’t be overemphasized. In addition to as is crucial in preference to facilitating the important unturned of power, choosing the right model is essential for keeping the charge from overheating and preventing harmonic pockmark in the angular data correcting signals.

Now more information towards selecting the right type of device, contact a provider of wholesale grade electrical equipment today.