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# amplitude definition physics

B. Amplitude . It makes more sense to separate loudness and harmonic quality to be parameters controlled independently of each other. am‧pli‧tude /ˈæmplɪtjuːd \$ -tuːd/ noun [ uncountable] technical the distance between the middle and the top or bottom of a wave such as a sound wave Examples from the Corpus amplitude • The duration and amplitude of rebound pressure increased as the distension volume increased. This is because the value is different depending on whether the maximum positive signal is measured relative to the mean, the maximum negative signal is measured relative to the mean, or the maximum positive signal is measured relative to the maximum negative signal (the peak-to-peak amplitude) and then divided by two (the semi-amplitude). The wavelength of a wave is the distance between a point on one wave and the same point on the next wave. Similarly, the amplitude can be measured from the rest position to the trough position. The frequency of a wave can be calculated using the equation: $\text{frequency f =}~\frac{\text{number of waves to pass a point}}{\text{time taken in seconds}}$. The square of the amplitude is proportional to the intensity of the wave. Observe that whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared. The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The advent of microprocessor controlled meters capable of calculating RMS by sampling the waveform has made true RMS measurement commonplace. Amplitudes are always measured as positive numbers (for example: 3.5, 1, 120) and are never negative (for example: -3.5, -1, -120). Get the huge list of Physics Formulas here. Here’s a brief overview of It is used to understand gravity, electromagnetism, and many other properties of nature. In a sense, the amplitude is the distance from rest to crest. For electromagnetic radiation, the amplitude of a photon corresponds to the changes in the electric field of the wave. It is important to note that the amplitude is not the distance between the top and bottom of a wave. The symbol for wavelength is the Greek letter lambda, λ. Our tips from experts and exam survivors will help you through. The amplitude of the wave motion is defined as the maximum displacement of a particle in the wave. Semi-amplitude means half of the peak-to-peak amplitude. With appropriate circuitry, peak-to-peak amplitudes of electric oscillations can be measured by meters or by viewing the waveform on an oscilloscope. The amplitude of a pendulum swinging through an angle of 90° is 45°. ‘The correlation between changes in the kinetics of synaptic current and quantal amplitude remains strong for the corrected values as well.’ ‘Practical researchers are only too aware, however, that the optical output can frequently vary significantly in amplitude and spatial quality from point to … 1. variable noun In physics, the amplitude of a sound wave or electrical signal is its strength. In general, the use of peak amplitude is simple and unambiguous only for symmetric periodic waves, like a sine wave, a square wave, or a triangle wave. The amplitude of a sound wave can be defined as the loudness or the amount of maximum displacement of vibrating particles of the medium from their mean position when the sound is produced. For alternating current electric power, the universal practice is to specify RMS values of a sinusoidal waveform. The logarithm of the amplitude squared is usually quoted in dB, so a null amplitude corresponds to −∞ dB. Definition. Breadth or range, as of intelligence. Amplitude is also the greatest height of a graph (= drawing) of the relationships of a sine or cosine. However, radio signals may be carried by electromagnetic radiation; the intensity of the radiation (amplitude modulation) or the frequency of the radiation (frequency modulation) is oscillated and then the individual oscillations are varied (modulated) to produce the signal. Strictly speaking, this is no longer amplitude since there is the possibility that a constant (DC component) is included in the measurement. Position = amplitude × sine function (angular frequency × time + phase difference) x = A sin ($$\omega t + \phi$$) Derivation of the Amplitude Formula. The amplitude of an ocean wave is the maximum height of the wave crest above the level of calm water, or … The amplitude of a wave is its maximum disturbance from its undisturbed position. Otherwise, the amplitude is transient and must be represented as either a continuous function or a discrete vector. As waves travel, they set up patterns of disturbance. One half the full extent of a vibration, oscillation, or wave. Modern laser sources routinely achieve intensities as high as 10²² W/cm². Pulse amplitude is measured with respect to a specified reference and therefore should be modified by qualifiers, such as average, instantaneous, peak, or root-mean-square. The wave can be reconstructed by repeating a section of the wave. In electrical engineering, the usual solution to this ambiguity is to measure the amplitude from a defined reference potential (such as ground or 0 V). For audio, transient amplitude envelopes model signals better because many common sounds have a transient loudness attack, decay, sustain, and release. Peak-to-peak amplitude (abbreviated p–p) is the change between peak (highest amplitude value) and trough (lowest amplitude value, which can be negative). For example, the average power transmitted by an acoustic or electromagnetic wave or by an electrical signal is proportional to the square of the RMS amplitude (and not, in general, to the square of the peak amplitude).[6]. Meaning, pronunciation, picture, example sentences, grammar, usage notes, synonyms and more. It could be a little ripple or a giant tsunami. Another word for amplitude. In the harmonic motion harmonic motion, regular vibration in which the acceleration of the vibrating object is directly proportional to the displacement of the object from … They might have the exact same frequency and wavelength, but the amplitudes of the waves can be very different. amplitude (ăm`plĭto͞od'), in physics, maximum displacement from a zero value or rest position. If the amplitude is small th… Imagine a wave in the ocean. That's because distance can only be greater than zero or equal to zero; negative distance does not exist. The amplitude is a measure of the strength or intensity of the wave. It is often easiest to measure this from the trough of one wave to the trough of the next wave, or from thecrest of one wave to the crest of the next wave. Scientists measure the amplitude of waves at two points. The amplitude of a pendulum swinging through an angle of 90° is 45°. How to use amplitude in a sentence. amplitude (countable and uncountable, plural amplitudes) 1. Amplitude. The amplitude of a wave is shown below: For any transverse wave, the amplitude is measured the highest point of any point of the displacement when the string is in rest. This maximum displacement is measured from … This means that something is pulled away from an equilibrium position, moves back, then through the other side. For other uses, see. In older texts, the phase of a period function is sometimes called the amplitude.[1]. The starting point of the measurement is the flat, calm surface of the water. In telecommunication, pulse amplitude is the magnitude of a pulse parameter, such as the voltage level, current level, field intensity, or power level. The amplitude of a pendulum is thus one-half the distance that … Pulse amplitude also applies to the amplitude of frequency- and phase-modulated waveform envelopes.[7]. What is important to remember — frequency, cycle and wavelength remain constant, however, the hight of the wave form is dynamic based on the power of the wave. Learn about how waves are measured according to amplitude, wavelength and frequency. It is the distance between crest or trough and the mean position of the wave. The amplitude or peak amplitude of a wave is a measure of how big its oscillation is. There are various definitions of amplitude (see below), which are all functions of the magnitude of the differences between the variable's extreme values. It is equal to one-half the length of the vibration path. 3. The amplitude of an ocean wave, for example, is the maximum height of the wave crest above the level of calm water, or the maximum depth of the wave trough below the level of calm water. Some common voltmeters are calibrated for RMS amplitude, but respond to the average value of a rectified waveform. The example of transverse amplitude can be generated by stuck the string. The amplitude or peak amplitude of a wave is a measure of how big its oscillation is. Find more ways to say amplitude, along with related words, antonyms and example phrases at Thesaurus.com, the world's most trusted free thesaurus. most people cannot hear a high-pitched sound above 20 kHz; radio stations broadcast radio waves with frequencies of about 100 MHz; most wireless computer networks operate at 2.4 GHz. To do so, harmonic amplitude envelopes are frame-by-frame normalized to become amplitude proportion envelopes, where at each time frame all the harmonic amplitudes will add to 100% (or 1). The amplitude of a wave is its maximum disturbance from its undisturbed position. [8]. Greatness of size; magnitude. 4. The RMS calibration is only correct for a sine wave input since the ratio between peak, average and RMS values is dependent on waveform. Amplitude is the highest or maximum displacement for a moving particle which is vibrated and propagating along positive and the negative axis. That's because distance can only be greater than zero or equal to zero; negative distance does not exist. This remains a common way of specifying amplitude, but sometimes other measures of amplitude are more appropriate. 1.1.1. With waveforms containing many overtones, complex transient timbres can be achieved by assigning each overtone to its own distinct transient amplitude envelope. A general sine wave (y-displacement versus time) is y=A*sin (wt) where A is the amplitude (since sin is restricted from -1 to 1) and 2pi/w turns out to be the period. Amplitude is a measure of how big the wave is. The peak-to-peak value is used, for example, when choosing rectifiers for power supplies, or when estimating the maximum voltage that insulation must withstand. The frequency of a wave is the number of waves produced by a source each second. Unfortunately, this has the effect of modulating the loudness of the sound as well. If the amplitude of a sound wave is large then the loudness of sound will be more. It can be noted from Figure 1 that the wave repeats. It is also the number of waves that pass a certain point each second. Read about our approach to external linking. Root mean square (RMS) amplitude is used especially in electrical engineering: the RMS is defined as the square root of the mean over time of the square of the vertical distance of the graph from the rest state;[5] Amplitudes are always measured as positive numbers (for example: 3.5, 1, 120) and are never negative (for example: -3.5, -1, -120). Universe of Light: What is the Amplitude of a Wave? If the wave shape being measured is greatly different from a sine wave, the relationship between RMS and average value changes. It is equal to one-half the length of the vibration path. The amplitude is a nonnegative scalar measure of a wave's magnitude of oscillation, that is, the magnitude of the maximum disturbance in the medium during one wave cycle.. i.e. The measure of something's size, especially in terms of width or breadth; largeness, magnitude. The amplitude of sound waves and audio signals (which relates to the volume) conventionally refers to the amplitude of the air pressure in the wave, but sometimes the amplitude of the displacement (movements of the air or the diaphragm of a speaker) is described. [8], In Sound Recognition, max amplitude normalization can be used to help align the key harmonic features of 2 alike sounds, allowing similar timbres to be recognized independent of loudness. Amplitude physics is one of the many branches of mathematics. Amplitude of a wave is shown in Figure 1. Except for a numerical factor in front, the amplitude to go from [math] to [math] is [math] where [math], and [math] is the momentum which is related to the energy [math] by the relativistic equation [math] or the nonrelativistic equation [math] Equation (3.7) says in effect that the particle has wavelike properties, the amplitude propagating as a wave with a wave number equal to the momentum divided by [math]. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. It is common for kilohertz (kHz), megahertz (MHz) and gigahertz (GHz) to be used when waves have very high frequencies. If the reference is zero, this is the maximum absolute value of the signal; if the reference is a mean value (DC component), the peak amplitude is the maximum absolute value of the difference from that reference. AMPLITUDE "Amplitude is the height, force or power of the wave" - The CWNA definition of Amplitude v106. The amplitude of a sound wave is the measure of the height of the wave. This way, the main loudness-controlling envelope can be cleanly controlled. Other parameters can be assigned steady state or transient amplitude envelopes: high/low frequency/amplitude modulation, Gaussian noise, overtones, etc. The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process. Fullness; copiousness. of one wave to the crest of the next wave. For example, when looking at a sound wave, the amplitude will measure the loudness of the sound. A more general representation of the wave equation is more complex, but the role of amplitude remains analogous to this simple case. Amplitude, wavelength and frequency - CCEA, Reflection and refraction of waves - CCEA, Home Economics: Food and Nutrition (CCEA). Amplitude All waves involve an oscillation of some kind. It is the most widely used measure of orbital wobble in astronomy and the measurement of small radial velocity semi-amplitudes of nearby stars is important in the search for exoplanets (see Doppler spectroscopy).[4]. Some scientists[3] use amplitude or peak amplitude to mean semi-amplitude. One property of root mean square voltages and currents is that they produce the same heating effect as a direct current in a given resistance. "Additive Sound Synthesizer Project with CODE! In audio system measurements, telecommunications and others where the measurand is a signal that swings above and below a reference value but is not sinusoidal, peak amplitude is often used. … For example, changing the amplitude from 1 unit to 2 units represents a 2-fold increase in the amplitude and is accompanied by a 4-fold (2 2) increase in the energy; thus 2 units of energy becomes 4 times bigger - 8 units. Amplitude is an ocean wave is the maximum height of the wave crest above the level of calm water, or the maximum depth of the wave trough below the level of calm water. Amplitude is the maximum displacement of points on a wave, which you can think of as the degree or intensity of change. Loudness is directly proportional to the amplitude of the sound. The displacement y is the amplitude of the wave. (since sin (x) repeats itself after 2pi radians, sin (wt) would repeat itself at 2pi/w radians) Amplitude (wave motion) synonyms, Amplitude (wave motion) pronunciation, Amplitude (wave motion) translation, English dictionary definition of Amplitude (wave motion). 2. Amplitude definition is - the extent or range of a quality, property, process, or phenomenon: such as. A steady state amplitude remains constant during time, thus is represented by a scalar. But it doesn't matter where you measure it - as long as it is the same point on each wave. True RMS-responding meters were used in radio frequency measurements, where instruments measured the heating effect in a resistor to measure a current. For an asymmetric wave (periodic pulses in one direction, for example), the peak amplitude becomes ambiguous. For waves on a string, or in a medium such as water, the amplitude is a displacement. For complicated waveforms, especially non-repeating signals like noise, the RMS amplitude is usually used because it is both unambiguous and has physical significance. The cathedral of Lincoln […] is a magnificent structure, prop… Many digital voltmeters and all moving coil meters are in this category. What you are actually seeing are waves with different amplitudes. We call the amount of movement from equilibrium displacement. The higher power, or amplitude, the higher the wave form peeks. n. 1. [9][10], This article is about amplitude in classical physics. (Can we date this quote by Fuller and provide title, author's full name, and other details?) [2] the RMS of the AC waveform (with no DC component). It is often easiest to measure this from the, of one wave to the trough of the next wave, or from the. Amplitude. See more at wave. Definition of amplitude noun in Oxford Advanced Learner's Dictionary. For symmetric periodic waves, like sine waves, square waves or triangle waves peak amplitude and semi amplitude are the same. Amplitude Formula. The energy of the wave also varies in direct proportion to the amplitude of the wave. Peak-to-peak is a straightforward measurement on an oscilloscope, the peaks of the waveform being easily identified and measured against the graticule. Amplitude, in physics, the maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position. (Remember kilo k = 103 mega M = 106 giga G = 109). amplitude [ ăm ′plĭ-tōōd′ ] Physics One half the full extent of a vibration, oscillation, or wave. The units of the amplitude depend on the type of wave, but are always in the same units as the oscillating variable. Loudness is related to amplitude and intensity and is one of the most salient qualities of a sound, although in general sounds it can be recognized independently of amplitude. In older texts, the phase of a period function is sometimes called the amplitude. The latter is described by the wavefunction {\displaystyle \psi (\mathbf {r})=e^ {ikz}+f (\theta) {\frac {e^ {ikr}} {r}}\;,} It is important to note that the amplitude is not the distance between the top and bottom of a wave. ", "Sound Sampling, Analysis, and Recognition", "I wrote a Sound Recognition Application", https://en.wikipedia.org/w/index.php?title=Amplitude&oldid=992835230, Wikipedia articles incorporating text from the Federal Standard 1037C, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License, This page was last edited on 7 December 2020, at 10:27. There are various definitions of amplitude (see below), which are all functions of the magnitude of the differences between the variable's extreme values. 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### Does the Hubble constant depend on redshift?

• I know there are lot of questions on the Hubble constant already, but I am curious to know if it changes with redshift? If at current redshift, $$z=0$$, we know its value to be 0.7, will it be different at higher redshift ($$z=0.1$$)? If so, is there any relationship with redshift?

4 years ago

Yes, definitely.

The Hubble constant describes the expansion rate of the Universe, and the expansion may, in turn, may be decelerated by "regular" matter/energy, and accelerated by dark energy.

It's more or less the norm to use the term Hubble constant $H_0$ for the value today, and Hubble parameter $H(t)$ or $H(a)$ for the value at a time $t$ or, equivalently, a scale factor $a = 1/(1+z)$, where $z$ is the redshift.

The value is given by the Friedmann equation:
$$\frac{H^2(a)}{H_0^2} = \frac{\Omega_\mathrm{r}}{a^4} + \frac{\Omega_\mathrm{M}}{a^3} + \frac{\Omega_k}{a^2} + \Omega_\Lambda,$$
where
$\{ \Omega_\mathrm{r}, \Omega_\mathrm{M}, \Omega_k, \Omega_\Lambda \} \simeq \{ 10^{-3},0.3,0,0.7 \}$
are the fractional energy densities in radiation, matter, curvature, and dark energy, respectively.

For instance, you can solve the above equation at $z=0.1$ and find that the expansion rate was 5% higher than today.

Since everything but dark energy dilutes with increasing $a$, $H(a)$ will asymptotically converge to a value $H_0\sqrt{\Omega_\Lambda} \simeq 56\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$.

The figure below shows the evolution of the Hubble parameter with time:

As noted by KenG, the fact that $H$ decreases with time may seem at odds with the accelerated expansion of the Universe. But $H$ describes how fast a point in space at a given distance recedes. Later, that point will be farther away, and so will recede faster. From the definition of the Hubble parameter, $H\equiv\dot{a}/a$, multiplying by the scale factor shows the acceleration $da/dt$:

And just to stave off any possible confusion surrounding that wonderful answer, when people talk about the expansion "accelerating," they are talking about what is happening to the expansion speed H times a, not the expansion rate H itself. So your result shows that while H is dropping with or without dark energy, dark energy makes H times a rise with a, whereas matter alone makes H times a drop with a.

@KenG Yes, that's an important point.

If an animal grew by 1% per year, a gedanken microbe on its skin might claim that the expansion was accelerating!

@JohnDuffield Gedanken microbe :D

Good to see this one is finally settled.

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# Global Numbering for Equation

I have a problem opposite to most of the questions posted regarding the equation numbering. I would like to have a global numbering for equations, regardless of the section. With or without the package chngcntr, in the equation, it shows the global numbering, as wished.

However, when I try to refer these equation, the first one is correctly referred, but the second one includes the section number. See below for the minimal working example and the result.

Does anyone know why this is happening?

Thanks!

\documentclass[12pt,amsfonts,graphicx,subfigure,letterpaper]{article}
\usepackage{amsmath}

\usepackage{chngcntr}
\counterwithout{equation}{subsection}
\counterwithout{equation}{section} % undo numbering system provided by phstyle.cls

\begin{document}

\section{Section}
\label{sec:1}

\subsection{A}

\subsection{B}

\begin{align}
a &= b\notag\\
&= c.\label{eq:a}
\end{align}

Equation (\ref{eq:a})
\begin{align}
a &= b
\end{align}\label{eq:b}
Equation (\ref{eq:b})

\end{document}

• Welcome to TeX.SX! Try to put the \label into the align environment and not out (that references the subsection). – TeXnician Mar 26 '17 at 15:48
• I have posted it as answer, so you can accept it. – TeXnician Mar 26 '17 at 16:04

You have one mistake in your code: The \label you want to tag the second equation with is outside of it. That causes LaTeX to use the subsection-counter to be referenced.
If you want to reference the equation move the \label into the align environment.


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### 0378

Chirp-Encoded 3D GRE and MPRAGE sequences
Kamlesh Pawar1,2, Zhaolin Chen1,3, Jingxin Zhang3,4, N Jon Shah1,5, and Gary F Egan1,2

1Monash Biomedical Imaging, Monash University, Melbourne, Australia, 2School of Psychological Sciences, Monash University, Melbourne, Australia, 3Department of Electrical and Computer System Engineering, Monash University, Melbourne, Australia, 4School of Software and Electrical Engineering, Swinburne University of Technology, Melbourne, Australia, 5Institute of Medicine, Research Centre Juelich, Juelich, Germany

### Synopsis

Non-Fourier encoding such as random, noiselets and chirp encoding have demonstrated image quality improvement in accelerated compressive sensing (CS) MRI applications. However, implementation of the non-Fourier encoding schemes in 2D spin echo sequence limits its use in practice, due to the fact that spin echo is inherently a slow sequence. In this work, we present a novel implementation of chirp encoding in fast 3D gradient echo (GRE) and MPRAGE sequences. The chirp encoding scheme is compared with conventional Fourier encoding for compressive sensing and susceptibility weighted imaging applications. The evaluation demonstrates that chirp encoding is able to preserve spatial resolution better than the Fourier encoding.

### Introduction

Non-Fourier encoding schemes such as random,1 chirp 2, toeplitz 3 and noiselets 4 have shown performance improvement in compressive sensing and parallel imaging. Despite the advantages provided by non-Fourier encoding, there are certain practical limitations with the non-Fourier encoding. One of the limitations is the implementation in an inherently slow spin echo sequence. Further, non-Fourier encoding schemes are not compatible with the turbo spin echo sequence that limits its use for practical purposes, as the turbo spin echo sequence is normally used in the clinical setting. An alternative is to implement non-Fourier encoding in GRE sequences, although slice selection is one of the problems with the 2D GRE sequences. Therefore, in this work we present an implementation of chirp encoding for the fast 3D GRE sequence where slice selection can be achieved by oversampling in readout direction. A 3D chirp-encoded pulse sequence is designed using real time spatially selective RF pulses, and the application of chirp encoding in compressive sensing (CS) and susceptibility weighted imaging (SWI) is evaluated.

### Methods

Pulse sequence design

A 3D GRE pulse sequence is designed as shown in Fig.1, that encodes the primary phase encoding (PE) direction with the chirp and other two directions with Fourier. For N PE, a total of N different spatially selective RF excitation pulses is required to acquire the chirp-encoded k-space data. Due to the hardware constraint on the scanner, which limits the number of different RF pulses to 128, the RF pulses are prepared in real time for each PE. The gradient required during the excitation RF pulse is given by:

$$G_y= \frac{1}{\gamma \hspace{3pt} FOV_{pe} \hspace{3pt} \Delta t_{p} }$$

where γ is gyromagnetic ratio, FOVpe is field of view in PE direction, Δtp is dwell time of RF pulse.

CS Simulation

Simulations were performed on a phantom image to determine the effect of chirp encoding in CS reconstruction. The k-space of a phantom image was under-sampled by a factor of 4 and 6, and the reconstructed images using multichannel compressive sensing (MCS)4 was evaluated. To simulate chirp-encoded data, the PE direction was encoded with chirp basis and readout was encoded with Fourier basis. The chirp basis is constructed by modulating each row of the Fourier basis with a modulation function defined as:

$$C(n) = e^{j (\Delta c \hspace{1 pt} n^2 + \Delta c)}$$

where n is phase encode number and Δc is chirp encoding factor.

Multichannel compressive sensing reconstruction

The acquired data were under-sampled using complete random under-sampling for chirp and variable density under-sampling for Fourier. The images were reconstructed with MCS4 reconstruction by solving the following optimization:

$$Min_{\widehat{x}} \hspace{3mm} \lambda_1\|\Psi \widehat{x}\|_{l_1} + \lambda_2\| TV \widehat{x}\|_{l_1} + \|Y-E \widehat{x}\|_{l_2}^2$$

where, ψ is the wavelet transform operator, TV is the total variation operator and λ1, λ2 are the regularization parameters.

Imaging Parameters

Two sequences, 3D GRE and MPRAGE were evaluated for chirp and Fourier encoding. The imaging parameters for flow compensated GRE were: TE/TR = 20/30 ms, flip angle = 100, FOV = 256x256x192 mm3 and resolution 1 mm isotropic. For the MPRAGE sequence the TE/TR = 5/2000 ms, and the other parameters were same as GRE. The data were acquired in a 3T MAGNETOM Skyra, Siemens Healthineers, Erlangen, Germany.

### Results

Fig.2 shows the simulation results for the acceleration factors of 4 and 6. It is evident form the enlarged images and line plot that chirp encoding is able to preserve the spatial resolution better than the Fourier encoding. Fig.3 shows the result of MCS reconstruction from the acquired in-vivo data that is consistent with the simulation results and shows improved resolution in the reconstructed images. Minimum intensity projection (mIP) across 20 axial slices from fully sampled GRE dataset is shown in Fig.4, highlighting that the veins are more prominent and sharper in the chirp-encoded mIP image.

### Discussion

The improved resolution in the chirp-encoded images can be attributed to the fact that chirp encoding spreads the energy in the measurement domain. A variable density under-sampling performed in the Fourier-encoded compressive sensing acquisition results in insufficient capture of high frequency components and hence a loss in resolution. However, for chirp encoding the energy is completely spread out in the measurement domain. A complete random under-sampling of chirp-encoded data results in the preservation of sufficient high frequency components and hence provides better resolution in the reconstructed images.

### Conclusion

An implementation of the chirp-encoded MRI is presented for the 3D GRE and MPRAGE sequences. The application of chirp encoding to CS and SWI demonstrates that chirp encoding outperforms Fourier encoding in preserving spatial resolution for anatomical scans and is an excellent candidate for accelerated imaging.

### Acknowledgements

No acknowledgement found.

### References

1. Justin P Haldar, Diego Hernando, and Zhi-Pei Liang. Compressed-sensing MRI with random encoding. IEEE transactions on Medical Imaging, 30(4):893–903, 2011.
2. Gilles Puy, Jose P Marques, Rolf Gruetter, Jean-Philippe Thiran, Dimitri Van De Ville, Pierre Vandergheynst, and Yves Wiaux. Spread spectrum magnetic resonance imaging. IEEE transactions on medical imaging, 1(3):586–598, 2012.
3. Dong Liang, Guangwu Xu, Haifeng Wang, Kevin F King, Dan Xu, and Leslie Ying. Toeplitz random encoding MR imaging using compressed sensing. In Biomedical Imaging: From Nano to Macro, 2009. ISBI’09. IEEE International Symposium on, pages 270–273. IEEE, 2009.
4. Kamlesh Pawar, Gary Egan, and Jingxin Zhang. Multichannel compressive sensing MRI using noiselet encoding. PloS one, 10(5):e0126386, 2015.

### Figures

Fig.1: Pulse sequence for chirp encoding where Gpe, Gse, Gro are the gradient in phase, slice and read encoding directions respectively. The RF pulse along with gradient excites a chirp profile in PE direction.

Fig.2: Simulation results of MCS reconstruction for chirp and Fourier encoding schemes where bottom right corner of each panel is the acceleration factor. (a), (c) are reconstructed images from Fourier encoding; (b), (d) are reconstructed images from chirp encoding; (e) shows the line plot through one vertical line in the reconstructed image for acceleration factor of 6. It can be seen from the enlarged images and line plot that chirp encoding preserves spatial resolution better than the Fourier encoding.

Fig.3: MCS reconstructed images from in-vivo scans, the left bottom corner shows the acceleration factor; Ref: Reference image; CE: chirp encoding direction. It is evident from the enlarged images that chirp-encoded MCS reconstruction outperforms the Fourier encoding in preserving spatial resolution.

Fig.4: Minimum intensity projection (mIP) across 20 slices of processed SWI images; (a) mIP from Fourier encoded data; (b) mIP from chirp-encoded data. The red arrow shows the veins that are properly delineated in chirp encoding reconstruction while not properly delineated in Fourier encoding reconstruction.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)
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# 10 sec challenge 8

Calculus Level 4

Let $$f: \mathbb R \rightarrow \mathbb R$$ be a continuous function $$:$$

$f(x)=\int_{0}^x f(t) \, dt.$

Find$$f(\ln5).$$

Here are more 10 sec challenge.

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text:
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## Energy conservation vs momentum conservation

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## Sign conventions

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## Heat & conservative forces

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