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56-651: If, within one TR, either one of the gradient moments of magnetic gradients along three logical directions, including slice selection direction (G ss ), phase encoding (G pe ) and readout (G ro ), is not zero, then spins along such direction obtain different phases , making the signal intensity (SI) of a single voxel the vector sum of magnetizations therein. It causes some inevitable loss of signal. Such situations belong to ordinary SSFP imaging, with its commercial names listed below. Otherwise, if all gradient moments are zero within one TR, i.e. gradients of opposite polarities cancel out, then there are no additional effects on

112-451: A phase reversal or phase inversion implies a 180-degree phase shift. When the phase difference φ ( t ) {\displaystyle \varphi (t)} is a quarter of turn (a right angle, +90° = π/2 or −90° = 270° = −π/2 = 3π/2 ), sinusoidal signals are sometimes said to be in quadrature , e.g., in-phase and quadrature components of a composite signal or even different signals (e.g., voltage and current). If

168-924: A simple harmonic oscillation or sinusoidal signal is the value of φ {\textstyle \varphi } in the following functions: x ( t ) = A cos ⁡ ( 2 π f t + φ ) y ( t ) = A sin ⁡ ( 2 π f t + φ ) = A cos ⁡ ( 2 π f t + φ − π 2 ) {\displaystyle {\begin{aligned}x(t)&=A\cos(2\pi ft+\varphi )\\y(t)&=A\sin(2\pi ft+\varphi )=A\cos \left(2\pi ft+\varphi -{\tfrac {\pi }{2}}\right)\end{aligned}}} where A {\textstyle A} , f {\textstyle f} , and φ {\textstyle \varphi } are constant parameters called

224-449: A zero at the origin of the complex frequency plane. The gain of its frequency response increases at a rate of +20  dB per decade of frequency (for root-power quantities), the same positive slope as a 1 order high-pass filter 's stopband , although a differentiator doesn't have a cutoff frequency or a flat passband . A n -order high-pass filter approximately applies the n time derivative of signals whose frequency band

280-400: A cycle. This concept can be visualized by imagining a clock with a hand that turns at constant speed, making a full turn every T {\displaystyle T} seconds, and is pointing straight up at time t 0 {\displaystyle t_{0}} . The phase φ ( t ) {\displaystyle \varphi (t)} is then the angle from

336-507: A full period. This convention is especially appropriate for a sinusoidal function, since its value at any argument t {\displaystyle t} then can be expressed as φ ( t ) {\displaystyle \varphi (t)} , the sine of the phase, multiplied by some factor (the amplitude of the sinusoid). (The cosine may be used instead of sine, depending on where one considers each period to start.) Usually, whole turns are ignored when expressing

392-411: A full turn: φ = 2 π [ [ τ T ] ] . {\displaystyle \varphi =2\pi \left[\!\!\left[{\frac {\tau }{T}}\right]\!\!\right].} If F {\displaystyle F} is a "canonical" representative for a class of signals, like sin ⁡ ( t ) {\displaystyle \sin(t)}

448-421: A microphone. This is usually the case in linear systems, when the superposition principle holds. For arguments t {\displaystyle t} when the phase difference is zero, the two signals will have the same sign and will be reinforcing each other. One says that constructive interference is occurring. At arguments t {\displaystyle t} when the phases are different,

504-417: A periodic soundwave recorded by two microphones at separate locations. Or, conversely, they may be periodic soundwaves created by two separate speakers from the same electrical signal, and recorded by a single microphone. They may be a radio signal that reaches the receiving antenna in a straight line, and a copy of it that was reflected off a large building nearby. A well-known example of phase difference

560-532: A quarter cycle, the sine and cosine components , respectively. A sine wave represents a single frequency with no harmonics and is considered an acoustically pure tone . Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to the fundamental causes variation in the timbre , which is the reason why the same musical pitch played on different instruments sounds different. Sine waves of arbitrary phase and amplitude are called sinusoids and have

616-431: A shifted and possibly scaled version G {\displaystyle G} of it. That is, suppose that G ( t ) = α F ( t + τ ) {\displaystyle G(t)=\alpha \,F(t+\tau )} for some constants α , τ {\displaystyle \alpha ,\tau } and all t {\displaystyle t} . Suppose also that

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672-503: A shifted version G {\displaystyle G} of it. If the shift in t {\displaystyle t} is expressed as a fraction of the period, and then scaled to an angle φ {\displaystyle \varphi } spanning a whole turn, one gets the phase shift , phase offset , or phase difference of G {\displaystyle G} relative to F {\displaystyle F} . If F {\displaystyle F}

728-409: A single line. This could, for example, be considered the value of a wave along a wire. In two or three spatial dimensions, the same equation describes a travelling plane wave if position x {\displaystyle x} and wavenumber k {\displaystyle k} are interpreted as vectors, and their product as a dot product . For more complex waves such as the height of

784-423: A sonic phase difference occurs in the warble of a Native American flute . The amplitude of different harmonic components of same long-held note on the flute come into dominance at different points in the phase cycle. The phase difference between the different harmonics can be observed on a spectrogram of the sound of a warbling flute. Phase comparison is a comparison of the phase of two waveforms, usually of

840-447: A sum of sine waves of various frequencies, relative phases, and magnitudes. When any two sine waves of the same frequency (but arbitrary phase ) are linearly combined , the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and

896-476: A water wave in a pond after a stone has been dropped in, more complex equations are needed. French mathematician Joseph Fourier discovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, including square waves . These Fourier series are frequently used in signal processing and the statistical analysis of time series . The Fourier transform then extended Fourier series to handle general functions, and birthed

952-474: Is a periodic wave whose waveform (shape) is the trigonometric sine function . In mechanics , as a linear motion over time, this is simple harmonic motion ; as rotation , it corresponds to uniform circular motion . Sine waves occur often in physics , including wind waves , sound waves, and light waves, such as monochromatic radiation . In engineering , signal processing , and mathematics , Fourier analysis decomposes general functions into

1008-495: Is a "canonical" function for a class of signals, like sin ⁡ ( t ) {\displaystyle \sin(t)} is for all sinusoidal signals, then φ {\displaystyle \varphi } is called the initial phase of G {\displaystyle G} . Let the signal F {\displaystyle F} be a periodic function of one real variable, and T {\displaystyle T} be its period (that is,

1064-581: Is a "canonical" function of a phase angle in 0 to 2π, that describes just one cycle of that waveform; and A {\displaystyle A} is a scaling factor for the amplitude. (This claim assumes that the starting time t 0 {\displaystyle t_{0}} chosen to compute the phase of F {\displaystyle F} corresponds to argument 0 of w {\displaystyle w} .) Since phases are angles, any whole full turns should usually be ignored when performing arithmetic operations on them. That is,

1120-528: Is a function of an angle, defined only for a single full turn, that describes the variation of F {\displaystyle F} as t {\displaystyle t} ranges over a single period. In fact, every periodic signal F {\displaystyle F} with a specific waveform can be expressed as F ( t ) = A w ( φ ( t ) ) {\displaystyle F(t)=A\,w(\varphi (t))} where w {\displaystyle w}

1176-514: Is defined the same way, except with "360°" in place of "2π". With any of the above definitions, the phase φ ( t ) {\displaystyle \varphi (t)} of a periodic signal is periodic too, with the same period T {\displaystyle T} : φ ( t + T ) = φ ( t )  for all  t . {\displaystyle \varphi (t+T)=\varphi (t)\quad \quad {\text{ for all }}t.} The phase

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1232-631: Is either identically zero, or is a sinusoidal signal with the same period and phase, whose amplitude is the difference of the original amplitudes. The phase shift of the co-sine function relative to the sine function is +90°. It follows that, for two sinusoidal signals F {\displaystyle F} and G {\displaystyle G} with same frequency and amplitudes A {\displaystyle A} and B {\displaystyle B} , and G {\displaystyle G} has phase shift +90° relative to F {\displaystyle F} ,

1288-410: Is for all sinusoidal signals, then the phase shift φ {\displaystyle \varphi } called simply the initial phase of G {\displaystyle G} . Therefore, when two periodic signals have the same frequency, they are always in phase, or always out of phase. Physically, this situation commonly occurs, for many reasons. For example, the two signals may be

1344-576: Is nearly always much, much shorter than T1 or T2, the exponential terms containing TR can be disregarded. SSFP is beneficial as a localizer sequence, such as for initial images of the anal canal in order to align the planes of subsequent T2-weighted images to be cross-sections and longitudinal sections of the canal. A particular SSFP used for this purpose is one termed TRUE FISP by Siemens, FIESTA by GE, and balanced FFE by Philips. SSFP protocols have different names among different MRI manufacturers. Phase (waves) In physics and mathematics ,

1400-1181: Is significantly lower than the filter's cutoff frequency. Integrating any sinusoid with respect to time can be viewed as dividing its amplitude by its angular frequency and delaying it a quarter cycle: ∫ A sin ⁡ ( ω t + φ ) d t = − A ω cos ⁡ ( ω t + φ ) + C = − A ω sin ⁡ ( ω t + φ + π 2 ) + C = A ω sin ⁡ ( ω t + φ − π 2 ) + C . {\displaystyle {\begin{aligned}\int A\sin(\omega t+\varphi )dt&=-{\frac {A}{\omega }}\cos(\omega t+\varphi )+C\\&=-{\frac {A}{\omega }}\sin(\omega t+\varphi +{\tfrac {\pi }{2}})+C\\&={\frac {A}{\omega }}\sin(\omega t+\varphi -{\tfrac {\pi }{2}})+C\,.\end{aligned}}} The constant of integration C {\displaystyle C} will be zero if

1456-402: Is the test frequency , and the bottom sine signal represents a signal from the reference. If the two frequencies were exactly the same, their phase relationship would not change and both would appear to be stationary on the oscilloscope display. Since the two frequencies are not exactly the same, the reference appears to be stationary and the test signal moves. By measuring the rate of motion of

1512-400: Is the length of shadows seen at different points of Earth. To a first approximation, if F ( t ) {\displaystyle F(t)} is the length seen at time t {\displaystyle t} at one spot, and G {\displaystyle G} is the length seen at the same time at a longitude 30° west of that point, then the phase difference between

1568-772: Is zero at the start of each period; that is φ ( t 0 + k T ) = 0  for any integer  k . {\displaystyle \varphi (t_{0}+kT)=0\quad \quad {\text{ for any integer }}k.} Moreover, for any given choice of the origin t 0 {\displaystyle t_{0}} , the value of the signal F {\displaystyle F} for any argument t {\displaystyle t} depends only on its phase at t {\displaystyle t} . Namely, one can write F ( t ) = f ( φ ( t ) ) {\displaystyle F(t)=f(\varphi (t))} , where f {\displaystyle f}

1624-499: The amplitude , frequency , and phase of the sinusoid. These signals are periodic with period T = 1 f {\textstyle T={\frac {1}{f}}} , and they are identical except for a displacement of T 4 {\textstyle {\frac {T}{4}}} along the t {\textstyle t} axis. The term phase can refer to several different things: Sinusoid A sine wave , sinusoidal wave , or sinusoid (symbol: ∿ )

1680-402: The bounds of integration is an integer multiple of the sinusoid's period. An integrator has a pole at the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1 order low-pass filter 's stopband, although an integrator doesn't have a cutoff frequency or

1736-399: The phase (symbol φ or ϕ) of a wave or other periodic function F {\displaystyle F} of some real variable t {\displaystyle t} (such as time) is an angle -like quantity representing the fraction of the cycle covered up to t {\displaystyle t} . It is expressed in such a scale that it varies by one full turn as

Steady-state free precession imaging - Misplaced Pages Continue

1792-434: The 12:00 position to the current position of the hand, at time t {\displaystyle t} , measured clockwise . The phase concept is most useful when the origin t 0 {\displaystyle t_{0}} is chosen based on features of F {\displaystyle F} . For example, for a sinusoid, a convenient choice is any t {\displaystyle t} where

1848-414: The clock analogy, each signal is represented by a hand (or pointer) of the same clock, both turning at constant but possibly different speeds. The phase difference is then the angle between the two hands, measured clockwise. The phase difference is particularly important when two signals are added together by a physical process, such as two periodic sound waves emitted by two sources and recorded together by

1904-419: The clock analogy, this situation corresponds to the two hands turning at the same speed, so that the angle between them is constant. In this case, the phase shift is simply the argument shift τ {\displaystyle \tau } , expressed as a fraction of the common period T {\displaystyle T} (in terms of the modulo operation ) of the two signals and then scaled to

1960-416: The difference between them is a whole number of periods. The numeric value of the phase φ ( t ) {\displaystyle \varphi (t)} depends on the arbitrary choice of the start of each period, and on the interval of angles that each period is to be mapped to. The term "phase" is also used when comparing a periodic function F {\displaystyle F} with

2016-793: The field of Fourier analysis . Differentiating any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle: d d t [ A sin ⁡ ( ω t + φ ) ] = A ω cos ⁡ ( ω t + φ ) = A ω sin ⁡ ( ω t + φ + π 2 ) . {\displaystyle {\begin{aligned}{\frac {d}{dt}}[A\sin(\omega t+\varphi )]&=A\omega \cos(\omega t+\varphi )\\&=A\omega \sin(\omega t+\varphi +{\tfrac {\pi }{2}})\,.\end{aligned}}} A differentiator has

2072-403: The form: Since sine waves propagate without changing form in distributed linear systems , they are often used to analyze wave propagation . When two waves with the same amplitude and frequency traveling in opposite directions superpose each other, then a standing wave pattern is created. On a plucked string, the superimposing waves are the waves reflected from the fixed endpoints of

2128-406: The fractional part of a real number, discarding its integer part; that is, [ [ x ] ] = x − ⌊ x ⌋ {\displaystyle [\![x]\!]=x-\left\lfloor x\right\rfloor \!\,} ; and t 0 {\displaystyle t_{0}} is an arbitrary "origin" value of the argument, that one considers to be the beginning of

2184-438: The frequencies are different, the phase difference φ ( t ) {\displaystyle \varphi (t)} increases linearly with the argument t {\displaystyle t} . The periodic changes from reinforcement and opposition cause a phenomenon called beating . The phase difference is especially important when comparing a periodic signal F {\displaystyle F} with

2240-830: The function's value changes from zero to positive. The formula above gives the phase as an angle in radians between 0 and 2 π {\displaystyle 2\pi } . To get the phase as an angle between − π {\displaystyle -\pi } and + π {\displaystyle +\pi } , one uses instead φ ( t ) = 2 π ( [ [ t − t 0 T + 1 2 ] ] − 1 2 ) {\displaystyle \varphi (t)=2\pi \left(\left[\!\!\left[{\frac {t-t_{0}}{T}}+{\frac {1}{2}}\right]\!\!\right]-{\frac {1}{2}}\right)} The phase expressed in degrees (from 0° to 360°, or from −180° to +180°)

2296-415: The general form: y ( t ) = A sin ⁡ ( ω t + φ ) = A sin ⁡ ( 2 π f t + φ ) {\displaystyle y(t)=A\sin(\omega t+\varphi )=A\sin(2\pi ft+\varphi )} where: Sinusoids that exist in both position and time also have: Depending on their direction of travel, they can take

Steady-state free precession imaging - Misplaced Pages Continue

2352-422: The last decade modern scanners have overcome these limitations making bSSFP a viable and useful sequence on most mid- and high-field systems. When the echo is recorded close to the middle of the interval (TE ≈ TR/2, as is usually the case), the final term e−TE/T2 depends on T2, not T2*. Thus, bSSFP sequences behave more like spin echo than gradient echo sequences in that they do not have T2*-dependence. Also, since TR

2408-436: The origin for computing the phase of G {\displaystyle G} has been shifted too. In that case, the phase difference φ {\displaystyle \varphi } is a constant (independent of t {\displaystyle t} ), called the 'phase shift' or 'phase offset' of G {\displaystyle G} relative to F {\displaystyle F} . In

2464-544: The phase from gradients; that is to say, SI of each voxels is the contributions of a series of RF pulses and relaxation phenomena. Although the principles underlying echo formation in balanced SSFP have long been known, widespread clinical implementation has been slow due to stringent technical requirements. bSSFP sequences demand a very high level of magnetic field homogeneity and control over gradient switching and shaping. The refocusing mechanism fails if intravoxel dephasing exceeds over ±180º manifest by band-like artifacts. During

2520-731: The phase; so that φ ( t ) {\displaystyle \varphi (t)} is also a periodic function, with the same period as F {\displaystyle F} , that repeatedly scans the same range of angles as t {\displaystyle t} goes through each period. Then, F {\displaystyle F} is said to be "at the same phase" at two argument values t 1 {\displaystyle t_{1}} and t 2 {\displaystyle t_{2}} (that is, φ ( t 1 ) = φ ( t 2 ) {\displaystyle \varphi (t_{1})=\varphi (t_{2})} ) if

2576-478: The phases of two periodic signals F {\displaystyle F} and G {\displaystyle G} is called the phase difference or phase shift of G {\displaystyle G} relative to F {\displaystyle F} . At values of t {\displaystyle t} when the difference is zero, the two signals are said to be in phase; otherwise, they are out of phase with each other. In

2632-417: The same nominal frequency. In time and frequency, the purpose of a phase comparison is generally to determine the frequency offset (difference between signal cycles) with respect to a reference. A phase comparison can be made by connecting two signals to a two-channel oscilloscope . The oscilloscope will display two sine signals, as shown in the graphic to the right. In the adjacent image, the top sine signal

2688-716: The smallest positive real number such that F ( t + T ) = F ( t ) {\displaystyle F(t+T)=F(t)} for all t {\displaystyle t} ). Then the phase of F {\displaystyle F} at any argument t {\displaystyle t} is φ ( t ) = 2 π [ [ t − t 0 T ] ] {\displaystyle \varphi (t)=2\pi \left[\!\!\left[{\frac {t-t_{0}}{T}}\right]\!\!\right]} Here [ [ ⋅ ] ] {\displaystyle [\![\,\cdot \,]\!]\!\,} denotes

2744-500: The string. The string's resonant frequencies are the string's only possible standing waves, which only occur for wavelengths that are twice the string's length (corresponding to the fundamental frequency ) and integer divisions of that (corresponding to higher harmonics). The earlier equation gives the displacement y {\displaystyle y} of the wave at a position x {\displaystyle x} at time t {\displaystyle t} along

2800-740: The sum F + G {\displaystyle F+G} is a sinusoidal signal with the same frequency, with amplitude C {\displaystyle C} and phase shift − 90 ∘ < φ < + 90 ∘ {\displaystyle -90^{\circ }<\varphi <+90^{\circ }} from F {\displaystyle F} , such that C = A 2 + B 2  and  sin ⁡ ( φ ) = B / C . {\displaystyle C={\sqrt {A^{2}+B^{2}}}\quad \quad {\text{ and }}\quad \quad \sin(\varphi )=B/C.} A real-world example of

2856-567: The sum and difference of two phases (in degrees) should be computed by the formulas 360 [ [ α + β 360 ] ]  and  360 [ [ α − β 360 ] ] {\displaystyle 360\,\left[\!\!\left[{\frac {\alpha +\beta }{360}}\right]\!\!\right]\quad \quad {\text{ and }}\quad \quad 360\,\left[\!\!\left[{\frac {\alpha -\beta }{360}}\right]\!\!\right]} respectively. Thus, for example,

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2912-533: The sum of phase angles 190° + 200° is 30° ( 190 + 200 = 390 , minus one full turn), and subtracting 50° from 30° gives a phase of 340° ( 30 − 50 = −20 , plus one full turn). Similar formulas hold for radians, with 2 π {\displaystyle 2\pi } instead of 360. The difference φ ( t ) = φ G ( t ) − φ F ( t ) {\displaystyle \varphi (t)=\varphi _{G}(t)-\varphi _{F}(t)} between

2968-401: The test signal the offset between frequencies can be determined. Vertical lines have been drawn through the points where each sine signal passes through zero. The bottom of the figure shows bars whose width represents the phase difference between the signals. In this case the phase difference is increasing, indicating that the test signal is lower in frequency than the reference. The phase of

3024-441: The two signals will be 30° (assuming that, in each signal, each period starts when the shadow is shortest). For sinusoidal signals (and a few other waveforms, like square or symmetric triangular), a phase shift of 180° is equivalent to a phase shift of 0° with negation of the amplitude. When two signals with these waveforms, same period, and opposite phases are added together, the sum F + G {\displaystyle F+G}

3080-426: The value of the sum depends on the waveform. For sinusoidal signals, when the phase difference φ ( t ) {\displaystyle \varphi (t)} is 180° ( π {\displaystyle \pi } radians), one says that the phases are opposite , and that the signals are in antiphase . Then the signals have opposite signs, and destructive interference occurs. Conversely,

3136-427: The variable t {\displaystyle t} goes through each period (and F ( t ) {\displaystyle F(t)} goes through each complete cycle). It may be measured in any angular unit such as degrees or radians , thus increasing by 360° or 2 π {\displaystyle 2\pi } as the variable t {\displaystyle t} completes

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