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62-500: The MS-10 is well known for its huge sounding electro bass sounds. The MS-10 synthesizer features a single voltage-controlled oscillator (VCO) offering four waveforms: triangle, sawtooth, pulse, and white noise, and incorporates the same 12dB/octave low-pass filter found in the original MS-20. It is equipped with one low-frequency oscillator (LFO), known as the "modulation generator," which has two controls for rate and shape, and two outputs: pulse and sloped. The shape control adjusts

124-870: A , possibly including some points of the boundary line Re( s ) = a . In the region of convergence Re( s ) > Re( s 0 ) , the Laplace transform of f can be expressed by integrating by parts as the integral F ( s ) = ( s − s 0 ) ∫ 0 ∞ e − ( s − s 0 ) t β ( t ) d t , β ( u ) = ∫ 0 u e − s 0 t f ( t ) d t . {\displaystyle F(s)=(s-s_{0})\int _{0}^{\infty }e^{-(s-s_{0})t}\beta (t)\,dt,\quad \beta (u)=\int _{0}^{u}e^{-s_{0}t}f(t)\,dt.} That is, F ( s ) can effectively be expressed, in

186-421: A Mellin transform , to transform the whole of a difference equation , in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power. Laplace also recognised that Joseph Fourier 's method of Fourier series for solving the diffusion equation could only apply to

248-421: A Borel measure locally of bounded variation), then the Laplace transform F ( s ) of f converges provided that the limit lim R → ∞ ∫ 0 R f ( t ) e − s t d t {\displaystyle \lim _{R\to \infty }\int _{0}^{R}f(t)e^{-st}\,dt} exists. The Laplace transform converges absolutely if

310-658: A Laplace transform into known transforms of functions obtained from a table and construct the inverse by inspection. In pure and applied probability , the Laplace transform is defined as an expected value . If X is a random variable with probability density function f , then the Laplace transform of f is given by the expectation L { f } ( s ) = E ⁡ [ e − s X ] , {\displaystyle {\mathcal {L}}\{f\}(s)=\operatorname {E} \left[e^{-sX}\right],} where E ⁡ [ r ] {\displaystyle \operatorname {E} [r]}

372-441: A VCO. Generally, low phase noise is preferred in a VCO. Tuning gain and noise present in the control signal affect the phase noise; high noise or high tuning gain imply more phase noise. Other important elements that determine the phase noise are sources of flicker noise (1/ f noise) in the circuit, the output power level, and the loaded Q factor of the resonator. (see Leeson's equation ). The low frequency flicker noise affects

434-405: A control voltage. Any reverse-biased semiconductor diode displays a measure of voltage-dependent capacitance and can be used to change the frequency of an oscillator by varying a control voltage applied to the diode. Special-purpose variable-capacitance varactor diodes are available with well-characterized wide-ranging values of capacitance. A varactor is used to change the capacitance (and hence

496-414: A generalization of the Laplace transform connected to his work on moments . Other contributors in this time period included Mathias Lerch , Oliver Heaviside , and Thomas Bromwich . In 1934, Raymond Paley and Norbert Wiener published the important work Fourier transforms in the complex domain , about what is now called the Laplace transform (see below). Also during the 30s, the Laplace transform

558-455: A limited region of space, because those solutions were periodic . In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space. In 1821, Cauchy developed an operational calculus for the Laplace transform that could be used to study linear differential equations in much the same way the transform is now used in basic engineering. This method was popularized, and perhaps rediscovered, by Oliver Heaviside around

620-455: A long distance, since the frequency will not drift or be affected by noise. Oscillators in this application may have sine or square wave outputs. Where the oscillator drives equipment that may generate radio-frequency interference, adding a varying voltage to its control input, called dithering , can disperse the interference spectrum to make it less objectionable (see spread spectrum clock ). Laplace transform In mathematics ,

682-445: A purely algebraic manner by applying a field of fractions construction to the convolution ring of functions on the positive half-line. The resulting space of abstract operators is exactly equivalent to Laplace space, but in this construction the forward and reverse transforms never need to be explicitly defined (avoiding the related difficulties with proving convergence). If f is a locally integrable function (or more generally

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744-780: A stable single-frequency clock. A digitally controlled oscillator based on a frequency synthesizer may serve as a digital alternative to analog voltage controlled oscillator circuits. VCOs are used in function generators , phase-locked loops including frequency synthesizers used in communication equipment and the production of electronic music , to generate variable tones in synthesizers . Function generators are low-frequency oscillators which feature multiple waveforms, typically sine, square, and triangle waves. Monolithic function generators are voltage-controlled. Analog phase-locked loops typically contain VCOs. High-frequency VCOs are usually used in phase-locked loops for radio receivers. Phase noise

806-854: A system. The Laplace transform's key property is that it converts differentiation and integration in the time domain into multiplication and division by s in the Laplace domain. Thus, the Laplace variable s is also known as an operator variable in the Laplace domain: either the derivative operator or (for s ) the integration operator . Given the functions f ( t ) and g ( t ) , and their respective Laplace transforms F ( s ) and G ( s ) , f ( t ) = L − 1 { F ( s ) } , g ( t ) = L − 1 { G ( s ) } , {\displaystyle {\begin{aligned}f(t)&={\mathcal {L}}^{-1}\{F(s)\},\\g(t)&={\mathcal {L}}^{-1}\{G(s)\},\end{aligned}}}

868-462: A timing signal to synchronize operations in digital circuits. VCXO clock generators are used in many areas such as digital TV, modems, transmitters and computers. Design parameters for a VCXO clock generator are tuning voltage range, center frequency, frequency tuning range and the timing jitter of the output signal. Jitter is a form of phase noise that must be minimised in applications such as radio receivers, transmitters and measuring equipment. When

930-427: A voltage determined by a musical keyboard or other input. A voltage-to-frequency converter ( VFC ) is a special type of VCO designed to be very linear in frequency control over a wide range of input control voltages. VCOs can be generally categorized into two groups based on the type of waveform produced. A voltage-controlled capacitor is one method of making an LC oscillator vary its frequency in response to

992-415: A voltage input for fine control. The temperature is selected to be the turnover temperature : the temperature where small changes do not affect the resonance. The control voltage can be used to occasionally adjust the reference frequency to a NIST source. Sophisticated designs may also adjust the control voltage over time to compensate for crystal aging. A clock generator is an oscillator that provides

1054-413: A voltage-controlled crystal oscillator can be varied a few tens of parts per million (ppm) over a control voltage range of typically 0 to 3 volts, because the high Q factor of the crystals allows frequency control over only a small range of frequencies. A temperature-compensated VCXO ( TCVCXO ) incorporates components that partially correct the dependence on temperature of the resonant frequency of

1116-525: A wider selection of clock frequencies is needed the VCXO output can be passed through digital divider circuits to obtain lower frequencies or be fed to a phase-locked loop (PLL). ICs containing both a VCXO (for external crystal) and a PLL are available. A typical application is to provide clock frequencies in a range from 12 kHz to 96 kHz to an audio digital-to-analog converter . A frequency synthesizer generates precise and adjustable frequencies based on

1178-406: Is stable if every bounded input produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region Re( s ) ≥ 0 . As a result, LTI systems are stable, provided that the poles of the Laplace transform of the impulse response function have negative real part. This ROC is used in knowing about the causality and stability of

1240-530: Is a complex frequency-domain parameter s = σ + i ω {\displaystyle s=\sigma +i\omega } with real numbers σ and ω . An alternate notation for the Laplace transform is L { f } {\displaystyle {\mathcal {L}}\{f\}} instead of F , often written as F ( s ) = L { f ( t ) } {\displaystyle F(s)={\mathcal {L}}\{f(t)\}} in an abuse of notation . The meaning of

1302-571: Is a complex number . It is related to many other transforms, most notably the Fourier transform and the Mellin transform . Formally , the Laplace transform is converted into a Fourier transform by the substitution s = i ω {\displaystyle s=i\omega } where ω {\displaystyle \omega } is real. However, unlike the Fourier transform, which gives

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1364-408: Is a real number so that the contour path of integration is in the region of convergence of F ( s ) . In most applications, the contour can be closed, allowing the use of the residue theorem . An alternative formula for the inverse Laplace transform is given by Post's inversion formula . The limit here is interpreted in the weak-* topology . In practice, it is typically more convenient to decompose

1426-401: Is a special type of VCO designed to be very linear over a wide range of input voltages. Modeling for VCOs is often not concerned with the amplitude or shape (sinewave, triangle wave, sawtooth) but rather its instantaneous phase. In effect, the focus is not on the time-domain signal A sin( ωt + θ 0 ) but rather the argument of the sine function (the phase). Consequently, modeling

1488-409: Is controlled by a voltage input. The applied input voltage determines the instantaneous oscillation frequency. Consequently, a VCO can be used for frequency modulation (FM) or phase modulation (PM) by applying a modulating signal to the control input. A VCO is also an integral part of a phase-locked loop . VCOs are used in synthesizers to generate a waveform whose pitch can be adjusted by

1550-433: Is either of the form Re( s ) > a or Re( s ) ≥ a , where a is an extended real constant with −∞ ≤ a ≤ ∞ (a consequence of the dominated convergence theorem ). The constant a is known as the abscissa of absolute convergence, and depends on the growth behavior of f ( t ) . Analogously, the two-sided transform converges absolutely in a strip of the form a < Re( s ) < b , and possibly including

1612-431: Is named after mathematician and astronomer Pierre-Simon, Marquis de Laplace , who used a similar transform in his work on probability theory . Laplace wrote extensively about the use of generating functions (1814), and the integral form of the Laplace transform evolved naturally as a result. Laplace's use of generating functions was similar to what is now known as the z-transform , and he gave little attention to

1674-471: Is not necessary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform . When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually intended. The Laplace transform can be alternatively defined as the bilateral Laplace transform , or two-sided Laplace transform , by extending the limits of integration to be

1736-462: Is often done in the phase domain. The instantaneous frequency of a VCO is often modeled as a linear relationship with its instantaneous control voltage. The output phase of the oscillator is the integral of the instantaneous frequency. For analyzing a control system, the Laplace transforms of the above signals are useful. Tuning range, tuning gain and phase noise are the important characteristics of

1798-487: Is simple to prove via Poisson summation , to the functional equation. Hjalmar Mellin was among the first to study the Laplace transform, rigorously in the Karl Weierstrass school of analysis, and apply it to the study of differential equations and special functions , at the turn of the 20th century. At around the same time, Heaviside was busy with his operational calculus. Thomas Joannes Stieltjes considered

1860-459: Is the expectation of random variable r {\displaystyle r} . By convention , this is referred to as the Laplace transform of the random variable X itself. Here, replacing s by − t gives the moment generating function of X . The Laplace transform has applications throughout probability theory, including first passage times of stochastic processes such as Markov chains , and renewal theory . Of particular use

1922-754: Is the ability to recover the cumulative distribution function of a continuous random variable X by means of the Laplace transform as follows: F X ( x ) = L − 1 { 1 s E ⁡ [ e − s X ] } ( x ) = L − 1 { 1 s L { f } ( s ) } ( x ) . {\displaystyle F_{X}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}\operatorname {E} \left[e^{-sX}\right]\right\}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}{\mathcal {L}}\{f\}(s)\right\}(x).} The Laplace transform can be alternatively defined in

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1984-407: Is the most important specification in this application. Audio-frequency VCOs are used in analog music synthesizers. For these, sweep range, linearity, and distortion are often the most important specifications. Audio-frequency VCOs for use in musical contexts were largely superseded in the 1980s by their digital counterparts, digitally controlled oscillators (DCOs), due to their output stability in

2046-613: Is useful for converting differentiation and integration in the time domain into much easier multiplication and division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering , mostly as a tool for solving linear differential equations and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic polynomial equations , and by simplifying convolution into multiplication . Once solved,

2108-427: Is usually no easy characterization of the range. Typical function spaces in which this is true include the spaces of bounded continuous functions, the space L (0, ∞) , or more generally tempered distributions on (0, ∞) . The Laplace transform is also defined and injective for suitable spaces of tempered distributions. In these cases, the image of the Laplace transform lives in a space of analytic functions in

2170-628: Is where μ is a probability measure , for example, the Dirac delta function . In operational calculus , the Laplace transform of a measure is often treated as though the measure came from a probability density function f . In that case, to avoid potential confusion, one often writes L { f } ( s ) = ∫ 0 − ∞ f ( t ) e − s t d t , {\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0^{-}}^{\infty }f(t)e^{-st}\,dt,} where

2232-438: The Laplace transform , named after Pierre-Simon Laplace ( / l ə ˈ p l ɑː s / ), is an integral transform that converts a function of a real variable (usually t {\displaystyle t} , in the time domain ) to a function of a complex variable s {\displaystyle s} (in the complex-valued frequency domain , also known as s -domain , or s -plane ). The transform

2294-491: The continuous variable case which was discussed by Niels Henrik Abel . From 1744, Leonhard Euler investigated integrals of the form z = ∫ X ( x ) e a x d x  and  z = ∫ X ( x ) x A d x {\displaystyle z=\int X(x)e^{ax}\,dx\quad {\text{ and }}\quad z=\int X(x)x^{A}\,dx} as solutions of differential equations, introducing in particular

2356-474: The gamma function . Joseph-Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions , investigated expressions of the form ∫ X ( x ) e − a x a x d x , {\displaystyle \int X(x)e^{-ax}a^{x}\,dx,} which resembles a Laplace transform. These types of integrals seem first to have attracted Laplace's attention in 1782, where he

2418-865: The region of convergence . The inverse Laplace transform is given by the following complex integral, which is known by various names (the Bromwich integral , the Fourier–Mellin integral , and Mellin's inverse formula ): f ( t ) = L − 1 { F } ( t ) = 1 2 π i lim T → ∞ ∫ γ − i T γ + i T e s t F ( s ) d s , {\displaystyle f(t)={\mathcal {L}}^{-1}\{F\}(t)={\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{\gamma -iT}^{\gamma +iT}e^{st}F(s)\,ds,}    ( Eq. 3 ) where γ

2480-592: The LFO to modulate the pulse width of the VCO or the amplitude of the VCA. It also has a noise generator with white and pink noise outputs, CV in and out, and an external signal input, not to be confused with the "External Signal Processor" of the MS-20. Voltage-controlled oscillator A voltage-controlled oscillator ( VCO ) is an electronic oscillator whose oscillation frequency

2542-482: The advantages of having no off-chip components (expensive) or on-chip inductors (low yields on generic CMOS processes). Commonly used VCO circuits are the Clapp and Colpitts oscillators. The more widely used oscillator of the two is Colpitts and these oscillators are very similar in configuration. A voltage-controlled crystal oscillator ( VCXO ) is used for fine adjustment of the operating frequency. The frequency of

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2604-511: The bilateral Laplace transform is B { f } {\displaystyle {\mathcal {B}}\{f\}} , instead of F . Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero. This means that, on the range of the transform, there is an inverse transform. In fact, besides integrable functions, the Laplace transform is a one-to-one mapping from one function space into another in many other function spaces as well, although there

2666-423: The crystal. A smaller range of voltage control then suffices to stabilize the oscillator frequency in applications where temperature varies, such as heat buildup inside a transmitter . Placing the oscillator in a crystal oven at a constant but higher-than-ambient temperature is another way to stabilize oscillator frequency. High stability crystal oscillator references often place the crystal in an oven and use

2728-485: The current available to each inverter stage, or the capacitive loading on each stage. VCOs are used in analog applications such as frequency modulation and frequency-shift keying . The functional relationship between the control voltage and the output frequency for a VCO (especially those used at radio frequency ) may not be linear, but over small ranges, the relationship is approximately linear, and linear control theory can be used. A voltage-to-frequency converter (VFC)

2790-490: The decomposition of a function into its components in each frequency, the Laplace transform of a function with suitable decay is an analytic function , and so has a convergent power series , the coefficients of which give the decomposition of a function into its moments . Also unlike the Fourier transform, when regarded in this way as an analytic function, the techniques of complex analysis , and especially contour integrals , can be used for calculations. The Laplace transform

2852-607: The earlier Heaviside operational calculus . The advantages of the Laplace transform had been emphasized by Gustav Doetsch , to whom the name Laplace transform is apparently due. The Laplace transform of a function f ( t ) , defined for all real numbers t ≥ 0 , is the function F ( s ) , which is a unilateral transform defined by F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t , {\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,}    ( Eq. 1 ) where s

2914-656: The entire real axis. If that is done, the common unilateral transform simply becomes a special case of the bilateral transform, where the definition of the function being transformed is multiplied by the Heaviside step function . The bilateral Laplace transform F ( s ) is defined as follows: F ( s ) = ∫ − ∞ ∞ e − s t f ( t ) d t . {\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.}    ( Eq. 2 ) An alternate notation for

2976-400: The face of temperature changes during operation. Since the 1990s, musical software has become the dominant sound-generating method. Voltage-to-frequency converters are voltage-controlled oscillators with a highly linear relation between applied voltage and frequency. They are used to convert a slow analog signal (such as from a temperature transducer) to a signal suitable for transmission over

3038-415: The frequency) of an LC tank. A varactor can also change loading on a crystal resonator and pull its resonant frequency. For low-frequency VCOs, other methods of varying the frequency (such as altering the charging rate of a capacitor by means of a voltage-controlled current source ) are used (see function generator ). The frequency of a ring oscillator is controlled by varying either the supply voltage,

3100-471: The integral ∫ 0 ∞ | f ( t ) e − s t | d t {\displaystyle \int _{0}^{\infty }\left|f(t)e^{-st}\right|\,dt} exists as a proper Lebesgue integral. The Laplace transform is usually understood as conditionally convergent , meaning that it converges in the former but not in the latter sense. The set of values for which F ( s ) converges absolutely

3162-754: The integral can be understood to be a (proper) Lebesgue integral . However, for many applications it is necessary to regard it as a conditionally convergent improper integral at ∞ . Still more generally, the integral can be understood in a weak sense , and this is dealt with below. One can define the Laplace transform of a finite Borel measure μ by the Lebesgue integral L { μ } ( s ) = ∫ [ 0 , ∞ ) e − s t d μ ( t ) . {\displaystyle {\mathcal {L}}\{\mu \}(s)=\int _{[0,\infty )}e^{-st}\,d\mu (t).} An important special case

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3224-410: The integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on [0, ∞) . For locally integrable functions that decay at infinity or are of exponential type ( | f ( t ) | ≤ A e B | t | {\displaystyle |f(t)|\leq Ae^{B|t|}} ),

3286-455: The inverse Laplace transform reverts to the original domain. The Laplace transform is defined (for suitable functions f {\displaystyle f} ) by the integral L { f } ( s ) = ∫ 0 ∞ f ( t ) e − s t d t , {\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,} where s

3348-450: The lines Re( s ) = a or Re( s ) = b . The subset of values of s for which the Laplace transform converges absolutely is called the region of absolute convergence, or the domain of absolute convergence. In the two-sided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence: this is a consequence of Fubini's theorem and Morera's theorem . Similarly,

3410-423: The lower limit of 0 is shorthand notation for lim ε → 0 + ∫ − ε ∞ . {\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\varepsilon }^{\infty }.} This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it

3472-521: The phase noise because the flicker noise is heterodyned to the oscillator output frequency due to the non-linear transfer function of active devices. The effect of flicker noise can be reduced with negative feedback that linearizes the transfer function (for example, emitter degeneration ). VCOs generally have lower Q factor compared to similar fixed-frequency oscillators, and so suffer more jitter . The jitter can be made low enough for many applications (such as driving an ASIC), in which case VCOs enjoy

3534-529: The pulse width and the contour of the sloped output, with the midpoint setting yielding a 50% pulse width and a triangle wave shape. The LFO's influence on the VCO frequency and filter cutoff is adjustable via dedicated knobs on the control panel. Additionally, the MS-10 features a single envelope generator, complete with controls for Hold, Attack, Decay, Sustain, and Release. Although the MS-10 has normalized connections, they can be modified with patch cables. This allows

3596-415: The region of convergence, as the absolutely convergent Laplace transform of some other function. In particular, it is analytic. There are several Paley–Wiener theorems concerning the relationship between the decay properties of f , and the properties of the Laplace transform within the region of convergence. In engineering applications, a function corresponding to a linear time-invariant (LTI) system

3658-405: The set of values for which F ( s ) converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the region of convergence (ROC). If the Laplace transform converges (conditionally) at s = s 0 , then it automatically converges for all s with Re( s ) > Re( s 0 ) . Therefore, the region of convergence is a half-plane of the form Re( s ) >

3720-559: The turn of the century. Bernhard Riemann used the Laplace transform in his 1859 paper On the Number of Primes Less Than a Given Magnitude , in which he also developed the inversion theorem. Riemann used the Laplace transform to develop the functional equation of the Riemann zeta function , and this method is still used to related the modular transformation law of the Jacobi theta function , which

3782-494: Was following in the spirit of Euler in using the integrals themselves as solutions of equations. However, in 1785, Laplace took the critical step forward when, rather than simply looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form ∫ x s φ ( x ) d x , {\displaystyle \int x^{s}\varphi (x)\,dx,} akin to

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3844-476: Was instrumental in G H Hardy and John Edensor Littlewood 's study of tauberian theorems , and this application was later expounded on by Widder (1941), who developed other aspects of the theory such as a new method for inversion. Edward Charles Titchmarsh wrote the influential Introduction to the theory of the Fourier integral (1937). The current widespread use of the transform (mainly in engineering) came about during and soon after World War II , replacing

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