Dictionary:Impulse (δ(t))

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{{#category_index:I|impulse (δ(t))}} (im’ puls)

1. The limit of a pulse of unit area as its width approaches zero and its height approaches infinity. Also called Dirac function and delta function and symbolized by δ(t).

The Dirac delta function is not mathematically a function but is a different category of mathematical object called a distribution. The introduction of the delta function is attributed to physicist Paul A. M. Dirac, but the necessity of such an object was alluded to in the discussion of the cascade of forward and inverse Fourier transforms in Theorie de Chaleur by Joseph Fourier. [1] The formal mathematical theory was introduced by mathematician Laurent Schwartz.[2] Another direction of representing the Dirac delta function as a generalized function, as presented by Lighthill, 1964.[3]


2. The fundamental properties of the Dirac delta function include the sifting property


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^{\infty} \phi(t) \delta(t - t_0 ) \; dt = \phi(t_0) }


implying that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta(t - t_0) } only has support at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t = t_0. } The notion of support in this case is that the limits of integration must contain the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t = t_0 } for the integral to be nonzero.

The function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(t) } is called a test function and is any function such that this integral exists (i.e. doesn't blow up). The most general class of these functions are called the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_0^\infty } which are the infinitely differentiable functions that vanish smoothly at infinity.

3. The second property of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta(t - t_0) } is that it has a unit integral

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^{\infty} \delta(t - t_0) \; dt = 1 .}

Indeed, the definite integral

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^{t} \delta(\tau - t_0) \; d\tau = \begin{cases} 0 &\text{if } t < t_0 \\ 1 &\text{if } t > t_0 \end{cases} = H(t- t_0) }

thus, the integral of the delta function is the Heaviside step function. Conversely, the derivative of a step function is a delta function. Hence, distributions extend our ability to define differentiation to cases for which the derivative is not defined classically.

4. While it appears in many engineering and less formal texts that the value of delta function has an infinite value where its argument is zero, this is not correct, because the delta function has no intrinsic meaning outside of integration with a test function.

What is more correct is to consider defining the delta function as the limit of a sequence of strongly peaked functions that have, in the limit, support at the desired value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_0 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{n \rightarrow \infty} \int_{-\infty}^{\infty} S_n(t - t_0 ) \; dt = 1 }


and the limit of the sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \left\{ S_n(t) \right\} } must exhibit the sifting property


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{n \rightarrow \infty} \int_{-\infty}^{\infty} S_n(t - t_0 ) \phi(t) \; dt = \phi(t_0). }


There are many functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_n(t - t_0) } that have this property, including functions for which the limit of the function does not exist at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t = t_0. }

5. We may define, formally the derivative of a delta function (called delta prime) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta^{\prime} ( t - t_0) } via integration by parts with a test function


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^{\infty} \phi(t) \delta^{\prime}(t - t_0 ) \; dt = \left. \phi(t)\right|_{-\infty}^{\infty} - \int_{-\infty}^{\infty} \phi^{\prime}(t) \delta(t - t_0 ) \; dt = - \phi^{\prime} (t_0), }


which follows because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(t) = 0 } at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \pm \infty. }

Higher derivatives are defined by successive integration by parts.


See impulse response.


6. A pulse that is of sufficiently short time-duration that its waveshape is of no consequence.


7. A complex impulse δ*(t) or complex delta function is defined as an analytic signal, through the construction of an imaginary part via the Hilbert transform as


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta^*(t)=\delta(t)+\left(\frac{i}{\pi}\right) t } .

8. See Kroenecker delta.


References

  1. Fourier, Joseph. Theorie analytique de la chaleur, par M. Fourier. Chez Firmin Didot, père et fils, 1822.
  2. Schwartz, Laurent. "Theory des distributions." Hermann, Paris (1950).
  3. Lighthill, Michael J. An introduction to Fourier analysis and generalised functions. Cambridge University Press, 1964.


External links

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