In the language of mathematics, the impulse response of a linear transformation is the image of Dirac's delta function under the transformation.
In control theory the impulse response is the response of a system to a Dirac delta input. This proves useful in the analysis of dynamic systems: The Laplace transform of the delta function is 1, so the impulse response is equivalent to the inverse Laplace transform of the system's transfer function.
The Laplace transform of the impulse response function is known as the transfer function. It is usually easier to analyze systems using transfer functions as opposed to impulse response functions. The Laplace transform of a system's output may be determined by the multiplication of the transfer function with the input function in the complex plane, also known as the frequency domain. An inverse Laplace transform of this result will yield the output function in the time domain.
To determine an output function directly in the time domain requires the convolution of the input function with the impulse response function. This requires the use of integrals, and is usually more difficult than simply multiplying two functions in the frequency domain.
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