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# Impulse response

The impulse response of a system is its output when presented with a very brief input signal, an impulse. Mathematically, an impulse can be modeled as a Dirac delta function for continuous-time systems, or as the Kronecker delta for discrete-time systems. The Dirac delta represents the limiting case of a pulse made very short in time while maintaining its area or integral (thus giving an infinitely high peak). While this is impossible in any real system, it is a useful idealization. In Fourier analysis theory, such an impulse comprises equal portions of all possible excitation frequencies, which makes it a convenient test probe.

Any system in a large class known as linear, time-invariant (LTI) is completely characterized by its impulse response. That is, for any input function, the output function can be calculated in terms of the input and the impulse response. (See LTI system theory.)

## Mathematical applications

In the language of mathematics, the impulse response of a linear transformation is the image of Dirac's delta function under the transformation.

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.

## Practical applications

In practical systems, it is not possible to produce a perfect impulse to serve as input for testing; therefore, a brief pulse is sometimes used as an approximation of an impulse. Provided that the pulse is short enough compared to the impulse response, the result will be close to the true, theoretical, impulse response. In many systems, however, driving with a very short strong pulse may drive the system into a nonlinear regime, so instead the system is driven with a pseudo-random sequence, and the impulse response is computed from the input and output signals.

### Loudspeakers

An application that demonstrates this idea was the development of impulse response loudspeaker testing in the 1970s. Loudspeakers suffer from phase inaccuracy, a defect unlike other measured properties such as frequency response. Phase inaccuracy is caused by small delayed sounds that are the result of resonance, energy storage in the cone, the internal volume, or the enclosure panels vibrating. Measuring the impulse response, which is a direct plot of this "time-smearing," provided a tool for use in reducing resonances by the use of improved materials for cones and enclosures, as well as changes to the speaker crossover. The need to limit input amplitude to maintain the linearity of the system led to the use of inputs such as pseudo-random maximum length sequences, and to the use of computer processing to derive the impulse response.

### Digital filtering

Impulse response is a very important concept in the design of digital filters for audio processing, because these differ from 'real' filters in often having a pre-echo, which the ear is not accustomed to.

### Electronic processing

Impulse response analysis is a major facet of radar, ultrasound imaging, and many areas of digital signal processing. An interesting example would be broadband internet connections. Where once it was only possible to get 4 kHz speech signal over a local telephone wire, or data at 300 bit/s using a modem, it is now commonplace to pass 2 Mb/s over these same wires, largely because of 'adaptive equalisation' which processes out the time smearing and echoes on the line.

### Control systems

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.

## References

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