Impulse response
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In simple terms, the impulse response of a system is its output when presented with a very brief signal; an impulse. While an impulse is a difficult concept to imagine, and an impossible thing in reality, it represents the limit case of a pulse made infinitely 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 concept as an idealization.
Mathematically, an impulse can be modeled as a Dirac delta function. Suppose that T is a (discrete) system, i.e. something that takes an input x[n] and produces an output y[n]:
<math>y\left[ n\right] =T\left[ x\left[ n\right] \right] </math>
So T is an operator acting on sequences (over the integers) and producing sequences. Beware that T is not the system but a mathematical representation of the system. Now, T can be non-linear, e.g. <math>T\left[ x\left[ n\right] \right] =x^{2}\left[ n\right]</math> or linear e.g. <math>T\left[ x\left[ n\right] \right] =x\left[ n-1\right]</math>. Suppose that T is linear. Then
<math>T\left[ x\left[ n\right] +y\left[ n\right] \right] =T\left[ x\left[ n\right] \right] +T\left[ y\left[ n\right] \right]</math>
and
<math>T\left[ \lambda x\left[ n\right] \right] =\lambda T\left[ x\left[ n\right] \right]</math>
Suppose also that T is invariant under translation i.e. if <math>y\left[ n\right] =T\left[ x\left[ n\right] \right]</math> then <math>y\left[ n-k\right] =T\left[ x\left[ n-k\right] \right]</math>. In such a system any output can be calculated in terms of the input and a very special sequence called impulse response which characterizes the system completely. This can be seen as follows: Take the identity
<math>x\left[ n\right] =\sum_{k}x\left[ k\right] \delta \left[ n-k\right]</math>
and take the T of both sides
<math>T\left[ x\left[ n\right] \right] =T\left[ \sum_{k}x\left[ k\right] \delta \left[ n-k\right] \right]</math>
Of course this has a meaning only if <math>\sum_{k}x\left[ k\right] \delta \left[ n-k\right]</math> lies in the domain of T. Now, since T is linear and invariant under translation we may write
<math>T\left[ x\left[ n\right] \right] =\sum_{k}x\left[ k\right] T\left[ \delta \left[ n-k\right] \right]</math>
Since the output y[k] is given by <math>y\left[ k\right] =T\left[ x\left[ k\right] \right]</math> we may write
<math>y\left[ n\right] =\sum_{k}x\left[ k\right] T\left[ \delta \left[ n-k\right] \right]</math>
Putting
<math>h\left[ n-k\right] =T\left[ \delta \left[ n-k\right] \right]</math>
we have finally
<math>y\left[ n\right] =\sum_{k}x\left[ k\right] h\left[ n-k\right]</math>
The sequence <math>h\left[ n\right]</math> is the impulse response of the system represented by T. As can be seen from the above, h[n] is the output of the system when its input is the discrete Dirac delta. Similar results hold for continuous time systems.
As a practical example consider a room and a balloon in it at point p. The balloon pops up. Here the room is a system T which takes the "pow" sound and diffuses it through multiple reflections. The input <math>\delta_{p}[n]</math> is the "pow", which is nothing but a Dirac delta and the output h[n,p] is the sequence of the damped sound. Here h[n,p] depends from the point p of the balloon. If we know h[n,p] for every p of the room, then we actually know the impulse response of the room and hence its behaviour to any sound produced in it.
In practical applications, things are not nearly so obscure, because the impulse response of a system can be determined by simply using a short pulse as the input and examining the output. Provided that the pulse is short compared to the impulse response, the result will be near enough to the true, theoretical, impulse response.
A very useful real application that demonstrates this idea was the development of impulse response loudspeaker testing in the 1980's which led to big improvements in loudspeaker design. Loudspeakers suffer from colouration, a defect that has nothing to do with the normal measured properties like frequency response because it is the result of small delayed sounds that are the result of resonance, or energy storage in the cone, the internal volume, or the enclosure panels. These 'smear' the sound, giving reduced 'clarity' or 'transparency' to the sound. 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. Initially, short pulses were used, but the need to limit their amplitude to maintain the linearity of the system meant that the resulting output was very small and hard to distinguish from the noise. Later techniques therefore moved towards the use of other types of input, like maximal length sequences, and using computer processing to derive the impulse response. Recently this led to the very graphic three dimensional waterfall plots that can often be seen in test reviews, of delayed response shown against time for each frequency.
Impulse response is also 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.
Other applications would be radar, ultrasound imaging, and many areas of electronic processing. An interesting example would be broadband internet connections. Where once it was only possible to get 4kHz speech signal over a local telephone wire, or data at 300 bit/s using a modem, it is now commonplace to pass 2Mb/s over these same wires, largely because of 'adaptive equalisation' which processes out the time smearing and echoes on the line.
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.
See also
- Dirac delta function
- Unit impulse function
- Green's function
- frequency response
- LTI system theory
- system analysis
- transfer function
