Impulse response
n 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 . While this is impossible in any real system, it is a useful concept as an idealization.
Encyclopedia
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 . While this is impossible in any real system, it is a useful concept as an idealization.
Mathematical basis
Mathematically, an impulse can be modeled as a
Dirac delta function. Suppose that T is a system, i.e. something that takes an input x[n] and produces an output y[n]:
So T is an operator acting on sequences 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. or linear e.g. . Suppose that T is linear. Then
and
Suppose also that T is invariant under translation i.e. if then . 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
and take the T of both sides
Of course this has a meaning only if lies in the domain of T. Now, since T is linear and invariant under translation we may write
Since the output y[k] is given by we may write
Putting
we have finally
The sequence 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 conceptual example consider a room and a balloon in it at point p. The balloon pops and makes a "pow" sound. Here the room is a system T which takes the "pow" sound and diffuses it through multiple reflections. The input is the "pow", which is similar to a Dirac delta, and the output h[n,p] is the sequence of the damped sound. Here h[n,p] depends on the location 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. It is then possible to predict its response to any sound produced in it.
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 real, practical systems, it is not possible to produce a perfect impulse to serve as input for testing. Therefore, a brief pulse is used as an approximation of an impulse. Provided that the pulse is short compared to the impulse response, the result will be near enough to the true, theoretical, impulse response.
Loudspeakers
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.
Digital filtration
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 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.
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.
See also
