Impulse response

The Impulse response from a simple audio system. Showing the original impulse, with high frequencies boosted, then with low frequencies boosted.

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.

Mathematical basis

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]:

$y\left[n\right]=T\left[x\left[n\right]\right]$

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.

$T\left[x\left[n\right]\right]=x^{2}\left[n\right]$

or linear e.g.

$T\left[x\left[n\right]\right]=x\left[n-1\right]$.

Suppose that T is linear. Then

$T\left[x\left[n\right]+y\left[n\right]\right]=T\left[x\left[n\right]\right]+T\left[y\left[n\right]\right]$

and

$T\left[\lambda x\left[n\right]\right]=\lambda T\left[x\left[n\right]\right]$

Suppose also that T is invariant under translation i.e. if $y\left[n\right]=T\left[x\left[n\right]\right]$ then $y\left[n-k\right]=T\left[x\left[n-k\right]\right]$. 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

$x\left[n\right]=\sum _{k}x\left[k\right]\delta \left[n-k\right]$

and take the T of both sides

$T\left[x\left[n\right]\right]=T\left[\sum _{k}x\left[k\right]\delta \left[n-k\right]\right]$

Of course this has a meaning only if

$\sum _{k}x\left[k\right]\delta \left[n-k\right]$

lies in the domain of T. Now, since T is linear and invariant under translation we may write

$T\left[x\left[n\right]\right]=\sum _{k}x\left[k\right]T\left[\delta \left[n-k\right]\right]$

Since the output y[k] is given by

$y\left[k\right]=T\left[x\left[k\right]\right]$

we may write

$y\left[n\right]=\sum _{k}x\left[k\right]T\left[\delta \left[n-k\right]\right]$

Putting

$h\left[n-k\right]=T\left[\delta \left[n-k\right]\right]$

we have finally

$y\left[n\right]=\sum _{k}x\left[k\right]h\left[n-k\right]$

The sequence $h\left[n\right]$ 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 $\delta _{p}[n]$ is the "pow", which is similar (due in part to its short duration) 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 (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. 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 1980s which led to big improvements in loudspeaker design. Loudspeakers suffer from colouration, a defect unlike normal measured properties like frequency response. Colouration 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. Colouration 'smears' the sound which reduces the 'clarity' and 'transparency.' 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 has led to graphical spectrogram plots that show delayed response against time for each frequency.

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.

See also

• Kronecker delta
• Dirac delta function
• Unit impulse function
• Green function
• frequency response
• LTI system theory
• system analysis
• transfer function
• Convolution reverb
• transient
• transient response