What are Impulses?

Signals and Systems

A system is an abstract representation of something that takes a signal as input and produces a signal as output. Many systems work with analog signals, such as amplifiers and band-pass filters. However, systems can be more abstract, such as a room in which sound bounces around off the walls. The input is the musical instrument, and the output is the way the sound is transformed by the time it reaches the microphone.

The simplest type of system to work with is a linear, time-invariant (LTI) system, which as the name implies has two important properties. Linear means that adding two input signals together results in the adding of the output signals. Time-invariant means that the system doesn't change the phase of the input signal, meaning that the phase difference between two input signals is maintained after passing through the system. Most systems can be approximated as LTI systems, including the air that sound passes through.

What is an Impulse?

An impulse signal is a near-instantanteous input to a system across all frequencies. If we want to characterize how the system behaves at every frequency, for example to see how attenuation varies across frequencies, then we can analyze the impulse response (or impulse response function) of the system.

For continuous-time systems, the Dirac delta function is used as the impulse to the system. This function is very strange in that it isn't really a "function" in the normal sense. The integral of the function is equal to 1 if the range includes $x = 0$, but the integral is equal to 0 if range does not include $x = 0$. The Diract delta function can be approximated using a normal distribution that gets thinner and taller as its standard deviation decreases. The equation for the Diract delta function is shown below

$$\delta(x) = \lim_{a \to 0} \frac{1}{|a|\sqrt{\pi}} e^{-(x/a)^2}$$

Module 1

Image

The conversion of a normal distribution into a Dirac delta function.

Code

The code used to generate the image.

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For discrete-time systems, the Kronecker delta function is used as the impulse to the system. It is defined by the following equation.

$$\delta(x) = \begin{cases} 1 \quad \text{if} \quad x = 0 \\ 0 \quad \text{if} \quad x \neq 0 \\ \end{cases}$$

The Kronecker delta function is sometimes referred to as the unit sample function.

If you apply the discrete Fourier transform to the Kronecker delta function, the spectrum is spread out equally among all frequency bins.

Module 2

Image

The time and frequency domain for a signal that oscillates at 0.5 Hz. When the signal is an impulse the DFT shows energy in all frequencies.

Code

The code used to generate the image.

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Acoustic Response

The acoustics of a room describe how sound travels throughout the room. Some rooms are designed to minimize echoes/reverberations as much as possible, while others are designed to give a certain "music hall" feeling to the sound.

We can use the impulse response from an impulse in the room to "capture" the acoustical properties of the room. We do this by creating an impulse (e.g. using a computer to produce sound at all frequencies and play it through a speaker) then looking at the frequency spectrum of the recorded sound from a microphone. The recorded spectrum is called a transfer function since it describes how the impulse is transferred to the room.

Module 3

Audio

A song.

An impulse response that has a lot of echoes.

The song with the echo impulse response applied.