Sine sweep

A sine sweep is a sine function that gradually changes frequency over time.

The sine sweep can also be called "sinusoidal sweep," "frequency sweep", or "chirp".

The functions

General formula for a linear sine sweep


General formula for an exponential sine sweep

for example, change frequency from f0 to f1 over the time T. The first function is called a linear sine sweep, as the derivative of the frequency term inside the sine with respect to the time t is linear. Similarly, the second function is an exponential sine sweep.

Both functions have an initial phase of zero. A different initial phase can be added as with any other sine function, if necessary. In both functions, replacing t with k / fs, where k = 0, 1, 2, … and fs is the sampling frequency produces the discrete time equivalent of the functions.

Example sine sweep

The following is an exponential sine sweep that changes frequency from 1 Hz to 50 Hz over 2 seconds.

Graphical representation of an example exponential sine sweep

Sine sweep sound

The following is a linear sine sweep, generated with the formula above, where the beginning frequency f0 is 50 Hz, the end frequency is f1 is 1000 Hz, and the time T is 1 second.

Click Play to hear the linear sine sweep.

Play the linear sine sweep

The following is an exponential sine sweep with the same parameters – sweeping the frequencies between 50 Hz and 1000 Hz over 1 second.

Click Play to hear the exponential sine sweep.

Play the exponential sine sweep

Using the sine sweep

Sine sweeps are useful when obtaining the impulse response of natural reverberations to design an impulse reverb. In practice, one can play a sine sweep in a room, record the result, and deconvolve the result with the sine sweep inverted in time to obtain the impulse of the reverb.

Take the following impulse response a(t).

An example impulse response of a simple reverb to be used for testing the convolution of an impulse response and a sine sweep

This graph imitates the impulse response of some reverb. It was produced by running an impulse through a feedforward comb filter and running the output of the comb filter through a Shroeder all pass filter (see All pass filter). This is a 300-point impulse response over the sampling frequency 500 Hz.

Convolve this impulse response with an exponential sine sweep between 1 Hz and 200 Hz. The convolution h(t), which by itself is not very interesting, is as follows.

The convolution of an exponential sine sweep with an example impulse response – a potential recording of natural reverberations of a sine sweep

In practice, when recording natural reverberations of a sine sweep, we will have this result and we will know the sine sweep employed, but we will not know the impulse response. To get the impulse response, we can convolve this result with the same sine sweep but inverted in time. If we do so in this example, we will get the following impulse response.

An impulse response of an example reverb obtained by deconvolving the recorded natural reverberations of the sine sweep with the time inverted sine sweep

This impulse response is quite close to the initial impulse response.

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