(compMus2) Convolution

Convolution is a fundamental process in digital audio processing. Even if you do not specifically know that the process is happening, you know the effects of the process. Filtering, reverberation, and cross synthesis all illustrate convolution. For example, a filter convolves its impulse response (IR) with the input signal to produce filtered output. Sampling reverbs convolve impulse responses of physical spaces with input signals to produce the effect of playing a sound in the physical space.

Musical Uses of Convolution: Reverberation and Filtering/Cross Synthesis

Convolution can be used to simulate an arbitrary signal being played back in a specific physical space by sampling the impulse response of a space and convolving that IR with an arbitrary signal. Sampling a room requires a signal with (preferably) all frequencies. Often, a balloon popping or a starter pistol is used. The best way is to use a quick sine tone sweep. Convolution can be used to filter signals, either for cross synthesis purposes, or to simulate the characteristics of an audio system, such as a microphone or guitar amp.

The Math of Convolution

For fun, the equation:

For every sample in one signal a (the arbitrary signal), multiply it by every sample in the IR b, and sum the results offset by each sample in a. The length of the output file will equal the length of signals a + b – 1 sample. Convolution is not multiplication. Multiplication in the time domain is amplitude modulation. (For each sample, multiply one sample from a times one sample from b.) Convolution of two audio signals is a series of multiplications, and a summation of those results. Each sample in one signal is multiplied by the entire set of samples in the second signal, offset in the original signal by the location of the sample being multiplied.

Implementation of Convolution

Implementing convolution in the time domain is very computationally expensive, and not practical as a process. To implement convolution as a digital signal process we rely on the Law of Convolution. The Law of Convolution states that convolution in the time domain is equal to multiplication in the frequency domain, and vice versa. Both signals are converted to the frequency domain via an FFT, and their resulting frequency spectra are multiplied.

Understanding convolution as multiplication of the frequency spectra is the easiest way to understand how convolution can be used to filter a signal. Shared frequency content between two signals will be resynthesized, but any frequency not found in both signals will be silenced (multiplying any number by 0 equals 0). Understanding convolution in the time domain is the easiest way to understand how convolution works as reverberation, since each sample in one signal will be scale and repeated for every sample in the other signal. The result of this operation is time smearing. It should be noted that however you understand convolution, the process of convolution is acting as both a filter and time smearing operation. Therefore, if your purpose is weighted towards filtering, your impulse should be very short. If you wish to simulate reverb, your impulse should be of a duration that matches with typical reverb times (0.8 and above, with a decaying amplitude envelope).