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Deconvolution

 

On any analogue signal the characteristic acoustics of a specific room can be mapped by convoluting it with the impulse response of that room. It is a fundamental rule of digital signal processing that the convolution of two waves corresponds to the multiplication of their spectrums, and vice versa: that the multiplication of two waves corresponds to the convolution of their spectrums. This relation also explains the occurrence of alias frequencies when the signals are sampled since sampling means a multiplication of the analogue audio signal with the set of unit pulses.

In mathematical terms, convolution is described as follows:


N is the length of the sequence a in sample values and k fields along the entire length of b. Every sample value a(n) serves as a weight function for the delayed “copy” of b(t). These weighted and delayed copies are added.

Convolution is easy to understand when looking at the theoretical process of sampling. With each sampling the analogue signal (a) is convoluted with the unit pulse n. As long as this pulse is only once (t)=1 for t=0 and for all other values of t (t)=0, the result is:


The result of equation (2) is a set of output values corresponding to the input values. Thus, we can say: Every convolution with a single unit pulse does not influence the original signal.

When introducing a constant c, the equation is:


Equation (3) demonstrates that in convolution constant c is a scale factor with which the absolute value of the individual samples is multiplied. When introducing a time constant, which delays the unit pulse by t, this can be put into the following equation:


The output signal of the operation corresponds to the input signal, which is moved on the time axis because of the convolution with t.

The mathematical sign * indicates convolution and should not be confused with the sign for multiplication. The output of the convolution of two signals differs fundamentally from the output of multiplication. Multiplication means: Each sample value of a is multiplied with the corresponding sample value of b. For example, if applying multiplication instead of convolution in equation (2), the result will be 0 for all output values, except for the input value where the unit pulse is 1. In contrast, to convolute two signals means: each sample value of a is multiplied with each sample value of b. This is comparable to generating an array, whereas the sum of these arrays constitutes the result of the convolution of the signals.