Convolution: filtering

The function which describes the relation between the input and the output of a system is called the transfer function of a system. A handclap in an empty hall, for instance, produces an `echo': in other words, the handlap is the input, the echo is the output, and the hall itself is a system with certain acoustic properties, i.e. a response. If the input signal were a pure instantaneous impulse, the output would be the impulse response of the system. The handclap approximates intuitively to the impulse response, but is in fact more complex than a pure impulse. A system of this kind is a filter.

The transfer function is, in this case, defined by convolution of the input signal with the impulse response of the system. Convolution has the following properties in the general case:

1. The output signal is delayed in time and may be longer in time than the input signal.
2. The output signal has a different amplitude envelope from the input.
3. The output signal has a different timbre from the input.

In the equation, h may be taken to represent the system and x to represent the input signal (though the operation is symmetrical, so this may be reversed); y stands for the output signal.

where .

The interesting features of the operation are:

1. The operation is calculated over the sum of the lengths of each signal setgment (note the time-lengthening effect of the echo).
2. One of the signals is reversed, by subtracting the counter variable from the position reached in the other signal.

  
Figure: Sine wave (high frequency).

  
Figure 22: Filter impulse response.

  
Figure 23: Convolution (filtering).