Inside Collection (Textbook): Signals, Systems, and Society

Filters are devices which are commonly found in electronic gadgets. When you adjust the *bass* (low frequency) or *treble* (high frequency) settings on your MP3 player, you are adjusting the characteristics of a filter. A more technical name for a filter is a *linear system* . A filter is represented by a box having a single input (usually

We can denote the operation the filter has on the input using the following notation:

The types of filters we will consider in this book are *linear* and *time-invariant*.
A filter is time-invariant if given that

Equation Equation 2 is often referred to as the *superposition principle*. We can use linearity and time invariance to derive the mathematical operation which the filter performs on the input,

The signal *impulse response* of the filter. From time invariance, we have

Now we can use linearity to find the filter output when the input is

We can extend the linearity property further by noting that

where we can assume that the constants

Using the sifting property of the unit impulse in the right side of Equation 7 gives

So it follows that the filter performs the following operation on the input,

The integral in Equation 9 is called the *convolution integral*. A change of variables can be used to show that

which means that the order in which two signals are convolved is unimportant. A short-hand notation for convolution is

- « Previous module in collection Fourier Transform of Periodic Signals
- Collection home: Signals, Systems, and Society
- Next module in collection » Properties of Convolution Integrals