Any sequence can be expressed as a sum of scaled and shifted unit samples. This no doubt seems like a lot of trouble to go to, just to get back the same sequence that we originally started with, but in fact, we will very shortly be able to use equation 1 to perform a marvelous trick, so make sure that you understand it. Thus such a system or transform is a function from sequences to sequences. Systems or transforms come in a wide variety of types. One important class is known as linear systems.
Linearity means the same thing as applied to sequences.
The only thing T operates on is the set of shifted unit impulses, which is independent of x . However, there is one annoyance - we still must calculate for every shift k . This would be unnecessary if the response of T to a shifted input was just an output shifted by the same amount. This property, which is called shift invariance , holds of many interesting systems. The way of combining two sequences specified by equation 9 is known as convolution .