# The Z-transform

### Signals as a Sum of Delta Functions

Any discrete signal can be written as an infinite sum of scaled Kronecker delta functions. You can easily see that all terms where are zero, because the Kronecker delta is zero in that case. Only the term for is non-zero, in which case the Kronecker delta is one, so the result is just . This is a consequence of the sifting property of the delta function, covered in the previous page.

### DTLTI Transformations as Convolutions

You can express the output of any discrete-time linear time-invariant system as the convolution of the input with the impulse response of the system, :

#### Proof

The proof itself is very simple: We just decompose the input as a sum of delta functions, as described in the previous paragraph, and then we use the linearity and time-invariance to bring the operator inside of the summation. The symbol in the last step is called the convolution operator, and it is defined as the sum in the step before it.