Digital Signal Processing Reference
In-Depth Information
one above the other. This also works with lter coecients. Next, multiply the
rst value of the second signal by each value of the rst, and write the result below,
starting from the left. Repeat this for each of the second signal's values, indenting
the results. Finally, add the values along each column.
For example, suppose signal x = [1; 2; 3; 4; 5] were the input to an FIR lter with
coecients b = [6; 7; 8]. What is the output y? Following the algorithm given above,
we generate this:
1 2 3 4 5
6 7 8
-------------
6 12 18 24 30
7 14 21 28 35
8 16 24 32 40
-------------------
6 19 40 61 82 67 40.
Therefore, we nd y = [6; 19; 40; 61; 82; 67; 40]. Note that we get the same answer
no matter which of the two signals we decide to put in the rst line. In other words,
we arrive at the same values for y if we convolve [6; 7; 8] with [1; 2; 3; 4; 5] instead of
convolving [1; 2; 3; 4; 5] with [6; 7; 8]. This can be conrmed below.
6 7 8
1 2 3 4 5
-------------
6 7 8
12 14 16
18 21 24
24 28 32
30 35 40
-------------------
6 19 40 61 82 67 40
3.3
Causality, Linearity, and Time-Invariance
There are three notable properties of a system:
1. causality
2. linearity
