Let's input some signal to the filter that we designed last time. Table 1 shows the input of constant signal. Constant signal will be boring, but, We start with a simple one.

Table 1 Constant input

To make sure, I will show you how to compute the Table 1's red output. Here y_n and n=1,

In this case, the filter gets the first three inputs. The inputs are all the same (= 1), therefore all the outputs are also the same. Let's compute the transfer function. We compute all the time the transfer function in digital filter.

What, Jojo, You! Be surprised. (it's a bit old and does anybody know Jojo's strange adventure by Araki Hirohiko?) The ratio of input and output is equal to the transfer function!

Here is a small details. In the Table, y_n has a value at n=0,8 since we can compute the cos value. But there usually is no n=-1 value in the real data acquisition (If we start with n=0, then no data at n=-1). Therefore, it is also possible to say there is no output value at y_0.

I think this example is a bit boring since input is constant and thus

the output is also constant. The followings are dynamic examples in

Table 2 and 3.

Table 2 cos π/3 input

Table 3 cos 2π/3 input

Let me show you how to compute the red output in Table 2.

The transfer function is the following.

Transfer function represents that how much input is transfered to the output. Here the value is 1, this means, the input and output are the same. The signal of this frequency go through this filter without change. You can clearly see the correspondence of the input and the output, they are the same. Transfer function is great. In the case of Table 3,

The transfer of this is:

Needless to say, this frequency can not pass this filter.

Conclusion

The mathematics used in here is not so complex. If you can accept the Euler's equation, the high school math can handle the rest of them. You can Euler's equation is also explained by (infinite) series expansion of e^x, sin(x), cos(x), then you can compare the sin(x), cos(x) and e^x. This gives you a hint of relationship between them. At least one can see some kind of relationship, I presume.

The transfer function is an eigenvalue of the sampling filter function. These are ideal cases, but it is fun to see the theory works pretty well.