Filter Classifications in the Time Domain

When a signal $x[n]$ is applied as an input to an LTI system, the output $y[n]$ is given by the convolution between the input and the impulse response $h[n]$. This output $y[n]$ is different than the input $x[n]$. But how?

Filters

Filtering is a process of discrimination and selection.

• A water filter blocks impurities and lets the water pass through.
• An air filter stops dust particles while allowing air into the system.

In a similar manner, a filter is a system that suppresses an undesired feature in an input signal while allowing the remaining to proceed. Furthermore, a filter can also be designed for restoration purposes, e.g. deblurring an image.

For example, most filters allow a certain portion of the frequency domain to pass through and block the rest. Although filtering in general is concerned with the frequency domain, it can involve some other aspects of the signal (e.g., phase).

Digital filters are the reason behind the rise of DSP over the past few decades. They can implement a surprisingly complicated level of functionality at an even more surprisingly low cost.

These filters can be divided into two categories:

• Finite impulse response filters
• Infinite impulse response filters

Now, we will briefly introduce these digital filters.

Finite impulse response filters

As the name implies, a filter with an impulse response is of finite duration is termed a finite impulse response (FIR) filter. For example, a system with the following impulse response is a FIR filter:

$$h[n]=\{1, -3, 1, 2, -1\}$$

When an input is applied to this filter, the output is given by the familiar convolution sum.

$$\begin{align*} y[n]&=\sum_m h[m]x[n-m]\\ &= \cdots + h[0]x[n]+h[1]x[n-1]+h[2]x[n-2]+\cdots \end{align*}$$

A block diagram for its implementation is drawn below: