#### Theoretical analysis of digital filter. 1 accurate article make it clear!

Digital filter play a role in filtering the input signal.

## Table of Contents

If the input and output of the filter are both discrete-time signals, then the impulse response h(n) of the filter must also be discrete. Such filters are digital filters. When implementing a digital filter in hardware, the required components are delays, multipliers, and adders. If implemented on a computer using software, it is a linear convolution program.

## low pass filters

Low-Pass Filter (LPF for short) is an electronic filter that allows low-frequency signals to pass while suppressing high-frequency signals. In signal processing, communications, audio processing and other fields, low-pass filter have a wide range of applications. The main characteristics of higher-order low-pass filters are that they have more poles and a more complex frequency response. The signal decays faster. This property makes high-order filters very useful in applications that require precise control of frequency response, such as audio processing, communication systems, and data acquisition.

Compared with low-order filter, high-order filter have a steeper roll-off characteristic near the cutoff frequency, which means that it can suppress high-frequency noise and interference more effectively.

A high-order low-pass filter is a circuit composed of multiple first-order or second-order links (i.e., first-order RC filter or second-order LC filter) connected in cascade or parallel. Used to smooth the input signal, allowing low-frequency signals to pass while suppressing high-frequency signals.

Compared with the first-order low-pass filter, the high-order filter has a steeper cutoff frequency attenuation slope and better selectivity, and can achieve more accurate signal frequency separation in a wider frequency range.

## Analog filter and digital filter

Analog filter is a linear time-invariant system that performs linear filtering on analog signals.

Usually the requirements put forward by users for analog filter include:

a. Filter performance indicators, including cutoff frequency, upper and lower boundary frequencies, passband ripple, stopband attenuation, etc.

b. Type of filter, usually Butterworth filter, Chebyshev filter, etc.

The design of digital filter is one of the core issues in signal processing. Like analog filters, the main function of digital filter is to process digital signals. The most common processing is to retain the useful frequency components of the digital signal and remove the useless frequency components in the signal. According to time domain characteristics, digital filter can be divided into two categories: infinite impulse response digital filters and finite impulse response digital filters.

In digital filters, high-order low-pass filters can be implemented through difference equations, IIR (infinite impulse response) or FIR (finite impulse response) structures. IIR structures are generally computationally efficient, but may introduce phase distortion and nonlinear effects. The disadvantage is that some errors and distortions may be introduced during the conversion process. The FIR structure has linear phase characteristics and better stability, but has higher computational complexity.

The IIR design filter digital simulation is designed using the mature theory of analog filters. Therefore, some excellent amplitude characteristics of typical analog filters are retained and only the amplitude characteristics are considered in the design. Considering the design and characteristics of the phase, the designed filter characteristics should be nonlinear.

In order to obtain linear increase, the phase characteristic network must be added, which complicates the grid filter design. While the FIR filter ensures amplitude characteristic technology, it is also required to strictly achieve very linear phase characteristics.

The design of FIR digital filter is mainly based on the ideal filter frequency characteristics and then approximates them in some way. These methods include window function method, frequency sampling method and best consistent approximation method.

Since the windowing process is the product of the unit impulse response of the ideal filter and the window function in the time domain, according to the complex convolution theorem, the frequency response of the windowed filter is the frequency response and window function of the frequency of the ideal filter. convolution of the frequency response. What affects the amplitude function of the actual FIR filter frequency response is the amplitude function of the window function frequency response.

## High order low pass filter

The design of higher-order filter is usually based on some classic filter prototypes, such as Butterworth, Chebyshev, Elliptic, etc. In addition, high-order filters also have better out-of-band suppression capabilities, that is, the signal attenuates faster in the frequency range above the cutoff frequency. These filter prototypes have different frequency response characteristics and can be selected based on specific application needs.

Butterworth filter: It has a flat passband and the slowest roll-off at the cut-off frequency. Its maximum amplitude response is 0dB and has a roll-off speed of 20n octave after the cut-off frequency.

Chebyshev filter: divided into type I (maximum flatness) and type II (minimum distortion), which allows certain fluctuations in the passband in exchange for steeper performance in the transition band.

Elliptical filter (Cauer filter): It is also possible to obtain a very steep transition band, but at the cost of ripples in the passband and stopband. The transfer function of the elliptic filter is relatively complex and is usually represented by zero-pole pairs.

The design of high-order low-pass filters generally uses filter design theories in analog electronics technology, such as Butterworth design, Chebyshev design or elliptical filter design methods. Combined with specific engineering requirements, the first-order or The parameters of the second-order filter are then cascaded or connected in parallel to obtain the final high-order filter structure.

In the field of digital signal processing, the above continuous time domain filters can be converted into discrete time domain digital filters through digital conversion of analog filters, such as bilinear transformation method or impulse response invariant method.