The basic concept of filters

Basic concepts

A filter, as the name suggests, is a device that filters waves. "Wave" is a very broad physical concept, and in the field of electronic technology, "wave" is narrowly limited to describe the process of the value of various physical quantities that fluctuates with time. This process is converted into a time function of voltage or current through the action of various sensors, which is called the time waveform of various physical quantities, or the signal. Because the independent variable time' is continuous, it is called a continuous-time signal, and it is customary to call it an analog signal. With the emergence and rapid development of digital electronic computer technology (generally referred to as computer), in order to facilitate the processing of signals by computers, a complete theory and method of transforming continuous-time signals into discrete-time signals under the guidance of the sampling theorem have been produced. That is to say, the original signal can be expressed without losing any information only with the sample value of the original analog signal on a series of discrete time coordinate points, and since the concepts of wave, waveform, and signal express the changes of various physical quantities in the objective world, they are naturally the carriers of various information on which modern society depends. Information needs to be disseminated, relying on the transmission of waveform signals. The signal may be distorted in every link of its generation, conversion and transmission due to the existence of environment and interference, and even in quite a few cases, this distortion is so serious that the signal and the information it carries are deeply buried in the noise.

Classification of filters

A filter is a device or circuit that can process a signal, enhance useful components, and suppress interference components. Depending on the field of application, frequency range, characteristics, etc., filters can be divided into the following main types:

1. Classification by frequency range

1. Low-pass Filter: Allows signals below a certain cut-off frequency to pass through, blocking signals above that frequency.
2. High-pass Filter: Allows signals above a certain cut-off frequency to pass through, blocking signals below that frequency.
3. Band-pass Filter: Allows signals in a certain frequency range to pass through and blocks signals outside that range.
4. Band-stop Filter: Blocks the passage of signals in a certain frequency range and allows signals outside of that range to pass through, also known as notch filters.
5. All-pass Filter: Allows signals of all frequencies to pass through, but changes the phase of the signal.

2. Classification according to the mode of implementation

1. Analog Filter: A filter that is implemented by simulating electronic components such as resistors, capacitors, and inductors.

• Passive Filter: Consists only of passive components such as resistors, capacitors, and inductors, and does not require an external power supply.
• Active Filter: In addition to passive components, active components such as amplifiers are also included, and an external power supply is required.

Digital Filter: A filter implemented by a digital signal processing algorithm (e.g., FIR, IIR), typically running on a digital processor.

3. Classification according to application fields

1. Audio Filter: Used for audio signal processing, such as bass and treble adjustment in the sound system.
2. RF Filter: Used in wireless communication systems to process RF signals.
3. Image Filter: Used for image processing, such as smoothing and sharpening images.

Fourth, according to the filter characteristics

1. 线性滤波器（Linear Filter）：输出信号是输入信号的线性函数。
2. Nonlinear Filter: The output signal is not a linear function of the input signal, and is commonly used in image processing.

5. Classification according to circuit structure

1. Butterworth Filter: Has a flat frequency response curve with no ripples.

   Butterworth (Flattest Response)The Butterworth response maximizes the passband flatness of the filter. The response is very flat, close to the DC signal, and then slowly decays to a cut-off frequency of -3dB, eventually approaching an attenuation rate of -20ndB/decade, where n is the order of the filter. Butterworth filters are particularly suitable for low frequency applications, where they are important for maintaining the flatness of the gain.

2. Chebyshev Filter: There are ripples in the passband or stop band, but the transition band is narrow.

   In some applications, the most important factor is the speed at which the filter cuts off unwanted signals. If you can accept that the passband has some ripple, you can get faster attenuation than with the Butterworth filter. Appendix A contains a table of parameters required to design up to 8th order filters with Butterworth, Bessel, and Chebeshev responses. Two tables are used for Chebyshev responses: one for 0.1dB maximum passband ripple;

3. 椭圆滤波器（Elliptic Filter）：通带和阻带内都有波纹，但过渡带最窄。

4. Bessel Filter: It has a linear phase response and is suitable for applications where a waveform needs to be maintained.

   In addition to varying the amplitude of the frequency-dependent input signal, the filter introduces a delay to it. The delay distorts the non-sinusoidal signal from frequency-based phase shifts. Just as the Butterworth response maximizes the flatness of the amplitude using the passband, the Bessel response minimizes the phase nonlinearity of the passband.

The above are the main classifications of filters, and each filter has its own specific application scenarios and characteristics.

滤波器的主要参数（Definitions）

中心频率（Center Frequency）：

The frequency of the filter passband f0 is generally taken as f0=(f1+f2)/2, and f1 and f2 are the side frequencies of the bandpass or bandstop filter that are relative to the left and right of the filter falling by 1dB or 3dB. Narrowband filters often calculate the passband bandwidth at the center frequency of the minimum insertion loss.

Cutoff frequency: refers to the frequency on the right side of the passband of the low-pass filter and the left-hand side of the passband of the high-pass filter. It is usually defined as a standard of 1dB or 3dB relative loss points. The reference datum for relative loss is as follows: low-pass is based on the insertion loss at DC, and high-pass is based on the insertion loss at sufficient high-pass band frequencies without parasitic stop bands.

通带带宽（BWxdB）：指需要通过的频谱宽度，BWxdB=（f2-f1）。 f1、f2为以中心频率f0处插入损 耗为基准，下降X（dB）处对应的左、右边频点。 通常用X=3、1、0.5 即BW3dB、BW1dB、BW0.5dB 表征滤波器通带带宽参数。 分数带宽（fractional bandwidth）=BW3dB/f0×100[%]，也常用来表征滤波器通带带宽。

Insertion loss: The attenuation of the original signal in the circuit due to the introduction of the filter is characterized by the loss at the center or cut-off frequency, such as the full-band interpolation loss needs to be emphasized.

Ripple: The peak-to-peak value of the interpolation loss fluctuating with frequency on the basis of the mean loss curve within the 1dB or 3dB bandwidth (cut-off frequency).

Passband Riplpe: The amount by which the insertion loss in the passband varies with frequency. The in-band fluctuation within the 1dB bandwidth is 1dB.

In-band Standing Wave Ratio (VSWR): An important measure of whether a filter's in-band signal is well matched for transmission. Ideal match VSWR = 1:1, mismatch VSWR >1. For an actual filter, the bandwidth of VSWR<1.5:1 is generally less than BW3dB, and its proportion of BW3dB is related to the filter order and insertion loss.

Return Loss: The number of decibels (dB) of the ratio of the input power of the port signal to the reflected power, which is also equal to |20Log10ρ|, where ρ is the voltage reflection coefficient. The return loss is infinity when the input power is fully absorbed by the port.

Stopband rejection: An important indicator to measure the performance of filter selection. The higher the index, the better the rejection of out-of-band interference signals. There are usually two formulations: one is how many dB dB is required to be suppressed for a given out-of-band frequency fs, which is calculated as the attenuation at fs As-IL; The other is to propose an index to characterize the closeness of the amplitude-frequency response of the filter to the ideal rectangle-rectangle coefficient (KxdB>1), KxdB=BWxdB/BW3dB, (X can be 40dB, 30dB, 20dB, etc.). The higher the order of the filter, the higher the rectangularity – i.e. the closer K is to the ideal value of 1, the more difficult it is, of course.

Delay (Td): Refers to the time it takes for the signal to pass through the filter, numerically the derivative of the diagonal frequency of the transmission phase function, i.e., Td=df/dv.

In-band phase linearity: This metric characterizes the magnitude of the phase distortion introduced by the filter to the transmitted signal within the passband. The filter designed according to the linear phase response function has good phase linearity.

Transfer functions

The transfer function of a filter is a mathematical expression that describes the relationship between the input signal and the output signal, and is commonly used to analyze and design filters. Transfer functions are usually expressed as complex variables ss (Laplace transform domain) or zz (Z transform domain). Here are the definitions of transfer functions and the transfer functions of some common types of filters: