Design of Distortionless Analog Filter Based on Matlab

O Introduction

The design of analog filters generally includes two aspects: determining the filter transfer function H(s) based on the technical indicators, i.e. the amplitude-frequency characteristics of the filter; and designing the actual network to implement this transfer function. The key to designing the filter H(s) is to find an approximation function, and there are currently many approximation functions. However, no matter which approximation function is used, very tedious calculations are required, and a table lookup is required based on the calculation results.

Matlab is a simple and efficient high-level language. It is a rich and powerful analysis tool. Its application covers almost all scientific and engineering computing fields. Matlab provides a wealth of functions for analog filter design. Through programming, it is easy to realize low-pass, high-pass, band-pass, and band-stop filters, and can draw the amplitude-frequency and phase-frequency characteristic curves of the filter, which greatly simplifies the design of analog filters. This paper introduces the design and implementation of a distortion-free analog filter using Matlab, and gives the simulation results of the amplitude-frequency and phase-frequency characteristics and the eye diagram after the signal passes through the distortion-free filter.

1 Distortionless filter design based on Matlab

Design requirements: A digital baseband signal with a frequency of 8 kHz is converted into an analog signal through digital/analog (D/A) conversion. The analog signal is input into the distortion-free filter, and the attenuation is required to be ~6 dB at 8 kHz; after 16 kHz (cut-off frequency), the attenuation reaches more than -60 dB. The purpose of this is to suppress harmonic interference and reduce the fluctuation of the waveform after the cut-off frequency after the analog signal passes through the distortion-free filter.

The distortion-free filter consists of three parts, namely, low-pass filter, band-stop filter and phase shift circuit. The low-pass filter is used to pass low-frequency signals within 8 kHz; the band-stop filter is used to suppress 16 kHz signals. After being cascaded with the previous low-pass filter, it can make the attenuation of the signal after 16 kHz reach more than 60 dB, effectively suppressing the interference of each harmonic and reducing the fluctuation of the waveform after 16 kHz; the phase shift circuit is used to compensate for the phase offset of the previous two circuits, so that the phase characteristic of the entire distortion-free filter becomes a straight line.

1.1 Low-pass filter design

The low-pass filter uses a Butterworth filter because it has a relatively flat amplitude-frequency characteristic and a good linear phase-frequency characteristic, and is often used as the primary of the filter. In Matlab, the [b, a] = butter(n, wn, 's') statement is used to implement the design of the Butterworth filter. Among them, [6, a] are the coefficients of the filter transfer function polynomial; n is the order of the filter; wn is the angular frequency at 3 dB; 's' represents the analog filter.

In this design, an 8th order Butterworth filter is used, and the frequency at -3 dB is 8 kHz. The main procedures are as follows:

Finally, the amplitude-frequency and phase-frequency characteristics of the low-pass filter are plotted as shown in Figure 1(a). The actual circuit uses the Max291 chip to implement the Butterworth filter, which is equivalent to an 8th-order Butterworth filter, as shown in Figure 1(b).

1.2 Band-stop filter design

The function of a band-stop filter (also known as a notch filter) is to suppress a certain frequency signal and pass other frequency signals. In Matlab, the statement [b, a] = butter (n, Wn, 'stop', 's') is used to implement a band-stop filter, where [b, a] are the coefficients of the filter transfer function polynomial; n is the order of the filter; wn = [fl*2*pi fu*2*pi] are the angular frequencies at the lower end of the stop band - 3 dB and the upper end - 3 dB; 's' represents the analog filter. The main program is as follows:

Similar to the amplitude-frequency and phase-frequency characteristics of the low-pass filter, the amplitude-frequency and phase-frequency characteristics of the band-stop filter are shown in Figure 2(a). In this design, the actual circuit uses a VCVS type second-order band-stop filter as shown in Figure 2(b). The values ​​of resistors R1, R2, R3, and capacitor C1 need to be determined, and their calculation formulas can be obtained through Matlab programming, and finally the corresponding nominal values ​​are selected according to the calculation results.

1.3 Cascade of low-pass filter and band-stop filter

Cascade the low-pass and band-stop filters designed previously so that the amplitude-frequency characteristics of the circuit after cascading can meet the design requirements, that is, the attenuation at 8 kHz is -6 dB; the attenuation after 16 kHz (cut-off frequency) reaches more than -60 dB. If the requirements cannot be met, the Q value in the band-stop filter is continuously modified according to the simulation results until the design requirements are met. In Matlab, the conv function is used to realize the cascade of the two circuits. The main program is as follows:

b3 and a3 are the polynomial coefficients of the transfer function after the low-pass and band-stop filters are cascaded. Similar to the amplitude-frequency and phase-frequency characteristics of the low-pass filter, the amplitude-frequency and phase-frequency characteristics of the cascade circuit are shown in Figure 3. It can be seen that when Q is 0.7, the amplitude-frequency characteristic meets the design requirements, but the phase-frequency characteristic is not a straight line within 8000-Hz. In order to prevent the output signal from being distorted, a phase shift filter must be added to make the total phase shift characteristic a straight line.

1.4 Phase-shift filter design and phase-frequency characteristics of distortion-free filters

In this design, a first-order inverting gain all-pass filter circuit is used, and its transfer function is:

The circuit diagram is shown in Figure 4. The low-pass, band-stop, and phase-shift circuits are cascaded to form a distortion-free filter. In Matlab, the phase-frequency characteristic curve of the phase-shift and distortion-free filter can be obtained by programming.

The main procedures are as follows: