Design of ideal low-pass filter based on MATLAB

For different filters, the signal strength of each frequency is different. When used in audio applications, it is sometimes referred to as a high frequency shear filter, or a treble cancellation filter. The low-pass filter concept has many different forms, including electronic circuits (such as the hass filter used in audio equipment), digital algorithms for smoothing data, acoustic barriers (acousTIc barriers), image blurring, etc. A smooth form of the signal is provided by eliminating short-term fluctuations and preserving long-term trends.

The role of the low-pass filter in signal processing is equivalent to that of other fields such as the moving average in the financial field; there are many types of low-pass filters, the most common of which is the Butterworth filter and the cut. Bechev filter.

Digital filter design flow chart

Digital filter design steps

The design steps of the digital filter: firstly design the transitional analog filter according to the technical specifications of the digital filter to obtain the system function Ha(s), and then convert Ha(s) into a certain method (the bilinear transformation method in this experiment) The system function H(z) of the digital filter. Specifically:

(1) Determine the technical specifications of Butterworth digital low-pass filter: passband boundary frequency ωp, stopband cutoff frequency ωs, passband maximum attenuation аp, stopband minimum attenuation аs.

(2) Convert the technical specifications of the digital filter into the technical specifications of the analog filter. This refers to the transformation of ωp and ωs while аp and аs remain unchanged. This problem uses the bilinear transformation method, and its conversion formula is:

(3) Find the order of the filter according to the technical specifications Ωp, Ωs, ωp, and ωs using the following formula.

(4) The normalized pole kp and the normalized low-pass prototype system function Ga(p) are obtained from the following table according to N.

(5) Normalize Ga(p), Substituting Ga(p) gives the actual filter system function:

Here Ωc is the 3dB cutoff frequency.

(6) The analog filter Ha(s) is converted into a digital low pass filter system function H(z) by a bilinear transformation method. The conversion formula is

Implementing Butterworth low-pass digital filter with matlab

The Matlab program is as follows:

Fs=500;

t=0:1/fs:1;

x=sin(2*pi*20*t)+2*sin(2*pi*100*t)+5*sin(2*pi*200*t);

Wp=2*30/fs; ws=2*60/fs;

Rp=1; As=30;

Subplot(311);

Plot(t,x);

TItle ('input signal');

[N, wc]=buttord(wp,ws,Rp,As);

[B,A]=butter(N,wc);

[H, W] = freqz (B, A);

y=filter(B,A,x);

Subplot(312);

Plot(W,abs(H));

TItle ('low pass filter');

Subplot(313);

Plot(t,y)

TItle('30Hz');

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