Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 10, No. 3, 2024 

 

106 

Design	and	Simulation	of	Butterworth	Low‐pass	Filter	
Hao Feng, Linjie Sun 

Taiyuan University of Technology, China 

 

Abstract: Filters are required to filter the signal to improve the signal-to-noise ratio. It is known that the maximum frequency 
of the human ear is about 20 kHz, and filtering out the signal components above 20 kHz can greatly improve the audio signal 
quality. Therefore, the design requires the cut-off frequency of 20 kHz. In this design, the normalized transfer function of the 
fourth-order Butterworth filter is obtained by using the normalization process, and the denormalization process is carried out 
according to the cutoff frequency of 20kHz, the transfer function of the fourth-order Butterworth filter is obtained.The idea of 
circuit cascade is applied to derive the parameters of the components, so that a filter that meets the design requirements is 
obtained, and the filter is used in the Multisim and Matlab simulation software to verify the filtering effect. 

Keywords: Filter, cut-off frequency, normalization process, Butterworth. 
 

1. Introduction 
The Butterworth filter, also known as the maximum flat 

filter, is characterized by a high degree of flatness of the 
waveform curve in the passband, small ripple, the waveform 
curve can be kept monotonically decreasing, and the 
amplitude has the same trend as the diagonal frequency. 
Compared with Bessel and Chebyshev filters, Butterworth 
filters have the advantage of balanced characteristics in the 
three aspects of linear phase, decay slope and loading 
characteristics.In practice, Butterworth filters have been 
listed as the first choice, which is widely used in the field of 
communications, motor testing, radar signal processing field, 
high-quality audio and so on. 

In this paper, the specific parameters of the fourth-order 
Butterworth filter are calculated by normalization, according 
to the design requirement of the cutoff frequency of 20kHz, 
and simulated and verified by Multisim, and the fourth-order 
Butterworth filter that can be applied to audio devices is 
designed. 

2. The Butterworth Low-pass Filter 
Design Principle 

Amplitude squared function of the Butterworth low-pass 
filter|𝐻 𝑗𝜔 | can be expressed by the following equation: 

 
|𝐻 𝑗𝜔 |               (1) 

 
Where N represents the order of the filter, and Ω   is the 

cutoff frequency of the filter, the frequency when the 
amplitude decayed by 3𝑑𝐵.Around 𝛺 𝛺 ，the amplitude 
decays rapidly as the Ω increases. The speed of the amplitude 
decrease is related to the order N. The larger N, the flatter the 
pass band, the narrower the transition band, the faster the 
amplitude decrease of the transition band and the resistance 
band, and the smaller the error between the total frequency 
sound characteristics and the ideal low-pass filter. 

With 𝑠 instead of 𝑗Ω,write the amplitude square function 
|𝐻 𝑗Ω |  as a function about 𝑠[1] : 

 

 𝐻 𝑠 𝐻 𝑠           (2) 

 
The amplitude square function has 2N poles,and the 

pole𝑠  is represented by the following equation: 
 

 𝑠 𝛺 𝑒                (3) 

 
In the equation 𝑘 0,1,2, ⋯ ,2𝑁 1。The distribution of 

2N poles is equally spaced on a circle of radius 𝛺 .In order 
to ensure that the filter is causally stable, the N poles in the 
left half plane of the  𝑠  in 2N poles are taken to form 
𝐻 𝑠 ，and the remaining N poles in the right half plane of 
the s form 𝐻 𝑠 . The expression of 𝐻 𝑠  is: 

 

 𝐻 𝑠
∏

            (4) 

 
Due to the different boundary frequency and filter 

amplitude frequency characteristics corresponding to 
different technical indicators, the frequency is normalized to 
unify the design formula and chart. The Butterworth filter is 
normalized by the 3𝑑𝐵 cutoff frequency, and the normalized 
system function is: 

 

 𝐺
∏            (5) 

 

Order 𝑝  , 𝜆   , 𝜆  is called the normalized 

frequency，𝑝 is called the normalized complex variable, and 
now we get the system function of the normalized 
Butterworth low-pass filter: 

 

 𝐺 𝑝
∏            (6) 

 
Write in a polynomial form as: 
 

 𝐺 𝑝
⋯

 (7) 

 
In this funtion 𝑝  is the pole after normalization: 



 

107 

 𝑝 𝑒               (8) 

 

Here, the polynomial coefficient table and the polynomial 
factorization table corresponding to the Butterworth filter 
order N are given as shown below[2]: 

 
Table 1. Table of polynomial coefficients 

N 𝑏  𝑏  𝑏  𝑏  𝑏  𝑏  𝑏  𝑏  𝑏  

1 1.0000         
2 1.0000 1.4142        
3 1.0000 2.0000 2.0000       
4 1.0000 2.6131 3.4142 2.6131      
5 1.0000 3.2361 5.2361 5.2361 3.2361     
6 1.0000 3.8637 7.4641 9.1416 7.4641 3.8637    
7 1.0000 4.4940 10.0978 14.5918 10.0978 4.4940    
8 1.0000 5.1258 13.1371 21.8462 25.6884 21.8642 13.1371 5.1258  
9 1.0000 5.7588 16.5817 31.1634 41.9864 41.9864 31.1634 16.5817 5.7588 

 
Table 2. Table of polynomial factorization 

N polynomial form 
1 𝑝 1  
2 𝑝 1.414𝑝 1  
3 𝑝 𝑝 1 𝑝 1  
4 𝑝 0.7654𝑝 1 𝑝 1.8478𝑝 1  
5 𝑝 0.6180𝑝 1 𝑝 1.6180𝑝 1 𝑝 1  
6 𝑝 0.5176𝑝 1 𝑝 1.4142𝑝 1 𝑝 1.9319𝑝 1  
7 𝑝 0.4450𝑝 1 𝑝 1.2470𝑝 1 𝑝 1.8019𝑝 1 𝑝 1  
8 𝑝 0.3902𝑝 1 𝑝 1.1111𝑝 1 𝑝 1.6629𝑝 1 𝑝 1.9616𝑝 1  
9 𝑝 0.3473𝑝 1 𝑝 𝑝 1 𝑝 1.5321𝑝 1 𝑝 1.8974𝑝 1 𝑝 1  

 
In this way, check the system function of the Butterworth 

low-pass filter normalized in Table 2 according to the filter 
order N, and then according to: 

 
 𝑠 𝛺 𝑝                  (9) 

 
Realize the normalization, get 𝐻 𝑠 ,and finally calculate 

each component parameters according to the circuit structure. 

3. Design Process of Butterworth Low-
pass Filter 

3.1. The process of deriving the expression for 
parameter calculation 

Figure 1 shows the circuit structure of a typical second-
order Butterworth filter, and its transfer function is: 

 

 𝐻 𝑠           (10) 

 

 
Figure 1. Circuit diagram of a second-order Butterworth filter 

 
According to the derivation: 
 

  𝐴𝛺               (11) 

 

 𝐵𝛺             (12) 

 

 𝐾                 (13) 

 
Sorted out: 

                                                  

 𝑅                (14) 

 

 𝑅          (15) 

 

 𝑅             (16) 

 
𝐶   and 𝐶  are arbitrary values. Each resistor is measured 

in Ω,The unit of each capacitor is F, therefore, when K,A,B 
and 𝛺  are given, First, arbitrarily select the values of the 𝐶  



 

108 

and 𝐶 , The values of each resistance of the circuit are then 
calculated. There is a rule of thumb for the capacitance 
selection of capacitors: The 𝐶   selected is an approximate 

μF nominal value, which 𝐶  satisfied: 

 

 𝐶                (17) 

 

3.2. Filter order selection 
Figure 2 shows the amplitude-frequency characteristics of 

different order filters: 
 

N=1

N=2

N=3

N=4

N=5

N=6

N=7

N=8

N=9

N=10

Am
pl
it

ud
e/

dB

Frequency/Hz  
Figure 2. Amplitude-frequency characteristics of filters of different orders 

 
When the order of the filter is greater than the fourth order, 

the stopband descent rate does not change obviously, and the 
larger the order, the more complex the design[3]，All things 
considered, the design adopts a fourth-order filter. 

3.3. Filter parameter calculations 
Looking up the Table 2, the normalized transfer function is: 
 

 𝐺 𝑝
. .

       (18) 

 
Its four poles are released as shown in Figure 3: 

 

Real part

i
ma
g
i
na

ry
 p

a
r
t

 
Figure 3. Zero pole plot 

 
The four poles of the transfer function are all in the left half 

plane of the imaginary axis, and the system is stable[4]. 
𝐺 𝑝  is the cascade of Eq. (19) and (20): 
 

 𝐺 𝑝
.

          (19) 

 

 𝐺 𝑝
.

           (20) 

 
According to Eq. (9), the 𝐺 𝑝   and 𝐺 𝑝   are 

denormalized to obtain:    
                                                                                                       

 𝐻 𝑠
.

. .
        (21) 

 

 𝐻 𝑠
.

. .
       (22) 

 
The first-stage filter parameters are calculated according to 

Eq. (10) and Eq. (21).：𝐾 1,𝐴 0.77,𝐶 1,select 𝐶
𝜇F 500𝑝𝐹 , 𝐶 50pF , 𝑅   83𝑘𝛺 , 𝑅

83𝑘𝛺，𝑅 41𝑘𝛺.  
Calculate the second-stage filter parameters：𝐾 1，𝐴

1.85，𝐵 1,select 𝐶 𝜇F 500𝑝𝐹，𝐶 110pF, 

𝑅 34.4𝑘𝛺, 𝑅 34.4𝑘𝛺，𝑅 17.2𝑘𝛺. 
 
 



 

109 

 

 
Figure 4. Multisim Simulation circuit diagram 

 

4. Simulation and Analysis of Fourth-
order Butterworth filter 

The amplitude-frequency characteristics of the filter 
obtained by Multisim simulation are as follows: 

 

 
Figure 5. Multisim simulation amplitude-frequency 

characteristic plot 
 

From the amplitude and frequency characteristics obtained 
by the simulation, it can be seen that the attenuation is -
2.083dB at the cut-off frequency of 20kHz, which meets the 
design requirements. 

Five voltage sources with different frequencies are used to 
simulate sound signals with different frequencies in series, as 
shown in Figure 6, the filtering experiment is carried out, and 
the frequencies of the five voltage sources are 0.5kHz, 1kHz, 
3kHz, 25kHz, and 30kHz respectively. 

 

 
Figure 6. Multisim sound signals of different frequencies 

 

The prefiltered signal waveform is shown in Figure 7: 
 

t/s

U/
V

 
Figure 7. Waveform of the signal before Multisim filtering 

 
The filtered signal waveform is shown in Figure 8: 
 

U/
V

t/s  
Figure 8. Waveform of the signal after Multisim filtering 

 
In order to better observe the changes of different 

frequency components before and after filtering, MATLAB is 
used to decompose the output signal to obtain its frequency 
domain characteristics[5]: 

The filter front is shown in Figure 9: 
 

 
Figure 9. Frequency domain characteristic diagram of 

MATLAB filter front 



 

110 

There are components at both 25kHz and 30kHz before 
filtering. 

After filtering, it is shown in Figure 10: 
 

 
Figure 10. Frequency domain characteristic diagram after 

MATLAB filtering 
 

The components at 25kHz and 30kHz are close to 0 after 
filtering, which proves that the filter has good filtering 
characteristics. 

5. Conclusion 
Through the study of the design principle of Butterworth 

low-pass filter, the system function of the normalized 
Butterworth low-pass filter is obtained, the zero pole is 
obtained in Matlab to verify the stability, and the amplitude-
frequency characteristics of Butterworth filter of different 

orders are compared, and the fourth-order Butterworth low-
pass filter is determined to be used in this design, and the 
resistance values and capacitance values in the circuit model 
are deduced by checking the polynomial table of the 
corresponding order, and the circuit model is built in mutisim, 
and the amplitude-frequency characteristics meet the design 
requirements. Five sound signals with different frequencies 
are used to simulate the sound signal in reality, and the signal 
before and after filtering is Fourier decomposed, and the 
performance of the filter is analyzed at the frequency. 

References 
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[2] LIN Kaisi,ZHANG Lu,LIN Kaiwu. Optimal design and 
simulation of Butterworth low-pass filter[J]. Journal of 
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[3] YANG Hui. Design and simulation analysis of Butterworth 
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[4] HUANG Hao,LI Rui. Simulation and application of 
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technology,2023,43(6):92-95.  

[5] Wang Qirui,Wang Guihua,Wang Yongjun. Design and 
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