Low pass active filters are covered in this presentation. For those people who have never worked with filters, a short overview of terminology relating to filter characteristics and terminology precedes the actual discussion. The Sallen-Key topology and Butterworth low pass filter are discussed, operating parameters are calculated and then verified via experimentation.

**Catalog**

Article Core | Low Pass Filter |

Introduction | Terminology Working Principle |

Low-pass Filtering | |

Low Pass Filter | |

Low Pass Filtering Circuit | Introduction Electrical Diagrams |

Cutoff Frequency of Low Pass Filter | |

Filtering Circuit | 1. Passive Filter Circuit 2. Active Filter Circuit |

Low Pass Filter Identification Methods | |

Functions of Low-pass Filter | Summary Example Expression |

Low-pass Filter Design | FIR Filter Design IIR Filter Design Comparison of IIR and FIR |

**Introduction**

Low-pass filter is a kind of filtering devices. Specifically, a low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter design. Its rule is that the low-frequency signal can pass normally, but the high-frequency signal exceeding the set threshold is blocked and weakened. However, the magnitude of the block and attenuation will vary depending on the frequency and the different filtering procedures (in other words, different filtering purposes). In addition, it is sometimes called high frequency removal filtering or highest removal filtration in some ways.

**Working Principle**

Use the principle of capacitors to pass the high frequency and block the low frequency, passing through the low frequency and blocking high frequency with inductance effect. For the high frequency that needs to be cut off, it is blocked by the method of capacitance absorption and inductance obstruction; for the required low frequency, it is made to pass the high resistance with capacitors and pass through the low resistance with inductor.

**Low****-****pass Filtering**

Low-pass filtering can be simply thought of: setting a frequency point, when the signal frequency is higher than this frequency, it cannot pass. In the digital signal, this frequency point is also called the cutoff frequency. When the frequency domain is higher than the cutoff frequency, then all the assigned to 0. Because the low frequency signal is passed all through this process, it is called low pass filtering.

The concept of low-pass filtering exists in a variety of different fields, such as electronic circuits, data smoothing, acoustic blocking, image blurring, and so on. For example, in the field of digital image processing, low-pass filtering can smooth and de-noise the image.

**Low Pass Filter**

For different filters, the attenuation of the signal at each frequency is different. When used in audio applications, it is sometimes referred to as a high-cut filter, or a treble-cut 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, image blurring, etc.) And eliminating short-term fluctuations and retaining long-term trends provides a smooth form of signal.

The role of the low-pass filter in signal processing is equivalent to the moving average in the financial field. There are many types of low-pass filters, among them, the most common ones are Butterworth filters and Chebyshev filters.

**Low ****P****ass ****F****ilter****ing**** ****C****ircuit**

The low-pass filter is a circuit that allows low-frequency signals to pass through without passing medium and high-frequency signals in the car amplifier. Its function is to filter out the mid-range and high-pitched components in the audio signal, and enhance the bass component to drive the woofer of the speaker. Since the car power amplifying part is a full-band power amplifier, the class AB amplification design is usually adopted, and the power loss is relatively large. So filtering out the signal of the low frequency band pushes the middle and high frequency speaker to be the best choice for saving power and ensuring sound quality.

An ideal low-pass filter that allows low-frequency signals to pass through the filter without loss. When the signal frequency exceeds the cutoff frequency, the signal decays to infinity.

The first-order active low-pass filter circuit is the simplest filter circuit, and it is also the smallest unit that constitutes the second-order or the high-order active low-pass filter circuit.

**Electrical diagrams: common low-pass filtering circuit**

The first-order filter reduces the signal intensity by half (about - 6dB) when the frequency doubles (increasing octave). The first-order filter amplitude Bode plot is a horizontal line below the cutoff frequency and a diagonal line above the cutoff frequency. There is also a "knee curve" at the boundary between the two straight lines.

Second-order filter can play a higher role in reducing high-frequency signals. The Bode plot of this type of filter is similar to that of the first-order filter, but its roll-off rate is faster. For example, a second-order Butterworth filter (which is a critical attenuation RLC circuit without spikes) reduces the signal strength to an initial quarter (12 dB per frequency doubling) when the frequency doubles. The initial roll-off speed of other second-order filters may depend on their Q-factor, but the final speed is - 12dB per frequency doubling.

Third-order and higher-order filters are similar. In a word, the roll-off rate of the last n-order filter is 6ndB per octave.

**C****utoff**** Frequency of Low Pass Filter**

In electronics, the cutoff frequency is the frequency at which the output signal power of a circuit (eg, wire, amplifier, electronic filter) exceeds or falls below the conducted frequency.

When the signal frequency is lower than the cutoff frequency, the signal passes; when the signal frequency is higher than the cutoff frequency, the signal output is greatly attenuated. This cutoff frequency is defined as the boundary between the passband and the stopband.

cutoff frequency calculation formula：

**Filtering Circuit**

Commonly used filter circuits are passive filtering and active filtering. If the filtering circuit component consists of only passive components (resistors, capacitors, inductors), it is called a passive filter circuit. The main forms of passive filtering are capacitor filter, inductance filter and complex filter (including inverted L-type, LC filter, LCπ-type filter and RCπ-type filter, etc.). If the filter circuit is composed of not only passive components but also active components (bipolar, monopole, and integrated op amps), it is called an active filter circuit. The main form of active filtering is active RC filtering, also known as electronic filters.

1. Passive filter circuit

The passive filter circuit has a simple structure and is easy to design, but its passband amplification factor and cutoff frequency vary with load, therefore, it is not suitable for applications where signal processing is required to be high. Passive filter circuits are commonly used in power circuits, such as DC power supply rectification, or LC (inductor, capacitor) circuit filtering for high current loads.

2. Active filter circuit

The load of the active filter circuit does not affect the filter characteristics, so it is often used in applications where high signal processing is required. The active filter circuit is generally composed of an RC network and an integrated operational amplifier, and must be used with a suitable DC power supply, and can also be amplified. But the composition and design of the circuit is also more complicated. Active filter circuits are not suitable for high voltage and high current applications, and are only suitable for signal processing.

According to the characteristics of the filter, the amplitude-frequency characteristic of its voltage amplification can accurately describe whether the circuit belongs to a low-pass, high-pass, band-pass or band-stop filter, so if the passband and stopband can be qualitatively analyzed in the frequency band, the type of filter can be determined.

**Low Pass Filter Identification Methods**

If the signal frequency tends to zero, there is a certain voltage amplification factor, and when the signal frequency tends to infinity, the voltage amplification factor tends to zero, which is a low-pass filter; on the contrary, if the signal frequency tends to infinity, there is a certain voltage amplification factor. And when the signal frequency tends to zero, the voltage amplification factor tends to zero, which is a high-pass filter; if the signal frequency tends to zero and the infinity, the voltage amplification factor tends to zero, then it is a band-pass filter; otherwise, if the signal frequency when the voltage is zero and infinity, the voltage amplification has the same certain value, and the voltage amplification factor tends to zero in a certain frequency range, which is a band-stop filter.

**Functions of Low-pass Filter**

1. A low pass filter is a component or combination of electronic circuits that allows signals that pass below the cutoff frequency to pass, while signals that are above the cutoff frequency cannot pass.

2. When the low-pass filter is used in the speaker's splitter, the bass in the signal can be separated and install a separate amplifier, allowing the subwoofer to work.

3. In radio transmitters, a low pass filter can be used to block harmonic emissions that may cause interference with other communications.

4. In network transmission, the DSL splitter uses a low-pass and high-pass filter to separate the DSL and POTS signals that share the twisted pair.

5. In the electronic analog music device, the music analog synthesizer, the low-pass filter works.

**Example：**

PWM simulates DAC through RC low-pass filter

When the circuit requires a DAC and the microcontroller does not have a DAC peripheral, PWM can be used to simulate the DAC function through an RC low-pass filter.

When using a low-pass filter to simulate a DAC, the PWM frequency should be much larger than the cutoff frequency of the RC low-pass filter circuit fc = 1/2 πRC (more than 10 times). The output voltage is Vout=Vcc*Duty.

*Note:*

1. Under normal circumstances, when the capacitance C is small and the resistance R is large, the output voltage loss is small and the ripple is large; when the capacitance C is large and the resistance R is small, the output voltage loss is large and the ripple is small. Therefore, in order to obtain accurate DA conversion with higher linearity, a smaller capacitor is generally used and an electrolytic capacitor is not used as much as possible.

2. In order to improve the driving capability of the output, a high-performance voltage follower is usually added after the RC low-pass filter, and a filter electrolytic capacitor is added to the output of the follower to further increase the output voltage smoothly. However, it should be noted that the output voltage at this time may contain more AC harmonic components. If not handled properly, the voltage follower may damaged. The solution is to use a small tantalum capacitor. Moreover, the order of the capacitors here must be that the electrolytic capacitor is in front and the tantalum capacitor is in the back.

3. If the output voltage accuracy and linearity requirements are not high, but the ripple requirements are high, or when the voltage is relatively fixed, a larger filter combination can be used. Although the DC loss of large capacitors is large, we can achieve the required output voltage by adjusting the PWM duty cycle, or achieve accurate fixed voltage output through the feedback of the primary AD conversion. It is only need to add a voltage follower to facilitate the use of the post-stage acquisition circuit, and the AD acquisition point is placed at the follower output.

4. If the primary RC low-pass filter does not work, a multi-stage RC low-pass filter can be used to further improve output smoothness.

**Low-pass Filter Design**

1) FIR filter design

The design of the FIR filter is relatively simple, that is, to design a digital filter to close to an ideal low-pass filter. Usually this ideal low-pass filter is a rectangular window in the frequency domain. According to the Fourier transform, we can see that this function is a sampling function in the time domain, in general, the expression of this function is: sa(n)=sin(n∩)/n∏.

This sampling sequence is infinite and cannot be calculated by the computer. Therefore, we need to truncate this sampling function. That is to add a window function, in other words, the legendary window. It means that the time domain sampling sequence is multiplied by a window function, and the infinite time domain sampling sequence is cut into a finite sequence value. However, with the window function, the frequency domain of the sample sequence is also affected: the frequency domain at this time is not an ideal rectangular window, but a low-pass filter with a transition band and a ripple. Generally, depending on the added window function, after the window is sampled, the stopband attenuation of the low-pass filter obtained in the frequency domain is also different. Usually we choose a suitable window function based on this stop-band attenuation, such as Rectangular window, Hanning window, Hamming window, BLACKMAN window, Caesar window and so on. After selecting a specific window function, the required order and the expression of this window function are calculated according to the parameters of the designed filter, then use this window function to multiply the sample sequence to get the impulse response of the actual filter.

2) IIR filter design ( bilinear transform method )

The design concept of IIR is as follows: the transfer function of an analog filter is determined according to the parameters of the filter to be designed, and then the design of the digital filter is performed by bilinear transformation or impulse response invariance according to the transfer function. Its design is more complicated, and the complexity lies in the determination of its analog filter transfer function H(s), but this can be done by software. Its specific implementation steps: First you need to determine what kind of filter you need, Butterworth type, Chebyshev type, or any other type of filter. When you select a model, you can determine the order and transfer function expression based on the design parameters and the calculation formula for this filter. Usually there is a problem of pre-distortion in this process (this is only a problem that needs to be paid attention to by the bilinear transform method, while the impulse response invariant method does not have such a problem). After determining the H(S), the difference equation of its digital domain can be obtained by bilinear transformation.

3) Comparison of IIR and FIR

In terms of performance, the IIR filter transfer function includes two sets of adjustable factors, zero and pole, and the only limit on the pole is in the unit circle. Therefore, high selectivity can be obtained with a lower order number, less memory cells, less computation, and higher efficiency. But this high efficiency comes at the expense of phase nonlinearity. The better the selectivity, the more severe the phase nonlinearity. The pole of the FIR filter transfer function is fixed at the origin and is immovable. It can only change its performance by changing the zero position. Therefore, to achieve high selectivity, a higher order must be used; for the same filter design index, the order required by the FIR filter may be 5-10 times higher than the IIR filter. As a result, the cost is higher, and the signal is higher. The delay is also large; if the linear phase is required, the IIR filter must be added to the all-pass network for phase correction, which also greatly increases the order and complexity of the filter, and the FIR filter can get a strict linear phase.

Structurally, the IIR filter must use a recursive structure to configure the poles and ensure that the poles are in the unit circle. Due to the finite word length effect, the coefficients are rounded during the operation, causing a pole offset. This situation sometimes causes stability problems and even parasitic oscillations. On the contrary, as long as the FIR filter adopts a non-recursive structure, there is no stability problem in both theoretical and practical finite precision operations, and thus the frequency error is also small. In addition, the FIR filter can use the fast Fourier transform algorithm, and the operation speed can be much faster under the same order.

In addition, it should be noted that although the design of the IIR filter is simple, it is mainly used to design filters with segmentation constant characteristics, such as low-pass, high-pass, band-pass, and band-stop, which often cannot be separated from the form of analog filter. The FIR filter is much more flexible, especially because it is easy to adapt to certain special applications, such as forming a digital differentiator or a Hiller converter, so it has greater adaptability and a wide range of applications.

From the mentioned comparison above, we can see that the IIR and FIR filters have their own strengths, so in practical applications, we should choose from many aspects. From the point of view of the use requirements, in the case of insensitivity to phase requirements, such as language communication, it is more appropriate to use IIR, so that it can fully utilize its economical and efficient features; for image signal processing, data transmission and other systems that carry information by waveform, then the linear phase is required to be higher. If possible, it is better to use a FIR filter. Of course, more factors may be considered in practical applications.

Regardless of IIR and FIR, the higher the order, the larger the signal delay. And meanwhile, in the IIR filter, the higher the order, the higher the accuracy requirement of the coefficient, otherwise it is easy to cause the error of the finite word length to move the pole to the unit outside the field. Therefore, it is considered comprehensively in order selection.

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