Frequency Domain Design of Multi-Band FIR Digital Filter

Frequency Domain Design of Multi-Band FIR Digital Filter Time: 15:34:47
Source: modern electronic technology
Author: Zheng jiachun 0

With
With the advent of the information age and the digital world, digital signal processing has become an extremely important discipline and technology field. Digital Signal Processing in communication, voice, image, automatic control, radar, military, aerospace,
Medical and household appliances have been widely used in many fields. In digital signal processing applications, digital filters are very important and have been widely used. The digital filter can be divided
Two types are available: an Infinite Long Impulse Response (IIR) filter and a Finite Long Impulse Response (FIR) filter. The characteristics of the IIR filter are the unit pulse response with an infinite duration; the unit of the FIR Filter
Pulse response can only last for a certain period of time. It is widely used in engineering because it can easily implement linear phase characteristics and is easy to implement. There are many methods to design FIR digital filters, such as window function design.
Method, frequency sampling method, and optimal design method. Most of the existing literature only introduces the design ideas and methods, and does not consider the design of multi-Band FIR filters from the perspective of practical application and implementation.

The filtering operation of the FIR filter. When the input sequence is finite, the FFT rapid convolution is used for computation as long as two FFT operations are performed, and one IFFT operation can complete linear convolution (filtering) computation. For infinite long order
Columns can be converted into finite-length convolution by means of overlapping addition or overlap retention. However, if H (k) can be directly obtained during filter design, one FFT and one IFFT can be used to complete the filter.
It is called the FFT Algorithm of FIR filter. The following describes how to directly calculate H (k) in the frequency domain of an FIR filter.

1. Frequency Domain Design of FIR Filter
1.1 H (k) Formula

According to the design of the FIR filter sampling method in the frequency domain, if the filter type and Order N are determined, the N-point FFT of H (n) can be determined in the frequency domain, expressed by H1 (k. Therefore, there are:

It can be seen that when L is an integer multiple of N, H (k) can be obtained by interpolation H1 (k) to O and proportional amplification ).
(2) assuming that the filter has linear phase characteristics, the samples of H1 (k) available amplitude function and phase function are represented:

The above analysis shows that H (k) can be calculated as long as the Order N and H1 (k) of the filter are determined ). The following describes the methods for determining the performance.
1.2 method for determining the order and H1 (k) of the filter


Before designing a filter, the technical indexes of the filter must be determined according to the actual engineering needs. Digital filters are often used for frequency selection. Therefore, the form of indicators generally gives the amplitude and phase
In addition, for the design of the FIR filter, the phase response indicator usually requires the system to have a linear phase in the pass band. After the technical indicators are determined, a target digital filter model can be created. Confirm Filter
The order of the wave generator and H1 (k ).

The following is an example to describe the calculation process. Suppose we want to design a multi-Band FIR linear phase data filter. The technical indicator is: the period T = O.000 1 for sampling analog signals.
S, the attenuation at the fp = [1 250, 2] Hz is less than 2 dB, and the attenuation at the fr = [1, 1] Hz is greater than 40
DB. Then:
(1) filter technical indicators
The filter's technical indicators are as follows:
Cut-off frequency:

(2) Target Filter Model
The ideal filter model is used as the target filter model to be designed. Select the range model shown in 1 as required.

(3) determine the order of the filter and H1 (k)
According to the index requirements of the transition zone, the Order N = 40 of the filter can be determined from the transition zone of the ideal filter model to 2 π/n. According to the constraints of linear phase, n is an even number, and the amplitude function of the filter requires odd symmetry, that is, h1k satisfies: h1k =-HL (n-k ).
Frequency by border:
The boundary frequency sampling point Kp = [4, 8, 12] can be obtained.
The sample point of the amplitude function is:

(4) Performance Analysis of Design Results
Based on the interpolation formula, the frequency response of the designed filter can be obtained. Here, result 2 is obtained through matlab programming analysis.

As shown in figure 2, the boundary frequency of the filter conforms to the requirements. However, the impedance band attenuation only reaches 16 dB, and therefore must be optimized.
1.3 Optimization Design
1.3.1 simplified Optimization Design Method


The interpolation formula shows that the frequency response of the frequency sampling filter is equal to the value of the sampling point. Therefore, you can modify the sampling value of the boundary frequency point without increasing the order. Based on metric requirements: Pass
The bandwidth attenuation is less than AP = 2 dB, and the bandwidth attenuation is greater than AR = 40 dB. Calculate the value of the boundary sampling points of the pass band and the drag band according to formula (10) and formula (11:

According to formula (10) and formula (11), the AP and Ar are substituted into matlab programming. The corresponding statements are as follows:

Calculation results: HP = 0.794 3, HR = 0.010 o, optimized formula (7), and new design results are obtained. The result shown in Figure 3 is displayed. As shown in figure 3, the boundary frequency of the filter meets the requirements. However, the minimum attenuation of the impedance band has exceeded 20 dB, which has been significantly improved.

L.3.2 adaptive search optimization

The above method is simple but not optimal, and the performance indicators are not high. The following uses the adaptive search algorithm to further optimize the transition point. The adaptive search algorithm adopts the mean square error minimum criterion. For an FIR digital filter with a linear phase, its performance mainly depends on the amplitude function. Therefore, the error function E (ω) is defined:

Formula: HD (ω) is the amplitude function of the designed target filter; H (ω) is the amplitude function of the designed filter, and its expression is:

The mean square error is:

Formula medium: m is the number of sampling points in the frequency domain, the amount should be large, take
Set: Hi as the transition point, and ETA as the search step. You can export the algorithm for adjusting the transition point:

It can be proved that the algorithm converges when o <≈ <2/n.
According to the above algorithm, the transition point is adjusted using Matlab programming, and Result 4 is displayed.

As shown in figure 4, the boundary frequency of the filter conforms to the requirements, and the impedance band attenuation exceeds 40 dB. The filter conforms to the index requirements, but the transition zone increases. After N is doubled, the design result 5 shows that the filter indicators meet the requirements.

2 closed


The fft quick algorithm of the fir filter can be implemented only after one FFT and one IFFT. In the case of a high order, it has a high computational efficiency. The direct design in the frequency domain can be very effective.
Then obtain the frequency-domain coefficient of the required filter. When L is an integer multiple of N, As long as H1 (k) is interpolated to 0, and then proportional amplification is performed, H (k) can be obtained. Therefore, it can be used as L, N Parameter Selected
Exam. In this paper, the boundary frequency point amplitude sampling and adaptive search algorithms are determined based on the technical indicators of the filter to be designed, which can be used for Filter Optimization in teaching and scientific research design.