When designing a FIR absorber, what is the effect of windowing on the system frequency?

If the ideal frequency response function of the filter is HD(EJ), then the corresponding unit impulse response is HD(N) = The basic principle of the window function design method is to force HD(N) with a finite-length unit impulse response sequence h(n). Since hd(n) tends to be infinitely long and non-causal, the window function is used. w(n) truncated hd(n) and weighted:

h(n)=hd(n)w(n)h(n)h(n) is the unit impulse response sequence of the FIR digital filter actually designed, and its frequency response function h(ej) is h(ej) = the performance of the filter designed by the window function method depends on the type of the window function w(n) and the value of the window length n. During the design process, the appropriate window function type and window length n should be selected according to the requirements for the minimum attenuation of the stop band and the width of the transition band.

Generally, the linear phase filter is selected, that is, the filter order m is an even number, and the procedure is as follows:

wp=;ws=;ap=1;as=100;

dev=[rp rs];

m,wc,beta,ftype]=kaiserord(f,a,dev);

m=mod(m,2)+m;

plot(omega/pi,20*log10(abs(mag)))

If the attenuation of the same stopband does not meet the requirements, the filter can also be optimized, and the equal ripple FIR is generally used for optimization.

In order to improve the performance of the FIR filter, the main lobe width of the window function is required to be as narrow as possible to obtain a narrow transition band. The relative value of the sidelobes is as small as possible, and the number is as small as possible, so as to obtain the characteristics of small passband ripple, large stopband attenuation, and stable in both the passband and stopband, so that the actual frequency response of the filter can better approximate the ideal frequency response.

Minimum stopband attenuation Transition band bandwidth w

Rectangular window: Hanning window:

Heming window: Blackman window:

1.Frequency domain leakage error of the window function: The window function shows a certain main lobe width and peak attenuation in the frequency domain, resulting in the frequency response of the filter not fully meeting the design requirements.

The width of the main lobe limits the frequency resolution of the filter, while the peak attenuation determines the rejection ability of the filter. Select the appropriate window function type and length to minimize frequency-domain leakage errors within the required frequency range of the design.

2.Frequency Selection Error: When designing a FIR filter using the window function method, it is necessary to determine the desired frequency range of the passband, stopband, and transition band.

Selecting an inappropriate window function and filter length can lead to a mismatch between the actual frequency response and the design requirements, resulting in frequency selection errors.

3.Amplitude error of the window function: The window function is applied in the time domain to the ideal response of the FIR filter, but the amplitude error may be introduced in the actual application.

This is because the amplitude of the window function is not a constant equal to 1, but a non-zero attenuation factor. This amplitude error causes the filter's frequency response to vary so that it does not exactly match the response that is ideally required.

4.Phase error of window function: When designing FIR filters by window function method, the time-domain characteristics of window functions affect the phase response of the filter. Especially for applications with strict nonlinear phase requirements, the phase characteristics of the window function can introduce phase errors into the energy cycle.

5.Truncation error: In the design of the FIR filter, the infinitely long ideal response of the filter is usually truncated to obtain a finite-length filter. This truncation introduces a truncation error, resulting in a difference between the actual frequency response and the ideal response.

These errors** need to be carefully considered when designing and implementing FIR filters, and reasonable trade-offs and optimizations need to be made in the design to meet specific requirements and performance metrics. The specific error effects are related to the design parameters, the selection of the window function, and the length of the filter.

The main errors** in designing FIR filters using the window function method are the width of the main lobe and the amplitude of the side lobes. The main lobe width refers to the width of the main lobe in the frequency response of the filter, which depends on the shape of the window function, mainly including rectangular window, Hanning window, Hamming window, Blackman window, etc., among which the main lobe width of rectangular window is the largest, while the main lobe width of other window functions is relatively small. The wider the main lobe width, the worse the filter's performance in frequency selection, so you need to choose the appropriate window function to control the main lobe width.

The amplitude of the sidelobes refers to the amplitude of the sidelobes in the frequency response of the filter, and it also depends on the shape of the window function. In general, the smaller the sidelobe amplitude of the window function, the better the performance of the filter. However, decreasing the amplitude of the side lobes leads to an increase in the width of the main lobe, so the family needs to make a trade-off between the width of the main lobe and the amplitude of the side lobes.

In addition to the sensitivity width and sidelobe amplitude in the main lobe, there are some other errors** in the design of FIR filters by the window function method, such as frequency jitter, filter order, etc. Frequency jitter refers to the fluctuation of the frequency response between the passband and stop band of a filter, and in general, the smaller the frequency jitter of the window function, the better the performance of the filter. The order of the filter refers to the order of the filter, and in general, the higher the order, the better the performance of the filter, but it will also lead to an increase in the amount of computation.

Therefore, there are a variety of factors that need to be considered when designing FIR filters, and the appropriate window function and order are selected to meet the performance requirements of the specific application.

When designing a FIR filter using a window function, the main errors** are frequency domain leakage and frequency response distortion.

Frequency-domain leakage refers to the truncation effect of the window function, which causes the frequency response of the filter to deviate from the ideal filter, that is, the frequency response is not 0 outside the theoretical cut-off frequency, which leads to the distortion of the waveform. The shorter the window function, the greater the leakage, and conversely, the longer the window function, the smaller the leakage.

Frequency response distortion refers to the deviation between the actual amplitude-frequency mind-line characteristics and phase-frequency characteristics of the filter and the theoretical amplitude-frequency characteristics and phase-frequency characteristics. This distortion of Allah is due to the fact that the frequency response of the window function is not flat, but fluctuates.

In addition, when designing FIR filters using window functions, there is also an issue of inaccurate cut-off frequencies. Because the design of the window function is based on the frequency response of the ideal filter, in practical applications, if the cut-off frequency of the filter is inconsistent with the theoretical value, the performance of the filter will be degraded, or even unable to meet the design requirements. Kai Gao Wei.

Therefore, in the design of FIR filters, it is necessary to select the appropriate window function according to the actual application situation to reduce the leakage and frequency response distortion in the frequency domain, and to achieve a more accurate cut-off frequency by adjusting the parameters of the window function. At the same time, the designed FIR filter needs to be tested and evaluated in practice to ensure that it meets the design requirements.

Frequency response error: The window function method truncates the frequency response of an ideal filter in the frequency domain, so a frequency response error is introduced. This error usually manifests itself in the presence of ripples at the cut-off frequency of the filter or a wide transition bandwidth.

Errors in the window function itself: The window function used in the window function method can also introduce errors in itself. Different window functions have different properties, such as the rectangular window function has a wide main lobe and a high side lobe base judgment, while the Hanning window function has a smaller side lobe and a wider main lobe.

Choosing an inappropriate window function can also lead to a decrease in filter performance.

Systematic error: In practical application, due to the influence of various factors, such as quantization error, filter coefficient calculation error, etc., will lead to the existence of systematic error. This error usually manifests itself as a deviation between the output of the filter and the desired output.

In order to reduce these errors, some improved window function design methods, such as Kaiser window function, Chebyshev window function, etc., can also use other FIR filter design methods, such as least squares method, frequency sampling method, etc. At the same time, in practical applications, it is also necessary to select appropriate filter design methods and parameters according to specific needs and application scenarios to achieve the optimal filtering effect.

The main error** in designing FIR filters using the window function method is frequency domain leakage. The window function method is a widely used FIR filter design method, the basic idea of which is to multiply the frequency response of the RIR filter by a window function so that the FIR filter has a limited impulse response length. The error of the FIR filter designed by the window function method mainly comes from the leakage in the frequency domain, that is, in the region where the spectrum of the pure permeation function of the window is not zero, there will also be a non-zero response in the region where the frequency response of the ideal filter is not zero.

This results in increased errors at these frequencies, which can affect the performance of the filter. Therefore, it is very important to select a suitable window function for the design of FIR filter by window function method.

In addition to frequency domain leakage, there are some other factors that will affect the accuracy of the design of FIR filters by the window function method, such as the selection of cut-off frequency and the selection of filter order. Therefore, in practical application, it is necessary to select appropriate design methods and parameters according to the requirements and constraints of specific pants to achieve the best filter performance.

The main error** in designing FIR filters using the window function method is due to the nature of the window function. When designing the FIR filter, the window function method limits the time domain length of the signal by multiplying a window function in the time domain to achieve the purpose of controlling the frequency domain characteristics of the filter. However, this way of limiting the time domain length of the signal also introduces some errors, mainly of the following two types:

1.Cut-off bandwidth error: Due to the existence of the cut-off width of the window function, the filter will produce an error at the cut-off bandwidth, so that the actual cut-off bandwidth of the filter is deviated from the theoretical cut-off bandwidth of the design.

2.Pre-fluctuation error: Due to the different shapes of the window function, the filter will cause pre-fluctuation near the cut-off bandwidth, which will cause the gain of the filter to fluctuate near the cut-off bandwidth, which will affect the performance of the filter.

In addition to the window function method, there are other methods that can be used for the design of FIR filters, such as least squares method, frequency sampling method, etc. These methods have their own advantages and disadvantages, and it is necessary to choose the appropriate method for filter design according to the actual situation.

The window function method is a commonly used method for designing the coincidence of finite impulse response (FIR) filters. The main error** of this method is spectral leakage, which is the difference between the actual frequency response of the filter and the ideal frequency response. This error is due to the fact that when the window function is truncated, the edge of the window function produces a waveform distortion caused by the window function truncation.

This waveform distortion causes energy from the high-frequency section to leak into the low-vertical search section, resulting in spectral leakage errors.

In order to reduce the spectrum leakage error, the selection of the window function and the adjustment of the parameters can be used. Commonly used window functions are rectangular windows, Hamming windows, Hanning windows, Blackman windows, etc. The same window function has different properties, such as main lobe width, side lobe attenuation, etc., and the appropriate window function can be selected according to the needs of specific applications.

In addition, when designing FIR filters using the window function method, it is also necessary to pay attention to the selection of cut-off frequency. If the cut-off frequency is too high, the order of the filter will increase, which will increase the computational complexity and storage space, and will also increase the spectral leakage error. Therefore, it is necessary to select the appropriate cut-off frequency and filter order according to the needs of the actual application and the limitation of computing resources.

The window function method is a commonly used fir filter design method, which truncates the wide frequency response of the ideal filter, and then multiplies it with the window function to obtain the actual fir filter coefficient per acre. However, there are some major errors with the window function method**.

1.Error caused by the main lobe: Due to the application of the window function to truncate the ideal filter, the main lobe will be introduced, resulting in a deviation between the frequency response of the filter and the ideal filter, which is the most important error in the window function method.

2.Errors due to auxiliary lobes: The window function method also introduces auxiliary lobes, i.e., other frequency components other than the main lobe, which causes the frequency response of the filter to fluctuate additionally around the cut-off frequency.

3.Error caused by transition band: Transition band refers to the frequency range between the cut-off frequency of the ideal filter and the pass band and stop band, and the window function method will also produce errors in the truncation of the transition band and the change of the window function.

In order to reduce these errors, some improved window functions, such as Kaiser window, Chebyshev window, etc., can be used, as well as methods such as adjusting window function parameters and filter order.

It should be noted that although there are errors, the window function method is still a simple and effective design method for FIR filters, especially for the situation where low-pass, high-pass, band-pass, and band-stop filters need to be designed in the field of signal processing.