The second design method described herein explain the FIR filter - sampling frequency.
We have also summarized at the end of the previous over, window function method is a design method starting from the time domain, not intuitive. why? Because we are more concerned about filter design is its frequency characteristics (at least at this stage is so), is it possible departure from its frequency performance to design directly filter it? Of course, this is this article, we want to say "sampling frequency."
Three frequency sampling method
1. Design Principles
As the name suggests, the frequency of sampling, that is, in the frequency domain sampling, who take samples? Of course, to take the filter frequency response, that is to say:
After that N discrete sample values obtained, how to do the following?
We know, our ultimate goal of this chapter filter design (Ha stage, the theoretical ultimate goal) is to give the system described filter function H (z) of the numerator and denominator polynomial coefficients. For the FIR filter, the denominator is 1, the coefficient is its molecular unit impulse response h (n). So, the next step is to find h (n).
Come to H (k), how to get h (n) do? We learned the magic DFT grand debut, but this is the inverse transform: IDFT.
Well, at this stage to achieve the ultimate goal. It seems very simple, but ...... take this h (n) as our ultimate goal, fly it?
So we need to analyze how the filtering characteristics of the filter h (n) represented in the end? You can not achieve the design requirements?
Through the previous learning window function method, we know that H (n) obtained by the window function method is to be truncated to the ideal filter HD (N), so the filtering performance is poor than the ideal filter. So, we get through the front of the process h (n), it is the ideal filter hd (n), then what does it matter? Or, the actual frequency of the system is FIR h (n) response on behalf of H (e ^ jw), representative of the desired filter frequency hd (n) in response Hd (e ^ jw), is the relationship? The following were analyzed by two methods:
2. approximation error
The first analysis: Analysis of the relationship between h (n) and hd (n) of
Our "correspondence between the frequency domain when the ultimate" - a domain of discrete sampling, another domain extension period, but also debut. Look:
figure 1
The frequency domain, the Hd (e ^ jw) to 2Π / N discrete sampling interval, corresponds to the time domain, is to hd (n) in N periodic extension. DFT time domain and the frequency domain values have taken the main section, the words used to describe this formula, is:
See here, we have thought, hd (n) is infinitely long, it will certainly be periodic extension aliasing ah, then take h (n) obtained primary value interval, and its value is certainly hd (n) different. It is also because of this "aliasing", which leads to the FIR frequency response we obtain H (e ^ jw), certainly not ideal filter Hd (e ^ jw).
The second analysis: Analysis of the relationship between H (e ^ jw) and Hd (e ^ jw) of
Continuing with the previous sentence, H (e ^ jw) is better Hd (e ^ jw), where better to do in the end? H (e ^ jw) said they did not convinced, "said I can not give evidence!"
We know, Hd (e ^ jw) is the ideal frequency response of the filter, it looks like we are very clear, we can write an expression to a low-pass, for example, is the following way:
figure 2
Our own children H (e ^ jw), what is it? Expressions can not find out?
We return to Fig 1: H (e ^ jw) is h (n) is the Fourier transform (DTFT), and h (n) is the DFT of H (k), according to previously learned, a DFT cycle is DTFT ( [0 ~ 2Π]) of N equally spaced sample value. The "frequency domain interpolation recovery", has the following formula learned sampling frequency domain to represent H (e ^ jw):
image 3
Sorry, I know this formula is very annoying, but do not put it out, can not let their own children H (e ^ jw) convinced ah.
Having said that, rationalize it: the idea of the frequency of sampling is: to perfect people's children Hd (e ^ jw), the N discrete sampling points to obtain H (k), then H (k) seeking an N-point IDFT, get h (n), which is our own children, its frequency response H (e ^ jw) What is it? That we are represented in Figure 3 interpolation formula.
The following description intuitive graphical their own children and other people's children gaps, a low-pass filter as an example.
Figure 4
In FIG. 4, dotted blue line is "people's children" - over the low-pass filter amplitude-frequency characteristic, the cutoff frequency of π / 3, according to the sampling frequency (the number of points N = 15, as shown in the blue open circles FIG point, because of which only shows the inner FIGS range of 0 ~ π, so only 7 points).
It is emphasized that the description, H (k) is a low pass over the spectrum of sample values (including amplitude and phase), and we draw, for simplicity, only the amplitude function is given.
The actual filter (that is, their own children) in accordance with these discrete frequency sample points interpolated amplitude-frequency characteristics are what it? FIG 4 is a solid black line.
Wow, better than not know, a more than surprised. The gap between their own children at a glance. A first disadvantage: the emergence of transitions between passband and stopband; second disadvantage: the fluctuation appears in the passband and stopband.
Both disadvantages can be overcome it?
First look at the first drawback, the transition zone can minimize it? Obviously it is possible, from the transition zone width ≈ two adjacent sampling points, i.e., 2π / N. So, increasing N, the transition zone can be narrowed.
N we increased from 15 to 75 points (FIG red solid dots, is between 38 0-π), then the frequency response using the 75 frequencies interpolated sample values of the FIR filter, such as red solid line in FIG. 5, as expected, the transition zone is narrowed.
Figure 5
However, the red solid lines we compared the solid black line in FIG. 5, will find that, although the inside away from the passband and stopband cutoff frequency fluctuation solid red line is less than the solid black line, but the undulation and the first side lobe not significantly changed, i.e., increased N does not reduce the minimum stopband attenuation.
Old mother at the time just to sigh, and desperate to find their own child's second disadvantage is caused by a gene, can not be changed.
Do not despair, though not completely eradicated, but it can not improve it a little? This is our third question: improvements.
3. Improvement
To improve, first find the cause. Cause of ups and downs, because the sample values H (k) suddenly changed from 1 to 0 (break point), if we make it a little slower, is expected to reduce the ups and downs. That is to say:
Discontinuity point in the frequency response near the transition interval inserting one or several sampling points, so that the discontinuous point becomes slow transition band. Figure 5 shows an example of inserting transition sampling point.
Figure 5
Insertion of m transition points, transition band width of approximately: (m + 1) × 2Π / N
M the number of transition points and the relationship between the filter stopband minimum attenuation data have some experience in the following table.
Further, the value of the transition sampling points required to optimize the design, different values will affect the stop-band attenuation.
It seems, the transformation of their own children, approaching toward other people's children, is a glorious and arduous, almost impossible task ah.
Old mothers how to do it? First, select the method, the method used to design your own children, window function, or frequency sampling method, or a method of optimization that we have not mentioned here ......; second, since election given this method, indicating that it must have to attract your advantage, but at the same time, you can readily accept disadvantage of this method meant to be. If you find that you can not accept this shortcoming, sorry, you have to start over again, re-use other methods to design your filter.
Of course, the filter can be redesigned, but can only raise a child, like launching satellites, launch failures once, and had to re-create one of the. , Read, original title: Digital Signal Processing Series construes section 19 (digital filter bis) - FIR filters (3): sampling frequency (1)
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