JPH05259815A - Filter device for band-division encoding - Google Patents

Abstract

PURPOSE: To improve attenuation and alias removal characteristics by obtaining an interpolation filter which is given complete reconstitution and a linear phase and has taps that are two couples more than those of a decimation filter without having a tap sum equal to a power of 2. CONSTITUTION: This device is equipped with couples of matched 1/2-band linear phase FIR decimation and interpolation filters; and the decimation filters 12 and 16 have tap coefficients which are specific integers and the interpolation filters 14 and 18 have more taps than the decimation filters and are given complete reconstitution. The tap coefficients of the matched interpolation filters 14 and 18 are determined by repeatedly solving a couple of simultaneous equations derived from a 0 term of a 1/2-band filter generated by convoluting two filters.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to band division coding (sub-coding).
The present invention relates to a filter device for band coding, an image processing device including the filter device, and a method of configuring the filter device.

[0002]

Band division coding is a term for certain coding techniques that use a set of filters to decimate (divide and downsample) an input signal into several narrow bands. .. These separated bands are processed separately for transmission, storage, etc. Interpolate the decimated signal at the reconstruction (synthesis) stage
After filtering, the signals are added and reproduced as an original signal or a signal close to the original signal. The decimate means decreasing the sampling rate (sample rate) in the filter, and the interpolating rate means increasing the sampling rate in the filter. Band-division coding is commonly applied to the processing of audio and / or video signals.

It goes without saying that it is desirable to have the best possible reconstruction. Reconstruction close to what is called "perfect reconstruction" is possible with various band division coding devices.

FIG. 1 is a block diagram showing a simple example of a band division coding filter device. This filter device is
Low-pass decimation filter (A) 12, low-pass interpolation filter (B) 14, high-pass decimation filter (C) 16, high-pass interpolation filter (D) 18, and adder 20
Consists of The filter device 10 is for storing or transmitting 22 the received signal at half its sample rate.

FIG. 2 is a diagram showing an expanded version of the apparatus of FIG. 1 for decimating an input signal into 8 bands and storing or transmitting 22 the same. In the apparatus of FIG. 2, transmission, storage, etc. can be performed on each of the eight signal paths at 1/8 of the original sample rate of the input signal.

The mathematical requirements for "perfect reconstruction" in the filter device of FIG. 1 are as follows. 1) A * B + C * D = 1 (however, * indicates convolution (convol
ution). 2) C is the complementary filter of B and D is the complementary filter of A (ie, C and D have the same coefficient values as B and A, but with the majority of each coefficient being alternately inverted) .). 3) The convolution of A * B and C * D is divided into two (1
/ 2) Must be a bandpass filter. 4) The highest bit precision should be maintained. Regarding requirement 4), in any actual device, the highest possible accuracy should be maintained within the constraints of the device. 12 which was usually rounded appropriately
Bit precision is used. Therefore, it is impossible to obtain absolutely perfect resolution in a practical device, usually due to some rounding effect. here,
The term "full resolution" is used in its ordinary technical sense. That is, in consideration of the limits of the actual device, the word "perfect" is not used to mean absolutely perfect.

The definition of a perfect reconstruction filter pair used for band division coding is as follows. "First filter F
Given a 1 and a second filter F2, the product of these frequency responses is a 1/2 band filter. "

When two of such pairs are applied to construct a high pass and a low pass in a band division encoder, the following is obtained. F1 and F for lowpass
2 Use the filter as is. For high-passages,
The F1 and F2 filters are modified to bring Nyquist to the center (every other tap has nothing). The F2 filter is used for decimation and the F1 filter is used for interpolation.

The F1 and F2 filters may be replaced if desired, but the modification must be made in both the lowpass and highpass paths. That is, the F1 filter cannot be used for decimation in both the low-pass and high-pass paths.

The two-division (1/2) bandpass filter has a half
It is defined as having a response of 6 dB at the Nyquist frequency and having a symmetrical response on both sides of the 1/2 Nyquist point. This leads to a filter whose setting taps have the following form (shown in FIG. 3). ... F0E0D0C0BAB0C0D0E0F ... (that is, all even taps except zero are 0)

As can be seen by convolving the conceptual filter F1 with itself, the results of such a convolution do not all result in a zero value at even tap positions, so that two exactly identical filters result in 1/2 It is impossible to construct a bandpass filter. IEEE Minutes of International Conference on Speech and Signal Processing, April 1980 (IEEE reference number ...
CH1559-4 / 80 / 0000-029C198
0) “Quadrature (quadratur)” announced on pages 291-294.
e) a filter group designed for
Jay Dee Johnson reviews filters with equal decimation and interpolation filters. Johnson acknowledges the existence of reconstruction errors and strives to minimize them. Another way of looking at this problem is that there is no finite length filter with the square root response of a 1/2 bandpass filter.

Perfect reconstruction is a desirable property since it is to minimize any error in the band division encoder (which is subject to the rounding or quantization errors mentioned above) as much as possible. From the above discussion, it can be concluded that the prerequisite for achieving perfect reconstruction is different decimation and interpolation filters.

So far, the following 2 which gives a perfect reconstruction:
Two types of filters have been announced. IEEE Bulletin 4 on sound, voice and signal processing, published June 1986.
In a paper entitled "Precise Reconstruction Techniques for Band-Division Encoders with Tree Construction" on pages 34-441, N. Smith and T. Burnwell first designed a product filter and then reported the results. For example, separate decomposition and synthesis filter banks (terminal groups) for the minimum and maximum phase components
We propose a design method to achieve perfect reconstruction by factoring into. The Smith and Barnwell method is
It gives a good filtering effect but introduces a non-linear phase. Therefore, this method does not produce errors with low-level coding errors, but when distortion is introduced into the coding path, non-linear phase errors occur, which may cause image degradation during digital image processing. is there.

IEEE Minutes of the International Conference on Voice and Signal Processing in April 1988 (IEEE Reference Number ...
CH2561-9880000-0761, 1988).
In a paper entitled "Band Division Coding of Digital Video Using Symmetric Short Kernel Filters and Arithmetic Coding Techniques" published on pages 761 to 765, Dee Riggal and A Tabatabai described the perfect reconstruction capability. Symmetric short kernel filter
It describes a filter device that realizes a decomposition / synthesis filter bank using kernel filters). Riggal and Tabatabai in their article describe a method of designing a filterbank device that includes a symmetric short kernel FIR filter. Their design method is based on QMF (quadrature mirro) for factorization of the product filter into linear phase components.
r factorisation). However, this method is applied at a relatively simple level, and the decimation and interpolation filters have slow attenuation (roll-off) characteristics and poor aliasing (folding distortion) removal.

Our co-pending UK patent application No. GB
-9111782.0 (filed May 31, 1991)
Again, a new type of filter has been proposed. This new class of filters relates to symmetric linear phase filters with a 1/2 band F1 filter (ie 6 dB at 1/2 Nyquist frequency) and a higher order matched F2 filter.

Broadly speaking, the above-mentioned Japanese Patent Application No. GB-911.
No. 1782.0 proposes a filter device for band division coding consisting of a first and a second filter forming a pair of matched linear phase FIR decimation and interpolation filters giving perfect reconstruction. It is a thing. In the filter device, the first filter is a 1/2 band type having a predetermined integer tap coefficient, the second filter has more taps than the first filter, and the tap coefficient of the second filter is , Determined by the solution of a set of simultaneous equations derived from the zero terms of the 1/2 band filter created by the convolution of the first and second filters. This filter device provides perfect reconstruction and linear phase.

The above-mentioned Japanese Patent Application No. GB-9111782.0 also discloses that the second filter has at least one pair more taps than the first filter, and the selected taps of the second filter are predetermined 1/2 band filters. Of the corresponding taps of the second filter, the remaining tap coefficients of the second filter being derived from the zero terms of the 1 / 2-band filter formed by the convolution of the first and second filters. It also proposes a filter device determined by This preferred type of filter has a faster second filter attenuation and correspondingly improved antialiasing.

[0018]

SUMMARY OF THE INVENTION An object of the present invention is to provide a band division coding filter device of the type described above.
Even if the second filter does not have a tap sum equal to a power of 2 in an order higher than that of the first filter by 2 or more, it is possible to manufacture the matched second filter.

[0019]

SUMMARY OF THE INVENTION According to a first aspect of the invention, a pair of matched linear phase F's providing perfect reconstruction.
A band division coding filter arrangement is provided which comprises first and second filters forming an IR decimation and interpolation filter. In the filter device, the first filter has 2 with a predetermined integer tap coefficient.
Of the split (1/2) band type, the second filter has at least one more pair of taps than the first filter, and the tap coefficients of the second filter are convolutions of the first and second filters. ) Defined by solving a set of simultaneous equations derived from the zero terms of the 1/2 band filter formed by), and performing iterative testing and error handling with the selected tap coefficients of the second filter set. The remaining tap coefficients of the second filter are determined by solving the simultaneous equations using the selected selected tap coefficients that have been set.

According to a second aspect of the present invention, there is provided a method of constructing first and second filters for forming a pair of matched linear phase FIR decimation and interpolation filters for band division coding. It In the method, the second filter shall have at least one more pair of taps than the first filter. The method includes the following steps.

A) setting a tap coefficient for the first filter of the 1/2 band type, b) convolutio of the first filter and a second filter with an undetermined tap coefficient.
n) deriving a set of simultaneous equations using the zero terms of the 1 / 2-band filter formed by c), c) iteratively solving the set of simultaneous equations to derive the tap coefficients of the second filter, The iterative step comprises: c) (i) setting selected taps of the interpolation filter; and c) (ii) solving the system of equations using the set selected taps and leaving the rest of the second filter. D) setting the tap coefficient of the second filter using the result of step C.

[0022]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS The present invention will be specifically described below with reference to the drawings. For one given 1/2 band response (first) filter, there is (almost always) matched second (second) filter. As described in Japanese Patent Application No. GB-9111782.0, the band-division coding filter device designs an appropriate first filter, and tap coefficients of the first filter and unknown taps of the second filter. Can be made by generating a set of simultaneous equations using the zero terms of a 1/2 bandpass filter formed by convolution with coefficients, and solving the simultaneous equations to derive a matched second filter. it can.

The filter described in Japanese Patent Application No. GB-9111782.0 is similar to the filter discussed in the above-mentioned Regal and Tabatabai paper, but of a higher order type. In other words, they are 1
Symmetric linear phase filter with a / 2 band first filter (i.e., 6 dB at 1/2 Nyquist frequency) and a matched second filter that produces a convolution response of 1/2 band but higher order.

The classical LGT filter pair is: First filter ... 2,1 Second filter ... 6,2, -1 Tap set of convoluted filter ... 16,9,
0, -1 (in the following, the filters are all odd symmetric and therefore only the center and right coefficients are shown.
The full set of tap coefficients for the LGT filter is: First set 1,2,1 Second set -1,2,6,2, -1 Convolved filter set-1,0,9,16,9,0,
-1)

As mentioned above, the drawbacks of the LGT filter are:
Both filters have a gradual attenuation characteristic and aliases are not well removed. It was decided that for a given 1/2 band first filter of higher order than Riggal and Tabatabai thought, but almost always there is a second filter that matches this.

Using the techniques outlined above and considering a second filter that is one order higher than the first filter, a matching second filter can always be created.

As described above, the definition of the perfect reconstruction filter pair for band division coding is that "if the first filter F1 and the second filter F2 are given, the product of their frequency responses is divided into two (1 / 2) It becomes a bandpass filter. " So the convolution of these two filters
ion) must have a 1/2 band characteristic. This means that a set of simultaneous equations must be generated in which some of the convolution terms are zero. In order to solve the system of equations, special conditions that give one more equation must be met. That is, the second filter must have a zero response at Nyquist (frequency) (and therefore the sum of the alternating taps must be zero).

Next, three examples of filter pairs designed according to this technique will be described with reference to FIG. The examples shown in FIGS. 4, 5 and 6 relate to 4th, 6th and 8th order first filters having the following tap values respectively. 16,9,0, -1 (Σ = 32) 4th order 128,77,0, -16,0, -3 (Σ = 256) 6th order 512,315,0, -79,0,26, 0, -6 (Σ = 1024) 8th order

The method of calculating the tap value of the second filter will be described in detail below by taking the fourth progressive first filter as an example. The tap value of the first filter is 16, 9, 0,-
1, and the matching tap values of the second filter are A, B,
Assuming C, D, and E, using the convolution shown in FIG. 7, A,
Solve for B, C, D and E. From the convolution shown in FIG.
It can be seen that the following simultaneous equations hold. D = 0 16E + 9D-B = 0, that is, 16E-B = 0 9D + 16C + 9B-B = 0, that is, 16C + 8B =
0

Since there are 5 variables and the number of equations is 3, two more equations are required. These are obtained as follows. If E = 1, B = 16 E = 16, C = −
(8/16) B = -8 If the Nyquist response is defined as 0 (that is, A-2B +
2C-2D + 2E = 0), A = 2B-2C + 2D-2E
= 32 + 16-2 = 46 The solution is A = 46, B = 16, C = -8, D = 0, E =
1 and the matched second filter response to the fourth order filter is 46, 16, -8, 0, 1 (Σ = 64).

The sum of tap coefficients for the second filter is 2
Note that it is a power of.

The combination of the first filter with tap coefficients of 16, 9, 0, -1 and the second filter with tap coefficients of 46, 16, -8, 0, 1 forms a perfect reconstruction pair. .. However, the second filter has a drawback that the anti-aliasing characteristic is not so good because the attenuation (roll-off) characteristic is not so fast.

As described in the above-mentioned Japanese Patent Application No. GB-9111782.0, the above-mentioned technique can be similarly applied to a higher-order first filter. The characteristics of the sixth progressive first filter and the matched second filter produced by the above method are shown in FIG. 5, respectively. The tap values that give the frequency response of FIG. 5 are as follows. (First filter) 128, 77, 0, -16, 0, 3 (Σ =
256) (Second filter) 358, 128, -61, 0, 13, 0, -3 (Σ =
512)

FIG. 6 shows characteristics of the eighth progressive first filter, the matched second filter produced by the above method, and the product thereof.
Shown in. The tap values that give the frequency response of FIG. 6 are as follows. (First filter) 512, 315, 0, -79, 0, 26, 0, -6
(Σ = 1024) (Second filter) 1418, 512, -236, 0, 53, 0, -20,
0,6 (Σ = 2048)

Note that the second filter always has a sum equal to a power of two. However, using the above method has the disadvantage that the second filter frequency response exhibits a poor response which causes aliasing, regardless of the order of the first filter.

The above-mentioned Japanese Patent Application No. GB-9111782.0 is followed by a certain type of filter, especially the first and second filters.
The technique described above has been developed for a filter device having taps in which both filters reach a power of two. this is,
The case where the 1/2 Nyquist response to such a filter is simple, that is, the 1/2 Nyquist response to the first filter is 0.5 and the 1/2 Nyquist response to the second filter is 1.0. Is.

However, there are filters whose 1/2 Nyquist response is not that simple, and the sum of the taps of the second filter is related to the 1/2 Nyquist response of the interpolator. The relationship is plural and depends on the aspect of the second filter and the first filter.

Now, the filter tap group 10, 4, -1
Consider a first filter with When the tap group to be matched is determined using the above technique, the tap groups 56, 25,-
A second filter with 4, -1 is obtained. That is, (first filter) 10, 4, -1 (Σ = 16) (second filter) 56, 25, -4, -1 (Σ = 96) The sum of tap coefficients of the second filter is not a power of 2. Please note that.

Taking the above example, 1 / of the first filter
The response at 2 Nyquist is 3/4. Therefore, the 1/2 Nyquist response of the second filter is 2 in total.
In order to satisfy the divided band response, it must be 2/3. The sum of the second filter is 96, which is 1
Since the size at 1/2 Nyquist is 64, the required 2
Give a / 3 response. This second filter, which has just one tap more than the first filter, generally has poor response damping. However, as long as the technique of the above-mentioned Japanese Patent Application No. GB-9111782.0 is used, a matching second filter having a sum of tap groups whose order is higher than that of the first filter by 2 or more and which is not equal to a power of 2 is produced. Is impossible.

The filter device according to the present invention uses, for example, a first group having tap groups 10, 4 and -1 by the technique described below.
It can be obtained using a filter. Here, it is assumed that the order of the second filter is larger than the first filter by 1, 3, 5 or an arbitrary odd number. If the order is even, then the last term in the second filter will only equal zero. In the example described below, a second filter whose order is larger than the first filter by 5 is considered. That is, (first filter) 10, 4, -1 (second filter) A, B, C, D, E, F, G, H

The convolution of these first and second filters is shown in FIG. By setting the zero term of the two-divided band response equal to zero by the convolution processing and setting the Nyquist response to zero, the following equation is obtained. -G + 4H = 0 ‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ (1) -E + 4F + 10G + 4H = 0 ‥‥‥‥‥‥‥‥‥‥‥‥ (2) -C + 4D + 10E + 4F-G = 0 ‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ (3) −A + 4B + 10C + 4D−E = 0 ‥‥‥‥‥‥‥‥‥‥‥‥ (4) A-2B + 2C-2D + 2E-2F + 2G-2H = 0 ‥‥ (5)

For the equation obtained from the convolution process, there are more unknowns than the equation, so there is no single solution.
Therefore, to solve these equations,
Use error handling. In the above example, the process includes the following steps. H is set and G is calculated from the equation (1).
F is set and E is calculated from the equation (2). A value is set for D and the remaining equations (3) to (5) are solved.

When the solution is obtained in this way, the frequency response of the obtained filter is modeled, its attenuation characteristic is observed, and whether or not it is acceptable is examined. If the frequency response is unacceptable, one or more variable values set by the user (eg, H, F or D) are changed and the above technique is repeated. It has been found that while setting and repeating one of these values, one tendency is easily detected that allows the equation to converge to a particular solution. However, it should be noted that there is not always the right solution. Also,
It is also very easy to generate extremely large coefficients, but it is not suitable for practical implementation. In this regard,
The first filter is a simple fractional 1 such as 2/3, 3/4
It has been found that the coefficients of the second filter are generally much simpler with a / 2 Nyquist response. The coefficients of the second filter are highly dependent on the coefficients of the first filter, especially those outside it. Also, as the difference between the tap order of the second filter and the tap order of the first filter increases, it becomes increasingly difficult to find a tendency to converge to a reasonable solution.

However, by using the above technique,
It should be noted that a very useful inventive filter device can be determined. Next, a specific example of such a filter will be described. 9 to 11 show the filter response characteristics of these filters.

FIG. 9 shows the frequency response for the filter pair of the first filter device. The filters that generate their frequency response are: (First filter) 10, 4, -1 (Σ = 16, 1/2 Nyquist = 0,
Nyquist = 0) (Second filter) 272, 135, -20, -16, 4, 1 (Σ = 48)
0,1 / 2 Nyquist = 320, Nyquist = 0)

FIG. 10 shows the frequency response for the filter pair of the second filter device. The filters that generate their frequency response are: (First filter) 38, 18, -4, -2, 1 (Σ = 64, 1/2 Nyquist = 48, Nyquist = 0) (Second filter) 1924, 992, -140, -139, 46, 13 ，
-4, -2 (Σ = 3456, 1/2 Nyquist = 230
4, Nyquist = 0).

FIG. 11 shows the frequency response for the filter pair of the third filter device. The filters that generate their frequency response are: (First filter) 142, 76, -11, -14, 5, 2, -1 (Σ = 2
56,1 / 2 Nyquist = 176, Nyquist = 0) (2nd filter) 9628,4932, -906, -779,364,8
9, -54, -21, 6, 3 (Σ = 16896, 1/2
Nyquist = 12288, Nyquist = 0).

[0048]

Since the filter device according to the present invention provides perfect reconstruction and linear phase, it is suitable for use in processing digital image signals. The decimation filter is
The interpolation filter is used to store image signals in a video storage medium for transmission, etc.
It is used to reconstruct or subsequently transmit the original image signal from the storage medium.

In summary, the first filter can be easily determined by simple low pass filter design software, -3
It has a 1/2 Nyquist response close to dB. The second filter also results in a 1 near -3 dB.
/ 2 Nyquist response. The tap group of the second filter is preferably set to be several elements longer than the first filter in order to improve antialiasing.

Usually, the first filter is, for example, a Sony CX.
It can be implemented with a single FIR digital filter chip such as the D2200 IC chip. The second filter generally requires higher resolution factors, which means two CXD2200 chips in parallel, one for lower order bits,
One can be achieved by using the higher order bits and summing their outputs in an adder. These filters generally
It can be applied for the purpose of data reduction.

[Brief description of drawings]

FIG. 1 is a block diagram showing a simple example of a band division encoding filter device.

FIG. 2 is a block diagram showing an example of a band division encoding filter device divided into eight bands.

FIG. 3 is a curve diagram showing characteristics of a two-division band filter.

FIG. 4 is a curve diagram showing Example 1 of response characteristics of the first and second filters of the first type.

FIG. 5 is a curve diagram showing an example 2 of response characteristics of the first and second filters of the first type.

FIG. 6 is a curve diagram showing an example 3 of response characteristics with respect to first and second filters of the first type and their products.

FIG. 7 is a chart for explaining the creation of the first type filter.

FIG. 8 is a chart for explaining the creation of the following second type filter.

FIG. 9 is a curve diagram showing Example 1 of the response characteristics of the first and second filters of the second type according to the present invention.

FIG. 10 is a curve diagram showing an example 2 of the response characteristics of the first and second filters of the second type according to the present invention.

FIG. 11 is a curve diagram showing Example 3 of the response characteristics of the first and second filters of the second type according to the present invention.

FIG. 12 is a block diagram showing an application example of the filter device of the present invention.

[Explanation of symbols]

 12 Low Pass Decimation Filter 14 Low Pass Interpolation Filter 16 High Pass Decimation Filter 18 High Pass Interpolation Filter

Claims (11)

[Claims] 1. A band-division coding filter device comprising first and second filters forming a pair of matched linear phase FIR decimation and interpolation filters providing perfect reconstruction, said filter device comprising: The first filter is of a predetermined type with an integer tap coefficient, the second filter has at least one more pair of taps than the first filter, and the tap coefficient of the second filter is the first filter. And solving a set of simultaneous equations derived from the zero terms of the bisection bandpass filter formed by the convolution of the second filter and setting a selected tap coefficient of the second filter.
A filter device for band-division coding, which is determined by error processing, and the remaining tap coefficients of the second filter are determined by solving the simultaneous equations using the selected selected tap coefficients that have been set. .. 2. The repetitive test / error processing is the second test.
The filter device of claim 1 including modeling the response of the filter and varying the selected tap coefficient to determine a change in the response of the second filter. 3. The filter device according to claim 1, wherein the predetermined filter is the decimation filter, and the derived filter is the interpolation filter. 4. The filter device according to claim 1, wherein the predetermined filter is the interpolation filter, and the derived filter is the decimation filter. 5. A high frequency pair of matched decimation and interpolation filters and a low frequency pair of matched decimation and interpolation filters, said high and low frequency decimation filters. Each of which has an input for receiving a common stream of input samples, said high and low frequency interpolation filters each having an output connected to a means for combining the signals output therefrom. The filter device according to any one of items. 6. A band-division coding filter device, or each decimation filter thereof, forms part of a coding stage for coding an image signal in an image processing device, or each of its interposers. The ration filter is
An image processing apparatus, which forms part of a decoding means for reconstructing an original image signal from the coded signal. 7. The image of claim 6 including image storage means for storing the output of said decimation filter, the output of said storage means being subsequently added to said interpolation filter to reconstruct the original image signal. Processing equipment. 8. A pair of matched linear phase FIR decimation and interpolation for band division coding.
A method of constructing first and second filters forming a filter, wherein the second filter has at least one pair more taps than the first filter, a) setting a tap coefficient for the first filter B) deriving a set of simultaneous equations using the zero terms of a two-part bandpass filter formed by the convolution of the first filter with a second filter with undetermined tap coefficients, c) the following c ) (I) and c) (ii) are repeated to find a solution to the set of simultaneous equations and to derive the tap coefficient of the second filter, c) (i) the interpolation filter is selected. C) (ii) solving the system of equations with the selected taps set and determining the remaining tap coefficients of the second filter, d) the step A method of constructing a filter, comprising: each step of setting a tap coefficient of the second filter by using the result of Cup. 9. The step of determining the iterative solution further comprises: c) (iii) modeling the response of the second filter; and c) (iV) varying the tap coefficient selected, the response of the second filter. 9. The method of claim 8 including the steps of determining a change in 10. The predetermined filter is the decimation filter, and the derived filter is the interpolation filter.
the method of. 11. The predetermined filter is the interpolation filter, and the derived filter is the decimation filter.
the method of.