JP4324676B2 - Adaptive filter - Google Patents

Description

  The present invention relates to an adaptive filter, and in particular, improves the tracking performance of a target signal.

The adaptive filter is a filter that performs an adaptive process so that an error between an output signal and a target signal (desired signal) is minimized. Usually, a finite impulse response (FIR) filter is used, The filter coefficient is adjusted each time an input signal is input using an adaptive algorithm.
FIG. 8 shows a conceptual diagram of the adaptive filter. The adaptive filter 10 performs a product-sum operation on the input signal x _{n at the} discrete time n using the filter coefficient C _n and outputs an output signal y _n . Adder 30 output signal y _n is input, and outputs an error signal e _n representing the difference between the output signal y _n and the desired signal d _n, the adaptive filter 10, as the error signal e _n is decreased, The filter coefficient is updated to C _{n + 1} . Then, using this filter coefficient C _{n + 1} , a product-sum operation is performed on the input signal x _{n + 1 at} the next discrete time n + 1, and this procedure is repeated.

FIG. 9 is a block diagram showing the configuration of the adaptive filter. The adaptive filter includes a transversal filter circuit 11 that performs a product-sum operation and a coefficient adjustment unit 12 that updates the coefficient of the transversal filter circuit 11 using an adaptive algorithm. The transversal filter circuit 11 includes an input signal x a plurality of delay elements 13 and 14 for delaying the _n, the input signal x _n and the input signal delayed x _n-1, x _n-2 coefficients c ₀ coefficient adjusting unit 12 is set for (n), c ₁ (n), comprises a multiplier 15, 16 and 17 for multiplying c ₃ (n) of each, and an adder 18 for outputting an output signal y _n by adding the outputs of the multipliers 15, 16, 17 ing.

As an adaptive algorithm (coefficient adjustment algorithm for an adaptive filter), the most basic least mean square (Least Mean Square: LMS) algorithm has been widely used. In this algorithm, it updates the coefficient is performed so as to minimize the mean square error E of the error signal ^{_{e n [(e n) 2}} ].

Ｔは転置である。また、係数は（数２）によりベクトルＣ_nで表す。

このとき、適応処理は、(数３)（数４)（数５）により記述される。

ここで、μは、正規化ＬＭＳアルゴリズムにおけるステップサイズパラメータであり、
０＜μ＜２ （数６）
のとき、係数ベクトルは収束する。この収束範囲は、ＬＭＳアルゴリズムにおける収束条件と違って、入力信号に依存しない。そのため、正規化ＬＭＳアルゴリズムは、ＬＭＳアルゴリズムに比べて扱い易い。この範囲内でμの値を大きく取れば時定数が低減し、係数ベクトルの収束速度が向上する。逆に、μの値を小さく取れば、時定数が増大し、係数ベクトルの収束速度が低下する。
また、βは、０による割り算を避けるために導入された小さな正の実数値であり、“安定化パラメータ”と呼ばれている。
なお、正規化ＬＭＳアルゴリズムについては、下記特許文献１あるいは下記非特許文献１に詳述されている。 In recent years, a normalized LMS algorithm improved by making the LMS algorithm coefficient vector convergence conditions easier to handle has also been used.
The adaptive process using the normalized LMS algorithm can be described by a mathematical formula as follows.
The input signal is represented by a vector X _{n according} to (Equation 1).

T is a transpose. Further, the coefficient is expressed by a vector C _n by (Equation 2).

At this time, the adaptive processing is described by (Equation 3) (Equation 4) (Equation 5).

Where μ is the step size parameter in the normalized LMS algorithm,
0 <μ <2 (Equation 6)
The coefficient vector converges. Unlike the convergence condition in the LMS algorithm, this convergence range does not depend on the input signal. Therefore, the normalized LMS algorithm is easier to handle than the LMS algorithm. If the value of μ is large within this range, the time constant is reduced and the convergence speed of the coefficient vector is improved. Conversely, if the value of μ is reduced, the time constant increases and the coefficient vector convergence speed decreases.
Β is a small positive real value introduced to avoid division by zero, and is called a “stabilization parameter”.
The normalized LMS algorithm is described in detail in Patent Document 1 or Non-Patent Document 1 below.

The adaptive filter coefficients are adaptively updated and the output of the adaptive filter follows the appropriately selected desired signal, so that automatic transmission line equalization, echo canceling in communication lines, and detection of signals buried in noise are detected. Conversely, it is possible to detect noise slightly mixed in the signal, predict the signal, and the like.
For example, FIG. 10 shows a simulation model of a system that performs automatic channel equalization using the adaptive filter 10. Here, H _n denotes the impulse response of the channel (the unknown system) 32, v _n are added by the adder 31 represents the additional noise added to the transmission signal s _n being transmitted.
In this system, in a training mode where the transmitting side and the receiving side can see the pseudo the same signal sequence, the transmitter transmits a transmission signal s _n, the receiving side, the desired signal of the same signal as the transmission signal s _n Using d _n , the filter coefficient is adjusted so that the output signal y _n of the adaptive filter 10 using the received signal as the input signal matches the desired signal d _n . In this simulation model, by delaying the transmission signal s _n at delay unit 33, and generates a signal s _nD of a desired signal d _n.

FIG. 11 shows a simulation model of a system that identifies the unknown system 32 using the adaptive filter 10. In this system, the same input signal x _n (white noise) is input to the unknown system 32 and the adaptive filter 10, and the signal obtained by adding the additional noise v _n to the output signal of the unknown system 32 is used as the desired signal d _n and adaptively applied. adjust the filter coefficient so that the output signal y _n of the filter 10 is consistent with the desired signal d _n, to identify unknown system 32 including the additive noise v _n.
JP 2004-64681 A Masaaki Ikehara and Tetsuya Shimamura, “Multimedia Signal Processing,” Baifukan, January 20, 2004, pp.182-207

However, in the conventional adaptive filter, it is difficult to sufficiently reduce the mean square error of the error signal e _n. As a result, when the time variation of the input signal is severe, the desired signal cannot be sufficiently tracked, and a situation in which only insufficient results can be obtained in a system that requires prediction accuracy occurs. ing.

The present invention is intended to solve such conventional problems, and its object is to provide an adaptive filter that the mean square error of the error signal e _n can be sufficiently reduced.

The adaptive filter of the present invention adds a plurality of delay elements for delaying an input signal , a plurality of multipliers for multiplying each of the input signal and the output signal from each delay element by a filter coefficient, and the output of each multiplier. a first transversal filter comprising an adder, the same input signal and the input signal to be input to the first transversal filter input, a plurality of delay elements for delaying the input signal from each delay element multiplying each filter coefficient of the output signal of the the first transversal filter of the same number of multipliers, and a second transversal filter comprising an adder for adding outputs of the multipliers, the first of the output signal from the adder of the transversal filter and adding means for outputting as an error signal the difference between the desired signal, off for update to reduce the error signal It calculates the filter coefficients, and a coefficient adjustment unit configured to update the filter coefficients to be multiplied by the multipliers of the first transversal filter and the same sequence of the second transversal filter with filter coefficients of the same value, In the coefficient adjustment unit, the step size μ when calculating the filter coefficient for update is
(1-μ) ² <1
It is set to be an output signal from the adder of the second transversal filter is outputted as the filter output.
The second transversal filter reduces the error in the first transversal filter by performing filtering again using the coefficient obtained from the first transversal filter.

In the adaptive filter of the present invention, the coefficient adjustment unit calculates the filter coefficient for updating by using the normalized least mean square algorithm.
In the normalized LMS algorithm, if the step size μ is set in a range of 0 <μ <2, the filter coefficient converges, and the error is reduced in the entire convergence range.

  The adaptive filter of the present invention can sufficiently reduce the mean square error of the error signal. As a result, the prediction accuracy is improved, and it is possible to follow the desired signal even when the temporal change of the input signal is severe.

FIG. 1 shows a conceptual diagram of an adaptive filter according to an embodiment of the present invention. The adaptive filter, outputs an input signal x _n and the first transversal filter circuit 11 filters and outputs an output signal y _n, the difference between the output signal y _n and the desired signal d _n as an error signal e _n an adder 30 which, on the basis of the normalized LMS algorithm calculates the filter coefficients that minimize the mean square error of the error signal e _n, the filter coefficients of the first transversal filter circuit 11 and a second transversal filter circuits 21 A normalized LMS coefficient adjustment unit 20 to be updated and a second transversal filter circuit 21 that performs a filtering process on the input signal x _n using the updated filter coefficient and outputs an output signal y _pn are provided.
In this adaptive filter, the output signal y _{n of} the first transversal filter circuit 11 is used only for updating the filter coefficient, and the output signal y _{pn of} the second transversal filter circuit 21 is output as the output signal of the adaptive filter. Is done.

FIG. 2 is a block diagram showing the configuration of this adaptive filter. Configuration of the first transversal filter circuit 11 is the same as FIG. 9, the input signal x _n and a plurality of delay elements 13 and 14 for delaying the input signal x _n and the input signal delayed x _n-1, x a multiplier 15, 16, 17 the normalized LMS coefficient adjusting unit 20 is multiplied by a coefficient set for _n-2, and outputs an output signal y _n by adding the outputs of the multipliers 15, 16, 17 And an adder 18.
The configuration of the second transversal filter circuit 21 is different from that of the first transversal filter circuit 11 in that one extra delay element 23 that delays the input signal x _n is provided. 1 is the same as the transversal filter circuit 11, the normalization LMS coefficient to the input signal x delay elements _24, 25 _n for delaying each input signal delayed x _n-1, x _n-2, x _n-3 Multipliers 26, 27, and 28 that multiply the coefficients set by the adjustment unit 20, and an adder 29 that adds the outputs of the multipliers 26 to 28 and outputs an output signal y _pn are provided.

Next, the operation of this adaptive filter will be described.
Now, the input signal x _n is input, and the filter coefficients set in the multipliers 15, 16, and 17 of the first transversal filter circuit 11 at this time are c ₀ (n) and c ₁ (n), respectively. , C ₂ (n). First transversal filter circuit 11 performs a product-sum operation, and outputs an output signal y _n shown below.
y _n = (c ₀ (n) · x _n + c ₁ (n) · x _n-1 + c ₂ (n) · x _n-2 )
In response, the adder 30 outputs an error signal e _n as shown below.
e _n = d _n −y _n

Normalized LMS coefficient adjusting unit 20, the filter coefficients c ₀ (n + ₁₎ which minimizes the mean square error of the error signal _{e n, c 1 (n +} 1), c 2 (n + 1) is calculated, and the first transversal filter circuit 11 and the filter coefficient of the second transversal filter circuit 21 are updated.
When the next input signal x _{n + 1} is input, the first transversal filter circuit 11 performs a product-sum operation using c ₀ (n + 1), c ₁ (n + 1), and c ₂ (n + 1). Output signal y _{n + 1} is output.
y _{n + 1} = (c ₀ (n + 1) · x _{n + 1} + c ₁ (n + 1) · x _n + c ₂ (n + 1) · x _n-1 )
This output signal y _{n + 1} is used for the next calculation of the error signal e _{n + 1} (= d _{n + 1} −y _{n + 1} ), and the mean square error of the error signal en _{n + 1} is minimized. Filter coefficients c ₀ (n + 2), c ₁ (n + 2), and c ₂ (n + 2) are calculated.
The operation of the first transversal filter circuit 11 is the same as the operation of the transversal filter circuit 11 of FIG. 9, and the processing is described by (Equation 3), (Equation 4), and (Equation 5).

従って、第２トランスバーサルフィルタ回路２１から出力される出力信号ｙ_pnの所望信号ｄ_nに対する誤差をｅ_pnとすると、この誤差は（数８）で表される。
ｅ_pn＝ｄ_n−ｙ_pn （数８） On the other hand, when the next input signal x _{n + 1} is input, the second transversal filter circuit 21 in which the filter coefficient is updated uses c ₀ (n + 1), c ₁ (n + 1), and c ₂ (n + 1). The sum operation is performed, and the following output signal y _pn is output.
y _pn = (c ₀ (n + 1) · x _n + c ₁ (n + 1) · x _n-1 + c ₂ (n + 1) · x _n-2 )
This can be described as a mathematical expression (Expression 7).

Therefore, when an error with respect to the desired signal d _n of the output signal y _pn output from the second transversal filter circuit 21 and e _pn, this error is expressed by equation (8).
e _pn = d _n −y _pn (Equation 8)

そして、（数５）より（数１０）の関係がある。

ここで、α_nは（数１１）のように定義している。

（数１０）を（数９）に代入し、さらに、（数４）の関係を代入することにより、ｙ_pnは（数１２）のように変形できる。

（数１２）を（数８）に代入することにより、ｅ_pnは（数１３）で表される。

この結果から、（数１４）の関係が導き出せる。
｜ｅ_pn｜²＝（１−μ）²｜ｅ_n｜² （数１４）
ここで、α_nについては、βが小さな正の実数値であるため、（数１５）に示すように１と仮定する。

このとき、（数１４）は（数１６）のようになる。

正規化ＬＭＳアルゴリズムでは、（数６）に示したように、ステップサイズμは、
０＜μ＜２ （数６）
の範囲に設定される。そのため、
（１−μ）²＜１ （数１７）
となり、｜ｅ_pn｜²は｜ｅ_n｜²より常に小さくなる。 The error e _pn, as described below, the error _{e n (= d n -y n} ) less than for the desired signal d _n of the output signal y _n of the first transversal filter circuit 11. That is, the final output of the adaptive filter, the y _n without With y _pn, it is possible to reduce the mean square error (MSE).
Now, explaining that the error e _pn is smaller than the error e _n theoretically.
y _pn can be transformed into (Equation 9) using ( _Equation 3) and (Equation 5).

There is a relationship of (Equation 10) from (Equation 5).

Here, α _n is defined as (Equation 11).

By substituting (Equation 10) into (Equation 9) and further substituting the relationship of (Equation 4), y _pn can be transformed into (Equation 12).

By substituting (Equation 12) into (Equation 8), e _pn is expressed by (Equation 13).

From this result, the relationship of (Equation 14) can be derived.
| E _pn | ² = (1-μ) ² | e _n | ² (Expression 14)
Here, since α _n is a small positive real value, α is assumed to be 1 as shown in (Expression 15).

At this time, (Equation 14) becomes (Equation 16).

In the normalized LMS algorithm, as shown in (Equation 6), the step size μ is
0 <μ <2 (Equation 6)
Is set in the range. for that reason,
(1-μ) ² <1 (Equation 17)
Therefore, | e _pn | ² is always smaller than | e _n | ² .

  As described above, the second transversal filter circuit 21 reduces the error in the first transversal filter circuit 11 by performing filtering again using the coefficient obtained from the first transversal filter circuit 11. "Sophistication function" is played.

Further, the refinement function of the second transversal filter circuit 21 has been confirmed from the result of computer simulation.
FIG. 3 shows a configuration when the adaptive filter of FIG. 1 is used as the adaptive filter 10 of the channel equalization model of FIG. FIG. 4 shows the convergence characteristics in the simulation models of FIGS. 3 and 10 in comparison.
In this model, the order of the adaptive filter (equalizer length) M = 9, the delay amount D = 4 of the delay unit 33, the stabilization parameter β = 0.05, individual trials 100 runs, the SN ratio 40 dB, The step size μ is set to 0.5.

一方、（数１６）から（数２１）が得られる。

（数２１）にμ＝０．５を代入すると（数２２）が得られる。

（数２２）及び（数２０）の算出結果は、ほぼ一致しており、第２トランスバーサルフィルタ回路２１の洗練機能による効果が確認できる。 The vertical axis in FIG. 4 indicates the mean square error (MSE) in dB, and the horizontal axis indicates the number of iterations (n) of the adaptive process. A solid line characteristic indicates a case where the refinement function of the second transversal filter circuit 21 is not used, and a dotted line characteristic indicates a case where the refinement function of the second transversal filter circuit 21 is used.
In FIG. 4, the MSE in the convergence state when the refinement function of the second transversal filter circuit 21 is not used is −33.9281 (dB), and the MSE when the refinement function of the second transversal filter circuit 21 is used. Is -39.8961 (dB). from this result,
| E _n | ² = 10 ^{−33.9281 / 10} = 4.0475 × 10 ⁻⁴ (Equation 18)
| E _pn | ² = 10 ^{−39.8961 / 10} = 1.0242 × 10 ⁻⁴ (Equation 19)
And the ratio of | e _pn | ² to | e _n | ² becomes (Equation 20).

On the other hand, (Equation 21) is obtained from (Equation 16).

Substituting μ = 0.5 into (Equation 21) yields (Equation 22).

The calculation results of (Expression 22) and (Expression 20) are almost the same, and the effect of the refinement function of the second transversal filter circuit 21 can be confirmed.

Note that the difference between (Equation 22) and (Equation 20) is influenced by the assumption that α _n is 1 in (Equation 15). α _n is smaller than 1, but close to 1 (for example, 0.98). Therefore, if 0 <μ <1, (1-μ) ² <(1-μα _n ) ² (Equation 23)
When 1 <μ <2 (1-μ) ² > (1-μα _n ) ² (Equation 24)
It becomes. Here, since μ = 0.5, the relationship of (Equation 23) is established, and the value of (Equation 22) appears smaller than the value of (Equation 20).

  FIG. 5 shows the convergence characteristics when the step size is set to μ = 0.8 and the other conditions are set the same as in FIG. The characteristic of the solid line is the case where the refinement function of the second transversal filter circuit 21 is not used, and the characteristic of the dotted line is the case where the refinement function of the second transversal filter circuit 21 is used.

FIG. 6 shows a simulation model when the adaptive filter of FIG. 1 is used as the adaptive filter 10 of the identification system of FIG. 11, and FIG. 7 shows the convergence characteristics in the simulation models of FIGS. In contrast.
In this model, the adaptive filter order (equalizer length) M = 3, stabilization parameter β = 0.05, individual trials 100 runs, SN ratio 40 dB, and step size μ = 0.5 are set. ing. The characteristic of the solid line is the case where the refinement function of the second transversal filter circuit 21 is not used, and the characteristic of the dotted line is the case where the refinement function of the second transversal filter circuit 21 is used.

  In this way, the second transversal filter circuit 21 fulfilling the “sophistication function” performs the filtering again using the coefficient obtained from the first transversal filter circuit 11, thereby allowing the adaptive filter to follow the desired signal. Can improve sex.

The order of the adaptive filter and the value of each parameter shown here are examples, and the present invention is not limited to them.
Although the case where the normalized LMS algorithm is used for calculating the filter coefficient has been described here, the present invention can also be applied to an adaptive filter that calculates the filter coefficient using another algorithm. However, in the case of the normalized LMS algorithm, since the step size mu is set to a range 0 <mu <2, holds the relationship of equation (17), | e _pn | ² is | always less than ² | e _n However, when using another algorithm, it is necessary to select and use a step size μ such that | e _pn | ² is smaller than | e _n | ² .

  The adaptive filter of the present invention has good follow-up to a desired signal, and can be widely used for an echo canceller, a noise canceller, a channel equalizer, signal prediction, and the like.

The conceptual diagram of the adaptive filter in embodiment of this invention The block diagram which shows the structure of the adaptive filter in embodiment of this invention The figure which shows the channel equalization model to which the adaptive filter in embodiment of this invention is applied The figure which shows the convergence characteristic of the adaptive filter in embodiment of this invention in contrast with the conventional adaptive filter (the 1) FIG. 2 is a diagram showing the convergence characteristics of an adaptive filter in the embodiment of the present invention in comparison with a conventional adaptive filter (part 2); The figure which shows the model of the identification system to which the adaptive filter in embodiment of this invention is applied FIG. 3 is a diagram showing the convergence characteristics of the adaptive filter in the embodiment of the present invention in comparison with a conventional adaptive filter (part 3); Conceptual diagram of adaptive filter Block diagram showing configuration of adaptive filter Diagram showing channel equalization model Diagram showing identification system model

Explanation of symbols

DESCRIPTION OF SYMBOLS 10 Adaptive filter 11 1st transversal filter circuit 12 Coefficient adjustment part 13 Delay element 14 Delay element 15 Multiplier 16 Multiplier 17 Multiplier 18 Adder 20 Normalization LMS coefficient adjustment part 21 2nd transversal filter circuit 23 Delay element 24 Delay element 25 delay element 26 multiplier 27 multiplier 28 multiplier 29 adder 30 adder 31 adder 32 unknown system 33 delay element

Claims (2)

A plurality of delay elements for delaying an input signal ; a plurality of multipliers for multiplying each of the input signal and the output signal from each delay element by a filter coefficient; and an adder for adding the outputs of the multipliers. A transversal filter,
The same input signal as the input signal input to the first transversal filter is input , a plurality of delay elements that delay the input signal, and each of the output signals from each delay element is multiplied by a filter coefficient, A second transversal filter comprising the same number of multipliers as the first transversal filter and an adder for adding the outputs of the multipliers ;
Adding means for outputting a difference between an output signal from the adder of the first transversal filter and a desired signal as an error signal;
An update filter coefficient for reducing the error signal is calculated, and the same filter coefficient is multiplied by each multiplier of the first transversal filter and the second transversal filter in the same order . A coefficient adjuster that updates with the filter coefficients of the value ;
Have
In the coefficient adjustment unit, the step size μ when calculating the filter coefficient for update is
(1-μ) ² <1
And is set to be, the adaptive filter output signal from the adder of the second transversal filter, characterized in Rukoto is output as the filter output.   The adaptive filter according to claim 1, wherein the coefficient adjustment unit calculates the filter coefficient for updating by a normalized least mean square algorithm.

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