JP3820331B2 - DIGITAL / ANALOG CONVERSION DEVICE AND DESIGN METHOD FOR DIGITAL FILTER USED FOR THE DEVICE - Google Patents

Description

続いて、この信号ｘ_ｄはＫ（ｚ）で表現されるデジタルフィルタ６で処理されることにより上記挿入された０信号が適宜な値に修正され、周期ｈ／Ｍで動作する０次ホールド７により連続時間信号ｕ_ｃとなる。最後に、この信号ｕ_ｃをアナログローパスフィルタ（ＬＰＦ）８により平滑化し復元信号ｚ_ｃを得る。
【００１３】
オーディオ分野では、Ｄ／Ａ変換部４における信号復元には時間遅延が許される。そこで、この時間遅れを考慮して信号復元の際の誤差系モデルを考えると、図２に示すようになる。図２では、下側の信号経路がサンプラ３及びＤ／Ａ変換部４による信号処理系であり、上側の信号経路がその信号処理系による時間遅れを考慮した遅延系である。時間遅れ要素９は帯域制限信号ｙ_ｃに上述した信号処理による時間遅れｍｈを与え、減算器１０により復元信号ｚ_ｃと遅延した帯域制限信号との誤差信号ｅ_ｃを取り出す。この誤差信号ｅ_ｃも連続時間信号であるから、次の(2)式のようにおく。
ｅ_ｃ（ｔ）＝ｚ_ｃ（ｔ）−ｙ_ｃ（ｔ−ｍｈ） …(2)
【００１４】
本発明では、この誤差信号ｅ_ｃができる限り小さくなるようにＩＩＲ型デジタルフィルタを構成する。即ち、安定な連続時間フィルタ（アンチエリアシングフィルタ２及びアナログＬＰＦ８）と正の整数ｍ、Ｎ、Ｍとが与えられている条件下で、ＩＩＲ型デジタルフィルタを設計する。そのために、連続時間信号ｗ_ｃから誤差信号ｅ_ｃへ変換するシステムをＴ_ｅｗとおいたとき、与えられたγに対し、次の(3)式を満たすようなＫ（ｚ）を求める。
【数２】

つまり、この(3)式が上述したＩＩＲ型フィルタを設計すべく設定した条件式である。ここでγは誤差の大きさを支配するものであり、小さいほどよい。Ｈ^∞制御では、これを繰り返し計算によって最小化する方法が採られる。
【００１５】
（２）単一レート系への変換
まず、図２に示したアップサンプラ及びむだ時間系を含む系（マルチレート系）を単一のサンプル周期の有限次元系（単一レート系）に変換する。そのためには、次の式(4)で定義される離散時間リフティングＬ_Ｍ及び逆リフティングＬ_Ｍ ^−１を導入する。
【数３】

但し、↓Ｍはダウンサンプラであり次の(5)式で定義される。
【数４】

【００１６】
上記離散時間リフティング及び逆リフティングを用いてＫ（ｚ）（↑Ｍ）、つまりアップサンプラ５とデジタルフィルタ６とによる処理を変換する。
【数５】

ここで、Ｋ’（ｚ）は１入力Ｍ出力の線形時不変（ＬＴＩ：Linear Time-Invariant）システムであり、Ｋ（ｚ）との関係は次の(7)式で与えられる。
【数６】

【００１７】
次に、(8)式で示される一般化ホールド〈Ｈ_ｈ〉’を導入する。
【数７】

【００１８】
このとき次の等式(9)が成り立つ。
【数８】

(6)式及び(9)式より、次の(10)式が成り立つ。
【数９】

即ち、図２に示したマルチレート系のモデルは図３に示す単一レート系のモデルに等価的に変換される。
【００１９】
続いて、連続時間むだ時間要素であるｅ^−ｍｈｓを有限次元化するために、系の入力をｍステップだけ遅らせるような変換を行う。これにより、所望の設計問題はＫ’（ｚ）の代わりに因果的なフィルタｚ^−ｍＫ’（ｚ）を設計する問題に変換される。また、これを更に有限次元離散時間系の設計問題に帰着させることもできる。その手法の詳細は、カルゴネカー、山本：「ディレイド・シグナル・リコンストラクション・ユージング・サンプルド−データ・コントロール」、プロシーディングス・オブ・３５ス・コンファレンス・オン・デシジョン・アンド・コントロール、１２５９頁〜１２６３頁（１９９６年）（P.P.Khargonekar and Y.Yamamoto:"Delayed signal reconstruction using sampled-data control"; Proc. of 35th Conf. on Decision and Control, pp.1259-1263(1996))に記載されている。但し、これには近似は入らないものの、後述するような強い制約条件が課せられており、その仮定は本デジタルフィルタの設計問題では余り満たされないことに注意しておく必要がある。従って、より実際的な離散時間問題への変換法が求められる。
【００２０】
（３）離散時間系への変換
上記事情に鑑み、原問題が制約条件のない近似的な離散時間系設計問題に帰着できることを示す。そのために、ここではＦＳＦＨ（ファーストサンプル・ファーストホールド）手法を適用する。ＦＳＦＨ手法はサンプル値制御系の性能を評価する一手法であって、ｈ周期のサンプル値系の連続時間入出力をｈ／Ｎ（Ｎは自然数）周期で動作するサンプラとホールドによって離散化し、十分に大きなＮに対する離散時間信号で連続時間信号を近似する方法である。なお、ＦＳＦＨ手法の詳細は、山本、マディエフスキ、アンダーソン：「コンピュテーション・アンド・コンバージェンス・オブ・フリクエンシ・レスポンス・ビア・ファスト・サンプリング・フォー・サンプルド−データ・コントロール・システムズ」、プロシーディングス・オブ・３６ス・コンファレンス・オン・デシジョン・アンド・コントロール、２１５７頁〜２１６２頁（１９９７年）（Y.Yamamoto, A.G.Madievski and B.D.O.Anderson: "Computation and convergence of frequency response via fast sampling for sampled-data control systems"; Proc. of 36th Conf. on Decision and Control, pp.2157-2162(1997))に記載されている。
【００２１】
設計のために図３を一般化プラント形式に描き直したものが図４である。この図４中に示したサンプル値系ｇ_ｓ は次の(11)式で定義される。
【数１０】

ＦＳＦＨ手法を用いれば、このサンプル値系ｇ_ｓの近似離散時間系は次の(12)式で与えられる。但し、ＦＳＦＨにおいてＮ＝Ｍｌ（ｌは自然数）としている。
【数１１】

Ｇ_ｄＮ（ｚ）の各行列及び作用素は次のように定義される。
【数１２】

【００２２】
上記近似離散時間系Ｇ_ｄＮを用いて上記(3)式は(13)式で近似され、(3)式を満たすようなＫ（ｚ）を求めるということは近似的に有限次元離散時間系の問題に帰着される。
【数１３】

但し、
【数１４】

である。即ち、図４は図５に示す有限次元離散時間系に変換されることになる。
【００２３】
ここで注意すべきことは、条件‖Ｄ_１１‖＜γの下で図４のサンプル値系とＨ^∞ノルム上界等価な離散時間系を計算する方法が従来提案されているが（藤岡、臼井、山本：「マルチレートフィルタバンクのサンプル値Ｈ∞設計−Ｍチャンネルの場合−」、第２７回制御理論シンポジウム（１９９８年）参照）、ここでは通常のサンプル値Ｈ^∞制御とは異なり、‖Ｄ_１１‖＜γは(3)式において非常に強い制約となる。これが上述したようなＦＳＦＨ近似を必要とする（また上記のカルゴネカー及び山本による文献に記載の、近似無しの離散時間系への変換手法を適用し難い）理由である。但し、
【数１５】

であり、（Ａ_Ｆ，Ｂ_Ｆ，Ｃ_Ｆ）はアンチエリアシングフィルタ２のＦ（ｓ）の実現である。
【００２４】
而して、式(13)を求め、ごく一般的な離散時間Ｈ^∞制御問題を解けば、所望のＩＩＲ型フィルタＫ（ｚ）が得られることになる。なお、この発明によるデジタルフィルタの設計方法はコンピュータで所定のプログラムを実行することにより実現される。そのプログラムの一部は後述の如く既存のものを利用することができる。
【００２５】
次に、この発明に係る上記手法に従った具体的なフィルタ設計の一例を説明する。
まず、図１に示す構成において、フィルタ２、８の特性Ｆ（s）、Ｐ（ｓ）を決定する。ここでは、アンチエリアシングフィルタ２の特性を
Ｆ（ｓ）＝１／（０．７０２２３ｓ＋１）（７．０２２３ｓ＋１）
とする。またサンプル周期は正規化しｈ＝１とする。従って、ナイキスト周波数はπ［ｒａｄ／ｓｅｃ］＝０．５［Ｈｚ］となる。このフィルタの周波数特性を図６に示す。その特性は０．１４２［ｒａｄ／ｓｅｃ］から１次特性で減衰し、更に１．４２［ｒａｄ／ｓｅｃ］から２次特性で減衰する。これをＣＤのサンプリング周波数ｆｓ＝４４．１ｋＨｚに換算すると、１ｋＨｚから１次特性で減衰し、更に１０ｋＨｚから２次特性で減衰するフィルタに相当する。
【００２６】
ＣＤは約２０ｋＨｚまでのオーディオ周波数帯域を有しているが、実際に収録されている音楽の周波数スペクトルはこの帯域内で平坦ではなく、高域において減衰している。実際、オーケストラなどの広帯域のソースで優秀録音と言われているものを観測しても、２０ｋＨｚまでほぼ平坦に伸びているものは殆どなく、１０ｋＨｚ以上でなだらかに減衰する特性となっている。従って、上記のような特性のアンチエリアシングフィルタは、音楽信号の自然なエネルギー分布を反映したものと考えられる。即ち、従来のように完全に帯域制限された信号を原音楽信号のモデルとするのではなく、このような周波数分布に対する重み付けを反映してＣＤに対応するデジタルフィルタを設計するのが、より自然なアプローチであると言える。また計算を簡単化するために、ホールド後の信号平滑化のためのＬＰＦ８の特性はＰ（ｓ）＝１としている。実際には１次遅れ型ＬＰＦなどが適当であるが、この場合でも減衰特性が緩やかであるため殆ど周波数特性には影響を与えず、本例と類似した結果が得られる。
【００２７】
上記条件の下で、アップサンプリング次数Ｍ＝４（つまり４倍オーバーサンプリング）、ＦＳＦＨによる近似のためのステップ数Ｎ＝４、信号復元に許容する遅れのステップ数ｍ＝２としてデジタルフィルタの計算を行った。
まず、Ｆ（ｓ）は次のような状態空間表現を持つ。
【数１６】

但し、ここで、
【数１７】

である。
【００２８】
また、式(12-2)を用いて、
【数１８】

とし、これから、
【数１９】

が得られる。即ち、これが図４の誤差系の一般化プラントｇ_ｓに対応する式である。
【００２９】
このようにしてＡ_ｃ、Ｂ_ｃ１等、全てのパラメータが得られるので、以降は式(12−1)によりＧ_ｄＮ（ｚ）を求める。この計算は、マトラブ（Ｍａｔｌａｂ）等、既存のＣＡＤを利用して行うことができる。更に、図５に従って離散時間Ｈ^∞制御を利用して解を求める。この計算も例えばＭａｔｌａｂのミューツールズ（μTools）やロバスト・コントロール・ツールボックス（Robust Control Toolbox）を利用して行うことができる。
【００３０】
このようにして得られたＩＩＲ型デジタルフィルタの伝達関数Ｋ_ＳＤ（ｚ）は次式のようになる。
【数２０】

このデジタルフィルタのインパルス応答をｎ＝３２まで計算すると、次の(14)式のようになる。
【数２１】

【００３１】
このフィルタの特性の計算結果を図７〜図１２に示す。図７は利得の周波数特性、図８は位相の周波数特性、図９はインパルス応答、図１０は方形波パルス応答、図１１は正弦波応答（ｓｉｎ０．５ｔ）、図１２は正弦波応答（ｓｉｎ０．８ｔ）である。このデジタルフィルタはＩＩＲ型ではあるものの、図９のインパルス応答を見ると２５ステップ以降は実際上０となっており、これはｎ＝２５又は２６で打ち切った２５タップ程度のＦＩＲ型フィルタで実現可能であることを意味している。このことは、実際にハードウエアで構成する場合に非常に有利な性質である。また、本例のフィルタでは、方形波及び正弦波において良好な応答が得られていることがわかる。
【００３２】
比較対象のため、従来のデジタルフィルタについても計算機による特性計算を行った。このフィルタはいわゆるジョンストンフィルタとして知られているものである。４倍のアップサンプリングに対するジョンストンフィルタは、２倍のアップサンプリングに対するジョンストンフィルタＫ_Ｊ（ｚ）を用いて次式で計算できる。
Ｋ_Ｊ４（ｚ）＝Ｋ_Ｊ（ｚ）Ｋ_Ｊ（ｚ^２）
ここで採用したジョンストンフィルタＫ_Ｊ（ｚ）は１６タップであり、４倍のアップサンプリングに対するジョンストンフィルタＫ_Ｊ４（ｚ）は４８タップになる。
【００３３】
このジョンストンフィルタの特性の計算結果を図１３〜図１８に示す。図１３は利得の周波数特性、図１４は位相の周波数特性、図１５はインパルス応答、図１６は方形波パルス応答、図１７は正弦波応答（ｓｉｎ０．５ｔ）、図１８は正弦波応答（ｓｉｎ０．８ｔ）である。
【００３４】
図１０と図１６とを比較するとわかるように、本発明によるデジタルフィルタでは立ち上がり及び立ち下がり時のリンギングがジョンストンフィルタよりも小さくなっている。また、図９と図１５との比較でわかるように、インパルス応答でも本発明によるデジタルフィルタはリンギングが小さい。既述したように、このような相違は特に音の「硬さ」や「刺激の強さ」等に影響を与えるものと考えられる。また、図８に示すように、本発明によるデジタルフィルタの位相特性はナイキスト周波数π近傍までほぼ直線状（つまり群遅延が一定）になっており、オーディオ用としてきわめて好ましい特性であることがわかる。また、図１１、図１２と図１７、図１８の比較でわかる通り、ジョンストンフィルタでは大きな歪みを生じているのに対し本発明によるデジタルフィルタでは殆ど歪みが見られない。
【００３５】
また、図１９は、図２に示したような誤差信号ｅ_ｃの利得の周波数数特性を上記実施例とジョンストンフィルタとについて示したグラフである。この誤差信号が小さいほど（図１９中で下に位置しているほど）原音（厳密にはアンチエリアシングフィルタの出力）に忠実に再生されていると言うことができ、本発明によるデジタルフィルタを用いれば広い周波数帯域において再現音の忠実性が高いことがわかる。これは、本発明によるフィルタと従来のジョンストンフィルタとにおける上記の時間応答の差異を反映した結果になっている。
【００３６】
以上はコンピュータによりソフトウエア的にフィルタを構成した場合の計算結果であるが、このフィルタをハードウエアで具現化する際には周知の構成とすればよい（例えば、三谷：「ディジタルフィルタデザイン」昭晃堂（１９８７年）等を参照）。実際に、汎用のデジタルシグナルプロセッサ（ＤＳＰ）やワンチップのＬＳＩで容易に構成することができる。
【００３７】
【発明の効果】
以上説明した通り、本発明に係るデジタル／アナログ変換装置によれば、デジタル／アナログ変換によるアナログ信号の復元誤差が広い周波数帯域に亘って小さくなるように保証されているので、きわめて原音（集音又は録音時の音）に忠実なオーディオ信号を得ることができる。特に、聴感上きわめて滑らかで、高音がきわめて素直に伸びた再生音を得ることができる。
【００３８】
また、本発明に係るデジタルフィルタの設計方法によれば、上述したような高い性能を有するデジタルフィルタを既存のツール等を用いて容易に設計することができると共に、フィルタの次数を少なくすることができるのでフィルタの構成が簡単になる。
【図面の簡単な説明】
【図１】 本発明に係るＤ／Ａ変換器を含む信号復元系モデルを示すブロック図。
【図２】 図１の信号復元系モデルに対する誤差系モデルを示すブロック図。
【図３】 図２の誤差系モデルを単一レート系モデルに変換したときのブロック図。
【図４】 誤差系の一般プラント形式のブロック図。
【図５】 図４の形式を有限次元離散時間系に変換したときのブロック図。
【図６】 本発明の一実施例によるデジタルフィルタを設計する際のアンチエリアシングフィルタの周波数特性を示す図。
【図７】 本発明の一実施例によるＩＩＲ型デジタルフィルタの利得の周波数特性を示す図。
【図８】 このＩＩＲ型デジタルフィルタの位相の周波数特性を示す図。
【図９】 このＩＩＲ型デジタルフィルタのインパルス応答を示す図。
【図１０】 このＩＩＲ型デジタルフィルタの方形波パルス応答を示す図。
【図１１】 このＩＩＲ型デジタルフィルタの正弦波応答（ｓｉｎ０．５ｔ）を示す図。
【図１２】 このＩＩＲ型デジタルフィルタの正弦波応答（ｓｉｎ０．８ｔ）を示す図。
【図１３】 従来のジョンストンフィルタの利得の周波数特性を示す図。
【図１４】 このジョンストンフィルタの位相の周波数特性を示す図。
【図１５】 このジョンストンフィルタのインパルス応答を示す図。
【図１６】 このジョンストンフィルタの方形波パルス応答を示す図。
【図１７】 このジョンストンフィルタの正弦波応答（ｓｉｎ０．５ｔ）を示す図。
【図１８】 このジョンストンフィルタの正弦波応答（ｓｉｎ０．８ｔ）を示す図。
【図１９】 本発明によるＩＩＲ型デジタルフィルタと従来のジョンストンフィルタとの誤差信号ｅ_ｃの利得の周波数数特性を示す図。
【符号の説明】
１…Ａ／Ｄ変換部
２…アンチエリアシングフィルタ
３…サンプラ
４…オーバサンプリングＤ／Ａ変換部
５…アップサンプラ
６…デジタルフィルタ
７…０次ホールド
８…アナログＬＰＦ
９…時間遅れ要素
１０…減算器[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a digital / analog (hereinafter abbreviated as “D / A”) conversion apparatus used in a digital audio apparatus and a method of designing a digital filter for oversampling included in the apparatus.
[0002]
[Prior art]
In a digital audio reproducing apparatus such as a compact disc (CD) player, a D / A converter is used to convert a PCM digital signal into an analog signal. As this type of D / A converter, a so-called oversampling D / A converter has become mainstream.
[0003]
Oversampling is the interval between samples (digital data) obtained discretely at intervals corresponding to the original sampling frequency fs (for example, 44.1 kHz for a CD) by a digital filter provided in front of the D / A converter. The sample obtained by the arithmetic processing is inserted into (for example, one sample if double oversampling), and the sampling frequency is apparently increased to nfs (n: integer). When such oversampling is performed, the frequency band of the aliasing component that causes noise in the sense of hearing moves to a region that is much higher than the audible frequency band (DC to 20 kHz). This aliasing component can be easily removed with an analog filter. Also, the quantization noise has an advantage that the level becomes 1 / n as much as the frequency band is increased n times, and as a result, the noise power in the audible frequency band is reduced to 1 / n.
[0004]
As described above, various interpolation methods are known when sample interpolation is performed using a digital filter. A generally known method is called a Lagrange interpolation method, which obtains a t-order polynomial passing through t + 1 samples and obtains an interpolation value based on this. A digital filter using an interpolation method using a spline function has also been put into practical use. In any case, in the oversampling D / A converter, the characteristics of the digital filter greatly affect the audible characteristics of the audio signal after D / A conversion.
[0005]
[Problems to be solved by the invention]
By the way, the sampling frequency fs is usually determined to be about twice as high as the various experimental results that the upper limit of the human audible frequency band is around 20 kHz. Conventionally, when designing a digital filter as described above, for the same reason, a frequency characteristic (passband characteristic) as flat as possible in the frequency range of DC to 20 kHz can be obtained, and abruptly attenuated above 20 kHz. The parameters such as the order and the coefficient are determined so that the characteristics to be obtained can be obtained. However, in general, when a steep attenuation characteristic is obtained, the ripple in the passband tends to increase. Further, in the time response, ringing known as Gibbs phenomenon tends to occur, and it appears remarkably especially when the sound rises and falls sharply. In general, the sound of digital audio such as CD is often said to be “hard”, “strongly stimulated”, or “cold”, but it can be inferred that the above-mentioned phenomenon is one of the causes.
[0006]
The present invention has been made in view of such points, and the object of the present invention is to return a digital audio signal to an analog audio signal as faithfully as possible to the original sound at the time of recording or PCM conversion. An object of the present invention is to provide a digital / analog converter capable of obtaining a natural music signal in terms of audibility. Another object of the present invention is to provide a digital filter design method capable of easily designing a digital filter constituting such a digital / analog converter.
[0007]
[Means for Solving the Problems]
The digital filter included in the digital / analog conversion apparatus according to the present invention is designed by a method that is completely different from the conventional method that is designed by paying attention to frequency characteristics and the like. That is, in the conventional method, it is assumed that the continuous-time signal to be sampled is completely band-limited, but in reality, the influence of aliasing cannot be ignored. Therefore, in order to achieve better signal restoration, it is necessary to design not only the digital audio signal sample as a discrete-time signal but also an analog characteristic included in the inter-sample response. For this reason, the present inventor has continued research for many years on a sample value control theory capable of handling continuous-time characteristics, more specifically, an attempt to introduce ^H∞ control into a D / A conversion technique for handling digital audio signals. It was.
[0008]
In the course of that research, the present inventors have clarified a method for designing an FIR type digital filter using H ^∞ control (for example, “sample values H ^∞ , H ² / H of a ^multirate D / A converter). ^∞ design "(see Journal of the Institute of Systems, Control and Information Engineers, Vol. 11, No. 10, pp585-592, 1998). However, in order to guarantee the optimality in the entire frequency band, there is a restriction that an FIR type digital filter with a limited order is targeted. In addition, due to the limitation, it is necessary to solve by solving to one of the convex optimization problems in mathematical programming instead of the general H ^∞ control solution, but since this is high-dimensional, H ^∞ It was difficult to solve easily with a publicly available tool that gives a control solution. On the other hand, as a result of further research, the present inventor found that the optimal design problem for general IIR type filters as well as FIR type is not a problem of finite dimensional discrete time systems. As a result, we have obtained the knowledge that an effective approximate optimization solution can be obtained by introducing the FSFH (First Sample First Hold) method into the design of the sample value control system. According to this method, it is possible to easily design an IIR type digital filter that is not limited to the FIR type digital filter and generally requires only a small order. In addition, optimality in a general form without restriction on the order of the filter is guaranteed, and a general H ^∞ control theory solution can be applied without using a special solution. Therefore, it is possible to design using various software that is generally easily available.
[0009]
That is, the digital / analog conversion device according to the present invention is a D / A conversion device that converts a digital audio signal read or received from a recording medium into an analog audio signal. n times the sampling frequency by interpolating the sampling points (n is an integer of 2 or more) a digital filter for oversampling to, the digital filter is obtained by band-limiting the original analog signal w _c and analog / digital / error signal e _c of the analog signal obtained through the analog conversion so as to be smaller than mean the design standard value of the H ^∞ norm gamma, which will be described later is set in order to design an IIR filter an unlimited degree ( 3) to condition by an equation using the sample data H ^∞ control theory, first sample - Fast ho Applying a field approximation is converted into a finite-dimensional discrete-time system to derive an approximate calculation formula, the approximate equation based on the design standard value γ to set conditions so as to minimize or near an in Conditional Expression It is characterized by comprising an IIR type filter whose coefficient is a parameter calculated by solving the above by H ^∞ control.
[0010]
In addition, a digital filter design method according to the present invention provides a new method between time-sequentially adjacent samples in a D / A converter that converts a digital audio signal read or received from a recording medium into an analog audio signal. A method of designing a digital filter for performing an oversampling process for interpolating sampling points to increase the sampling frequency by n times (n is an integer of 2 or more), and band-limiting the original analog signal w _c analog / digital / error signal e _c of the analog signal obtained through the analog conversion so as to be smaller than the design reference value γ in the sense of H ^∞ norm was set in order to design an IIR filter that is unlimited in order to condition by below equation (3), using sample data H ^∞ control theory, first samples - Firth Is converted to finite-dimensional discrete-time system derives an approximation formula by applying a hold approximation, the approximate equation based on the design standard value γ to set conditions so as to minimize or near an in Conditional Expression It is characterized in that the coefficient of the IIR filter is calculated by solving the above by H ^∞ control.
[0011]
DETAILED DESCRIPTION OF THE INVENTION
The principle of the digital filter design method in the present invention will be described below. That is, the following description makes clear the conditional expression set for designing the IIR filter and a method for approximately converting the conditional expression to a finite-dimensional discrete-time system.
[0012]
(1) Design Proposition Setting FIG. 1 is a block diagram showing a signal restoration system model including a D / A converter according to the present invention. Input w _c is a continuous-time signal, after being band-limited by an anti-aliasing filter 2 included in the A / D converter 1, is sampled at the sampler 3 at sample period h a discrete-time signal y _d. In oversampling type D / A converter 4, first, it converts the discrete-time signal y _d by the up-sampler 5 into discrete-time signal x _d sample period h / M. At this time, 0 signal y _d is inserted as in the following equation (1).
[Expression 1]

Subsequently, the signal _xd is processed by the digital filter 6 expressed by K (z), so that the inserted 0 signal is corrected to an appropriate value, and the 0th-order hold 7 operating at the period h / M. continuous time the signal _{u c} by. Finally, to obtain a smoothed restored signal _{z c} The signal _{u c} by an analog low-pass filter (LPF) 8.
[0013]
In the audio field, a time delay is allowed for signal restoration in the D / A converter 4. Therefore, when an error system model for signal restoration is considered in consideration of this time delay, it is as shown in FIG. In FIG. 2, the lower signal path is a signal processing system by the sampler 3 and the D / A converter 4, and the upper signal path is a delay system in consideration of a time delay due to the signal processing system. The time delay element 9 gives the band limit signal y _{c the} time delay mh due to the signal processing described above, and the subtractor 10 extracts the error signal e _c between the restored signal z _c and the delayed band limit signal. Since this error signal e _c is also a continuous time signal, put as the following equation (2).
e _c (t) = z _c (t) −y _c (t−mh) (2)
[0014]
In the present invention, constitute an IIR type digital filter to be as small as possible is the error signal e _c. That is, the IIR digital filter is designed under the condition that a stable continuous-time filter (anti-aliasing filter 2 and analog LPF 8) and positive integers m, N, and M are given. Therefore, when a system for converting the continuous time signal w _c to the error signal e _c is set as T _ew , K (z) satisfying the following expression (3) is obtained for a given γ.
[Expression 2]

That is, this equation (3) is a conditional equation set to design the IIR filter described above. Here, γ dominates the magnitude of the error, and the smaller the better. In ^H∞ control, a method of minimizing this by repeated calculation is employed.
[0015]
(2) Conversion to a single rate system First, the system including the upsampler and time delay system (multirate system) shown in FIG. 2 is converted to a finite dimension system (single rate system) having a single sample period. . For this purpose, discrete time lifting L _M and inverse lifting L _M ⁻¹ defined by the following equation (4) are introduced.
[Equation 3]

However, ↓ M is a down sampler and is defined by the following equation (5).
[Expression 4]

[0016]
The discrete time lifting and inverse lifting are used to convert K (z) (↑ M), that is, processing by the upsampler 5 and the digital filter 6.
[Equation 5]

Here, K ′ (z) is a 1-input M-output linear time-invariant (LTI) system, and the relationship with K (z) is given by the following equation (7).
[Formula 6]

[0017]
Next, a generalized hold <H _h > ′ represented by the equation (8) is introduced.
[Expression 7]

[0018]
At this time, the following equation (9) holds.
[Equation 8]

From the equations (6) and (9), the following equation (10) is established.
[Equation 9]

That is, the multi-rate model shown in FIG. 2 is equivalently converted to the single-rate model shown in FIG.
[0019]
Subsequently, in order to make e- ^mhs which is a continuous time dead time element into a finite dimension, a conversion is performed so as to delay the input of the system by m steps. This translates the desired design problem into a problem of designing a causal filter z ^−m K ′ (z) instead of K ′ (z). This can be further reduced to a design problem of a finite-dimensional discrete-time system. For details of the method, see Carnecar, Yamamoto: “Delayed Signal Reconstruction Using Sampled Data Control”, Proceedings of 35th Conference on Decision and Control, pages 1259 to 1263. (1996) (PPKhargonekar and Y. Yamamoto: “Delayed signal reconstruction using sampled-data control”; Proc. Of 35th Conf. On Decision and Control, pp. 1259-1263 (1996)). However, although this is not approximated, it must be noted that the following severe constraints are imposed and the assumption is not satisfied by the design problem of this digital filter. Therefore, a more practical conversion method to a discrete time problem is required.
[0020]
(3) Conversion to discrete time system In view of the above circumstances, the original problem can be reduced to an approximate discrete time system design problem without constraints. For this purpose, the FSFH (First Sample First Hold) method is applied here. The FSFH method is a method for evaluating the performance of the sample value control system, and the continuous time input / output of the sample value system of h period is discretized by a sampler and hold that operates at a period of h / N (N is a natural number). This is a method of approximating a continuous-time signal with a discrete-time signal for a large N. For details of the FSFH method, see Yamamoto, Madievski, Anderson: "Computation and Convergence of Frequency Response Beer Fast Sampling for Sampled-Data Control Systems", Proceedings of 36th Conference on Decision and Control, pages 2157 to 2162 (1997) (Y. Yamamoto, AGMadievski and BDOAnderson: "Computation and convergence of frequency response via fast sampling for sampled-data control systems"; Proc. Of 36th Conf. On Decision and Control, pp. 2157-2162 (1997)).
[0021]
FIG. 4 is a redraw of FIG. 3 into a generalized plant format for design purposes. The sample value system g _s shown in FIG. 4 is defined by the following equation (11).
[Expression 10]

If the FSFH method is used, the approximate discrete time system of this sample value system g _s is given by the following equation (12). However, N = Ml (l is a natural number) in FSFH.
[Expression 11]

Each matrix and operator of G _dN (z) is defined as follows.
[Expression 12]

[0022]
The above equation (3) is approximated by equation (13) using the above approximate discrete time system _GdN, and _obtaining K (z) that satisfies equation (3) is approximately that of a finite-dimensional discrete time system. The problem is reduced.
[Formula 13]

However,
[Expression 14]

It is. That is, FIG. 4 is converted into the finite-dimensional discrete time system shown in FIG.
[0023]
It should be noted here that a method for calculating a discrete time system equivalent to the sample value system of FIG. 4 and the H ^∞ norm upper bound under the condition ‖D ₁₁ ‖ <γ has been proposed (Fujioka, Usui). Yamamoto: “Multi-rate filter bank sample value H∞ design-M channel case”, 27th Control Theory Symposium (1998)), here, unlike normal sample value ^H∞ control, 制 御 D ₁₁ ‖ <γ is a very strong constraint in equation (3). This is the reason why the FSFH approximation as described above is required (and it is difficult to apply the conversion method to the discrete time system without approximation described in the above-mentioned papers by Cargo Necker and Yamamoto). However,
[Expression 15]

(A _F , B _F , C _F ) is an implementation of F (s) of the anti-aliasing filter 2.
[0024]
Thus, the desired IIR filter K (z) can be obtained by obtaining Equation (13) and solving a very general discrete-time ^H∞ control problem. The digital filter design method according to the present invention is realized by executing a predetermined program on a computer. Some of the programs can use existing programs as will be described later.
[0025]
Next, an example of a specific filter design according to the above method according to the present invention will be described.
First, in the configuration shown in FIG. 1, the characteristics F (s) and P (s) of the filters 2 and 8 are determined. Here, the characteristic of the anti-aliasing filter 2 is F (s) = 1 / (0.70223s + 1) (7.0223s + 1).
And The sample period is normalized and h = 1. Therefore, the Nyquist frequency is π [rad / sec] = 0.5 [Hz]. The frequency characteristics of this filter are shown in FIG. The characteristic is attenuated by a primary characteristic from 0.142 [rad / sec], and further attenuated by a secondary characteristic from 1.42 [rad / sec]. When this is converted to a sampling frequency fs of CD = 44.1 kHz, it corresponds to a filter that attenuates from 1 kHz with a primary characteristic and further attenuates from 10 kHz with a secondary characteristic.
[0026]
The CD has an audio frequency band up to about 20 kHz, but the frequency spectrum of the music actually recorded is not flat within this band but is attenuated in the high frequency range. In fact, even if a wide-band source such as an orchestra, which is said to be excellent recording, hardly has an almost flat extension up to 20 kHz, it has a characteristic that it gently attenuates above 10 kHz. Therefore, the anti-aliasing filter having the above characteristics is considered to reflect the natural energy distribution of the music signal. That is, it is more natural to design a digital filter corresponding to the CD reflecting the weighting for such a frequency distribution, instead of using a completely band-limited signal as a model of the original music signal as in the prior art. It can be said that this is an approach. In order to simplify the calculation, the characteristic of the LPF 8 for smoothing the signal after the hold is P (s) = 1. In practice, a first-order lag type LPF or the like is suitable, but even in this case, the attenuation characteristic is moderate, so that the frequency characteristic is hardly affected and a result similar to this example is obtained.
[0027]
Under the above conditions, the digital filter is calculated with the upsampling order M = 4 (that is, four times oversampling), the step number N = 4 for approximation by FSFH, and the delay step number m = 2 allowed for signal restoration. went.
First, F (s) has the following state space expression.
[Expression 16]

Where
[Expression 17]

It is.
[0028]
Also, using equation (12-2)
[Formula 18]

And from now on
[Equation 19]

Is obtained. That is, this is an expression corresponding to the generalized plant g _s of the error system in Fig.
[0029]
Since all parameters such as A _c and B _c1 are obtained in this way, G _dN (z) is obtained from the following equation (12-1). This calculation can be performed using existing CAD such as Matlab. Furthermore, a solution is obtained using discrete time ^H∞ control according to FIG. This calculation can also be performed using, for example, Matlab's MuTools or Robust Control Toolbox.
[0030]
The transfer function K _SD (z) of the IIR type digital filter obtained in this way is expressed by the following equation.
[Expression 20]

When the impulse response of this digital filter is calculated up to n = 32, the following equation (14) is obtained.
[Expression 21]

[0031]
The calculation results of the characteristics of this filter are shown in FIGS. 7 is a frequency characteristic of gain, FIG. 8 is a frequency characteristic of phase, FIG. 9 is an impulse response, FIG. 10 is a square wave pulse response, FIG. 11 is a sine wave response (sin 0.5 t), and FIG. .8t). Although this digital filter is an IIR type, the impulse response in FIG. 9 is actually 0 after the 25th step, which can be realized by an FIR type filter of about 25 taps cut off by n = 25 or 26. It means that. This is a very advantageous property when actually configured by hardware. In addition, it can be seen that the filter of this example has a good response in a square wave and a sine wave.
[0032]
For comparison, the characteristics of a conventional digital filter were also calculated by a computer. This filter is known as a so-called Johnston filter. The Johnston filter for 4 times upsampling can be calculated using the Johnston filter K _J (z) for 2 times upsampling:
K _J4 (z) = K _J (z) K _J (z ² )
The Johnston filter K _J (z) employed here has 16 taps, and the Johnston filter K _J4 (z) for upsampling 4 times has 48 taps.
[0033]
The calculation results of the characteristics of this Johnston filter are shown in FIGS. 13 is a gain frequency characteristic, FIG. 14 is a phase frequency characteristic, FIG. 15 is an impulse response, FIG. 16 is a square wave pulse response, FIG. 17 is a sine wave response (sin 0.5 t), and FIG. .8t).
[0034]
As can be seen from a comparison between FIGS. 10 and 16, the digital filter according to the present invention has a smaller ringing at the rising and falling times than the Johnston filter. Further, as can be seen from a comparison between FIG. 9 and FIG. 15, the digital filter according to the present invention has small ringing even in the impulse response. As described above, such a difference is considered to particularly affect the “hardness” and “stimulus strength” of sound. Further, as shown in FIG. 8, the phase characteristic of the digital filter according to the present invention is substantially linear (that is, the group delay is constant) up to the vicinity of the Nyquist frequency π. As can be seen from a comparison between FIGS. 11 and 12 and FIGS. 17 and 18, the Johnston filter produces a large distortion, whereas the digital filter according to the present invention hardly shows any distortion.
[0035]
Further, FIG. 19, the number of frequency characteristics of the gain of the error signal e _c as shown in FIG. 2 is a graph showing the the above Examples and Johnston filter. It can be said that the smaller this error signal (the lower the position in FIG. 19), the more faithfully reproduced it is with the original sound (strictly speaking, the output of the anti-aliasing filter). If used, it can be seen that the fidelity of the reproduced sound is high in a wide frequency band. This reflects the difference in time response between the filter according to the present invention and the conventional Johnston filter.
[0036]
The above is the calculation result when the filter is configured by software by a computer. When this filter is implemented by hardware, a known configuration may be used (for example, Mitani: “Digital Filter Design”) (See Tsujido (1987)). Actually, it can be easily configured with a general-purpose digital signal processor (DSP) or a one-chip LSI.
[0037]
【The invention's effect】
As described above, according to the digital / analog conversion device according to the present invention, the restoration error of the analog signal by the digital / analog conversion is guaranteed to be reduced over a wide frequency band. Or an audio signal faithful to the sound at the time of recording). In particular, it is possible to obtain a playback sound that is very smooth in terms of hearing and whose treble is very straightforward.
[0038]
Moreover, according to the digital filter design method of the present invention, it is possible to easily design a digital filter having high performance as described above using an existing tool or the like, and to reduce the order of the filter. This makes it easy to configure the filter.
[Brief description of the drawings]
FIG. 1 is a block diagram showing a signal restoration system model including a D / A converter according to the present invention.
FIG. 2 is a block diagram showing an error system model for the signal restoration system model of FIG. 1;
FIG. 3 is a block diagram when the error system model of FIG. 2 is converted into a single rate system model.
FIG. 4 is a block diagram of a general plant format of an error system.
FIG. 5 is a block diagram when the format of FIG. 4 is converted into a finite-dimensional discrete-time system.
FIG. 6 is a diagram showing frequency characteristics of an anti-aliasing filter when designing a digital filter according to an embodiment of the present invention.
FIG. 7 is a graph showing frequency characteristics of gain of an IIR digital filter according to an embodiment of the present invention.
FIG. 8 is a diagram showing the frequency characteristics of the phase of this IIR type digital filter.
FIG. 9 is a view showing an impulse response of this IIR type digital filter.
FIG. 10 is a diagram illustrating a square wave pulse response of the IIR type digital filter.
FIG. 11 is a diagram showing a sine wave response (sin 0.5 t) of the IIR type digital filter.
FIG. 12 is a diagram showing a sine wave response (sin 0.8 t) of this IIR type digital filter.
FIG. 13 is a diagram showing frequency characteristics of gain of a conventional Johnston filter.
FIG. 14 is a diagram showing the frequency characteristics of the phase of this Johnston filter.
FIG. 15 is a diagram showing an impulse response of the Johnston filter.
FIG. 16 is a diagram showing a square wave pulse response of the Johnston filter.
FIG. 17 is a diagram showing a sine wave response (sin 0.5 t) of the Johnston filter.
FIG. 18 is a diagram showing a sine wave response (sin 0.8t) of the Johnston filter.
Shows a number of frequency characteristics of the gain of the error signal e _c with IIR type digital filter and conventional Johnston filter according 19 this invention.
[Explanation of symbols]
DESCRIPTION OF SYMBOLS 1 ... A / D conversion part 2 ... Anti-aliasing filter 3 ... Sampler 4 ... Oversampling D / A conversion part 5 ... Up sampler 6 ... Digital filter 7 ... 0th-order hold 8 ... Analog LPF
9 ... Time delay element 10 ... Subtractor

Claims (2)

In a D / A converter that converts a digital audio signal read from or received from a recording medium into an analog audio signal, a sampling point is newly interpolated between adjacent samples in time series to increase the sampling frequency by n times (n Is a digital filter that performs oversampling to an integer of 2 or more), and the digital filter is an error signal e between the band-limited original analog signal w _c and the analog signal obtained through analog / digital / analog conversion. _The following conditional expression set to design an IIR filter that does not limit the order so that _c is smaller than the design reference value γ in the sense of H ^∞ norm

However, T _ew is converted to a finite-dimensional discrete-time system by applying the first sample-first hold approximation to the transfer function of the system that converts the analog signal w _c to the error signal e _c using the sample value H ^∞ control theory. Then, an approximate calculation formula is derived, and parameters calculated by solving the approximate calculation formula by H ^∞ control based on conditions set so that the design reference value γ is minimized or close to the condition formula in the conditional formula A digital / analog converter comprising an IIR type filter as a coefficient . In a D / A converter that converts a digital audio signal read from or received from a recording medium into an analog audio signal, a sampling point is newly interpolated between adjacent samples in time series to increase the sampling frequency by n times (n It is a method of designing a digital filter for performing oversampling to an integer of 2 or more), which has been band-limited the original analog signal w _c and the analog signal obtained through an analog / digital / analog converter the error signal e _c to less than mean the design standard value of the H ^∞ norm gamma, following condition set in order to design an IIR filter that is unlimited in order

However, T _ew is converted to a finite-dimensional discrete-time system by applying the first sample-first hold approximation to the transfer function of the system that converts the analog signal w _c to the error signal e _c using the sample value H ^∞ control theory. Then, an approximate calculation formula is derived, and the coefficient of the IIR filter is solved by solving the approximate calculation formula by H ^∞ control based on the condition set so that the design reference value γ is minimized or close to the condition formula. A method of designing a digital filter, wherein

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