搬送波周波数オフセット存在下のグラントフリーアクセスにおける短パイロットに基づくアクティブユーザ数推定

搬送波周波数オフセット存在下の グラントフリーアクセスにおける 短パイロットに基づく アクティブユーザ数推定 原 郁紀 東京理科大学 創域理工学部 電気電子情報工学科 Email: [email protected] T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

研究背景 2 n ファクトリーオートメーションや自動運転などでは， 低遅延性と多数同時接続性を満たすことが求められる[1] n 膨大な数の無線端末が散発的に通信を行うことが想定され， ランダムアクセスに基づいた伝送法が適切とされている[2] グラントフリー非直交多元接続(GF-NOMA)に注目[3] グラントの手続きなしに 即座にデータを送信 基地局 手続きの省略により 通信遅延を大幅に削減 [1] X. Chen et al., IEEE J. Sel. Areas Commun., vol. 39, no. 3, pp. 615–637, Mar. 2021. [2] L. Liu et al., IEEE Signal Process. Mag., vol. 35, no.5, pp. 88–99, Sep. 2018. [3] M. B. Shahab et al., IEEE Commun. Surveys Tuts., vol. 22, no. 3, pp. 1805–1838, 3rd Quart. 2020. T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

GF-NOMAの課題 3 n 基地局では，どのユーザがアクティブとなって伝送を 行っているのかを事前に把握していない – 高精度なアクティブユーザ検出が必須 – (高精度なチャネル推定，データ推定も求められる) n 一方，アクティブユーザ検出に関する多くの検討では， 各ユーザ-基地局間の完全な時間・周波数同期を仮定 – 安価な無線端末では，搬送波周波数オフセット(CFO)が 生じやすく，検出・推定の精度が劣化しやすい[4] 本研究では，CFO存在下のGF-NOMAを検討 [4] J. Xu et al., IEEE Internet Things J., vol. 5, no. 3, pp. 1449–1462, Jun. 2018. T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

CFO存在下のアクティブユーザ検出 4 n 最尤推定問題としてアクティブユーザ検出を行う手法[5,6] – CFOによる位相回転量の候補値を考慮し，既知のパイロット 系列からなる観測行列を拡張して定式化 – 更なる高精度化には，位相回転量の候補値数を十分に増やす 必要があり，計算量が増加しやすい n メッセージ伝播アルゴリズムに基づく手法[7,8] – 規格で許容される同期誤差の範囲を考慮し，効率的に推定可能 – ユーザのアクティブ率といった事前情報が必要 本研究では，ユーザのアクティブ率 (アクティブユーザ数)の推定を検討 [5] T. Hara et al., in Proc. IEEE ICC Workshops 2022, Virtual Conference, Jun. 2020. [6] W. Liu et al., in Proc. IEEE ICC2022., Seoul, Korea, Republic of, May 2022. [7] K. Ueda et al., in Proc. IEEE PIMRC2022, Kyoto, Japan, Sep. 2022. [8] G. Sun et al., IEEE Trans. Wireless Commun., vol. 21, no. 5, pp. 3365–3380, May 2022. T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

従来のアクティブユーザ数推定 5 n 従来手法１: 固有のパイロット系列を使用[9] – パイロット部の標本共分散行列の固有値を用いたランク推定 – アクティブユーザ数に対し，十分に長いパイロット系列および 多いアンテナ数が必要 (+固有値分解に要する計算量が増加) n 従来手法２: 共通パイロットを使用[10,11] – 全ユーザ共通の非常に短いパイロット系列を用いる – [10]では，共通パイロットと直交する系列を使用 – [11]では，最尤推定問題として定式化して解く – 要する計算量も低く，効率的なアクティブユーザ数推定が可能 [9] X. Shao et al., IEEE Trans. Signal Process., vol. 68, pp. 420–435, Dec. 2019. [10] H. Han et al., IEEE Internet Things J., vol.7, no.4, pp.3602–1462, Apr. 2020. [11] M. Zhu et al., “Rethinking grant-free protocol in mMTC,” Apr. 2024. https://arxiv.org/abs/2404.16152 T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

• https://arxiv.org/abs/2404.16152

従来のアクティブユーザ数推定 6 n 従来手法１: 固有のパイロット系列を使用[9] – パイロット部の標本共分散行列の固有値を用いたランク推定 – アクティブユーザ数に対し，十分に長いパイロット系列および 多いアンテナ数が必要 (+固有値分解に要する計算量が増加) n 従来手法２: 共通パイロットを使用[10,11] – 全ユーザ共通の非常に短いパイロット系列を用いる – [10]では，共通パイロットと直交する系列を使用 – [11]では，最尤推定問題として定式化して解く – 要する計算量も低く，効率的なアクティブユーザ数推定が可能 これらの手法は理想的な時間・周波数同期を仮定 T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

研究目的・方策 7 n 研究目的 – CFO存在下における効率的なアクティブユーザ数推定の実現 n 方策 – 文献[12]で提案したアクティブユーザ数推定法(Eig-sum)の NRMSE (Normalized Root-Mean-Square Error)特性を解析 – 解析結果より，共通パイロットではなく各要素が定振幅かつ 各要素の位相がランダムな系列を用いることを提案 数値結果により，共通パイロットを用いた従来手法よりも 高精度なアクティブユーザ数推定が可能であることを示す [12] 原 郁紀，信学技報, vol. 124, no. 166, RCS2024-126, pp. 54–59, 2024年8月. T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

システムモデル 8 n 上りリンクのGF-NOMA (時間同期は理想的) 基地局 (アンテナ数 M ) <latexit sha1_base64="wDC7kIYftq/mTFohBJsSlOOUmzg=">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</latexit> K アクティブユーザ <latexit sha1_base64="JG143GNTrMJ/EnwGJ3tJRl51rcs=">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</latexit> N ユーザ <latexit sha1_base64="U51Z7kd9AI4n6KHKN4aZM3/1bOk=">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</latexit> 共通パイロット s 固有パイロット <latexit sha1_base64="DCEPUSh/XGRfz6NtyhvIbssAAMw=">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</latexit> データ – 本研究では，共通パイロット部のみを考慮 – 伝送時間内においてアクティブとなるパターンとチャネルは 変動しないものとする T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

受信信号モデル

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n 基地局における共通パイロット部の受信信号
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– A : アクティブユーザの集合
送信電力制御
– pn : 送信電力
– n : ラージスケールフェーディング係数
– hn ⇠ CN (0M , IM ) : チャネルベクトル
– Z 2 C2⇥M : AWGNで各要素 i.i.d.の CN (0, z2 ) に従う
– !n 2 [ 2⇡✏max , 2⇡✏max ] : CFOによる回転量(一様分布を仮定)
– ✏max : 最大正規化CFO
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T. Hara (TUS)

第47回情報理論とその応用シンポジウム(SITA2024)

2024/12/13

本稿で解析する推定手法 (Eig-sum)[12]

10

n Eig-sumでは，受信信号の標本共分散行列 R = YY H /M の
固有値 max , min (< max )を用い，次式により K を推定
$
'
⇡
R1 + R2
max + min
2
2
K̂ =
K̂
=
z
z
2
2
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固有値の和がトレースと等しいため，
Rの対角成分( R1 , R2 )を用いて書き換え
<latexit sha1_base64="PgjAq+I8jG3o2RlmFg8pFhviEw0=">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</latexit>

<latexit sha1_base64="2RfhNMIlpjmXZzBungMxb8SzKDs=">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</latexit>

(
<latexit sha1_base64="UfFmJrGMAh28UniGlbs0VXeFBwU=">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</latexit>

R1 =

ky1 k22

M
ky2 k22
R2 = M

<latexit sha1_base64="6h2sxY7DDk7A7FzsDU+iuzW5VMo=">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</latexit>

y1 =

X

hT
n + z1 ,

y2 =

n2A

X

ej!n hT
n + z2

n2A

受信信号 Y の1行目と2行目に対応
<latexit sha1_base64="9x4A3N1OSozKav43qsLLT1GbQhA=">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</latexit>

[12] 原 郁紀，信学技報, vol. 124, no. 166, RCS2024-126, pp. 54–59, 2024年8月.
T. Hara (TUS)

第47回情報理論とその応用シンポジウム(SITA2024)

2024/12/13

NRMSE特性の解析(1/3) 11 n 文献[10]と同様に，次式で定義されるNRMSE特性を解析 r h r h i i 1 1 2 NRMSE = E (K̂ K) = E K̂ 2 K2 K K <latexit sha1_base64="iotJMQCOhEJt0kedA6t7kqSIQp4=">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</latexit> <latexit sha1_base64="+H16jIP0m49lP0syQIj9pF6Z7go=">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</latexit> ただし，K̂ に含まれる四捨五入の操作は無視して解析を行う <latexit sha1_base64="+H16jIP0m49lP0syQIj9pF6Z7go=">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</latexit> n まず，K̂ の期待値について考える – y1とy2 が CN (0M , (K + z2 )IM ) に従う変数とみなせるため， R1 とR2 は次式の確率密度関数を持つアーラン分布に従う M ◆M ✓ 2x M 1 K+ z x e M · f (x) = K + z2 (M 1)! <latexit sha1_base64="Dap0P8Ooo2MOEx6JxoCjzplbm58=">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</latexit> <latexit sha1_base64="ab0m6RnzL/usZ2Z9dp4BZ5jA1/k=">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</latexit> <latexit 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sha1_base64="jRiEtBB2EPQzDE3E5bbR0wKoUi8=">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</latexit> <latexit sha1_base64="VGjrEh54E9IeKKCVenvgGS1OvXw=">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</latexit> <latexit sha1_base64="ArvqnIjS/R6dDIejrFPilMltAcI=">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</latexit> 2 E[K̂] = K が得られる z であり， <latexit sha1_base64="Xyo/I9VjSVKvWVh0M4UnXxlw2gQ=">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</latexit> E[R1 ] = E[R2 ] = K + <latexit sha1_base64="nFSks1ImAhyRJwL0rBK/kj4Skbs=">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</latexit> [10] H. Han et al., IEEE Internet Things J., vol.7, no.4, pp.3602–1462, Apr. 2020 T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

NRMSE特性の解析(2/3) 12 n R12 , R22 , R1 R2 の期待値を求めることで， 最終的に次式が得られる(詳細な導出は割愛) s 1 K + K(K 1)↵2 + (K + NRMSE = K 2M <latexit sha1_base64="B6OnGWSP1u1RSh3xp4OxEY9ZJi4=">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</latexit> <latexit sha1_base64="glHEIc/h4yB3xgVhQj/ROaPktuk=">AAAEQHicjVFfa9NQFD9t/TPrn236IvgSHZWUreMmyBRhMB0DpXRsnd0GS1Nusts2LGmy5LZ0u9wv4BfwwScFEfFb6ItfwIf5CRQfRCYI4oOnSarMMesNSX73nPP7nXN/1wpcJ+KEHGSyuVOnz5wdO5c/f+HipfGJycvrkd8NbVazfdcPNy0aMdfpsBp3uMs2g5BRz3LZhrWzOMhv9FgYOX7nEd8LWN2jrY7TdGzKMdSYzHwoGJz1uSKWq5W1JTlvNENqC02KslSMaDfkwvAob1uWWJKGy5p8SzXalGO6VC6auhE6rTavy3yhqfaL83GFmmhUUGPaiJyWRxuJSOiJfSlNXSasollRDHvb50pC6JuiUtKkwkxRGimh9KUUKtYXr2PzOGdTVywuSzUZuCmIbKDCjDqUMIUuj8gUh5UP48piqoPJ/zYj9eK3EaUyzpYvVJNempxJkY5osK8OYH50mzhSni6rZTyhYlA3aFNTn078PcGToau6FLpSkbIxMUVmSbyU40BLwRSka8WfeAkGbIMPNnTBAwYd4IhdoBDhswUaEAgwVgeBsRCRE+cZSMgjt4tVDCsoRnfw28LdVhrt4H6gGcVsG7u4+IbIVKBA3pNX5JC8I6/JJ/LzRC0Rawxm2cO/lXBZ0Bh/fHXt+0iWh38O7T+sf87MoQl34lkdnD2II4NT2Am/t//kcO1utSBukufkM87/jByQt3iCTu+b/WKVVZ9CHi9A+9vu42Bdn9XmZudWb00t3E+vYgyuwQ1Q0e/bsAAPYAVqYGfvZVvZILube5P7mPuS+5qUZjMp5wocWbkfvwDUxjxR</latexit> 2 )2 z <latexit sha1_base64="NG0laOcYmFCwWehN34bckr8TvcY=">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</latexit> ただし，↵ = sinc(2⇡✏max ) であり，区間 [ 2⇡✏max , 2⇡✏max ] の 一様分布に従う確率変数の特性関数 n ✏max = 0.5 のとき，↵ = 0 となり，NRMSEが最小 <latexit sha1_base64="UHbERfJiSoupySsXBcoHvnnkTCc=">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</latexit> <latexit sha1_base64="T0HdsaiLw2SXojVk2bCgDNKThgQ=">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</latexit> <latexit sha1_base64="dGQbmnurvNoNNTpU3aJlu9rpuGk=">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</latexit> <latexit sha1_base64="uZnko15+Us8ExPUm5lTh+uK6GNo=">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</latexit> 区間 [ ⇡, ⇡] の一様な位相回転が生じる場合に Eig-sumは最小のNRMSEを達成可能 T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

NRMSE特性の解析(3/3) 13 n R12 , R22 , R1 R2 の期待値を求めることで， 最終的に次式が得られる(詳細な導出は割愛) s 1 K + K(K 1)↵2 + (K + NRMSE = K 2M <latexit sha1_base64="B6OnGWSP1u1RSh3xp4OxEY9ZJi4=">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</latexit> <latexit sha1_base64="glHEIc/h4yB3xgVhQj/ROaPktuk=">AAAEQHicjVFfa9NQFD9t/TPrn236IvgSHZWUreMmyBRhMB0DpXRsnd0GS1Nusts2LGmy5LZ0u9wv4BfwwScFEfFb6ItfwIf5CRQfRCYI4oOnSarMMesNSX73nPP7nXN/1wpcJ+KEHGSyuVOnz5wdO5c/f+HipfGJycvrkd8NbVazfdcPNy0aMdfpsBp3uMs2g5BRz3LZhrWzOMhv9FgYOX7nEd8LWN2jrY7TdGzKMdSYzHwoGJz1uSKWq5W1JTlvNENqC02KslSMaDfkwvAob1uWWJKGy5p8SzXalGO6VC6auhE6rTavy3yhqfaL83GFmmhUUGPaiJyWRxuJSOiJfSlNXSasollRDHvb50pC6JuiUtKkwkxRGimh9KUUKtYXr2PzOGdTVywuSzUZuCmIbKDCjDqUMIUuj8gUh5UP48piqoPJ/zYj9eK3EaUyzpYvVJNempxJkY5osK8OYH50mzhSni6rZTyhYlA3aFNTn078PcGToau6FLpSkbIxMUVmSbyU40BLwRSka8WfeAkGbIMPNnTBAwYd4IhdoBDhswUaEAgwVgeBsRCRE+cZSMgjt4tVDCsoRnfw28LdVhrt4H6gGcVsG7u4+IbIVKBA3pNX5JC8I6/JJ/LzRC0Rawxm2cO/lXBZ0Bh/fHXt+0iWh38O7T+sf87MoQl34lkdnD2II4NT2Am/t//kcO1utSBukufkM87/jByQt3iCTu+b/WKVVZ9CHi9A+9vu42Bdn9XmZudWb00t3E+vYgyuwQ1Q0e/bsAAPYAVqYGfvZVvZILube5P7mPuS+5qUZjMp5wocWbkfvwDUxjxR</latexit> 2 )2 z n CFOの回転量によらず，最小のNRMSEを達成するために 共通パイロットではなく，各ユーザで独立な系列を使用 ① 各要素の絶対値が1 ② 各要素の位相が互いに独立の位相が 区間 [0, 2⇡) の一様分布に従う <latexit sha1_base64="Mgx6kitNC1S5tNiKX5FIWWpzSoY=">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</latexit> 共通パイロット 固有パイロット 固有パイロット sn s データ T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) <latexit sha1_base64="Aw2McXyxwl89KgFwGYTpFCZDEOA=">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</latexit> <latexit sha1_base64="DCEPUSh/XGRfz6NtyhvIbssAAMw=">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</latexit> データ 2024/12/13

シミュレーション諸元 全ユーザ数 アクティブユーザ数 BSのアンテナ数 最大正規化CFO SNR 14 100 25 32 0.15 10 dB n 比較手法 – 直交系列を用いた手法[10](Orthogonal) – 最尤推定に基づいた手法[11](MLE) – 本研究で用いる手法[12](Eig-sum) n 評価指標: NRMSE [10] H. Han et al., IEEE Internet Things J., vol.7, no.4, pp.3602–1462, Apr. 2020. [11] M. Zhu et al., “Rethinking grant-free protocol in mMTC,” Apr. 2024. https://arxiv.org/abs/2404.16152 [12] 原 郁紀，信学技報, vol. 124, no. 166, RCS2024-126, pp. 54–59, 2024年8月. T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

• https://arxiv.org/abs/2404.16152

NRMSE特性(共通パイロットの場合) 最大正規化CFO対NRMSE BSのアンテナ数対NRMSE Eig-sumの解析結果とシミュレーション値が一致 T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13 15

最大正規化CFO対推定誤差特性 16 ランダムな系列を 用いることによる利得 T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

まとめ 17 n 研究目的 – CFO存在下における効率的なアクティブユーザ数推定の実現 n 方策 – 文献[12]で提案したアクティブユーザ数推定法の NRMSE (Normalized Root-Mean-Square Error)特性を解析 – 解析結果より，共通パイロットではなく各要素が定振幅かつ 各要素の位相がランダムな系列を用いることを提案 – ランダムな系列を用いることで共通パイロットを用いた 従来手法よりも優れた推定精度を達成 n 今後の課題 – 提案法を取り入れた高精度なアクティブユーザ検出の検討 T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

参考文献I 18 1. X. Chen et al., IEEE J. Sel. Areas Commun., vol. 39, no. 3, pp. 615–637, Mar. 2021. 2. L. Liu et al., IEEE Signal Process. Mag., vol. 35, no.5, pp. 88–99, Sep. 2018. 3. M. B. Shahab et al., IEEE Commun. Surveys Tuts., vol. 22, no. 3, pp. 1805–1838, 3rd Quart. 2020. 4. J. Xu et al., IEEE Internet Things J., vol. 5, no. 3, pp. 1449– 1462, Jun. 2018. 5. T. Hara et al., in Proc. IEEE ICC Workshops 2022, Virtual Conference, Jun. 2020. 6. W. Liu et al., in Proc. IEEE ICC2022., Seoul, Korea, Republic of, May 2022. 7. K. Ueda et al., in Proc. IEEE PIMRC2022, Kyoto, Japan, Sep. 2022. T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

参考文献II 19 8. G. Sun et al., IEEE Trans. Wireless Commun., vol. 21, no. 5, pp. 3365–3380, May 2022. 9. X. Shao et al., IEEE Trans. Signal Process., vol. 68, pp. 420–435, Dec. 2019. 10. H. Han et al., IEEE Internet Things J., vol.7, no.4, pp.3602– 1462, Apr. 2020. 11. M. Zhu et al., “Rethinking grant-free protocol in mMTC,” Apr. 2024. https://arxiv.org/abs/2404.16152 12. 原 郁紀，信学技報, vol. 124, no. 166, RCS2024-126, pp. 54–59, 2024年8月. T. Hara (TUS) 第47回情報理論とその応用シンポジウム(SITA2024) 2024/12/13

• https://arxiv.org/abs/2404.16152