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のための共通パイロット系列を用いたアクティブユーザ数推定に関する一検討,

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のための 共通パイロット系列を用いた アクティブユーザ数推定に関する一検討 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) 無線通信システム研究会(2024年8月) 2024/08/30

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) 無線通信システム研究会(2024年8月) 2024/08/30

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) 無線通信システム研究会(2024年8月) 2024/08/30

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[6,7] – 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) 無線通信システム研究会(2024年8月) 2024/08/30

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) 無線通信システム研究会(2024年8月) 2024/08/30

• 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) 無線通信システム研究会(2024年8月) 2024/08/30

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 – 短い共通パイロット系列および受信信号の標本共分散行列の 固有値を用いたアクティブユーザ数推定法を2つ提案 – 提案法1: 雑音分散の情報も利用し，高精度な推定 – 提案法2: 雑音分散ではなく最大正規化CFOの情報を利用 数値結果により，CFO存在下において従来手法よりも 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 T. Hara (TUS) 無線通信システム研究会(2024年8月) 2024/08/30

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) 無線通信システム研究会(2024年8月) 2024/08/30

受信信号モデル

9

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)

無線通信システム研究会(2024年8月)

2024/08/30

従来のアクティブユーザ数推定(1/2)

10

n 手法１: 共通パイロットと直交する系列を使用[10]
– 系列
<latexit sha1_base64="xQjdCXO9654ZJQmtT8iLGiEnjA4=">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</latexit>

s に直交する系列 s?を用いた次式により推定
(
<latexit sha1_base64="y+xbe2EeAH3k2kKEPeCoANxiyvE=">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</latexit>

K̂ =

<latexit sha1_base64="f8ouBOu6IkTj2qbxM8q/4R2iZ6Y=">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</latexit>

$

1

H
1

2

H
2

ksk42 M

'

<latexit sha1_base64="Kg30kL1YzLrPBUsn7aIzPPMWZ9o=">AAAFiHicjVFNTxNBGH6hVnH9APRi4qWBlECMZNoQUBMShAuJIeEblIHN7DLdTtgvd6ckMMwP0D/gwZMmxhgPXvXsxT/ggZ9gPGLixYNvd1tKZVuZpjvvPPM8z7wfVuiKWBJy3NObu5S/fKXvqnHt+o2b/QODt9bjoBbZfM0O3CDatFjMXeHzNSmkyzfDiDPPcvmGtTdXv9/Y51EsAn9VHoR822OOLyrCZhIhczA3VKRW4O7GBx5uioZVoU1V0kaRVplUT/Q0nRWOQ92KGwQRrUTMVpmCTHBHUY/JauSpea3vZ1DKWbpyu04repQcrYqKNT3aURMJaUGnqWEjhGtQizvCVzb2ItZGZjbTLZe2B5rwU02Fnx4sNYecEpXC43FhQRcozfIsd/Q0kcKj8CLeBuX+bjPxbm3vnv2ybhE6tQxnkFrFwvHYDiLmqdVhVqfLqexso4utkkamaVzzTOU3a7OZqx4jPX4eSRXihdajZwxtEdltXZSshoTA4w6rk8fGUiXOUibAaUnVMzWv1rvr63vNu2cah1PsmEu39y5mbw4Mk3GSrML5oNQIhqGxFoOB90BhFwKwoQYecPBBYuwCgxh/W1ACAiFi26AQizASyT0HDQZqa8jiyGCI7uHXwdNWA/XxXPeME7WNr7j4j1BZgCL5Tj6QE/KNfCQ/yJ+OXirxqOdygLuVanlo9r+8s/L7vyoPdwnVlqprzhIq8CDJVWDuYYLUq7BT/f7hq5OVR8tFNULekp+Y/xtyTL5iBf7+L/vdEl9+DQYOoPRvu88H6+Xx0uT45NLE8MxsYxR9cBeGYBT7PQUzMA+LsAZ27kXuU+5z7kveyJP8VP5hSu3taWhuQ9vKz/4FtHrNVQ==</latexit>

H
1⇥M
=
s
Y
2
C
1
H
1⇥M
=
s
Y
2
C
2
?
<latexit sha1_base64="pKxPR52oAvoBEHeafAsVHCSlJxY=">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</latexit>

※今回は s? =

n 手法２: 最尤推定に基づいて推定[11]
– 受信信号の共分散行列が ⌃ = KssH +
仮定し，次式の最尤推定問題を考える
<latexit sha1_base64="R/F8rIoVwf0SCe10aqysDlwhb5U=">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</latexit>

<latexit sha1_base64="ouG40lN1k1IQDlNglXt8fshJBcE=">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</latexit>

minimize log |⌃| + tr ⌃
K2[0,N ]

1

R



1
とする
1

2
z I2 であるものと

受信信号の標本共分散行列
YYH
2⇥2
<latexit sha1_base64="ZTwnxutSZlUVT9RY6Rlk3ndGAYU=">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</latexit>

R=

M

2C

[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)

無線通信システム研究会(2024年8月)

2024/08/30

• https://arxiv.org/abs/2404.16152

従来のアクティブユーザ数推定(2/2)

11

n 手法１: 共通パイロットと直交する系列を使用[10]
– 系列
<latexit sha1_base64="xQjdCXO9654ZJQmtT8iLGiEnjA4=">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</latexit>

s に直交する系列 s?を用いた次式により推定
<latexit sha1_base64="y+xbe2EeAH3k2kKEPeCoANxiyvE=">AAAE4HicjVHPaxNBFH5NVlvXH031IngplpQWMUxCqSIUar0UROjvVjvtMrudbIbuL3cngWY6d/EingQ9KYiIf4YX/wEPPXoUjxW8ePBlN00bm1Rn2Z033/u+N997a0eeSCQhBwO5vHHu/ODQBfPipctXhgsjV9eSsB47fNUJvTDesFnCPRHwVSmkxzeimDPf9vi6vfuglV9v8DgRYbAi9yK+5TM3EFXhMImQNZJrFqkdejvJno+bolFNaEuVtVmkNSbVQz1D54TrUq/qhWFMqzFzVE9BT3BbUZ/JWuyrea1v96BUeukq3Tqt6H56tKsq0XR/W02lpEc6s4aDEJ55zEDrndjCmjyOzm6nw+669ghe0seEflawt6xUIlyfbSNidUo1e3VQyWQnG+i4fqzHZ2hS9y0VUBGkqMM8dR/pydNYqggTWk+cKOiI2Omao2R1JIQ+d1mLPDmZKanNZQp0Wqqd6HlF61bu1lHuiabULPb1ctZ9/1feKoyREknX6Omg3A7GoL0WwsIHoLADIThQBx84BCAx9oBBgs8mlIFAhNgWKMRijESa56DBRG0dWRwZDNFd/Lp42myjAZ5bNZNU7eAtHr4xKkehSL6Sj+SQfCGfyHfyu28tldZoednD3c60PLKGn19f/vVPlY+7hNqx6kzPEqpwN/Uq0HuUIq0unEzfaL46XL63VFTj5B35gf7fkgPyGTsIGj+d94t86Q2Y+APKf4/7dLBWKZWnS9OLU2Ozc+1fMQQ34CZM4LzvwCzMwwKsgpP7lh/MF/Ijhm08M14YLzNqbqCtuQZdy3j9B1QRipI=</latexit>

K̂ =

<latexit sha1_base64="f8ouBOu6IkTj2qbxM8q/4R2iZ6Y=">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</latexit>

$

1

H
1

2

H
2

ksk42 M

'

CFOにより直交性が崩れ
推定精度が劣化

n 手法２: 最尤推定に基づいて推定[11]
– 最尤推定問題を解くことで K を推定
– 手法１よりもCFOによる推定精度の劣化はまだ少ない
<latexit sha1_base64="JsMzWIaCrHT4wDeFUEfiMX4t64Q=">AAAF03icjVJNT1NBFL3QVrF+ALoxcUMkJRBCM20IGpMmiBtMNeGrgDK0mfeYvo68L9+bEmGYjXFl4tqFK02MMf4MN/4AXfATjEtMTIwLb19LaeUVnKadO+eec+beOzV8W4SSkIO+/kQyde78wIX0xUuXrwwODV9dDb16YPKS6dlesG6wkNvC5SUppM3X/YAzx7D5mrF9r5Ff2+FBKDx3Re76fNNhliuqwmQSocpwwklnaI1JVdQFOicsi9pV2/MCWg2Yqajh2VvhroObon5N6IrK6ViwrKjDZC1w1LzWUzGUfJwu363Tiu5HR6OqQk33y2o6Ij3UzdKwZWFjvW0Kxp2my8JymC4UOzzaUddFkzRsUMvoXWnje7pNvx9de+pk4o2P4KWOm3t1hWNqWsXXclKWb8piZ/FIjxVoWHcqyqXCjVCT2eou0sOngVQ+JrQe7zA0RWB2TU+yOhI8h1usQZ6YaCqpwWUEtFuqdfS8onUjN3mUe6wpTWd61nLaff9nny5i0w/wD77FonE9w+TxUbi6kKc2r8rxYs9HDoRVkxNoYzHHYYVWm5EovmxeVk+OCz0yOEMxFSPRlaFRkiXRGjkZ5FrBKLTWgjf0AShsgQcm1MEBDi5IjG1gEOJnA3JAwEdsExRiAUYiynPQkEZtHVkcGQzRbfy18LTRQl08NzzDSG3iLTZ+A1SOQIZ8Ix/JIflCPpHv5E9PLxV5NGrZxd1oarlfGXx5ffnXmSoHdwm1Y9WpNUuowu2oVoG1+xHS6MJs6nf2Xh8u31nKqDHyjvzA+t+SA/IZO3B3fprvF/nSG0jjA+T+HffJYDWfzc1kZxanR2fnWk8xADfgJozjvG/BLMzDApTATHxN/E4mkslUKaVSz1MvmtT+vpbmGnSt1Ku/uzLqNA==</latexit>

<latexit sha1_base64="IOc0jmLyWE/chkyI8CvbvE3QYV8=">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</latexit>

<latexit sha1_base64="ouG40lN1k1IQDlNglXt8fshJBcE=">AAAJBnicvVVNaxNBGB4/E+NHW70IXoolIaGmTEJpRQjUilBJFW2tX5k2zG4nm7H75e6m2E7mLv4BD15UEBHBm3gVvPgHPPgTxKOCFw++O5ukSbNpgwc3JPvOO8/zvB/z7kZzTe4HGH87cPDQ4SNHE8ljqeMnTp4aGR07fcd3Gp7OVnTHdLx7GvWZyW22EvDAZPdcj1FLM9ldbeNKuH93k3k+d+zbwZbLVi1q2LzGdRqAqzp29EUqTeo0EGVZIvPcMIhZMx3HIzWP6oJojrnub1lwE8Stc1kVBRnrXBPEokHds8SClPkYSDGOV+zlSUGaaqnVhC9Jc01MK9B1GaUGJXMT8u1AwO4WXeaGRWWp3KXRsXoCTRI/hK6BdrXj35Yd+DUVds/OxAu33UtdkQdVBW2KpOJz6acVI1psL+7LTIn4DasqbMJt5dWpKS4D3H/kBcKFDSmzXYI69/Se7gW0AQDHYgYNwblcxCQaC5SjU1K9q+bbUoZ7k+29B5KQVHpgLnvFG05+z0MxYfDXqWrjYyDtLDmkD73LDzr3npYOqZ/v0ycmqwXNtq7PbV1mi8TlhLk+N+Fxi4g54nGjHjR7w2YIdV3PeTxebvVdlayJq1LJVtiaeLjTLRlprMag8jGwsG2LvckvdidfKhKDWhbdDZvsg4VRsuWBD5AKlwMZJVdqlaJI8SMRW1ZuH0ZMiTnZDpr5r1GjcS/3hFTDUpAwRP8QfQjy4ESGnqEh5mdndkjDXof/EAaPBSRTwRdurMrOqcNccItvM8hknJiO0Yx5JTcn2+jAk606+1EQutD9Bo0Kq45O4CmsrvF+o9AyJlDruumMvkEErSMH6aiBLMSQjQKwTUSRD58KKiCMXPCtIgE+Dyyu9hmSKAXcBqAYICh4N+DXgFWl5bVhHWr6iq1DFBO+HjDHURp/xW/xT/wFv8Pf8Z+BWkJphLlswV2LuMytjjw9u/x7X5YF9wDVd1h75hygGrqocuWQu6s8YRV6xN/cfvZz+dJSWmTwK/wD8n+Jv+HPUIG9+Ut/fYstPUcpOIDC7nb3G3eKU4WZqZlb0xNz862jSKJz6DzKQr9n0RxaQDfRCtITiUQ+MZOYTT5Jvk9+SH6MoAcPtDhnUM+V/PQX8xItmA==</latexit>

minimize log |⌃| + tr ⌃
K2[0,N ]

1

R

K̂ =

$

sH Rs
ksk42

2
z

ksk22

[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)

無線通信システム研究会(2024年8月)

2024/08/30

'

• https://arxiv.org/abs/2404.16152

提案手法の方針 12 n CFOの影響を考慮した受信信号の共分散行列を考える ⌃ = KssH + <latexit sha1_base64="R/F8rIoVwf0SCe10aqysDlwhb5U=">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</latexit> ˜ =E ⌃ <latexit sha1_base64="QWCeJoptZ5bFpeaU7hsH5E5ZaHs=">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</latexit> ⇥ H ym ym ⇤ = X 2 z I2 ⌧ (!n )⌧ (!n )H + n2A  K+ P = n2A e 2 z j!n ˜ の固有値を考える n 共分散行列 ⌃ <latexit sha1_base64="2NOLGu+2D5iRzE/0rh8upvZHOaY=">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</latexit> ( ⇤max = K + z2 + , ⇤min = K + z2 P 2 z I2 j!n e n2A K + z2 <latexit sha1_base64="oTNalaHEe/aVPRF6P66VBLDUL0o=">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</latexit> T. Hara (TUS) <latexit sha1_base64="atADps0eUt2nog19FwANxVhqzG4=">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</latexit> v CFOの影響を含む項 ! ! u X u X t j! = e n e j!n 無線通信システム研究会(2024年8月) n2A n2A 2024/08/30

提案手法1: 雑音分散が既知の場合

13

˜ の固有値の和を考えると…
n 共分散行列 ⌃
<latexit sha1_base64="2NOLGu+2D5iRzE/0rh8upvZHOaY=">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</latexit>

(
⇤max = K + z2 + ,
⇤min = K + z2
<latexit sha1_base64="oTNalaHEe/aVPRF6P66VBLDUL0o=">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</latexit>

<latexit sha1_base64="eI5eNVHYEMxfTG3vTjSuA6/wn6s=">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</latexit>

⇤max + ⇤min = 2 K +

2
z

n 上記の関係に基づき，次式により K を推定
<latexit sha1_base64="JsMzWIaCrHT4wDeFUEfiMX4t64Q=">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</latexit>

<latexit sha1_base64="T/UqkP87bMg30/VeirpaTfvLcUE=">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</latexit>

K̂ =

$

max +

min

2

2
z

'

<latexit sha1_base64="94LLbwkTrDhQmuswTuwSrfGB8Ng=">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</latexit>

max ,

min (<

max ) は

YYH
受信信号の標本共分散行列 R =
の固有値
M
T. Hara (TUS)

<latexit sha1_base64="mLe2E91Egy5I/KB388qE76fPLlg=">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</latexit>

無線通信システム研究会(2024年8月)

2024/08/30

提案手法2: 雑音分散が未知の場合(1/2)

14

˜ の固有値の差を考えると…
n 共分散行列 ⌃
<latexit sha1_base64="2NOLGu+2D5iRzE/0rh8upvZHOaY=">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</latexit>

(
⇤max = K + z2 + ,
⇤min = K + z2
<latexit sha1_base64="oTNalaHEe/aVPRF6P66VBLDUL0o=">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</latexit>

⇤max

⇤min = 2

<latexit sha1_base64="2jYSJjgv4bwNV795xf1sZ6H+SDY=">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</latexit>

n CFOを含む項である を近似
v
!
!
u
X
u X
=t
ej!n
e j!n
<latexit sha1_base64="wwvC+/zKcT8dAJv7PJLYv8JXZYc=">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</latexit>

<latexit sha1_base64="Q1xwgRx9JFcwLL+zGc+4G6dMZuE=">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</latexit>

n2A

v
u
u
= Kt
⇡K

p

T. Hara (TUS)

n2A

1 X j!n
e
K
n2A

!

E [ej! ] E [e j! ]

1 X j!n
e
K
n2A

!

（確率変数 ! の特性関数で表現）
<latexit sha1_base64="AdNNFYJX1DS8PrzGoo1EpyNgCvI=">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</latexit>

無線通信システム研究会(2024年8月)

2024/08/30

提案手法2: 雑音分散が未知の場合(2/2)

15

n 確率変数 ! の特性関数を考えると…
<latexit sha1_base64="AdNNFYJX1DS8PrzGoo1EpyNgCvI=">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</latexit>

<latexit sha1_base64="J+vOV918w0nm4fda13Bfhkq0E1s=">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</latexit>

⇥ j! ⇤
E e
=

1
4⇡✏max

Z 2⇡✏max

<latexit sha1_base64="yUdu9gPHulgqDEOm7VshvEr+SAw=">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</latexit>

j!

⇡K

e d!

2⇡✏max

p

E [ej! ] E [e j! ]

= K|sinc(2⇡✏max )|

= sinc(2⇡✏max )

! は区間 [ 2⇡✏max , 2⇡✏max ] の一様分布に従う
<latexit sha1_base64="B/W2k5D2Of4o2bxKMaWsoaoAy+4=">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</latexit>

※

<latexit sha1_base64="AdNNFYJX1DS8PrzGoo1EpyNgCvI=">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</latexit>

˜ の固有値の差の関係も考慮し，次式により K を推定
n⌃
<latexit sha1_base64="2NOLGu+2D5iRzE/0rh8upvZHOaY=">AAAFpnicjVFdTxNBFL1Qq1g/AH0x8YVISkoIZNoQNCZNEF8wRMNXAWVonV2m25H9cndKAsP8Af+ADz5pYojxZ/jii49q+AnGR0x88cG7u6W0soDTdOfOmXPuPfeO4dsilIQc9PRmLmQvXuq7nLty9dr1/oHBGyuh1wxMXjE92wvWDBZyW7i8IoW0+ZofcOYYNl81th5G96vbPAiF5y7LHZ9vOMxyRV2YTCJUG8xM5/K0waSa02U6IyyL2nXb8wJaD5ipqOHZm+GOg5uifkPomirqVLCqqMNkI3DUrNbjKZRSmq7UrdOK7sVHo65CTfeqajImPdaJNWxZ2Oi3TcG4M+mSsBymy3MdOdpRV6ExGkbUKuautfFd3aY/isueOZn0xEfwYkfl07rCMSWp0r2clJUSWeosnuqRMg2bTk25VLgxajJbPUB6+DKQyscLrQsdCU0RmF3Tk6yJBM/hFovIo6OJkhpcxkC7pUZHz8taR3djR3fPNKW5/Klezqr3f+lzVAp7k6uUh48ezGKOw8ot5zavy0K6E15VL45raxoIqyFHz1GMp0h0bWCYTJB4DZ0Miq1gGFpr3hvYBwqb4IEJTXCAgwsSYxsYhPhbhyIQ8BHbAIVYgJGI7zloyKG2iSyODIboFn4tPK23UBfPUc4wVptYxcZ/gMohyJOv5AM5JJ/JR/KD/Dk1l4pzRF52cDcSLfdr/a9uLf0+V+XgLqFxrDrTs4Q63Iu9CvTux0jUhZnot3dfHy7dX8yrEfKO/ET/b8kB+YQduNu/zPcLfPEN5PABiv+O+2SwUpooTk1MLUwOT8+0nqIPbsMdKOC878I0zMI8VMDM7Ge+ZL5lvmcL2SfZSnY1ofb2tDQ3oWtln/8F4iHahQ==</latexit>

<latexit sha1_base64="JsMzWIaCrHT4wDeFUEfiMX4t64Q=">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</latexit>

⇤max
<latexit sha1_base64="2jYSJjgv4bwNV795xf1sZ6H+SDY=">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</latexit>

⇤min = 2
<latexit sha1_base64="vHJtV1RN3V7Y0Nb/t+90tgfUgGQ=">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</latexit>

K̂ =

T. Hara (TUS)

$

max

min

2 |sinc(2⇡✏max )|

無線通信システム研究会(2024年8月)

'
2024/08/30

シミュレーション諸元 16 100 25 32 0.15 10 dB 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 n 比較手法 – 直交系列を用いた手法(Orthogonal, 雑音分散が未知) – 最尤推定に基づいた手法(MLE, 雑音分散が既知) n 評価指標: 正規化推定誤差 <latexit sha1_base64="EEtBTi0IfemW2i3ZGsMvYBfyiT0=">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</latexit> EK = E T. Hara (TUS) " |K̂ K| K # 無線通信システム研究会(2024年8月) 2024/08/30

最大正規化CFO対推定誤差特性 17 CFOを考慮した利得 (雑音分散が未知の場合) CFOを考慮した利得 (雑音分散が既知の場合) T. Hara (TUS) 無線通信システム研究会(2024年8月) 2024/08/30

BSのアンテナ数対推定誤差特性 18 アンテナ数増加に伴い 優劣関係が逆転 T. Hara (TUS) 無線通信システム研究会(2024年8月) 2024/08/30

アクティブユーザ数対推定誤差特性 アクティブユーザ数に対して 推定誤差がほぼ一定 T. Hara (TUS) 無線通信システム研究会(2024年8月) 2024/08/30 19

まとめ 20 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 – 短い共通パイロット系列および受信信号の標本共分散行列の 固有値を用いたアクティブユーザ数推定法を2つ提案 – 提案法1: 固有値の和と雑音分散の情報に基づく手法 – 提案法2: 固有値の差と最大正規化CFOの情報に基づく手法 – CFO存在下において従来手法よりも優れた推定精度を達成 n 今後の課題 – 提案法の推定精度の理論解析 – 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) 無線通信システム研究会(2024年8月) 2024/08/30

参考文献I 21 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) 無線通信システム研究会(2024年8月) 2024/08/30

参考文献II 22 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 T. Hara (TUS) 無線通信システム研究会(2024年8月) 2024/08/30

• https://arxiv.org/abs/2404.16152