使用週期靜態方法之擴充式演算法估計在無線電通道之頻率偏移及符號時間

國立交通大學

電機與控制工程學系

碩士論文

使用週期靜態方法之擴充式演算法估計

在無線電通道之頻率偏移及符號時間

Generalized Algorithms Using Cyclostationary

Approach for Frequency Offset and Symbol Timing

Recovery in Wireless Channel

研究生：陳逸帆

指導教授：鄭木火博士

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Generalized Algorithms Using Cyclostationary Approach

for Frequency Offset and Symbol Timing Recovery in

Wireless Channels

Student: YI-FAN CHEN

Advisor: Dr. Mu-Huo Cheng

Institute of Electrical and Control Engineering

National Chiao-Tung University

Abstract

It is important to estimate the frequency offset and timing offset in a digital communication system. Many approaches have been developed in literature. These approaches can be roughly categorized into two classes, data-aided and nondata-aided categories. The data-aided scheme uses preamble information such as pilot symbols or training sequences while the nondata-aided scheme may use cyclic prefix or use the property of cyclostationarity for estimation. This thesis focuses on the cyclostationary approach. The cyclostationary approach uses the statistic auto-correlation of received signal to estimate both frequency offset and time offset. One well-known algorithm developed by F. Gini and G. B. Giannakis is to estimate timing and frequency offsets in a single carrier system and the other popular algorithm, proposed by H. Bolcskei, is used in orthogonal frequency division multiplexing (OFDM) systems. But in these two algorithms there exists some pitfalls. The first algorithm only can work if the bandwidth of the pulse shaping filter is narrow enough to satisfy certain constraint which is derived in this thesis; the second algorithm can only work when the time offset is an integer. In this thesis, we formulate exactly the model and derive the conditions under which these existing algorithms can be applied; we further develop new generalized algorithms for estimating frequency offset and timing offset in single carrier systems as well as OFDM systems; these algorithms are general because the re-strictions of two existing methods can be alleviated. Computer simulations are also performed to illustrate the pitfalls of existing approaches and advantages of presented algorithms.

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3.2 Í$_ . . . 17 3.3 U‚ÓGÔ4 . . . 18 3.4 ,¿j¶ . . . 19 3.5 D H.Bolcskei FT|íj¶ªœ . . . 20 3.6 ÔyW . . . 21 3.7 ,=óÉ . . . 21 3.8 _Ò!‹ . . . 22 4 ! 29 A Ë“ 30 ¡5d. 33

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Ç 2.1 Gc(f ) ý<Ç . . . 7

Ç 2.2 Gc(β) £ Gc(β − (k − P )/T )ý<Ç . . . 8

Ç 2.3 Gc(β) £ Gc(β − (k + P )/T )ý<Ç . . . 9

Ç 2.4 (a) Bias and (b) MSE of ˆfe versus SNR , Uà¯ìý0§c$, fe = 0.3 ,  = 0.3 . 10

Ç 2.5 (a) Bias and (b) MSE of ˆfe versus fe , Uà¯ìý0§c$, SNR = 16dB . . . . 11

Ç 2.6 (a) Bias and (b) MSE of ˆ versus SNR , Uà¯ìý0§c$, fe = 0.3 ,  = 0.3 . 11

Ç 2.7 (a) Bias and (b) MSE of ˆ versus  , Uà¯ìý0§c$, SNR = 16dB . . . 12 Ç 2.8 (a) Bias and (b) MSE of ˆ versus SNR , Uà¯ìý0§c$, Uà [1]íj¶ . . . 12 Ç 2.9 (a) Bias and (b) MSE of ˆ versus  , Uà¯ìý0§c$, Uà [1]íj¶. . . 13 Ç 2.10 (a) Bias and (b) MSE of ˆfe versus SNR , Uà×ä í¯ìý0§c$˙šÂ,

fe = 0.3 ,  = 0.3 . . . 13

Ç 2.11 (a) Bias and (b) MSE of ˆfe versus fe , Uà×ä í¯ìý0§c$˙šÂ, SNR

= 16dB . . . 14 Ç 2.12 (a) Bias and (b) MSE of ˆ versus SNR , Uà×ä í¯ìý0§c$˙šÂ, fe = 0.3 ,  = 0.3 . . . 14

Ç 2.13 (a) Bias and (b) MSE of ˆ versus  , Uà×ä í¯ìý0§c$˙šÂ, SNR = 16dB . . . 15 Ç 2.14 (a) Bias and (b) MSE of ˆ versus SNR , Uà×ä í¯ìý0§c$˙šÂ, Uà [1]íj¶, fe= 0.3 ,  = 0.3 . . . 15

Ç 2.15 (a) Bias and (b) MSE of ˆ versus  , Uà×ä í¯ìý0§c$˙šÂ, Uà [1]íj¶, SNR = 16dB . . . 16 Ç 3.1 ¯ìý0§c$˙šÂ . . . 22

Ç 3.2 (a) Bias and (b) MSE of ˆne versus SNR , Uà¯ìý0§c$,512 ¯U ne = 3.3 ,

θe= 0.1 . . . 23

Ç 3.3 (a) Bias and (b) MSE of ˆfe versus SNR , Uà¯ìý0§c$,512 ¯U ne = 3.3 ,

θe= 0.1 . . . 23

Ç 3.4 (a) Bias and (b) MSE of ˆne versus SNR , Uà¯ìý0§c$,512 ¯U, Uà [2],

ne = 3.3 , θe = 0.1 . . . 24

Ç 3.5 (a) Bias and (b) MSE of ˆfe versus SNR , Uà¯ìý0§c$,512 ¯U, Uà [2]

ne = 3.3 , θe = 0.1 . . . 24

Ç 3.6 (a) Bias and (b) MSE of ˆne versus SNR , Uà¯ìý0§c$,512 ¯U, ne = 5 ,

θe= −0.2. . . 25

Ç 3.7 (a) Bias and (b) MSE of ˆfe versus SNR , Uà¯ìý0§c$,512 ¯U, ne = 5 ,

θe= −0.2. . . 25

Ç 3.8 (a) Bias and (b) MSE of ˆne versus SNR , Uà¯ìý0§c$,512 ¯U, Uà [2],

ne = 5 , θe= −0.2 . . . 26

Ç 3.9 (a) Bias and (b) MSE of ˆfe versus SNR , Uà¯ìý0§c$,512 ¯U, Uà [2],

ne = 5 , θe= −0.2 . . . 26

Ç 3.10 (a) Bias and (b) MSE of ˆne versus SNR , Uà¯ìý0§c$,128 ¯U, ne = 3.3

, θe = 0.3 . . . 27

Ç 3.11 (a) Bias and (b) MSE of ˆfe versus SNR , Uà¯ìý0§c$,128 ¯U, ne = 3.3

, θe = 0.3 . . . 27

Ç 3.12 (a) Bias and (b) MSE of ˆne versus SNR , Ö½˜¦−, Uà¯ìý0§c$,512

¯U, ne = 3.3 , θe= 0.3. . . 28

Ç 3.13 (a) Bias and (b) MSE of ˆfe versus SNR , Ö½˜¦−, Uà¯ìý0§c$,512

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1.1



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X

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w(l)g_c(tr)(t − T − lT ) + nc(t) (2.1)

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šÂ (matched filter) gc(rec)(t) (, mU xc(t) = rc(t)∗gc(rec)(t)

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spread) BµT 'ü, w2 Bµ H[ µc(t) íä 
/cqä0R fe óúk¯U0 (symbol rate) üí

Ö, ¤vâä0R¨Aí gc(rec)(t) .®º8$ªI, øOVz feT ≤ 0.2 ѯÜícq [1]
†%â¬

¦š (oversampling), w¦šä0Ñ P/T , P Ñøcb, ª×)¡Ní×àvÈ’e: x(n) = µ(n)ej(2π_PfeT n+θ)X

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 2 ı À-šÍ$í,¿j¶ w2 x(n) := xc(t)t=nT /P , µ(n) := µc(t)|t=nT /P , v(n) := nc(t)∗gc(rec)(t)|t=nT /P , ¢ g(n) :=

gc(t − T )|t=nT /P , gc(t) Ñf«DQY«˙šÂí!¯ gc(t) := gc(tr)(t)∗gc(rec)(t)
(2.3) 2í¡bBb

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w

• µ(n) ÑÓG (stationary) íµb˙åwAóÉì2Ñ m2µ(τ ) := E{µ(n)µ∗(n + τ )} , m2µ(τ )

íZ sž²˚d·  ä$ (Dopper spectrum)

• v(n) uø2ÓG4 (wide-sense stationary) íµb˙å

2.3

U‚ÓGÔ4

x(n) íóÉ}ÓOvÈZ‰, ì2 m2x(n; τ ) := E{x(n)x∗(n + τ )} , w2 τ uø_cbíôb

à‹ m2x(n; τ ) = m2x(n + kP ; τ ) ∀ n, k , †˚mU x(n) xU‚ÓGíÔ4
l (2.3) íóɪ ) m2x(n; τ ) = σ2wm2µ(τ )e −_j(2π P)feT τX l g(n − lP )g∗ (n + τ − lP ) + m2v(τ ) (2.4) Ñ7üwu´xU‚ÓGíÔ4, ø (2.4) 2í n J n + kP Hp m2x(n + kP ; τ ) = σw2m2µ(τ )e −_j(2π P)feT τ X l g(n + kP − lP )g∗ (n + kP + τ − lP ) + m2v(τ ) = σ2 wm2µ(τ )e −_j(2π P)feT τ X i g(n − iP )g∗ (n + τ − iP ) + m2v(τ ) = m2x(n; τ ) (2.5) úø_ ìí τ 7k,(2.5) ª„p m2x(n; τ ) ú n uJø_ P íU‚ʉ“, Ĥªl|…íZ s bí[bÑ M2x(k; τ ) := (1/P )PP −1n=0m2x(n; τ ) exp(−j(2π/P )kn) ú k ø_U‚ P , M2x(k; τ )

\˚=óÉ (cyclic correlation) / {2πk/P, k = −P/2, · · · , P/2 − 1} ˚Ñ=ä0 (cyclic fre-quencies)
* (2.4) ªRû| M2x(k; τ ) = σ2 w P m2µ(τ )e −j(2π P)feT τ X n g(n)g∗ (n + τ )e−j(2π P)kn+ m 2v(τ )δ(k) (2.6)

 2 ı À-šÍ$í,¿j¶ âk  ¥_¡bÊçBbø (2.1) žÑ (2.3) v˛%\ g(n)F¨Ö, Ñ7½héý| M2x(k; τ ) D  í

É[, à G(f ) :=P

ng(n) exp(−j2πf n) H[ g(n) íZ sž²1‚à Parseval’s relation

X n g(n)g∗ (n + τ )e−_j(2π P)kn = Z 0.5 −_0.5 G∗  β − k P  G(β)e2πβτdβ (2.7) è g(n) := gc(t − T )|t=nT /P , 1cq gc(t) íä  Bgük P/(2T ) ; 6ÿuz¦šä0 P/T Å— Nyquist ¦šìÜ, .}ßÞ>L (aliasing), ;W¦šÜª) G(f ) = 1 Ts ∞ X k=−∞ Gc( f Ts − k Ts )e−_j2π(f Ts− k Ts)T = 1 Ts ∞ X k=−∞ Gc( f − k Ts )e−_{j2π(f −k)P} (2.8) w2 Gc(f ) := R∞ −∞gc(t)e −_{j2πf t} dt , ¢ Ts= T /P ѦšU‚
ç³>Lv,(2.8) Ê |f | ≤ 0.5 í¸ˇ qª“Ñ G(f ) = 1 Ts Gc( f Ts )e−_{j2πf P} , for |f | ≤ 0.5 (2.9) ø (2.9) Hp (2.7), Í( (2.7) Hp (2.6), ç 0 ≤ k ≤ P/2 − 1 , ª) M2x(k; τ ) = σ2 w P m2µ(τ )e −j(2π P)feT τ Z 0.5 −_0.5 G∗  β − k P  G(β)e2πβτdβ + m2v(τ )δ(k) = σ 2 w P m2µ(τ )e −_j(2π P)feT τ Z 0.5 −_0.5+k/P G∗  β − k P  G(β)e2πβτdβ + Z −0.5+k/P −_0.5 G∗  β − k P + 1  G(β)e2πβτ_dβ ) + m2v(τ )δ(k) = σ 2 w P T2 s m2µ(τ )e−j( 2π P)feT τe−j2πk Z 0.5 −_0.5+k/P G∗ c  β − k/P Ts  Gc( β Ts )e2πβτdβ +ej2πP Z −_0.5+k/P −0.5 G∗ c  β − k/P + 1 Ts  Gc( β Ts )e2πβτdβ ) + m2v(τ )δ(k) = σ 2 w P T2 s m2µ(τ )e −_j(2π P)feT τe−j2πk Z 0.5 −0.5 G∗ c  β − k/P Ts  Gc( β Ts )e2πβτdβ +ej2πP Z 0.5 −_0.5 G∗ c  β − k/P + 1 Ts  Gc( β Ts )e2πβτdβ  + m2v(τ )δ(k) (2.10) ÇøjÞ, ç −P/2≤k < 0 v, M2x(k; τ ) = σ2 w P m2µ(τ )e −_j(2π P)feT τ Z 0.5 −_0.5 G∗  β − k P  G(β)e2πβτdβ + m2v(τ )δ(k)

 2 ı À-šÍ$í,¿j¶ = σ 2 w P m2µ(τ )e −j(2π P)feT τ ( Z 0.5+k/P −_0.5 G∗  β − k P  G(β)e2πβτdβ + Z 0.5 0.5+k/P G∗  β − k P − 1  G(β)e2πβτdβ  + m2v(τ )δ(k) = σ 2 w P T2 s m2µ(τ )e −_j(2π P )feT τe−j2πk ( Z 0.5+k/P −0.5 G∗ c  β − k/P Ts  Gc( β Ts )e2πβτdβ +e−_j2πP Z 0.5 0.5+k/P G∗ c  β − k/P − 1 Ts  Gc( β Ts )e2πβτdβ  + m2v(τ )δ(k) = σ 2 w P T2 s m2µ(τ )e −_j(2π P )feT τe−j2πk Z 0.5 −_0.5 G∗ c  β − k/P Ts  Gc( β Ts )e2πβτdβ +e−_j2πP Z 0.5 −_0.5 G∗ c  β − k/P − 1 Ts  Gc( β Ts )e2πβτdβ  + m2v(τ )δ(k) (2.11) ì2 G2(k; τ ) := (P/T )R−P/2T_P/2T G ∗ c(β − k/T )Gc(β)ej2πβτ T /Pdβ , † (2.10),(2.11) ªŸÑ M2x(k; τ ) = ( σ2 w P m2µ(τ )e −_j(2π P)feT τe−j2πk G₂(k; τ ) + ej2πPG₂(k − P ; τ )
, 0 < k ≤ P 2 − 1 σ2 w P m2µ(τ )e −_j(2π P)feT τe−j2πk G₂(k; τ ) + e−j2πPG₂(k + P ; τ )
, −P 2 ≤ k < 0 (2.12) Q-Vø½æÌìÊM«Á (slow fading) íÕ”, †ú—Düí τ (¸ˇâ·  Øà (Doppler spread) ²ì), m2µ(τ ) ≈ m2µ(0) Ñøõbíb, ˚…Ñ σµ2
²¦ k = K , KÑø£cb, / K ük P/2 , ÄÑ G2(−k; τ ) = G2 ∗ (k, τ ) , FJ M2x(−K; τ )ªZŸAà-: M2x(−K; τ ) = σ2 wσµ2 P e −j(2π P)feT τe−j2πK G₂(K; τ ) + ej2πPG₂(K − P ; τ )
∗ (2.13) °v M2x(K; τ ) = σ2 wσ2µ P e −_j(2π P)feT τe−j2πK G 2(K; τ ) + ej2πPG2(K − P ; τ )
(2.14) FJŽO (2.13) £ (2.14) ªJ,¿| ˆ fe= − P 4πT τ arg{M2x(K; τ )M2x(−K; τ )} (2.15) Ou  í,¿j¶ÿªœµÆ, ílùà,¿|í ˆfe, ì2 M(K; τ ) = P σ2 w ej(2π_P) ˆfeT τ_M 2x(K; τ ) (2.16) ‚às_.°í τ , Í(ªû|  G2(K; τ1) G2(K − P ; τ1) G2(K; τ2) G2(K − P ; τ2)   σ2 µe −j2πk σ2 µej2π(P −k)  = M(K; τ1) M(K; τ2)  (2.17)

 2 ı À-šÍ$í,¿j¶ Q-Vì2ø_ªW[b η à-η = ( 1 , if G2(K − P ; τ1) = G2(K − P ; τ2) = 0 G2(K−P ;τ1) G2(K−P ;τ2) , otherwise. (2.18) ‚à (2.17) D η ªJ,¿| ˆ  = − 1 2πk]  M(K, τ1) − η M(K, τ2) G2(K; τ1) − η G2(K; τ2)  (2.19)

2.4

ÔyWD“

粦í k D P Å—/<Ì„v, M2x(k; τ ) ªJ\“AÀí$, Ê (2.12) 2, ªœnÀíu Ö7 ej2πP_G 2(k − P ; τ ) D e−j2πPG2(k + P ; τ ) ¥sá, FJ«nSv¥sáí^‹kÉ, ZªJø “rÖ, à¤ZªJ×Ù±QlíµÆ
*ä$,Võ, cq Gc(f ) ø_ä Ñ Bg , Ç2.1 Ñ Gc(f ) íý<Ç − Ç 2.1: Gc(f ) ý<Ç l«n 0 < k ≤ P/2−1 8$, Ñ7ôI (2.12) 2í ej2πP_G 2(k −P ; τ ) , FJ.âé G2(k −P ; τ ) k É, ¢ G2(k−P ; τ ) = (P/T ) RP/2T −_P/2TG ∗ c(β−(k−P )/T )Gc(β)e −_{j2πβτ T /P} dβ , w2í G∗ c(β−(k−P )/T ) D Gc(β) íÇ$àÇ2.2 Fý hôÇ2.2¹ªêÛ, ç (k − P )/T + Bg ≤ −Bg v, G2(k − P ; τ )ÿ}kÉ, âk k í|× MÑ P/2 − 1 , cÜ(ª) P ≥ 4BgT − 2
yõ −P/2 ≤ k < 0 í¶M, Bbíñ™ué G2(k + P ; τ ) kÉ, ÄÑ G2(k + P ; τ ) = (P/T )RP/2T −_P/2T G ∗ c(β − (k + P )/T )Gc(β)e−j2πβτ T /Pdβ , hôÇ2.3, ªêÛç (k + P )/T − Bg ≥ Bg v¹ª, âk k |üMÑ −P/2, ĤFbÅ—í‘KÑ P ≥ 4BgT
FJ!¯c_ k í¸ˇ)ƒ|(í

 2 ı À-šÍ$í,¿j¶ − β β − β − − β − − + − Ç 2.2: Gc(β) £ Gc(β − (k − P )/T )ý<Ç ‘KÑ P ≥ 4BgT (2.20) ,!,Þí!‹ªJ¦Ñ|Éb P Å— (2.20), ÿªJôI¥ ej2πP_G 2(k−P ; τ ) D e −_j2πP G2(k+P ; τ ) ¥sá, |( (2.12) ÿ‰A M2x(k; τ ) = σ2 w P m2µ(τ )e −j(2π P)feT τe−j2πkG₂(k; τ ) , −P 2 < k ≤ P 2 − 1 (2.21) FJç P ²ÏíD×v, M2x(k; τ ) Z}uø_œÀí, /¥_D Gini and Giannakis[1]R

ûí!‹øš, OuFbíRû¬˙2<˜Ï, ³·<ƒÛbÅ— (2.20) í‘Kn?A , .¬Éb² í P —D×Å— (2.20), ZªJà Gini and Giannakis[1]d2Tƒíj¶V,¿7.}˜Ï, j¶à -ˆ fe = − P 4πT τ arg  M2x(k; τ )M2x(−k; τ ) G2(k; τ )G2(−k; τ )  (2.22) ˆ = − 1 2πkarg ( M2x(k; τ )ej(2π/P ) ˆfeT τ G2(k; τ ) ) (2.23)

 2 ı À-šÍ$í,¿j¶ − β β − β + − β − + − + Ç 2.3: Gc(β) £ Gc(β − (k + P )/T )ý<Ç C6BbªJz‘Kš[ øõ, Éb |k| ≤ P −2BgT , ZªJôIej2πPG2(k−P ; τ ) D e−j2πPG2(k+ P ; τ ) ¥sá
úk¥_¸ˇqí k ÿªJUà (2.22) £ (2.23) V,¿ fe D 

2.5

l=óÉ

è m2x(n; τ ) = E{x(n)x∗(n + τ )} , .¬BbÉ?àÌí¦šÌ (sample mean) V¡N, c

qBb L ª’e ˆ m2x(n; τ ) = P L L/P −1 X l=0 x(n + lP )x∗ (n + τ + lP ) , n = 0, 1, · · · P − 1 (2.24) † ˆ M2x(k; τ ) = 1 P P −1 X n=0 ˆ m2x(n; τ )e −_j2πkn P (2.25) C6òQl ˆ M2x(k; τ ) = 1 L L−1 X n=0 x(n)x∗ (n + τ )e−_j2π Pkn (2.26) é7qc, à‹L.D×, µZ}×íÏÏßÞ

 2 ı À-šÍ$í,¿j¶

2.6

_Ò!‹

2.6.1 Uà¯ìý0§c$˙šÂí,¿

²Ï α = 0.5 í¯ìý (raised cosine) Vd0§c$ (pulse shaping), 1‚à 512 _¯UVl M2x , ‘² P = 4 6ÿuûI¦šä0V¦š, ÄÑ˙šÂíä Ñ 0.75/T , FJ P = 4 ¹ªÅ— (2.20) í‘K, }‚àBbT|íj¶D [1]2íj¶V¿t, _Ò!‹u*ûìŸ_ÒíÌ8$V× ), ¿tàí fe Ñ 0.3,  6u0.3, lõSàBbíj¶í_Ò!‹, Ç2.4DÇ2.5Ñä0Rí,¿!‹, Ç2.6£Ç2.7ývÈRí,¿8$
Q-VõUà[1]T|j¶í,¿!‹, ä0R,¿âkj¶ø 5 10 15 20 25 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 SNR bias(fe) (a) 5 10 15 20 25 10−4 10−3 10−2 SNR MSE(fe) (b)

Ç 2.4: (a) Bias and (b) MSE of ˆfe versus SNR , Uà¯ìý0§c$, fe = 0.3 ,  = 0.3

š, FJ!‹6øš, Z.‹Jý, Ç2.8DÇ2.9[ývÈRí,¿!‹ 2.6.2 Uà×ä í0§c˙šÂví,¿ Ñ7b_Ò˙šÂä .¯¯ (2.20) í8$, .âblßÞø_ä œ×í0§c$ (pulse shaping) ˙_{šÂ, FJ‚àø_Êv,9òsIí¯ìý˙šÂV_Ò, à¤Zªøä [×sI, ²í α = 1 ,} FJc_ä Ñ 2/T , .Å— (2.20), }‚àBbT|íj¶D [1]2íj¶V,¿ fe D 
lõ‚àBbT|íj¶,¿í!‹, Ç2.10DÇ2.11éýä0R,¿í_Ò!‹, Ç2.14éývÈ R,¿í_Ò!‹
Q-Võ‚à [1]FTj¶,¿í8$, âkä0Rí,¿juøší, FJB bÉõvÈRí¶M, Ç2.12DÇ2.13}[ý  ,¿Mú SNR D  퉓8$

 2 ı À-šÍ$í,¿j¶ −0.2 −0.1 0 0.1 0.2 −0.01 −0.005 0 0.005 0.01 fe bias(fe) (a) −0.2 −0.1 0 0.1 0.2 10−4 10−3 10−2 fe MSE(fe) (b)

Ç 2.5: (a) Bias and (b) MSE of ˆfe versus fe , Uà¯ìý0§c$, SNR = 16dB

5 10 15 20 25 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 SNR bias( ε) (a) 5 10 15 20 25 10−5 10−4 10−3 10−2 SNR MSE( ε) (b)

 2 ı À-šÍ$í,¿j¶ −0.4 −0.2 0 0.2 0.4 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 ε bias( ε) (a) −0.4 −0.2 0 0.2 0.4 10−4 ε MSE( ε) (b)

Ç 2.7: (a) Bias and (b) MSE of ˆ versus  , Uà¯ìý0§c$, SNR = 16dB

5 10 15 20 25 −5 −4 −3 −2 −1 0 1 2 3 4 5x 10 −3 SNR bias( ε) (a) 5 10 15 20 25 10−6 10−5 10−4 10−3 SNR MSE( ε) (b)

 2 ı À-šÍ$í,¿j¶ −0.4 −0.2 0 0.2 0.4 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 ε bias( ε) (a) −0.4 −0.2 0 0.2 0.4 10−4 ε MSE( ε) (b)

Ç 2.9: (a) Bias and (b) MSE of ˆ versus  , Uà¯ìý0§c$, Uà [1]íj¶

5 10 15 20 25 −1.5 −1 −0.5 0 0.5 1 1.5x 10 −3 SNR bias(fe) (a) 5 10 15 20 25 10−6 10−5 10−4 10−3 SNR MSE(fe)

Ç 2.10: (a) Bias and (b) MSE of ˆfe versus SNR , Uà×ä í¯ìý0§c$˙šÂ, fe = 0.3 ,

 2 ı À-šÍ$í,¿j¶ −0.2 −0.1 0 0.1 0.2 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 fe bias(fe) (a) −0.2 −0.1 0 0.1 0.2 10−5 10−4 MSE(fe) (b)

Ç 2.11: (a) Bias and (b) MSE of ˆfe versus fe , Uà×ä í¯ìý0§c$˙šÂ, SNR = 16dB

5 10 15 20 25 −5 −4 −3 −2 −1 0 1 2 3 4 5x 10 −3 bias( ε) SNR (a) 5 10 15 20 25 0 1 2 3 4 5 6 7 8 9x 10 −5 SNR MSE( ε) (b)

Ç 2.12: (a) Bias and (b) MSE of ˆ versus SNR , Uà×ä í¯ìý0§c$˙šÂ, fe = 0.3 ,

 2 ı À-šÍ$í,¿j¶ −0.4 −0.2 0 0.2 0.4 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 ε bias( ε) (a) −0.4 −0.2 0 0.2 0.4 10−6 10−5 ε MSE( ε) (b)

Ç 2.13: (a) Bias and (b) MSE of ˆ versus  , Uà×ä í¯ìý0§c$˙šÂ, SNR = 16dB

5 10 15 20 25 0 1 2 3 4 5 6x 10 −3 SNR bias( ε) (a) 5 10 15 20 25 10−6 10−5 10−4 SNR MSE( ε) (b)

Ç 2.14: (a) Bias and (b) MSE of ˆ versus SNR , Uà×ä í¯ìý0§c$˙šÂ, Uà [1]íj ¶, fe = 0.3 ,  = 0.3

 2 ı À-šÍ$í,¿j¶ −0.4 −0.2 0 0.2 0.4 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 ε bias( ε) (a) −0.4 −0.2 0 0.2 0.4 10−6 10−5 10−4 ε MSE( ε) (b)

Ç 2.15: (a) Bias and (b) MSE of ˆ versus  , Uà×ä í¯ìý0§c$˙šÂ, Uà [1]íj¶, SNR = 16dB

 3 ı

£>ä0Ö½fÍ$í,¿j¶

3.1



¡V, £>ä0ä0Ö½fÍ$ÚÚ§ƒ½e, 6˜í@àÊrÖ@à,, dubPÚe bPà E«c˜ (DSL) £ø< äíÌ(_0,
£>ä0ä0Ö½fÍ$úkÖ½˜ (multipath) í à ßíJ}‰, Žâø’e}Ab_üíŸ-š (subcarrier) Vf£, U)©_Ÿ-šî§ƒä0«Á (flat fading) 7.uä0²Ï«Á (frequency selective fading), ओ7“Âíql
ª<íu£ >ä0ä0Ö½fÍ$ú°¥íÏÏœøOÀ-šÍ$Ü>rÖ, FJä0R£vÈÏÏí,¿Zó$ ½b
…ı‚àYƒmUóÉ£w¿ÖíU‚ÓG (cyclostationary) Ô4VªWÔñí,¿ (blind estimation), 3b-ZuÓ H.Bolcskei[2]íj¶, .¬ÄÑ[2]2íRûrÖ˜Ï, ._àkÝcbí vÈR, FJBb½hRû|ø_£üí_1TXø_híƶVªW,¿, 1‚àÚ7_ÒVð„ !‹

3.2

Í$_

úk¨Ö0§c$ (pulse shaping) í£>ä0ä0Ö½fÍ$, ^!ä (baseband) mUª[ ýÑ x[n] = N −1 X k=0 ∞ X l=−∞ ck,lg[n − lM ]ej(2π/N )k(n−lM ) (3.1)

w2 N ÑŸ-š (subcarriers) íb¾, M u¯UíÅ, g[n] [ýf« pulse sharping ˙šÂ, ck,l

[ý’e¯U
ç M > N vZH[¥_Í$¨Ö7=ô. (cyclic extension)

ÊQY«YƒímU}„øívÈRDä0R, ĤBbªJzQYƒímU[ýÑ r[n] = ej(2πθen+φ)

Z 0.5

−_0.5

 3 ı £>ä0Ö½fÍ$í,¿j¶ w2 X(ej2πν_{) Ñ x[n] í×àvÈZ sž² (DTFT), n}

e ∈ R ÑvÈR, θe ∈ [−1/2, 1/2)

[ý-šä0R, φ u–áíi, 7 ρ[n] u2 ì (wide-sense stationary) Æm˙å, /D’e¯U ck,l

óÖ , b·<ƒ, ne 1³ÌìÑcb, ç ne ∈ Z , † (3.2) ªJ“

r[n] = ej(2πθen+φ)_{x[n − n}

e] + ρ[n] (3.3)

Æm˙åíóÉ4ƒbÑ cρ[τ ] = E {ρ[n]ρ∗[n − τ ]} , ¯U cp,l u*̯UÍ$ (finite-alphabet) µ

brèÇ (constellation) 2צ, Å— E{ck,lc∗k0_,l0} = σ2_cδ[k − k

0

]δ[l − l0

]
Q-VBbcq©_Ÿ-š (subcarrier) uJ.°í?¾Êf], C6zBbUàŸ-š‹ž (subcarrier weighting), #싞ƒ b w[k]
Ĥf£ímUZŸÑ x[n] = N −1 X k=0 ∞ X l=−∞ ck,lw[k]g[n − lM ]ej(2π/N )k(n−lM ) (3.4)

3.3

_U‚ÓGÔ4

à°5‡FTƒ, U‚ÓG (cyclostationarity) uÔñí°¥j¶íÉœ, lì2Ýv0óÉ (cor-relation) ƒbÑ cr[n, τ ] = E{r[n]r∗[n − τ ]}, τ uø_cbíôb¡b, ¡ÎË“íRû, ‹ž OFDM

mU (3.4) íóÉÑ cr[n, τ ] = σ2 cej2πθeτ M N −1 X m=0 |w[m]|2 (_{M −1} X s=0 Z 0.5−_Ms −_0.5 ej2πsM (n−ne)ej2πντG ej2π(ν+ s M− m N)
G∗ ej2π(ν− m N)
dν + −₁ X s=−M +1 Z 0.5 −_0.5−s M ej2πsM (n−ne)ej2πντG ej2π(ν+ s M− m N)
G∗ ej2π(ν− m N)
dν ) + cρ[τ ] (3.5) w2 G(ej2πθ_{) =} P∞ n=−∞g[n]e −j2πnθ , hôªõ| cr[n, τ ] ú n uJU‚ M ʉ“, ²Æuz cr[n, τ ] = cr[n + M, τ ], Ĥx= ì4íÔ4
QOl…íZ sb[b, ;Wì2: Cr[k, τ ] = 1 M M −1 X n=0 cr[n, τ ]e −_{j(2π/M )kn} , k = 0, 1, · · · , M − 1. (3.6) ¡ÎË“í}pª), ç 0 ≤ k ≤ M − 1 Cr[k, τ ] = σ2 c Me j2πθeτ_e−_j2πk M ne N −1 X m=0 |w[m]|2 ( Z 0.5−k/M −_0.5 ej2πντ_G_ej2π(ν+_Mk−m N)  G∗ ej2π(ν−m_N)
_dν + ej2πne Z 0.5 0.5−k/M ej2πντGej2π(ν+Mk− m N)  G∗ ej2π(ν−mN)
dν  + cρ[τ ]δ[k] (3.7)

 3 ı £>ä0Ö½fÍ$í,¿j¶ Í(Bbø (3.7) 2í }ÇV, 1/šdcÜ(ª) Cr[k, τ ] = σ2 c Me j2πθeτ_e−_j2πk M ne  ΓN[τ ]A(g, g)  τ, k M   1 + k M(e j2πne_{− 1)}  +(ej2πne _{− 1)} ∞ X γ=−∞ γ6=τ ΓN[γ]A(g, g)  γ, k M  (−1)τ −γ j2π(τ − γ)  1 − e−_{j2π(τ −γ)}k M  + cρ[τ ]δ[k] (3.8) w2 ΓN[τ ] :=PN −1k=0 |w[k]|2ej(2π/N )kτ , 7 A(g, h)[k, θ) := P∞ n=−∞g[n]h[n − k]e −j2πnθ

3.4

,¿j¶

‚à Cr[k, τ ] ªJ,¿|ä0RDvÈR, ílì2ø_.Œí¡b Φ[k, τ ] = k M + P∞ γ=−∞ γ6=τ ΓN[γ]A (g, g)_γ, k M
(−1)τ −γ j2π(τ −γ)  1 − e−_{j2π(τ −γ)}k M  ΓN[τ ]A(g, g)τ,_Mk
(3.9) ¥_¡bíMuªJ9ll|Ví, Éb‘²í τ £ k Å— ΓN[τ ]A(g, g)τ,_Mk
6= 0, ÄÑ ΓN[τ ]  A(g, g)_τ, k M
ª‚à˛øí w[k] D g[n] °)
, ‚à (3.8) ª) Cr[k, τ ] = σ2 c Me j2πθeτ_e−j2πk M neΓ_N[τ ]A(g, g)  τ, k M  1 + (ej2πne − 1)Φ[k, τ ]
+ c ρ[τ ]δ[k] (3.10)

Q-Vyì2 C[k, τ ] , uøŸ…í Cr[k, τ ] £d“ (normalize) FßÞ,

à-C[k, τ ] = _σ2 Cr[k, τ ] c MΓN[τ ]A(g, g)τ, k M
= ej2πθeτ_e−_j2πk M ne 1 + (ej2πne − 1)Φ[k, τ ]
+ cρ[τ ]δ[k] σ2 c MΓN[τ ]A(g, g)τ, k M
(3.11) Q-VÇá,¿: 1. ,¿ ne : lz θe í à ¥, FJ‘² τ = 0 , † C[k, 0] = e−j 2πk M ne(1 + (ej2πne − 1)Φ[k, 0]) ÉÖ n_e ø_¡bu„øí, Q-V5? C[k, 0] D C[M − k, 0] í , ª) C[k, 0]C[M − k, 0] = e−_j2πne (1 + (ej2πne_{− 1)Φ[k, 0])(1 + (e}j2πne _{− 1)Φ[M − k, 0])} = ej2πne_{Φ[k, 0]Φ[M − k, 0] + (Φ[k, 0] + Φ[M − k, 0] − 2Φ[k, 0]Φ[M − k])} +e−_j2πne (1 + Φ[k, 0]Φ[M − k, 0] − Φ[k, 0] − Φ[M − k, 0]) (3.12)

 3 ı £>ä0Ö½fÍ$í,¿j¶ Ix = ej2πne_{Hp (3.12) ª)ø_ùŸ Ax}2 _{+ Bx + C = 0 , w2} A = Φ[k, 0]Φ[M − k, 0] (3.13) B = Φ[k, 0] + Φ[M − k, 0] − 2Φ[k, 0]Φ[M − k, 0] − C[k, 0]C[M − k, 0]) (3.14) C = 1 + Φ[k, 0]Φ[M − k, 0] − Φ[k, 0] − Φ[M − k, 0] (3.15) ¤ùŸí}s_j, w2íø_Ñ x í,lM, .¬b}z¥_M,¿|í neH x nªø −¨_Mnu£üí, ne í,lj¶à-ˆ ne= − M 2πk]  C[k, 0] 1 + (x − 1)Φ[k, 0]  (3.16) 2. ,¿ θe : ˆ θe= 1 2πτ] C[k, τ ] e−j2πk M nˆe(1 + (ej2π ˆne− 1)Φ[k, τ ]) ! (3.17)

3.5

D H.Bolcskei

FT|íj¶ªœ

è5‡Fû|í, C[k, τ ] = ej2πθeτ_e−j2πk M ne 1 + (ej2πne− 1)Φ[k, 0]
(3.18) ªD H.Bolcskei Rûí!‹dªœ, -ÑwRû!‹ C[k, τ ] = ej2πθeτ_e−_j2πk M ne (3.19) ªœ(ªõ|‡ú ne ∈ Z í8”, ej2πne = 1 ,(3.18) D (3.19) }‰ó, FJ¥v H.Bolcskei T|í _}u£üí
¢FT|í,¿j¶à-: ˆ θe = 1 4πτ]{C[k, τ ]C[M − k, τ ]} (3.20) ˆ ne = − M 2πk]{C[k, τ ]e −j2π ˆθeτ_{} (3.21)}

 3 ı £>ä0Ö½fÍ$í,¿j¶

3.6

ÔyW

çøÇáUàí g[n] D w[k] u/Ôyív, BbªJ“, à¤6ªJ“,¿íj¶, l «n³‹Ÿ-š‹ží8$, ²Æuz, w[k] = 1 úFí k , à¤øV ΓN[τ ] = NP ∞ s=−∞δ[τ − sN ] , ½hZŸ (3.8), /‘² τ = N Vl Cr[k, N ] = N σ2 c M e j2πθeN_e−_j2πk M ne  A(g, g)  N, k M   1 + k M(e j2πne _{− 1)}  +(ej2πne _{− 1)} ∞ X s=−∞ s6=1 A(g, g)  sN, k M  (−1)N −sN j2π(N − sN )  1 − e−_{j2π(N −sN )}k M  (3.22) Í(¦ k = M − k Cr[M − k, N ] = N σ2c M e j2πθeN_e−_j2π(M −k) M ne  A(g, g)  N,M − k M   1 + M − k M (e j2πne _{− 1)}  +(ej2πne _{− 1)} ∞ X s=−∞ s6=1 A(g, g)  sN,M − k M  (−1)N −sN j2π(N − sN )  1 − e−_{j2π(τ −sN )}M −k M  = N σ 2 c M e j2πθeN_ej2πk_M ne  A∗_{(g, g)}  N, k M   1 + k M(e −_j2πne − 1)  +(e−_j2πne − 1) ∞ X s=−∞ s6=1 A∗_{(g, g)}  sN, k M   − (−1) N −sN j2π(N − sN )   1 − ej2π(τ −sN )Mk  = C∗ r[k, N ]ej4πθeN (3.23) ‚à (3.23) ªJòQ,¿ θe , .àd5‡íj¶blø,|Ví ne HpV° θe , FJªJTòÄü, j¶à-ˆ θe= 1 4πN]Cr[k, N ]Cr[M − k, N ] (3.24)

3.7

,=óÉ

è cr[n; τ ] = E{r[n]r∗[n − τ ]} , .¬BbÉ?àÌí¦šÌ (sample mean) V¡N, cqBb

L ª’e ˆ cr[n; τ ] = M L L/M −1 X l=0 r[n + lM ]r∗ [n − τ + lM ] , n = 0, 1, · · · M − 1 (3.25)

 3 ı £>ä0Ö½fÍ$í,¿j¶ † ˆ Cr[k; τ ] = 1 M M −1 X n=0 ˆ cr[n; τ ]e −_j2πkn M (3.26) C6òQl ˆ Cr[k; τ ] = 1 L L−1 X n=0 r[n]r∗ [n − τ ]e−_j2π Mkn (3.27) é7qc, à‹L.D×, µZ}×íÏÏßÞ

3.8

_Ò!‹

_Ò¯ìý0§c$˙šÂ‹Ÿ-š‹ž‹=ô.íÕ”, ²ÏŸ-šb N = 8, ¯UÅ M = 16 , g[n] Ñ α = 0.5 í¯ìýƒbàÇ (3.1), Õ‹Ÿ-š‹žÑ w = [1.1 2.0 1.4 1.33 1.0 0.6 0.8 1.2]
}Uà,HFTíj¶£ H.Bolcskei[2]íj¶%â 400 ŸÖ ¿t!‹í$l’ed}&, ¢¿tàí ¦−ÑË‹ëÆm (AWGN) ¦−DÖ½˜ (multipath) ¦−
0 10 20 30 40 50 60 70 80 90 −0.2 0 0.2 0.4 0.6 0.8 1

1.2 Pulse shaping filter g[n]

n

g[n]

 3 ı £>ä0Ö½fÍ$í,¿j¶ 1. Ë‹ëÆm (AWGN) ¦−Uà (3.16)(3.17) ,¿, ‚à 512 _¯U, / ne ∈ R :

(qì ne= 3.3 , θe = 0.1 , Ç3.2 [ývÈR ne í,¿, Ç3.3[ýä0R θe í,¿) 0 10 20 30 −0.1 −0.09 −0.08 −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0 (a) SNR bias(ne) 0 10 20 30 10−3 10−2 10−1 (b) SNR MSE(ne)

Ç 3.2: (a) Bias and (b) MSE of ˆne versus SNR , Uà¯ìý0§c$,512 ¯U ne= 3.3 , θe = 0.1

0 10 20 30 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 (a) SNR bias(fe) 0 10 20 30 10−4 10−3 (b) SNR MSE(fe)

 3 ı £>ä0Ö½fÍ$í,¿j¶ 2. Ë‹ëÆm¦−Uà H.Bolcskei[2],¿, ‚à 512 _¯U, / ne∈ R :

(qì ne= 3.3 , θe = 0.1 , Ç3.4 [ývÈR ne í,¿, Ç3.5[ýä0R θe í,¿) 0 10 20 30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 (a) SNR bias(ne) 0 10 20 30 10−2 10−1 100 (b) SNR MSE(ne)

Ç 3.4: (a) Bias and (b) MSE of ˆne versus SNR , Uà¯ìý0§c$,512 ¯U, Uà [2], ne = 3.3 ,

θe = 0.1 0 10 20 30 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0 (a) SNR bias(fe) 0 10 20 30 10−3 (b) SNR MSE(fe)

Ç 3.5: (a) Bias and (b) MSE of ˆfe versus SNR , Uà¯ìý0§c$,512 ¯U, Uà [2] ne = 3.3 ,

 3 ı £>ä0Ö½fÍ$í,¿j¶ 3. Ë‹ëÆm¦−Uà (3.16)(3.17) ,¿, ‚à 512 _¯U, / ne ∈ Z : (qì ne= 5 , θe = −0.2 , Ç3.6 [ývÈR ne í,¿, Ç3.7[ýä0R θe í,¿) 0 10 20 30 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 (a) SNR bias(ne) 0 10 20 30 10−3 10−2 10−1 (b) SNR MSE(ne)

Ç 3.6: (a) Bias and (b) MSE of ˆne versus SNR , Uà¯ìý0§c$,512 ¯U, ne = 5 , θe = −0.2

0 10 20 30 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 (a) SNR bias(fe) 0 10 20 30 10−4 10−3 (b) SNR MSE(fe)

 3 ı £>ä0Ö½fÍ$í,¿j¶ 4. Ë‹ëÆm¦−Uà H.Bolcskei[2],¿, ‚à 512 _¯U, / ne∈ Z :

(qì ne= 5 , θe = −0.2 , Ç3.8 [ývÈR ne í,¿, Ç3.9[ýä0R θe í,¿) 0 10 20 30 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 (a) SNR bias(ne) 0 10 20 30 10−2 10−1 (b) SNR MSE(ne)

Ç 3.8: (a) Bias and (b) MSE of ˆne versus SNR , Uà¯ìý0§c$,512 ¯U, Uà [2], ne = 5 ,

θe = −0.2 0 10 20 30 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 (a) SNR bias(fe) 0 10 20 30 10−5 10−4 10−3 (b) SNR MSE(fe)

Ç 3.9: (a) Bias and (b) MSE of ˆfe versus SNR , Uà¯ìý0§c$,512 ¯U, Uà [2], ne = 5 ,

 3 ı £>ä0Ö½fÍ$í,¿j¶ 5. Ë‹ëÆm¦−Uà (3.16)(3.17) ,¿, ‚à 128 _¯U: (qì ne= 3.3 , θe = 0.3 , Ç3.10 [ývÈR ne í,¿, Ç3.11[ýä0R θe í,¿) 0 10 20 30 −0.2 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 SNR bias(ne) (a) 0 10 20 30 10−2 10−1 100 SNR MSE(ne) (b)

Ç 3.10: (a) Bias and (b) MSE of ˆne versus SNR , Uà¯ìý0§c$,128¯U, ne= 3.3 , θe = 0.3

0 10 20 30 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 SNR bias(fe) (a) 0 10 20 30 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10 −3 SNR MSE(fe) (b)

 3 ı £>ä0Ö½fÍ$í,¿j¶ 6. Ö½˜¦− (multipath) Uà (3.16)(3.17) ,¿, ‚à 512 _¯U:

(qì ne= 3.3 , θe = 0.3 , Ç3.12 [ývÈR ne í,¿, Ç3.13[ýä0R θe í,¿) Uàíuû¼íÖ½˜ Rayleigh-fading ¦−, w2 σ2 h = [ 0.9502 0.0473 0.0024 0.0001 ]
0 10 20 30 −0.2 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 (a) SNR bias(ne) 0 10 20 30 10−2 10−1 100 (b) SNR MSE(ne)

Ç 3.12: (a) Bias and (b) MSE of ˆne versus SNR , Ö½˜¦−, Uà¯ìý0§c$,512 ¯U,

ne = 3.3 , θe = 0.3 0 10 20 30 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 (a) SNR bias(fe) 0 10 20 30 10−3 10−2 10−1 (b) SNR MSE(fe)

Ç 3.13: (a) Bias and (b) MSE of ˆfe versus SNR , Ö½˜¦−, Uà¯ìý0§c$,512 ¯U,

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 A ı

Ë“

ílRû cr[n, τ ] cr[n, τ ] = E{r[n]r ∗ [n − τ ]} = E  ej2πθeτ Z 0.5 −_0.5 X(ej2πµ)ej2πµ(n−ne)_dµ Z 0.5 −_0.5 X∗ (ej2πν)e−_{j2πν(n−τ −ne)} dν  + cρ[τ ] = E ( ej2πθeτ Z 0.5 −_0.5 N −1 X k=0 w[k]Gej2π(µ−Nk)  ∞ X l=−∞ ck,le −_j2πµlM ej2πµ(n−ne)_dµ · Z 0.5 −0.5 N −1 X p=0 w∗ [p]G∗ ej2π(ν−Np)  ∞ X q=−∞ c∗ p,qej2πνqMe −_{j2πν(n−τ −ne)} dν ) + cρ[τ ] = σ_c2ej2πθeτ N −1 X k=0 |w[k]|2 Z 0.5 −0.5 Z 0.5 −0.5 ej2π(µ−ν)(n−ne)_ej2πντ_G  ej2π(µ−Nk)  G∗ ej2π(ν−Nk)  · ∞ X l=−∞ ej2πlM (ν−µ)dµdν + cρ[τ ] (A.1) lcì ν uø_b, ª)P∞ l=−∞ej2πlM (ν−µ) = 1 M P∞ s=−∞δ(µ − ν − s/M ) Ñ©/ (continuous) í δ ƒb, Hp (A.1) ª) cr[n, τ ] = σ2 cej2πθeτ M N −1 X m=0 |w[m]|2 (_{M −1} X s=0 Z 0.5−_Ms −_0.5 ej(2πsM )(n−ne)ej2πντG ej2π(ν+ s M− m N)
G∗ ej2π(ν− m N)
dν + −₁ X s=−M +1 Z 0.5 −0.5−s M ej(2πsM )(n−ne)ej2πντG ej2π(ν+ s M− m N)
G∗ ej2π(ν− m N
dν ) + cρ[τ ] (A.2)

 A ı Ë“ Q-VRû Cr[k, τ ] , úk 0 ≤ k ≤ M − 1 , ª) Cr[k, τ ] = 1 M M −1 X n=0 σ2 cej2πθeτ M N −1 X m=0 |w[m]|2 (_{M −1} X s=0 Z 0.5−_Ms −0.5 ej(2πsM )(n−ne)ej2πντG ej2π(ν+ s M− m N)
· G∗ ej2π(ν−mN)
dν + −₁ X s=−M +1 Z 0.5 −_0.5−s M ej(2πsM )(n−ne)ej2πντG ej2π(ν+ s M− m N)
G∗ ej2π(ν− m N
dν ) ·e−j2πk M n+ 1 M M −1 X n=0 cρ[τ ]e−j 2πk M n = M −1 X n=0 σ2 cej2πθeτ M2 N −1 X m=0 |w[m]|2 (_{M −1} X s=0 ej(2πsM )(n−ne)e−j 2πk M n Z 0.5−_Ms −_0.5 ej2πντG ej2π(ν+Ms− m N)
· G∗ ej2π(ν−m_N)
_{dν +} −₁ X s=−M +1 ej(2πs_M )(n−ne)_e−_j2πk M n Z 0.5 −_0.5−s M ej2πντ_{G e}j2π(ν+_Ms−m N)
·G∗ ej2π(ν−mN
dν  + cρ[τ ]δ[k] = σ 2 cej2πθeτ M2 N −1 X m=0 |w[m]|2 (_{M −1} X s=0 M −1 X n=0 e−_j2πn M (k−s)e−j(2πsM )ne Z 0.5−_Ms −_0.5 ej2πντG ej2π(ν+Ms − m N)
· G∗ ej2π(ν−mN)
dν + −₁ X s=−M +1 M −1 X n=0 e−_j2πn M (k−s)e−j( 2πs M )ne Z 0.5 −_0.5−s M ej2πντG ej2π(ν+Ms− m N)
·G∗ ej2π(ν−mN
dν  + cρ[τ ]δ[k] (A.3) w2PM −1 n=0 e −_j2πn M (k−s)= M δ[k − s + lM ] Ñ×à (discrete) í δ ƒb, Hp (A.3), ll 1 ≤ k ≤ M − 1 í8$, ) Cr[k, τ ] = σ2 cej2πθeτ M N −1 X m=0 |w[m]|2 ( e−_j(2πk M )ne Z 0.5−_Mk −0.5 ej2πντGej2π(ν+Mk− m N)  G∗ ej2π(ν−mN)
dν +e−_j(2π M)(k−M )ne Z 0.5 −_0.5−k−M M ej2πντ_G_ej2π(ν+k−M_M −m N)  G∗ ej2π(ν−m_N
dν  = σ 2 cej2πθeτ M N −1 X m=0 |w[m]|2e−_j(2πk M )ne ( Z 0.5−_Mk −0.5 ej2πντGej2π(ν+Mk− m N)  G∗ ej2π(ν−mN)
dν +ej2πne Z 0.5 0.5−k M ej2πντ_G_ej2π(ν+_Mk−m N)  G∗ ej2π(ν−m_N
dν  (A.4) ÇÕl Cr[0, τ ] = σ2 cej2πθeτ M N −1 X m=0 |w[m]|2 Z 0.5 −_0.5 ej2πντG ej2π(ν−mN)
G∗ ej2π(ν− m N)
dν + c ρ[τ ] (A.5)

 A ı Ë“ ¯9 (A.4) D (A.5) ª), úk k = 0, 1, · · · , M − 1 Cr[k, τ ] = σ2 c Me j2πθeτ_e−_j2πk M ne N −1 X m=0 |w[m]|2 ( Z 0.5−k/M −_0.5 ej2πντGej2π(ν+Mk− m N)  G∗ ej2π(ν−mN)
dν + ej2πne Z 0.5 0.5−k/M ej2πντGej2π(ν+Mk− m N)  G∗ ej2π(ν−mN)
dν  + cρ[τ ]δ[k] (A.6) Q-Vl (A.6) 2í }¶} Z 0.5−k/M −_0.5 ej2πντGej2π(ν+Mk− m N)  G∗ ej2π(ν−mN)
dν = Z 0.5−k/M −_0.5 ej2πντ ∞ X α=−∞ g[α]e−_j2π(ν+k M− m N)α ! ∞ X β=−∞ g[β]ej2π(ν−mN)β ! dν = ∞ X α=−∞ ∞ X β=−∞ g[α]g[β]ej2πmN (α−β)e−j 2πk M α Z 0.5−k/M −0.5 ej2π(β−α+τ )νdν = ∞ X α=−∞ ∞ X γ=−∞ g[α]g[α − γ]ej2πmN γe−j 2πk M α Z 0.5−k/M −_0.5 ej2π(τ −γ)νdν = ∞ X α=−∞ g[α]g[α − τ ]ej2πmN τe−j 2πk M α(1 + k M) + ∞ X α=−∞ ∞ X γ=−∞ γ6=τ g[α]g[α − γ]ej2πmN γe−j 2πk M α (−1) τ −γ 2π(τ − γ)  ej2πkM (τ −γ)− 1  (A.7) °Üª„ Z 0.5 0.5−k/M ej2πντGej2π(ν+Mk− m N)  G∗ ej2π(ν−mN)
dν = ∞ X α=−∞ g[α]g[α − τ ]ej2πmN τe−j 2πk M α k M + ∞ X α=−∞ ∞ X γ=−∞ γ6=τ g[α]g[α − γ]ej2πmN γe−j 2πk M α (−1) τ −γ 2π(τ − γ)  1 − ej2πkM (τ −γ)  (A.8) ø (A.7) D (A.8) Hp (A.6) ª)ƒ|(!‹

Cr[k, τ ] = σ2 c Me j2πθeτ_e−j2πk M ne  ΓN[τ ]A(g, g)  τ, k M   1 + k M(e j2πne_{− 1)}  +(ej2πne _{− 1)} ∞ X γ=−∞ γ6=τ ΓN[γ]A(g, g)  γ, k M  (−1)τ −γ j2π(τ − γ)  1 − e−_{j2π(τ −γ)}k M  + cρ[τ ]δ[k] (A.9) w2 ΓN[τ ] :=PN −1_k=0 |w[k]|2ej(2π/N )kτ , 7 A(g, h)[k, θ) :=P ∞ n=−∞g[n]h[n − k]e −_j2πnθ

¡5d.

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