A Structured Channel Estimation Algorithm Based ...

A Structured Channel Estimation Algorithm Based on Estimating the Time-Of-Arrivals (TOAs), with Applications to Digital TV Receivers a,b and Michael D. Zoltowskia ¨ Serdar Ozen a Purdue

University, School of Electrical & Computer Eng., W. Lafayette, IN 47907-1285, USA Electronics Corp., 2000 Millbrook Drive, Lincolnshire, IL 60069, USA

b Zenith

ABSTRACT In this paper we introduce a new structured channel impulse response (CIR) estimation method for sparse multipath channels. We call this novel CIR estimation method Time-Of-Arrival based Blended Least Squares (TOA-BLS) which uses symbol rate sampled signals, based on blending the least squares based channel estimation and the correlation and cleaning followed by TOA estimation in the frequency domain. TOA estimation in the frequency domain is accomplished by estimating the AR model parameters by solving the forward and forward-backward linear prediction equations in the least squares sense. Simulation examples are drawn from the ATSC digital TV 8-VSB system.1 The delay spread for digital TV systems can be as long as several hundred times the symbol duration; however digital TV channels are, in general, sparse where there are only a few dominant multipaths.

1. OVERVIEW OF DATA TRANSMISSION MODEL For the communications systems utilizing periodically transmitted training sequence, least-squares (LS) based channel estimation or the correlation based channel estimation algorithms have been the most widely used two alternatives. Both methods use a stored copy of the known transmitted training sequence at the receiver. The properties and the length of the training sequence are generally different depending on the particular communication system’s standard specifications. In the sequel, although the examples following the derivations of the blended channel estimator will be drawn from the ATSC digital TV 8-VSB system,1 to the best of our knowledge it could be applied with minor modifications to any digital communication system with linear modulation which employs a training sequence. The baseband symbol rate sampled receiver pulse-matched filter output is given by X X X y[n] ≡ y(t)|t=nT = Ik h[n − k] + ν[n] = Ik h[n − k] + η[k]q ∗ [−n + k], k

k

(1)

k

where Ik =

½

ak , dk ,

0≤k ≤N −1 N ≤ n ≤ N 0 −1,

¾

∈ A ≡ {α1 ,· · · , αM }

(2)

is the M -ary complex valued transmitted sequence, A ⊂ C1 , and {ak } ∈ C1 denote the first N symbols within a frame of length N 0 to indicate that they are the known training symbols; ν[n] = η[n] ∗ q ∗ [−n] denotes the complex (colored) noise process after the (pulse) matched filter, with η[n] being a zero-mean white Gaussian complex noise process with variance ση2 per real and imaginary part; h(t) is the complex valued impulse response of the composite channel, including pulse shaping transmit filter q(t), the physical channel impulse response c(t), and the receive filter q ∗ (−t), and is given by h(t)

= p(t) ∗ c(t) =

L X

ck p(t − τk ),

k=−K

¨ Further author information: (Send correspondence to: S.O.) ¨ E-mail: [email protected], or [email protected], Telephone: 847-941-8797, Fax 847-941-8555 S.O.: M.D.Z.: E-mail: [email protected],

(3)

and p(t) = q(t)∗q ∗ (−t) is the convolution of the transmit and receive filters where q(t) has a finite support of [−Tq /2, Tq /2], and the span of the transmit and receive filters, Tq , is integer multiple of the symbol period, T ; that is Tq = Nq T , Nq ∈ Z+ . {ck } ⊂ C1 denote complex valued physical channel gains, and {τk } denote the multipath delays, or the Time-Of-Arrivals (TOA). It is assumed that the time-variations of the channel is slow enough that c(t) can be assumed to be a static intersymbol interference (ISI) channel, at least throughout the training period with the impulse response c(t)

=

L X

ck δ(t − τk )

(4)

k=−K

for 0 ≤ t ≤ N T , where N is the number of training symbols. The summation limits K and L denote the number of maximum anti-causal and causal multi-path delays respectively. The multi-path delays τk are not assumed to be at integer multiples of the sampling period T . Indeed it is one of the main contributions of this work that we show a robust way of recovering the pulse shape back into the composite channel estimate when the multi-path delays are not at the sampling instants. Without loss of generality symbol rate sampled composite CIR h[n] can be written as a finite dimensional vector h =

[h[−Na ], h[−Na + 1], · · · , h[−1], h[0], h[1], · · · , h[Nc − 1], h[Nc ]]T

(5)

where Na and Nc denote the number of anti-causal and the causal taps of the channel, respectively, and Na + Nc + 1 is the total memory of the channel. In the sequel the notation of y [n1 :n2 ] with n2 ≥ n1 , will be adopted to indicate the column vector y [n1 :n2 ] = [y[n1 ], y[n1 + 1], · · · , y[n2 ]]T . Same notation will also be applied to the noise variables η[n] and ν[n]. Based on Equation (1) and assuming that N ≥ Na + Nc + 1, the pulse matched filter output corresponding only to the known training symbols can be written compactly as y [Nc :N −Na −1]

˜ + ν [N :N −N −1] = Ah ˜ + Qη ˜ [N −L :N −1−N +L ] , = Ah c a c q a q

(6)

where

˜ = T A

©



ª   [aNc+Na , · · ·, aN−1 ]T , [aNc+Na , · · ·, a0 ] =  

aNc+Na aNc+Na−1 aNc+Na+1 aNc+Na .. .. . . aN−1

aN−2

··· ··· .. .

a0 a1 .. .

···

aN−1−Na−Nc



  , 

(7)

˜ is (N − Na − Nc ) × (Na + Nc + 1) Toeplitz convolution matrix with first column [aNc +Na , · · · , aN −1 ]T and where A ˜ [N −L :N −1−N +L ] is the colored noise at the receiver matched first row [aNc +Na , · · · , a0 ], and ν [Nc :N −Na −1] = Qη c q a q filter output, with   T q 0 ··· 0  0 qT · · · 0    ˜ (8) Q =  . .. ..  ..  .. . . .  0 0 · · · q T (N−N −N )×(N−N −N +N ) a

q

=

c

a

c

q

T

[q[+Lq ], · · · , q[0], · · · , q[−Lq ]] ,

(9)

where q is the time reversed pulse matched filter. The pulse matched filter output which includes all the contributions from the known training symbols (which includes the adjacent random data as well) can be written as y [−Na :N +Nc −1]

= (A + D) h + ν [−Na :N +Nc −1] = Ah + Dh + Qη [−Na −Lq :N +Nc −1+Lq ] , = Ah + Hd + Qη [−Na −Lq :N +Nc −1+Lq ] ,

(10) (11)

where       A = T [a0 , · · · , aN −1 , 0, · · · , 0]T , [a0 , 0, · · · , 0] | {z }  | {z }    Na +Nc

(12)

Na +Nc

is a (N + Na + Nc ) × (Na + Nc + 1) Toeplitz matrix with first column [a0 , a1 , · · · , aN −1 , 0, · · · , 0]T , and first row [a0 , 0, · · · , 0], and     D = T [0, · · · , 0, dN , · · · , dNc+Na+N−1 ]T , [0, d−1 , · · · , d−Nc−Na ] , (13)  | {z }  N

is a Toeplitz matrix which includes the adjacent unknown symbols only, prior to and after the training sequence. The data sequence [d−1 , · · · , d−Nc−Na ] is the unknown information symbols transmitted at the end of the frame prior to the current frame being transmitted. Q is of dimension (N + Na + Nc ) × (N + Na + Nc + Nq ) and is given by  T  q 0 ··· 0  0 qT · · · 0    Q =  . . (14) .. ..  ..  .. . . .  0 0 · · · q T (N +N +N )×(N +N +N +N ) a

c

a

c

q

d = [d−N −Na , · · · , d−1 , dN , · · · , dN +Nc +Na −1 ]T is the random data vector which contains information symbols transmitted prior to and after the training sequence, and   T ¯ h 0 ··· 0  ¯T · · · 0    0 h  (15) H =  . ..  .. ..  . .  . .  . ¯T 0 0 ··· h (N+Nc+Na )×(N+2(Na+Nc ))

¯ = h

(16)

J

(17)

[h[Nc ], · · · , h[1], h[0], h[−1], · · · , h[−Na ]]T = J h   0 ··· 0 1  0 ··· 1 0   =  . .. ..   .. . . 1

H

= HS

0

···

0

(Na+Nc+1)×(Na+Nc+1)

T

(18)

¯ is the time reversed version of h (re-ordering is accomplished by the permutation matrix J ), and H is of where h dimension (N + Na + Nc ) × (2(Nc +Na )) with a “hole” inside which is created by the selection matrix S defined by ¸ · 0(Na +Nc )×N 0(Na +Nc )×(Nc +Na ) I Na +Nc . (19) S = 0(Na +Nc )×(Na +Nc ) 0(Na +Nc )×N I Na +Nc (2(N +N ))×(N+2(N +N )) c

a

a

c

1.1. Review of (Unstructured) Least-Squares Channel Estimation ˜ is a tall matrix and of full column Based on Equation (6) and assuming that N ≥ Na + Nc + 1, and as long as the matrix A rank, that is (i) N ≥ 2(Na + Nc ) + 1, ˜ = Na + Nc + 1 (ii) rank{A}

then the generalized least squares solution which minimizes the objective function ˜ H (K ν )−1 (y [N :N −N −1] − Ah) ˜ JGLS (h) = (y [Nc :N −Na −1] − Ah) c a

(20)

exists and unique, and is given by b LS h

H

H

H

H

=

˜ (ση2 Q ˜Q ˜ )−1 A) ˜ −1 A ˜ (ση2 Q ˜Q ˜ )−1 y [N :N −N −1] (A c a

=

˜ (Q ˜Q ˜ )−1 A) ˜ −1 A ˜ (Q ˜Q ˜ )−1 y [N :N −N −1] . (A c a

H

H

H

H

(21)

The covariance matrix of the colored noise vector ν [Nc :N −Na −1] = Qη [Nc −Lq :N −1−Na +Lq ] is denoted by K ν and is given by Cov{ν} = K ν ≡

1 ˜Q ˜H E{νν H } = ση2 Q 2

(22)

where ση2 is the variance of the noise sequence η[n]. For a single antenna 8-VSB digital TV receiver the problems associated with the (unstructured) least squares based ¨ CIR estimation is summarized by Ozen, et al.2, 3 More specifically the delay spreads the digital TV receivers would encounter are generally in excess of 40-50 nanoseconds, which implies that the length of the channel impulse response is in the order of 500, or more, symbol spaced taps. In such a scenario the requirement of N ≥ 2(Na + Nc ) + 1 is ˜ matrix, and hence the system of equations of (6) becomes underdetermined. To violated which leads to having a wide A remedy this problem a structured channel estimation algorithm called Blended Least Squares (BLS) has been introduced ¨ by Ozen, et al,2, 3 where the TOA information is estimated from the time-domain correlation/cleaning and thresholding operations, and then subsequently the TOA information is blended into the least squares equations. In the sequel, once an initial estimate is obtained, we estimate the TOAs in the frequency domain via forward and forward-backward linear prediction of the AR parameters. Hence we call this version of the BLS algorithm FDTOA-BLS. For the model of (10) or (11) the derivations of the unstructured Best Linear Unbiased Estimator (BLUE) of the channel impulse response has also been presented.3, 4 Although the details of the BLUE channel estimation algorithm is outside the scope of this work, the BLUE algorithm can be utilized to provide an initial channel estimate.

2. PROPOSED CIR ESTIMATION ALGORITHM: FDTOA-BLS We will first briefly overview the initial channel estimation which involves correlation, cleaning followed by TOA estimation in the frequency domain. Once the TOA’s are properly obtained we will then present the Blended Least Squares (BLS) algorithm.

2.1. Initial Channel Estimation Cross correlating the stored training sequence with the received sequence, which is readily available in digital receivers for the primary purpose of frame synchronization,5 yields a raw channel estimate ˆ u [n] h

=

N −1 1 X ∗ ak y[k + n], n = −Na , · · · , 0, · · · , Nc ra [0]

(23)

k=0

where ra [0] =

NP −1

kak k2 . Equivalently Equation (23) can be written as

k=0

ˆu h

=

1 AH y [−Na :N +Nc −1] . ra [0]

(24)

Recall that y [−Na :N +Nc −1]

= Ah + Dh + Qη [−Na −Lq :N +Nc −1+Lq ] , = Ah + Hd + Qη [−Na −Lq :N +Nc −1+Lq ] .

(25)

Substituting Equation (25) into (24) we get ˆu h

= =

³ ´ 1 AH Ah + Hd + Qη [−Na −Lq :N +Nc −1+Lq ] ra [0] ³ ´ 1 1 AH Ah + AH Hd + Qη [−Na −Lq :N +Nc −1+Lq ] . ra [0] ra [0]

(26)

In order to get rid of the sidelobes of the aperiodic autocorrelation we can simply invert the normalized autocorrelation matrix Raa of the training symbols, defined by Raa

=

1 AH A. ra [0]

(27)

ˆ c is obtained from Then the cleaned channel estimate h ˆc h

ˆ = R−1 aa hu H = (A A)−1 AH y [−Na :N +Nc −1] .

(28)

Note that the estimator of (28) is also recognized as the standard least squares channel estimator for the model of (11), or (25), where the zeros are inserted in place of the random information symbols which are transmitted prior to and after the training sequence. Substituting Equation (24) into (28) we get ˆc h

³ ´−1 ´ ³ = h + AH A AH Hd + Qη [−Na −Lq :N +Nc −1+Lq ] .

(29)

ˆ c has the contributions due to unknown symbols prior to and As can be seen from Equation (29) the channel estimate h after the training sequence, which are elements of the vector d, as well as the additive channel noise; only the sidelobes due ³ ´−1 ´ ³ to aperiodic auto-correlation is removed. The term AH A AH Hd + Qη [−Na −Lq :N +Nc −1+Lq ] is called baseline

noise in the raw channel estimate.3, 5

If all the symbols involved in the correlation of Equation (24) were perfectly known then the baseline noise in the estimation vector would have been due to finite correlation of known symbols only. Then the cleaning algorithm would have cleaned this deterministic noise perfectly and we wouldn’t have needed any thresholding on the cleaned estimation vector. But we always have unknown symbols involved in the correlation in most of the practical applications. So we ¨ need a thresholding algorithm after cleaning. Previously Ozen et al2, 6 used thresholding in the time domain to obtain the locations of the TOA’s as well as to decrease the contributions of the unknown symbols in the correlation output. The correlations of Equation (24) will ideally yield peaks at {Dka ∈ Z+ }, for k = −K, · · · , −1, 0, and at {Dkc ∈ Z+ }, for k = 1, · · · , L. These delays are the sampling instants closest to the locations of the actual physical channel multi-path TOAs τk , k = −K, · · · , −1, 0, 1, · · · , L, within a symbol interval. If we apply a uniform thresholding to the vector obtained by Equation (28) which is in the form of setting the estimated channel taps to zero if they are below a certain preselected threshold; that is ½ ˜ c [n]k < ε 0, if kh ˜ (30) set hth [n] = ˜ hc [n], otherwise, for n = −Na , · · · , −1, 0, 1, · · · , Nc , then in general we can choose the tap location with largest magnitude and denote it as the cursor (reference) path, and the TOAs prior to and after this reference TOA are denoted as pre- and post-cursor channel impulse responses. It has also been demonstrated that more sophisticated thresholding algorithms yield better TOA detection performance.3 These algorithms use the statistical characterization of the baseline noise ³ ´−1 ³ ´ AH Hd + Qη [−Na −Lq :N +Nc −1+Lq ] which is obtained in Equation (29). However, in this work we term AH A introduce the estimation of TOA’s in the frequency domain via linear prediction.

2.2. TOA Estimation by Estimating the AR model parameters via Forward and Forward-Backward Linear Prediction Consider the CIR which is given by h(t) = c(t) ∗ p(t) =

Ã

L X

!

cn δ(t − τn )

n=−K

∗ p(t).

(31)

!

(32)

If we take the Fourier transform of h(t) we obtain H(f )

= C(f )P (f ) =

Ã

L X

cn exp {−j2πf τn } P (f ).

n=−K

Evaluating H(f ) at Nf discrete frequency points, that is having f = k/Nf , and defining H[k] = H(k/Nf ), C[k] = C(k/Nf ), and P [k] = P (k/Nf ) we obtain à L ¾! ½ X j2πkτn H[k] = C[k]P [k] = P [k]. (33) cn exp − Nf n=−K

Since the pulse shape p(t) and is discrete Fourier transform P [k] is known, we can convert the channel estimation problem into a complex sinusoid estimation problem by writing the physical CIR discrete frequency response as C[k]

= H[k]/P [k], ∀k P [k] 6= 0 ¾ ½ L X j2πkτn = cn exp − Nf

(34)

n=−K

for those frequencies k where P [k] is nonzero. The rest of the complex sinusoid estimation problem can be accomplished by following a similar approach to the one outlined in7–9 : we can form the forward linear predictor (FLP) of order Np of each sample C[k] for k ≥ Np based on Np previous samples, and also forward-backward linear predictor (FBLP) of order Np of each sample C[k] for k ≥ Np . For FBLP we write both the forward and the predictors of order Np . The forward linear prediction error to be minimized in the least squares sense is given by ¯2 ¯ ¯ Nf −1 ¯ Np X X ¯ ¯ f ¯ , ¯C[k] − (35) C[k − n] ρ Epf = n ¯ ¯ ¯ n=1 k=Np ¯ where ρfn , n = 1, · · · , Np , denote FLP AR Model parameters.8 These unknown AR model parameters, {ρfn }, can be obtained by solving the set of equations (via any stable numerical algorithm such as Conjugate Gradients) H RH f R f ρ = R f cf

(36)

where Rf

= T {[C[Np − 1], · · · , C[Nf − 2]]T , [C[Np − 1], · · · , C[0]]},

ρ = cf

=

[ρf1 , · · ·

, ρfNp ]T ,

(37) (38)

T

[C[Np ], · · · , C[Nf − 1]] .

Similarly the forward-backward linear prediction error to be minimized in the least squares sense is given by ¯2 ¯ ¯2 ¯ ¯ ¯ Nf −1 ¯ Nf −1 ¯ Np −1 Np X X ¯ X X ¯ ¯ ¯ b f fb ¯ ¯ ¯ ρn C[k − n]¯¯ , ρn C[k − n]¯ + Ep = ¯C[k − Np ] − ¯C[k] − ¯ ¯ n=0 n=1 k=Np ¯ k=Np ¯

(39)

(40)

where ρbn , n = 0, · · · , Np − 1, denote backward LP AR Model parameters.8 It is also well known that ρbn = ρfN∗p −n for n = 0, 1, · · · , Np , and ρbNp = ρf0 = 1, which enables us to rewrite the FBLP error equation (40) in terms of the forward LP parameters {ρfn } only.8 Then Equation (40) can be written as (and can be solved via any stable numerical algorithm such as Conjugate Gradients10 ) H RH f b R f b ρ = R f b cf b

(41)

where Rf b Rb cf b

=

·

Rf Rb

¸

,

(42)

= H{[C ∗ [1], · · · , C ∗ [Nf − Np ]]T , [C ∗ [Nf − Np ], · · · , C ∗ [Nf − 1]]}, =

∗

∗

T

[C[Np ], · · · , C[Nf − 1], C [0], · · · , C [Nf − Np − 1]] .

(43) (44)

where Rf and ρ are defined as in Equations (37) and (38) respectively, and H{v col , v row } denotes Hankel matrix with first column v col and last row v row . n Once the unknown AR model parameter vector, ρ, is obtained then the complex sinusoids {exp{− j2πkτ Nf }} which is also equivalent to estimating the TOAs {τn }, is accomplished via thresholding the discrete power spectrum

Ψ[m]

=

Epf b ¯ ¯2 , ¯ ¯ Np P ¯ ¯ ρ∗k e−j2πmk/Nft ¯ ¯1 + ¯ ¯ k=1

(45)

for 0 ≤ m ≤ Nft − 1, with Nft > Nf , and the thresholding is accomplished by set

τˆk

= kT Nf /Nft

if Ψ[k] > ε0

(46)

where ε0 can be set experimentally to a value which is a few standard deviations above the average value of Ψ[m]. However, it is important to note that the estimation of the AR parameters can only start from an initial CIR estimate which inherently includes baseline noise as demonstrated in Section 2.1.

2.3. Overview of Blended Least Squares Algorithm The channel estimation is performed in two steps using symbol-spaced received samples after the receiver pulse matched filter. In the first step, the received samples are correlated with the stored training sequence, and cleaning is applied, summarized by Equations (24,28) respectively; and the TOAs are determined as in Section 2.2. The purpose of the second step is to incorporate the transmitted pulse shape p(t) into the channel impulse response. To do this, we locate three copies of p(t) shifted by one-half of a symbol period around each multipath location and estimate complex scaling factors using a modified least squares approach. Then we can write the estimate of the z-transform of the baseband representation of the physical channel c(t) in general as b C(z)

c c c a a a ˜ L z −DL ˜ 2 z −D2 + · · · + α ˜0 + α ˜ 1 z −D1 + α = β˜K z DK + · · · + β˜2 z D2 + β˜1 z D1 + α ´ ³ c c a a = α ˜ 0 βK z DK + · · · + β1 z D1 + 1 + α1 z −D1 + · · · + αL z −DL

(47)

where βk ’s and αk ’s are the complex gains for the channel and {Dka ∈ Z+ ∪ {0} }, for k = −K, · · · , −1, 0, {Dkc ∈ Z+ }, for k = 1, · · · , L are the TOAs of the estimated channel corresponding to the pre-cursor (anti-causal) part, and the postcursor (causal) part respectively; and we defined αi = α ˜ i /˜ α0 , for 1 ≤ i ≤ L , and βk = β˜k /˜ α0 , for 1 ≤ k ≤ K. The adoption of the notation used in Equation (47) is based on the fact that, after the thresholding, there are usually very few dominant taps, or equivalently we can say that the channels we encounter are generally sparse. It is assumed that a c . Also by the construction of the channel estimates in , and similarly 1 ≤ D1a < · · · < DK 1 ≤ D1c < · · · < DL

a c Equations (23) we will have DK ≤ N − 1 and DL ≤ N − 1. The relationship between the actual TOA’s and the estimated TOAs is given by ³τ ´ k Dka = −round , for − K ≤ k ≤ 0, (48) T ³τ ´ k , for 1 ≤ k ≤ L (49) Dkc = round T

where the TOAs are first estimated by Equation (46) in particular, as shown in Section 2.2. After the correlation and cleaning takes place we will end up with a noisy raw estimate of the channel taps, which implies the fact that the tails of the pulse shape that are buried under the “noisy” correlation output. Now we will show to recover the pulse shape p(t) into the CIR estimate. The idea is that for every multi-path we would like to approximate the shifted and scaled copies of the pulse shape p(t) (shifted by τk and scaled by ck ) by a linear combination of three pulse shape functions shifted by half a symbol interval. More precisely  1 P (k)   γl p((n + Dka − 2l )T ), −K ≤ k ≤ 0  l=−1 ck p(nT − τk ) ≈ (50) 1 P  (k)   γl p((n − Dkc − 2l )T ), 1 ≤ k ≤ L l=−1

(k)

where {γl , −K ≤ k ≤ L}1l=−1 ⊂ C1 . To accomplish this approximation we introduce three vectors pk , for k = −1, 0, 1, each containing T spaced samples of the complex pulse shape p(t) shifted by kT /2, such that p−1

=

p0

=

p1

=

T T T T T ),· · ·, p(−T + ), p( ), p(T + ),· · ·, p(Nq T + )]T , 2 2 2 2 2 [p(−Nq T ), · · · , p(−T ), p(0), p(T ), · · · , p(Nq T )]T , T T T T T [p(−Nq T − ),· · ·, p(−T− ), p(− ), p(T − ),· · ·, p(Nq T − )]T , 2 2 2 2 2 [p(−Nq T +

(51) (52) (53)

or more compactly pk = [p(−Nq T −

kT kT T kT ),· · ·, p(− ),· · ·, p(Nq T − )] , 2 2 2

(54)

for k = −1, 0, 1, and by concatenating these vectors side by side we define a (2Nq + 1) × 3 matrix P by P

=

[p−1 , p0 , p1 ].

(55)

Then we form the matrix denoted by Γ whose columns are composed of the shifted vectors pk , where the shifts represent the relative delays of the multi-paths; that is   a −D a P 0(DK K−1 )×3   ..   0 a c . 0DK a ×3   (DK+DL )×3 P   a a +D c 0(DLc +DK− P 0 (56) Γ= , )×3 (D )×3 1 K L−1   ..   . P a +D c )×3   0DLc ×3 0(DK L 0(DLc −DLc−1 )×3 P

c a + 2Nq + 1) × 3(K + L + 1), and 0m×n denotes an m by n zero matrix. Recall + DL where Γ is of dimension (DK ˜ composed only of the known training symbols are that the observation vector y [Nc :N −Na ] , and the convolution matrix, A, defined by Equations (6, 7) respectively.

A simplified version of this idea applied for a real valued pulse shape p(t) is illustrated in Figure 1. Since it was assumed that q(t) spans Nq symbol durations, it implies that q[n] has Nq + 1 sample points, which in turn implies p[n] has 2Nq + 1 samples. Hence a + Nq Na = DK

and

c + Nq . Nc = DL

(57)

Multipath arrival

τk

DkT -T/2

DkT

DkT +T/2

T/2 T/2 Sampling instant

kT (k)

γ−1 (Dk − 1/2)T

kT

≈ ck p(t − τk ) (k)

γ0 Dk T

kT (k)

γ1 (Dk + 1/2)T

Figure 1. Simplified illustration of the BLS algorithm with a real valued p(t).

Table 1. Simulated channel delays in symbol periods, relative gains. First two paths are pre-cursor ghosts, and the six multipaths after the main path are post-ghosts (K=2, L=6)

k -2 -1 0 1 2 3 4 5 6

(k)

(k)

Delay {τk } -60.277 -0.957 0 3.551 15.250 24.032 29.165 221.2345 332.9810

Gain {|ck |} 0.55 0.7263 1 0.6457 0.9848 0.7456 0.8616 0.6150 0.4900

(k)

Defining γ (k) = [γ−1 , γ0 , γ1 ] for −K ≤ k ≤ L, we define γ

=

[γ (−K) , · · · , γ (0) , · · · , γ (L) ]T ,

(58)

(k)

as the unknown vector of the coefficients with {γn , n = −1, 0, 1; k = −K, · · · , 0, · · · , L}, of length 3(K + L + 1). Then we can represent the observation vector by the approximation y [Nc :N −Na ]

˜ ˜ [N −L :N −1−N +L ] . = AΓγ + Qη c q a q

Using the (generalized) least squares arguments again, we can estimate the unknown coefficient vector γ as ³ ´−1 ˜H (Q ˜Q ˜ H )−1 AΓ ˜ ˜H (Q ˜Q ˜ H )−1 y [N :N −N ] . b LS = γ ΓH A ΓH A c a

(59)

(60)

b BLS , where the pulse tails are recovered back, can simply be b LS is obtained, the new channel estimate h Once the vector γ obtained by ³ ´−1 b BLS = Γb ˜H (Q ˜Q ˜ H )−1 AΓ ˜ ˜H (Q ˜Q ˜ H )−1 y [N :N −N ] . h γ LS = Γ ΓH A ΓH A (61) c a ˜ be a tall matrix, that is it is required that b LS it is required that AΓ In order to have a unique γ a c N − DK − DL − 2Nq ≥ 3(K + L + 1).

(62)

3. SIMULATIONS We considered an 8-VSB1 receiver with a single antenna. 8-VSB system has a complex raised cosine pulse shape.1 The CIR we considered is given in Table 1. The phase angles of individual paths for all the channels are taken to be arg{ck } = where fc =

50 T

exp(−j2πfc τk ), k = 1, · · · , 6

(63)

and T = 92.9nsec.

The simulations were run at 28dB Signal-to-Noise-Ratio (SNR) measured at the input to the receive pulse matched filter, and it is calculated by P 2 Es k{c(t) ∗ q(t)}t=kT k ∀k , (64) SNR = ση2 where Es = 21 is the symbol energy for 8-VSB system. We set Nfft = 210 , Nft = 213 , and ε0 = Ψ[k] + 4σ{Ψ[k]} , that is we set the ε0 to 4 standard deviations above the average of the power spectrum Ψ[k]. Figure 3 shows the simulation results

for the test channel provided in Table 1. Part (a) shows the actual CIR; part (b) shows the CIR estimate, b hBLS , based on BLS with estimated TOAs; part (c) shows the CIR estimate, b hBLS , based on BLS with perfectly known TOAs. As can be seen in Figure 3 either the 4th or the 5th iterations of the CG algorithm can be taken as the solutions to the FBLP and FLP equations (41,36) to compute the spectrum Ψ[k]. The performance measure we used to compare the different channel estimates is the normalized least-squares error ˆ 2 hk . Finally we have observed a very promising performance by using estimated TOA which is defined by EN LS = Nkh− a +Nc +1 parameters for the BLS algorithm to compute the CIR estimate, where the normalized least-squares error is very close to that of the BLS when the true TOAs are used. Figure 3 shows the simulation results for the test channel provided in Table 1. Part (a) shows the actual CIR; part (b) shows the CIR estimate, b hBLS , based on BLS with estimated TOAs; part (c) shows the CIR estimate, b hBLS , based on BLS with perfectly known TOAs. Imaginary part of the CIR

Real part of the CIR

0.6 0.2

0.4

(a)

0.2

0

0

(b)−0.2

−0.2

−0.4

−0.4 −100

0

100

200

300

0

100

200

300

−100

0

100

200

300

0.4

0.6

0.2

0.4 0.2

(c)

−100

Imaginary part of the cleaned channel estimate h c

Real part of the cleaned channel estimate h c

(d)

0

0

−0.2

−0.2

−0.4

−0.4 −100

100

200

300

Spectral Estimate by FLP, Np=256, CG step 4

0

−20

(f)

−40

Magnitude

(e)

Magnitude

0

0

−60

−40

100

200

300

400

500

Spectral Estimate by FBLP, Np=256, CG step 4

0

Magnitude

Magnitude

(g)

−20

−60 0

0

Spectral Estimate by FLP, Np=256, CG step 5

−20

0

100

200

0

100

200

300

400

500

300

400

500

Spectral Estimate by FBLP, Np=256, CG step 5

−20

(h) −40 −60

−40 −60

0

100

200

300

400

500

Ω

Ω

˜ c; Figure 2. Parts (a) and (b) show the real and imaginary parts of the actual CIR; parts (c) and (d) show the cleaned CIR estimate, h parts (e) and (f) show the TOA estimates with forward prediction of order Np = 256 at CG iterations 4 and 5 respectively; parts (g) and (h) show the TOA estimates with forward-backward prediction of order Np = 256 at CG iterations 4 and 5 respectively.

ACKNOWLEDGMENTS This research was funded by the National Science Foundation under grant number CCR-0118842 and by the Zenith Electronics Corporation.

REFERENCES 1. ATSC Digital Television Standard, A/53, September 1995. ¨ 2. S. Ozen, M. D. Zoltowski, M. Fimoff, “A Novel Channel Estimation Method: Blending Correlation and LeastSquares Based Approaches,” Proceedings of ICASSP-2002. ¨ 3. S. Ozen, “Topics on Channel Estimation and Equalization for Sparse Channels with applications to Digital TV Systems,” Ph.D. Dissertation, Purdue University, School of Electrical and Computer Engineering, W. Lafayette, IN 47907-1285, USA, May 2003.

TOA−BLS CIR stimate via FBLP, at SNR=28dB 0.6

Normalized LS error =5.9309e−006 0.3 0.2

0.4

0.1 0

0.2

−0.1

(a)

(b)−0.2

0

−0.3 −0.2

−0.4 −0.5

−0.4 −100

0

100

200

300

−100

0

100

200

300

delay in symbols (T)

delay in symbols (T)

BLS estimate with true tap locations, at SNR=28dB

Normalized LS error =3.7754e−006

0.6

0.3 0.2

0.4

0.1 0

0.2

−0.1

(c)

(d)−0.2

0

−0.3 −0.2

−0.4 −0.5

−0.4 −100

0

100

200

delay in symbols (T)

300

−100

0

100

200

300

delay in symbols (T)

Figure 3. Parts (a) and (b) show the real and imaginary parts of the CIR estimate, b hBLS , based on BLS with estimated TOAs via FBLP; parts (c) and (d) show the CIR estimate, b hBLS , based on BLS with perfectly known TOAs.

¨ 4. C. Pladdy, S. Ozen, M. Fimoff, S. M. Nerayanuru, M. D. Zoltowski, “Best Linear Unbiased Channel Estimation for Frequency Selective Multipath Channels with Long Delay Spreads,” submitted to IEEE Vehicular Technology Conference, Orlando, FL, 2003. ¨ 5. M. Fimoff, S. Ozen, S.M. Nereyanuru, M.D. Zoltowski, W. Hillery, “Using 8-VSB Training Sequence Correlation as a Channel Estimate for DFE Tap Initialization,” Proceedings of the 39th Annual Allerton Conference in Communications, Control and Computing, September 2001. ¨ 6. S. Ozen, W. Hillery, M. D. Zoltowski, S. M. Nereyanuru, M. Fimoff, “Structured Channel Estimation Based Decision Feedback Equalizers for Sparse Multipath Channels with Applications to Digital TV Receivers,” to be published in the proceedings of Asilomar Signals and Systems Conference-2002. 7. A. Hewitt, W. Lau, J. Austin, E. Vilar, “An autoregressive approach to the identification of multipath ray parameters from field measurements,” IEEE Tr. on Communications, vol. 37, no. 11, pp. 1136-1143, November 1989. 8. J. G. Proakis, D. G. Manolakis, Digital Signal Processing: Principles, Algorithms and Applications, 3rd ed., Prentice Hall, 1996. 9. D. W. Tufts, R. Kumaresan, “Estimation of frequencies of multiple sinusoids: Making linear prediction perform like maximum likelihood,” Proceedings of the IEEE, vol. 70, no. 9, pp. 975-989, September 1982. 10. P. Chang, A. Willson, Jr., “Analysis of conjugate gradient algorithms for adaptive filtering,” IEEE Tr. on Signal Proc., v. 48, pp. 409–418, Feb. 2000.