Time-Varying FIR Equalization of Doubly-Selective Channels ∗ Imad Barhumi† , Geert Leus‡ and Marc Moonen K.U.Leuven-ESAT, Kasteelpark Arenberg 10, 3001 Leuven, Belgium Email: {imad.barhumi,geert.leus,marc.moonen}@esat.kuleuven.ac.be

Abstract— In this paper we propose a zero forcing (ZF) timevarying (TV) finite impulse response (FIR) equalizer for doublyselective (time- and frequency-selective) channels. We use the basis expansion model (BEM) to approximate the doubly-selective channel and to design the TV FIR equalizer. This allows us to turn a large TV problem into an equivalent small time-invariant (TIV) problem, containing only the BEM coefficients of the doublyselective channel and the TV FIR equalizer. It is shown that a ZF TV FIR equalizer only exists if there is more than one receive antenna. The ZF TV FIR equalizer approach we propose here unifies and extends many previously proposed serial equalization approaches. Through computer simulations we show that the performance of the proposed ZF TV FIR equalizer approaches the one of the ZF block equalizer, while the equalization as well as the design complexity is much lower.

I. I NTRODUCTION

T

HE wireless communication industry has experienced rapid growth in recent years, and digital cellular systems are currently being designed to provide high speed multimedia services, such as voice, Internet access and video conferencing. These services require access speeds ranging from a few hundred kbits/s for high mobility users up to a few Mbits/s for low mobility users. Such high data rates give rise to frequencyselective propagation, while mobility and carrier offsets introduce time-selectivity. This results in so-called doubly-selective channels. To combat these doubly-selective channel effects, equalizers are required. We can make a distinction between block equalizers and serial equalizers. For frequency-selective channels these equalizers have been extensively studied in literature. Block equalizers for frequency-selective channels only require a single receive antenna [1]. They are usually complex, since they have to invert a large matrix. However, since a frequencyselective channel can be diagonalized by means of the Fast Fourier Transform (FFT), they can be simplified, at the cost ∗ This research work was carried out at the ESAT laboratory of the Katholieke Universiteit Leuven, in the frame of the Belgian State, Prime Minister’s Office - Federal Office for Scientific, Technical and Cultural Affairs - Interuniversity Poles of Attraction Programme (2002-2007) - IUAP P5/22 (‘Dynamical Systems and Control: Computation, Identification and Modeling’) and P5/11 (‘Mobile multimedia communication systems and networks’), the Concerted Research Action GOA-MEFISTO-666 (Mathematical Engineering for Information and Communication Systems Technology) of the Flemish Government, Research Project FWO nr.G.0196.02 (‘Design of efficient communication techniques for wireless time-dispersive multi-user MIMO systems’) and was partially sponsored by IMEC (Flemish Interuniversity Microelectronics Center). The scientific responsibility is assumed by its authors. † Partly supported by the Palestinian European Academic Cooperation in Education (PEACE) ‡ Postdoctoral Fellow of the F.W.O. Vlaanderen.

of a slight decrease in performance. On the other hand, serial equalizers [2], more specifically finite impulse response (FIR) equalizers, for frequency-selective channels require more than one receive antenna, but are simpler to implement [3]. Recently, equalizers have also been developed for doublyselective channels. As for frequency-selective channels, block equalizers for doubly-selective channels only require a single receive antenna. However, since a doubly-selective channel can not be diagonalized, they can not be simplified and are always complex. This motivates us to look at serial equalizers for doubly-selective channels, which are simpler to implement. For simplicity, only zero-forcing (ZF) FIR equalizers are considered. Up till now, only ZF time-invariant (TIV) FIR equalizers for doubly-selective channels have been introduced [4]1 . However, a ZF TIV FIR equalizer only exists if there are many receive antennas. In this paper, we introduce ZF timevarying (TV) FIR equalizers for doubly-selective channels. We use the basis expansion model (BEM) to approximate the doubly-selective channel and to design the TV FIR equalizer. This allows us to turn a large TV problem into an equivalent small time-invariant (TIV) problem, containing only the BEM coefficients of the doubly-selective channel and the TV FIR equalizer. A ZF TV FIR equalizer only exists if there is more than one receive antenna. But this condition is much better than the one for a ZF TIV FIR equalizer [4]. This paper is organized as follows. In section II, we give a brief description of the channel model. The system model is described in section III. The ZF TV FIR equalizer is introduced in section IV. In section V, we show through computer simulations the performance of the proposed equalizer. Finally, our conclusions are drawn in section VI. Notations: We use upper (lower) bold face letters to denote matrices (column vectors). Superscripts ∗ , T , and H represent conjugate, transpose, and Hermitian, respectively. We denote the 1- and 2-dimensional Kronecker delta as δn and δn,m , respectively. We denote the N × N identity matrix as IN and the M × N all-zero matrix as 0M ×N . Finally, diag{x} denotes the diagonal matrix with x on the diagonal. II. C HANNEL M ODEL In this paper, we assume a doubly-selective (time- and frequency-selective) channel. We model the doubly-selective channel using the basis expansion model (BEM) [5], [6], [7], 1 Although [4] considers a set of ZF TIV FIR equalizers, each of which reconstructs the transmitted sequence with a different frequency offset, we will only consider one of these ZF TIV FIR equalizers.

[8], which is shown to accurately approximate the well-known Jakes’ model. In the BEM, the channel is modeled as an FIR filter where the taps are expressed as a superposition of complex exponential basis functions. Before we proceed to define the BEM for doubly-selective channels, we make the following assumptions: A1) The delay-spread is bounded by τmax ; A2) The Doppler-spread is bounded by fmax . Under assumptions A1) and A2), it is always possible to model the channel in discrete time as: h(n; ν) =

L X

δν−l

l=0

Q/2 X

hq,l ej2πqn/N ,

(1)

q=−Q/2

where N is the block size over which this channel model holds, and L and Q satisfy the following conditions: C1) LT ≥ τmax ; C2) Q/(N T ) ≥ 2fmax , where T is the symbol period. In this expansion model, L represents the delay-spread (expressed in multiples of T , the delay resolution of the model), and Q represents the Doppler-spread (expressed in multiples of 1/(N T ), the Doppler resolution of the model). Note that the coefficients hq,l remain invariant over a period of length N T . η(1) (n) y 1 (n)

h (1) (n; ν)

g (1)(n; ν) sˆ (n)

s(n)

η(Nr ) (n) y (Nr ) (n)

h (Nr ) (n; ν)

Fig. 1.

g (Nr ) (n; ν)

System Model

III. S YSTEM M ODEL The system under consideration is depicted in Figure 1. We assume a single-input multiple-output (SIMO) system, where Nr receive antennas are used. Suppose we want to transmit a symbol burst s(n), where s(n) 6= 0 for n ∈ / {0, 1, . . . , M − 1}, where M = N − Lzp and Lzp ≥ L. The received sample sequence at the rth receive antenna is given by: y (r) (n) =

∞ X

h(r) (n; ν)s(n − ν) + η (r) (n)

ν=−∞

=

L X

Q/2

X

(r)

ej2πqn/N hq,l s(n − l) + η (r) (n),

the received sample block at the rth receive antenna y(r) := [y (r) (0), . . . , y (r) (N − 1)]T , can be written as: y(r) = H(r) Tzp s + η (r) ,

where η (r) is similarly defined as y(r) , Tzp := [IM , 0M ×Lzp ]T , and H(r) is an N × N lower triangular matrix. Using (1), H(r) can be written as: H(r) =

where η (n) is the additive noise at the rth receive antenna. Since y (r) (n) only contains noise samples for n < 0 and n ≥ N , we can set y (r) (n) = 0 for n < 0 and n ≥ N . This input-output relation can also be written in block form. Defining the M ×1 symbol block as s := [s(0), . . . , s(M −1)]T ,

Q/2 L X X

(r)

hq,l Dq Zl

(3)

l=0 q=−Q/2

where Dq := diag{[1, · · · , ej2πq(N −1)/N ]T }, and Zl is an N × N lower triangular Toeplitz matrix with first column [01×l , 1, 01×(N −l−1) ]T . Substituting (3) in (2), the N × 1 received sample block at the rth receive antenna can be written as: y(r) =

Q/2 L X X

(r)

hq,l Dq Zl Tzp s + η (r) .

(4)

l=0 q=−Q/2

Stacking the Nr received [y(1)T , . . . , y(Nr )T ]T , we obtain

sample

blocks:

y

:=

y = HTzp s + η, where η is similarly defined as y and H := [H(1)T , . . . , H(Nr )T ]T . The traditional way to equalize the BEM channel is by using an M × Nr N block equalizer G := [G(1) , . . . , G(Nr ) ], where G(r) is the M × N block equalizer that operates on the rth receive antenna. Hence, an estimate of s is then computed as ˆs = Gy = GHTzp s + Gη ! Nr Nr X X (r) (r) = G(r) η (r) . G H Tzp s +

(5)

r=1

r=1

A possible ZF block equalizer can be obtained as the pseudoinverse of the Nr N × M channel matrix HTzp : G = (HTzp )† . This pseudo-inverse generally only requires a single receive antenna, but it is computationally expensive. In the next section, we will therefore develop a ZF TV FIR equalizer, with lower design and equalization complexity and only a slightly lower performance. In this case, to obtain a ZF solution we will require more than one receive antenna. IV. T IME -VARYING FIR E QUALIZATION In this section, we will apply a TV FIR equalizer g (r) (n; ν) on the rth receive antenna, as depicted in Figure 1. Hence, an estimate of s(n − d) is computed as

l=0 q=−Q/2

(r)

(2)

sˆ(n − d) =

Nr X ∞ X

g (r) (n; ν)y (r) (n − ν).

(6)

r=1 ν=−∞

where d is the synchronization delay. Since the doubly-selective channel h(r) (n; ν) was modeled by the BEM, it is also convenient to design the TV FIR equalizer g (r) (n; ν) using the BEM.

This will allow us to turn a large TV problem into an equivalent small TIV problem, containing only the BEM coefficients of the doubly-selective channel and the TV FIR equalizer. Using the BEM, we design each TV FIR equalizer g (r) (n; ν) to have L′ + 1 taps, where the time-variation of each tap is modeled by Q′ + 1 complex exponential basis functions: Q′ /2

′

g

(r)

(n; ν) =

L X

X

δν−l′

l′ =0

(r)

′

gq′ ,l′ ej2πq n/N ,

(7)

q ′ =−Q′ /2

Instead of continuing to work on the sample level, it’s easier to switch to the block level at this point. On the block level (6) corresponds to estimating s as in (5) but with G(r) constrained to Q′ /2 L′ X X (r) (r) G = Rzp gq′ ,l′ Dq′ Zl′ , (8) l′ =0 q ′ =−Q′ /2

where Rzp := [0M ×d IM 0M ×(Lzp −d) ], it is clear that this requires 0 < d < Lzp . In our simulations we always take Lzp = max(L, d) Since we focus on the ZF solution, we will just look at the estimate of s in the noiseless case, which can now be written as  Q′ /2 Nr L′ X X X (r)  ˆs = Rzp gq′ ,l′ Dq′ Zl′ r=1

q ′ =−Q′ /2 l′ =0

Q/2 L X X

×

q=−Q/2 l=0



(r) hq,l Dq Zl  Tzp s

(9)

Hence, a ZF solution is obtained if the following condition is satisfied:  Q′ /2 Nr L′ X X X (r)  g ′ ′ Dq′ Zl′ q ,l

r=1

q ′ =−Q′ /2 l′ =0



Q/2

×

L X X

q=−Q/2 l=0

(r)

hq,l Dq Zl  = Zd .

(10)

Defining p := q + q ′ , k := l + l′ , and using the property ′

Dq Zl′ = e−j2πql Zl′ Dq , (10) can be rewritten as  (Q+Q′ )/2 L+L′ Nr X X X 

Q′ /2

X

′ LX −n0

′

′

e−j2π(p−q )l /N

r=1 q ′ =−Q′ /2 l′ =−n0

p=−(Q+Q′ )/2 k=0

×

(r) (r) gq′ ,l′ hp−q′ ,k−l′

!

Dp Zk = IN .

(11)

(11) can on its turn be rewritten as Q/2+Q′ /2

X

′ L+L X

fp,k Dp Zk = Zd .

(13)

p=−Q/2−Q′ /2 k=0

Note that each term in the 2-dimensional function fp,k corresponds to a 2-dimensional convolution of the BEM coefficients related to the doubly-selective channel for the rth receive antenna and the ones related to the TV FIR filter for the rth receive antenna. The ZF solution for the Nr TV FIR equalizers is thus obtained if: fp,k = δp,k−d . (14) Hence, we basically have to solve a 2-dimensional deconvolution problem. In order to do so, we introduce some required defini(1) (N ) tions. Denoting hq,l := [hq,l , . . . , hq,l r ]T , we first define the ′ ′ Nr (L + 1) × (L + L + 1) block Toeplitz matrix   hq,0 . . . hq,L 0Nr ×L′   .. .. Tl,L′ +1 (hq,l ) :=  . . . 0Nr ×L′

hq,0

...

hq,L

e q := (D e q ⊗ IN )Tl,L′ +1 (hq,l ), where We then define H r e q := diag{[1, e−j2πq/N , . . . , e−j2πqL′ /N ]T }, and introduce D the Nr (Q′ + 1)(L′ + 1) × (Q + Q′ + 1)(L + L′ + 1) block Toeplitz matrix   e −Q/2 . . . H e Q/2 H 0  .. .. e q ) :=  Tq,Q′ +1 (H .  . . e e 0 H−Q/2 . . . HQ/2 T T T T Defining g := [g−Q ′ /2,0 , . . . , g−Q′ /2,L′ , . . . , gQ′ /2,L′ ] , where (1)

(N )

gq′ ,l′ := [gq′ ,l′ , . . . , gq′ ,lr′ ]T , we can finally formulate (14) as the following system of equations: e q ) = [0 . . . 0 1 0 . . . 0], gT Tq,Q′ +1 (H

(15)

where the 1 is in the (d(Q + Q′ + 1) + (Q + Q′ )/2 + 1)st position. The system (15) has (Q+Q′ +1)(L+L′ +1) equations and Nr (Q′ + 1)(L′ + 1) unknowns. To be able to solve it, we need to choose the number of receive antennas (Nr ), and the number of BEM coefficients of the equalizer (L′ and Q′ ) such that there are less or as many equations as unknowns, i.e., Nr (Q′ + 1)(L′ + 1) ≥ (Q + Q′ + 1)(L + L′ + 1), which is only possible if we use Nr ≥ 2 receive antennas. In our discussion so far we assumed that the BEM coefficients of the channel are known at the receiver. In practice, the coefficients have to be estimated. This can be done in a blind fashion [9], or through training [10]. Considering the MMSE criteria to design the TV-FIR equalizer is proposed in [11], [12]. This is beyond the scope of this paper.

Hence, introducing the 2-dimensional function fp,k :=

Nr X

Q′ /2

X

A. Unifying Frame Work

′

L X

(r) (r) e−j2π(p−q )l /N gq′ ,l′ hp−q′ ,k−l′ , ′

′

r=1 q ′ =−Q′ /2 l′ =0

(12)

The ZF TV FIR equalizer we propose in this paper unifies and extends many previously proposed serial equalization approaches, as illustrated next.

For the case of TIV FIR equalization of a purely frequencyselective channel (Q = 0 and Q′ = 0) we require the condition L L′ ≥ ⌈ N r−1 −1⌉. This coincides with the result obtained in [3]. On the other hand, for the case of TV 1-tap FIR equalization of a purely time-selective channel (L = 0 and L′ = 0) we require Q the condition Q′ ≥ ⌈ N r−1 − 1⌉. It is also worth to note that for Nr > Q + 1 it is possible to perfectly equalize a doubly-selective channel with a TIV FIR − 1⌉. This equalizer (Q′ = 0) if we choose L′ ≥ ⌈ N(Q+1)L r −(Q+1) result coincides with the result obtained in [4]. On the other hand, for Nr > L + 1, it is possible to equalize a doublyselective channel with a TV 1-tap FIR equalizer (L′ = 0) if we choose Q′ ≥ ⌈ N(L+1)Q − 1⌉. r −(L+1) B. Identifiability The question that arises at this point is under what conditions there exists a TV FIR equalizer that satisfies (15). It is clear that e q ) is the system (15) has a solution when the matrix Tq,Q′ +1 (H of full column rank. The following proposition gives sufficient conditions for this matrix to be of full column rank [13]: e q ) has full column Proposition 1: The matrix Tq,Q′ +1 (H rank if: C1) Q′ ≥ (L + L′ + 1)Q; PQ/2 e e q z −q is of full column rank L+L′ +1 C2) H(z) = q=−Q/2 H e −Q/2 and H e Q/2 are of full column rank L + L′ + 1. C3) H The second and third conditions are usually known as the irreducible and column reduced condition, respectively. It’s not easy to see how these conditions are related to hq,l . This is a topic of future research. C. Complexity In this section we will discuss the complexity of the ZF block equalizer and the ZF TV FIR equalizer proposed in this paper. Two types of complexity can be considered; one associated with designing the equalizer, and the other associated with implementation. Design complexity: The design complexity is the complexity associated with computing the equalizer. To design the ZF block equalizer, we need to compute the pseudo-inverse of the Nr N × M channel matrix HTzp , which implicitly requires the inversion of an M × M matrix. The complexity of inverting an M × M matrix is O(M 3 ). On the other hand, to compute the coefficients of the ZF TV FIR equalizer proposed in this paper, we require to compute the pseudo-inverse of the e q ), where K = Nr (Q′ + 1)(L′ + 1) × K matrix Tq,Q′ +1 (H ′ ′ (Q + Q + 1)(L + L + 1). This requires a matrix inversion of size K ×K, and hence the associated complexity is O(K 3 ). So provided that K < M (which is usually the case) the design complexity of the proposed ZF TV FIR equalizer is less than the design complexity of the ZF block equalizer. Equalization complexity: The run time (equalization) complexity is computed as the number of multiplications required to estimate the transmitted block. For the ZF block equalizer, estimating the transmitted block requires M N multiplications

per receive antenna. On the other hand, for the ZF TV FIR equalizer, the number of multiplications required to estimate the transmitted block is N (L′ + 1) per receive antenna. Hence, provided that the equalizer order L′ is less than the block size M , the run time complexity of the ZF TV FIR equalizer is less than the run time complexity of the ZF block equalizer. However, in practice and also in our simulations, the required equalizer order L′ ≪ N . V.

SIMULATIONS

In the simulations, we consider a SIMO system with Nr = 2, 4, and 6 receive antennas. The channel is assumed to be doubly-selective with the following parameters: • transmitted block size N = 800; • symbol period T = 25µsec; • Doppler spread fmax = 100Hz; • number of basis functions Q = 2⌈fmax N T ⌉ = 4; • maximum delay spread τmax = 75µsec • channel order L = ⌈τmax /T ⌉ = 3. The proposed number of equalizer parameters for different receive antennas is listed in table I. Nr Q′ L′

2 20 20

4 8 8

6 0 20

6 4 4

TABLE I F ILTER COEFFICIENTS FOR DIFFERENT RECEIVE ANTENNAS

The channel taps are simulated as i.i.d., correlated in time with a correlation function according to Jakes’ model rhh (τ ) = J0 (2πfd τ ). The doubly-selective channel is approximated using the BEM. The BEM channel coefficients are used to determine the equalizer (serial or block). We use both Jakes’ model and the approximated BEM to simulate propagation. In all simulations, QPSK signaling is used and the delay ′ d = ⌊ L+L 2 ⌋ + 1. The performance is measured in terms of the BER vs. SNR. We compare the performance of the proposed ZF TV FIR equalizer with the ZF block equalizer for both Jakes’ model and the BEM. As shown in Figure 2, using Nr = 2 receive antennas, the performance of the ZF TV FIR equalizer approaches the performance of the ZF block equalizer with Q′ = 20, and L′ = 20. We notice here a 4 dB SNR loss between the ZF TV FIR equalizer and the ZF block equalizer at BER = 10−2 . However, using Nr = 4 receive antennas, the ZF TV FIR equalizer performance with Q′ = 8 and L′ = 8 comes closer to the performance of the ZF block equalizer, where the gap is reduced to 1 dB at BER = 10−2 , as shown in Figure 3. In Figure 4, we use Nr = 6 receive antennas, which allows us to equalize the channel using a TIV FIR equalizer (see also [4]). We can see that the ZF TV FIR equalizer with Q′ = 4 and L′ = 4 significantly outperforms the ZF TIV FIR equalizer with Q′ = 0 and L′ = 20. Note that the design complexity (O(723 )) and equalization complexity (5M

0

the design complexity (O(1203 )) and equalization complexity (21M per receive antenna) of the ZF TIV FIR equalizer. This leads us to conclude that incorporating time variety in the equalizer really pays off.

10

ZF TV FIR, BEM ZF TV FIR, Jakes ZF Block, BEM ZF Block, Jakes N =2 r

Q=4 L=3 Q’=20 L’=20

−1

10

BER

VI. C ONCLUSIONS −2

10

−3

10

−4

10

0

5

Fig. 2.

10

15

20 SNR (dB)

25

30

35

40

BER vs. SN R for Nr = 2 receive antennas

0

10

ZF TV FIR, BEM ZF TV FIR, Jakes ZF Block, BEM ZF Block, Jakes

In this paper, we have proposed a ZF TV FIR equalizer to compensate for the doubly-selective channel distortions. We have used the BEM to approximate the doubly-selective channel and to design the TV FIR equalizer. This allows us to turn a large TV problem into an equivalent small TIV problem, containing only the BEM coefficients of the doubly-selective channel and the TV FIR equalizer. It is shown that a ZF TV FIR equalizer only exists if there is more than one receive antenna. It is also shown that by carefully choosing the number of BEM coefficients for our ZF TV FIR equalizer, the performance of the proposed equalizer approaches the performance of the ZF block equalizer. R EFERENCES

N =4 r

Q=4 L=3 Q’=8 L’=8

−1

BER

10

−2

10

−3

10

−4

10

0

5

Fig. 3.

10

15

20 SNR (dB)

25

30

35

40

BER vs. SN R for Nr = 4 receive antennas

0

10

ZF TIV FIR, BEM ZF TIV FIR, Jakes ZF TV FIR, BEM ZF TV FIR, Jakes ZF Block, BEM ZF Block, Jakes

−1

10

N =6 r

−2

10

TIV FIR Q’=0 L’=20

TV FIR Q’=4 L’=4

BER

Q=4 L=3

−3

10

−4

10

−5

10

0

5

Fig. 4.

10

15

20 SNR (dB)

25

30

35

40

BER vs. SN R for Nr = 6 receive antennas

per receive antenna) of the ZF TV FIR equalizer are less than

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