Frequency Domain Computation of the Feedback Filter of the Hybrid Decision Feedback Equalizer

1

Efficient Computation of the Feedback Filter for the Hybrid

Decision Feedback Equalizer in Highly Dispersive Channels

Maurizio Magarini, Member, IEEE, Luca Barletta, and Arnaldo Spalvieri

Dipartimento di Elettronica e Informazione, Politecnico di Milano,

Piazza Leonardo da Vinci, 32, I-20133 Milano (Italy),

{magarini,barletta,spalvier}@elet.polimi.it

Abstract

The hybrid decision feedback equalizer (DFE) is a combined time-frequency domain implementation of the conven-

tional time-domain DFE that is able to provide a good trade-off between performance and computational complexity

in single carrier transmission over severely frequency-selective channels. In the hybrid DFE the implementation of the

feedforward filter is done in the frequency domain, while the feedback filter (FBF) is implemented in the time-domain.

The computation of the coefficients for the two filters is usually done in the same domain where they are implemented.

A method for frequency-domain computation of the FBF is proposed in the paper. As known, the key operation in

the computation of the FBF is the spectral factorization. In the paper it is proposed to adopt the (cepstral) method

for spectral factorization due to Kolmogoroff, which can be efficiently implemented by using the fast Fourier transform

(FFT). The application of the method is considered for highly dispersive channels. By using simulations we show that

for this type of channels the performance of the proposed method is virtually the same as that obtained by using time-

domain approaches. The advantage of the proposed approach is that the efficient FFT gives a substantial reduction of

complexity compared to time-domain methods.

Keywords

Frequency domain equalization, decision feedback equalizers, intersymbol interference, spectral factorization.

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I. Introduction

Single carrier (SC) transmission with frequency-domain equalization is well recognized as an al-

ternative to orthogonal frequency-division multiplexing (OFDM) in combating the severe multipath

distortion effects that may arise in broadband wireless transmissions [1], [2]. An SC system with

frequency-domain equalization is often based on time-domain cyclic-extended data block transmission

as happens in many OFDM systems but, compared to OFDM, SC has the advantage of having a

lower peak-to-average power ratio. Moreover, as in OFDM, the block-based processing performed

at the receiver of an SC system with frequency-domain equalization allows us to take advantage of

the computational efficiency of the fast Fourier transform (FFT). Both linear and decision feedback

equalization algorithms have been considered in [3], [4]. Results show that the hybrid time-frequency

domain implementation of the conventional decision feedback equalizer (DFE), consisting of a forward

(linear) section and of a feedback (non-linear) section, in general provides better performance than the

frequency-domain implementation of the conventional linear equalizer.

The basic result of DFE is due to Monsen [5], who demonstrated that the optimal (unconstrained)

feedback section can be computed independently of the feedforward section by a spectral factorization.

Later, Belfiore and Park [6] considered a finite-impulse response (FIR) feedback section, regarding the

feedback section as a noise predictor. Again, Belfiore and Park obtained independence between the

feedback section and the feedforward section. These results lead to the conclusion that the feedforward

section can be implemented without losing optimality independently of the feedback section, the unique

condition being that no constraints are imposed on the impulse response of the feedforward filter

(FFF). As a consequence, in the hybrid DFE the feedforward filter can be implemented in the discrete

frequency-domain, thus taking advantage from the efficiency of the FFT. The reader is referred to [7]

for a tutorial review of frequency-domain filtering.

In the hybrid DFE, the feedback section is based on a feedback filter (FBF) that operates in the

time-domain. In the literature of DFE, two different forms are proposed for the feedback section of the

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DFE: intersymbol interference (ISI) cancellation and noise prediction. The block diagrams of the two

forms are reported in Fig. 1. For any given FBF, the two implementations are equivalent provided

that the FFF is unconstrained so that its transfer function can absorb the differences between the two

forms of the feedback section. Both in the hybrid DFE based on ISI cancellation of [3], [4] and in the

hybrid DFE based on noise prediction of [8], the coefficients of the impulse response of the FBF are

computed in the time domain according to the approach of [6].

The time-domain computation method requires the inversion of a matrix whose order is defined by

the length of the FBF. When transmission takes place over highly dispersive channels the length of

the optimal unconstrained FBF may be very high and its truncation to a shorter duration may lead to

performance degradation [9]. Hence, for this situation, a longer FIR FBF is required at the cost of an

increase of computational complexity that derives from the inversion of a high order matrix. The cost of

this higher complexity can be reduced by resorting to an iterative block DFE (IBDFE) implementation

[10], [11]. In the IBDFE a block-based processing is considered where both the FFF and the FBF are

implemented in the frequency domain. The structure realizes the adaptation of the FFF and of the

FBF directly in the frequency domain, thus avoiding matrix inversion in the time-domain. This allows

IBDFE to achieve a reduction in complexity compared to the hybrid DFE. An inverse FFT (IFFT)

is applied to generate the time-domain signal at the input of the decision device. However, due to

the iterative nature of the IBDFE convergence issues may arise in case of transmission over severe

frequency selective channels. In this case an approach based on a joint interleaved equalization and

decoding iterative process is required [12].

Another solution to avoid the matrix inversion is proposed in this paper and consists in computing

the FBF directly in the frequency-domain. When block processing making use of FFT is adopted,

the computation of the FBF based on FFT becomes attractive, because often the speed of the FFT

is so great that one can reuse the FFT kernel more times in the time taken by the processing of

one block. The frequency-domain computation of the FBF is based on the Kolmogoroff method of

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spectral factorization [13]. The Kolmogoroff method belongs to the class of cepstral methods and can

be efficiently implemented in the discrete frequency-domain by using FFT. As a method that operates

in the frequency-domain, the advantage over time-domain methods is substantial when the number

of coefficients of the feedback section is large and, therefore, for channels that are characterized by

strong multipath with long delay spread. This class of channels is encountered in many broadband

communication systems, such as high-speed wireless networks and high-data rate mobile radio in rural

and suburban areas. The potential drawback of the proposed approach is that, since in practice

the FBF is implemented in FIR form, the spectral factor from Kolmogoroff factorization must be

truncated. Unfortunately, the truncated impulse response may lose the minimum phase property,

leading to dramatic performance degradation compared to time-domain methods. Experimentally we

have found that, for channels that are characterized by a large delay spread truncation is not critical.

The main result in this paper is therefore that the application of the cepstral method to this class

of channels gives rise to the same performance as that obtained by using the time-domain approach.

Hence, due to the large number of taps of the FBF needed when this class of channels is considered, the

advantage taken from the efficiency of the FFT is substantial. This experimental result is very different

from that obtained considering severely frequency selective channels with impulse response of short

duration. For this class of channels, a countermeasure that mitigates the performance degradation due

to the loss of the minimum phase property can be adopted as proposed in [14].

The paper is organized as follows. The system model is presented in Section II. The background

material for decision feedback equalization is given in Section III. In Section IV time-domain compu-

tation and frequency-domain computation of the FBF are presented together with the method used

to mitigate the performance loss due to truncation of the impulse response of the spectral factor. The

complexity of frequency-domain computation, time-domain computation, and the performance of the

hybrid DFE are discussed in Section V. Experimental results are shown in Section VI and conclusion

are drawn in Section VII.

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II. System Model

We adopt the discrete time additive white Gaussian noise (AWGN) baseband equivalent model of

the observation, and use the z-transform to represent sequences (z−1 indicates a delay of one symbol

interval T ). Let

X(z) =∑

k

xkz−k = A(z)H(z) + W (z) (1)

be the z-transform of the observation {xk}, where all the quantities are assumed to be complex-valued.

In the above equation, H(z) is the z-transform of the impulse response of the channel {hk}, A(z) is

the z-transform of the independent identically distributed transmitted data sequence {ak}, and W (z)

is the z-transform of the AWGN sequence {wk}. Data are drawn from an (M − PAM)n constellation,

n = 1, 2 with variance

σ2a = n

M2 − 1

3. (2)

In the following, we denote by Ψs(z) the z-transform of the autocorrelation of the sequence repre-

sented by S(z), being understood that Ψs(z) = S(z)S∗(z−∗) =∑

k z−k∑

i s∗i si+k when

∑i |si|2 < ∞,

while Ψs(z) =∑

k z−kE{s∗i si+k} when S(z) represents a stationary random sequence. In the above

notation, the superscript ∗ denotes the complex conjugate and z−∗ = 1/z∗. The z-transform of the

autocorrelation of the AWGN is

Ψw(z) = N0. (3)

The z-transform of the autocorrelation of the observation is

Ψx(z) = σ2aH(z)H∗(z−∗) + N0. (4)

The signal-to-noise ratio is defined as

SNR =σ2

a

∑k |hk|2

N0

. (5)

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III. Background

A. MSE-DFE

Aiming for frequency-domain computation of the FFF, it is convenient to derive the MSE-DFE

feedforward and feedback filters by the approach of predictive DFE developed by Belfiore and Park

[6]. Let the z-transform of the error at the decision device be

V (z) = (1−B(z))(C(z)X(z)− A(z)) = D(z)(C(z)X(z)− A(z)), (6)

where B(z) is the z-transform of the impulse response of the FBF and C(z) represents the FFF of the

predictive DFE. The term

U(z) = C(z)X(z)− A(z) (7)

is the error that is predicted by the FBF. The MSE is

JMSE−DFE =T

2π

∫ π/T

−π/T

Ψv(ejωT )dω =

T

2π

∫ π/T

−π/T

Ψd(ejωT )Ψu(e

jωT )dω. (8)

Since Ψd(ejωT ) ≥ 0 and Ψu(e

jωT ) ≥ 0, one concludes that the optimal FFF minimizes, frequency-by-

frequency, the power spectral density of the error before the linear prediction, independently of the

FBF. Therefore the optimal FFF is the MMSE linear equalizer (MMSE-LE), that is

C(z) =σ2

aH∗(z−∗)

Ψx(z). (9)

With the MMSE-LE one has

Ψu(z) =σ2

aN0

Ψx(z), (10)

and

Ψv(z) =σ2

aN0Ψd(z)

Ψx(z). (11)

The computation of the FBF proceeds as follows. We wish to predict uk by linear prediction from the

past samples uk−1, uk−2, · · · . Let

vk = uk −µ∑

i=1

biuk−i, (12)

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be the error after linear prediction with µ taps. The coefficients b = (b1, b2, · · · , bµ) are obtained as

b = argminb

E{|vk|2}. (13)

The unique minimum is obtained by setting to zero the partial derivatives of the power of the error

with respect to the entries of b

∂

∂bi

E{|vk|2} = −E{vku∗k−i} = −E

{(uk −

µ∑j=1

bjuk−j

)u∗k−i

}

= −ψu,i +

µ∑j=1

bjψu,i−j = 0, i = 1, 2, . . . , µ, (14)

where ψu,i is the i-th sample of the autocorrelation of the sequence represented by U(z). Hence the

taps of the predictor are obtained by solving the set of linear equations

µ∑j=1

bjψu,i−j = ψu,i, i = 1, 2, . . . , µ, (15)

which are known as the Yule-Walker equations. The transfer function 1− B(z) is monic, causal, and

minimum phase. The conventional (nonpredictive) MSE-DFE with ideal FFF is obtained by using the

cascade of the MMSE-LE and 1−B(z) as FFF, and by feeding back through B(z) the past data.

B. MMSE-DFE

Ideal prediction is obtained when the power spectrum of the error after prediction is white, leading

to the MMSE-DFE. We write the white power spectrum of the error after prediction as the fraction

Ψv(z) =σ2

aN0

α. (16)

The scalar α can be computed as [15]:

α = exp

(T

2π

∫ π/T

−π/T

log(Ψx(e

jωT ))dω

). (17)

For the MSE of the MMSE-DFE one gets

JMMSE−DFE = σ2aN0α

−1 = σ2aN0 exp

(− T

2π

∫ π/T

−π/T

log(Ψx(e

jωT ))dω

). (18)

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Note that for the MMSE-DFE one has

Ψv(z) =σ2

aN0D(z)D∗(z−∗)Ψx(z)

=σ2

aN0

α, (19)

which demonstrates that D(z) is obtained from the spectral factorization

αD(z)D∗(z−∗) = Ψx(z), (20)

where the D(z) that is monic, causal, and minimum phase, is taken. When Ψv(z) is a finite impulse

response with 2ν + 1 taps, the FBF of the MMSE-DFE has ν taps.

IV. The Hybrid DFE

The block diagram of the hybrid noise predictive DFE is reported in Fig. 2. The basic element for

the computation of the filters of the hybrid DFE is the sampled spectrum

Ψu(ej2π k

N ) = Ψu(z)|z=ej2π k

N=

σ2aN0

Ψx(ej2π k

N ), (21)

where N is the size of the FFT. According to (9) and (10) the sampled frequency response of the

MMSE-LE is

C(ej2π kN ) =

1

N0

H∗(ej2π kN )Ψu(e

j2π kN ). (22)

A. Time-domain computation of the feedback filter

The transfer function B(z) of the FBF is computed in the time-domain by solving the Yule-Walker

equations given in (15). Note that the computation of the transfer function requires µ + 1 coefficients

of the autocorrelation Ψu(z). We observe that, in many cases of practical interest, an accurate approx-

imation of the coefficients ψu,i requires a number of IFFT points much larger than 2µ+1. In practice,

one is led to take the truncation of the IFFT of the spectrum Ψu(ej2π k

N ).

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B. Frequency-domain computation of the feedback filter

The desired transfer function is obtained from the spectral factorization given in (20). The cepstral

method of spectral factorization is implemented in the discrete frequency-domain as

√αD(ej2π k

N )= exp(FFT

(usk¯ IFFT

(log

(Ψx(e

j2π kN )

)))), (23)

where ¯ is the Schur-Hadamard (elementwise) product and

usk =

0, k < 0

1/2, k = 0

1, k > 0.

If desired, the samples {Ψu(ej2π k

N )} can be used in place of {Ψx(ej2π k

N )} by reversing the sign in front

of the logarithm and by properly adjusting the scale factor. The transfer function of the FBF is

obtained by truncating the IFFT of the spectral factor (23):

B(z) = −µ∑

n=1

dnz−n. (24)

As mentioned in the introduction, the duration of the impulse response of the spectral factor resulting

from (23) depends only on the duration of the impulse response of the channel, which, in the wireless

scenario, is not under the control of the designer. Truncation in the time-domain of the spectral factor

can induce substantial performance degradation when the factored z-spectrum has singularities close

to the unit circle. Specifically, while it is guaranteed that all the zeroes of D(z) lie on or inside the

unit circle, the transfer function 1−B(z) obtained by truncation of D(z) might not have this property.

Experimentally one can observe that the minimum phase property is often lost when Ψx(z) has zeros

that are close to the unit circle. Therefore one would like to bring these zeros far from the unit circle

before the factorization of the power spectrum. A method for mitigating this risk is reported in [14],

where the FBF is obtained by truncating the spectral factor obtained from a deliberately biased version

of the spectrum given by

αD(z)D∗(z−∗) = Ψx(z) + λ, (25)

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where λ ≥ 0 is up to the designer. The parameter λ adds a floor to the power spectrum, thus preventing

the presence of spectral zeros. Note that as λ → ∞, the feedback taps of the DFE tend to zero, and

the receiver is reduced to the pure MMSE-LE. After the spectral factorization and the IFFT, only

the first µ + 1 taps of the impulse response are retained. Unfortunately, it is difficult to derive by

analytical means the optimal λ for the given channel and for the desired µ. What one can do is to

apply the procedure for many values of λ and evaluate the MSE. This allows us to select the best λ

for the given channel and the desired µ.

V. Complexity and performance of the Hybrid DFE

As observed in [8], the noise predictive approach is more suited to hybrid implementation than

the ISI cancellation approach, the noise predictive approach being less demanding in determining

the FFF. Specifically, in the noise predictive approach the optimal FFF is the MMSE-LE, while in

the ISI cancellation approach the transfer function of the optimal FFF is the product between the

transfer function of the MMSE-LE and the transfer function 1 − B(z). In the case of time-domain

computation of the FBF, the ISI cancellation approach would therefore require the following operations

for determination of the FFF that are not required with the noise predictive approach:

• FFT of the impulse response of the FBF,

• product between the frequency response of the MMSE-LE and the frequency response of the FBF.

In the case of frequency-domain computation based on spectral factorization, the transfer function of

the FBF is obtained from the spectral factorization. Therefore the ISI cancellation approach requires

the product mentioned before, which is not required with the noise predictive approach. Hence,

also in this case, the noise predictive approach seems to be preferable as far as computation of the

FFF and of the FBF is concerned. Therefore, in the following we focus on the noise predictive

DFE. As far as evaluation of complexity is concerned, the FFF being common both to the frequency-

domain method and to the time-domain method, we focus on the complexity of the computation

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of the impulse response of the FBF, comparing the frequency domain method proposed here to the

time-domain method proposed in [8], that we take as a representative of time-domain methods. We

assume that the complexity in the computation of the impulse response of the FBF is dominated

by the number of complex multiplications. As in [8], we assume that the computation of an N -size

FFT requires (N/2) log2 N complex multiplications and that the channel estimation is made in the

frequency-domain and the samples of the power spectral density (21) are known to the receiver. The

method of [8] is based on the solution of the Yule-Walker equations (15). The efficient solution of such

equations is worked out by the Levinson recursion, that requires µ2 complex multiplications [16]. Note

that, since the method works in the time-domain and the samples of the spectrum are available in the

frequency domain, the cost of the N -size IFFT necessary to compute the first µ + 1 coefficients of the

autocorrelation Ψu(z) must be taken into account, thus obtaining the second term in the second column

of Table I. When our proposed method is considered, the computation of the spectral factorization

(23) requires one IFFT and one FFT, therefore a total of N log2 N complex multiplications. The exp

and the log functions are implemented by using two look-up tables. The impulse response of the FBF

is obtained by truncating the IFFT of the spectral factor in (23) to the first µ + 1 coefficients. By

taking into account also this operation the overall complexity of the computation, in terms of number

of complex multiplications, turns out to be 3(N/2) log2 N . Table I summarizes the overall complexity

for the computation of the FBF by the two approaches.

A. Performance of the hybrid DFE

We assume that a unique word is appended to the payload as suggested in [3], [4], [8]. Note that,

when the unique word is appended outside the FFT block, as proposed in [3], the noise predictive

approach fails, and only the ISI cancellation approach can be adopted. Also, as in [3], [4], [8], we

assume that the size of the unique word is not lower than the duration of the impulse response of the

time-discrete AWGN model (1), thus preventing inter-block interference. Moreover, we also assume

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that the size of the unique word is not lower than the duration of the impulse response of the FBF,

thus guaranteeing that the non-periodic convolution operated in the time-domain by the FBF can be

treated as if it were periodic. With these assumptions the MSE of the hybrid noise predictive DFE is

obtained by numerically evaluating (8) as

J =1

N

N−1∑

k=0

(σ2

a|1−H(ej2π kN )C(ej2π k

N )|2 +N0|C(ej2π kN )|2

)

·|1−µ∑

n=1

bne−j2π kn

N |2, (26)

where the impulse response of the FBF is determined by (15) in the case of time-domain computation,

while it is determined by (23) and (24) in the case of frequency-domain computation.

The performance of the hybrid DFE is conveniently measured by the first error event rate (FEER). To

compute the FEER one should derive the probability density function of ISI plus noise, for example

by the method proposed in [17]. However, a fairly accurate estimate is obtained by the Gaussian

approximation:

FEER ≈ 4(√

M − 1)√M

Q

(√2(σ2

a − J)

σ2aJ

), (27)

where

Q(x) =1√2π

∫ ∞

x

e−u2

2 du, (28)

and the unbiased version of the MSE-DFE is considered.

VI. Experimental Results

Simulation results are shown both for transmission over severely highly dispersive multipath fading

channels and over severely frequency selective channels. In the following it is assumed that the channel

is always perfectly known to the receiver and that, in the case of fading channel, it remains constant

at least for the duration of one block. The transmitting filter is a square root Nyquist filter with rolloff

factor 0.25. The shaped signal is passed through the multipath channel, filtered by a receive filter

matched to the transmit filter and sampled at the symbol-rate frequency.

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As a first example simulation results of QPSK transmission are reported for the static channels

termed “channel ]1” and “channel ]2” in [18]. For both these channels the size of the FFT used for

the FFF and for the spectral factorization is 2048, the size of the unique word is 256 and the duration

of the time-discrete impulse response is 256 for both channels. The choice of such a large number

of FFT points is motivated by the fact that these channels are characterized by very deep-nulls and,

consequently, their inverses exhibit non-negligible values for many symbol time intervals. Figures 3

and 4 show the FEER versus SNR of the proposed approach together with that obtained by the

time-domain approach. The performance of the MMSE-LE (µ = 0) is also reported in the mentioned

Figures. Since the duration of the impulse response of the channel is 256, the optimal FBF is obtained

with µ = 256, therefore it is an expected result that, for µ = 256, the performance of the two methods

is the same, since the unique effect of the truncation of the spectral factor to the first 256 taps is the

removal of a tail of zeroes. From Figs. 3 and 4 we see that, reducing the number of taps of the FBF

to µ = 128, leads to a substantial performance degradation, the degradation being more pronounced

with “channel ]2” . Also, we see from the mentioned Figures that the performance of the proposed

method remains virtually the same as that of the time-domain method.

As a second example of highly dispersive multipath fading we consider the fading channel B of

vehicular test environment for IMT-2000 [19]. In this case the size of the FFT used for the FFF and

for the spectral factorization is 1024, the size of the unique word is 128, and the symbol rate is 4 MHz.

The modulation format is again QPSK. Simulations are used to generate 107 realizations of the fading

channel. In this situation the SNR is the ratio between the average power of the useful signal to the

variance of the noise at the sampling instant. For each channel realization we compute the value of the

MSE given by (26) that is used in (27) to compute the instantaneous FEER. The FEER is obtained by

averaging the instantaneous FEER over the 107 channel realizations. Figure 5 reports the performance

of the two methods versus SNR. The result confirms those of Figs. 3 and 4. Specifically, we see that

the performance with µ = 64 and µ = 32 is substantially worse than the performance with µ = 128,

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hence the limitation of the number of taps of the FBF to 64 and to 32 is substantial. Nevertheless,

the performance of the two methods remains virtually the same.

In order to validate the effectiveness of method for mitigating the performance loss due to the

truncation of the spectral factor described in Section IV-B, computer simulations are presented for

two cases of severely frequency selective channels with short delay spread. As a first example we report

simulation results for the AWGN channel H(z) = 0.176+0.316z−1 +0.476z−2 +0.532z−3 +0.476z−4 +

0.316z−5 + 0.176z−6. It is worth noting that this channel is the worst channel with ν = 6 [20]. The

size of the FFT used for the FFF and for the spectral factorization is 512 and the length of the unique

word is 16. Simulation results for this channel were carried out by setting the memory of the FBF

to µ = 4 and by considering BPSK modulation. The preliminary operation of selecting the optimal

λ is required. In Fig. 6 it is reported the MSE versus λ computed according to equation (26). From

the Figure it can be observed that the MSE is a smooth function of λ, which is a desirable property

because the performance is not too sensitive to the choice of λ. It can be seen from the Figure that

good performance is obtained for λ = 0.08 in a wide range of SNRs. We observe that, for λ = 0, the

MSE obtained by the truncation method is sensitive to the SNR, so that at high SNR the performance

is dramatically degraded, making the truncation method not suitable. Conversely, a judicious choice

of λ brings the performance of the truncation method near to the performance of the MSE-DFE.

Figure 7 reports the zeros of the impulse responses obtained by the application of different methods

for SNR = 20 dB. Note that the bad performance obtained for λ = 0 by the frequency-domain method

is explained by observing that this method gives a maximum phase FBF (all the roots are outside the

unit circle), while the minimum phase property of the FBF is preserved by adding the bias. Figure 8

shows the bit error rate (BER) versus SNR, while Fig. 9 shows the first error event rate (FEER) of

the frequency-domain method for λ = 0 and λ = 0.08. The error performance has been measured by a

random sequence of 106 symbols. The performance of the time-domain method and of the MMSE-LE

is also reported. The two Figures confirm the observations made on Fig. 6 regarding the effectiveness

15

of the bias.

As last example the channel model A defined by ETSI BRAN for HiperLAN/2 [21] is considered.

The modulation format is QPSK with symbol rate of 20 MHz. The size of the FFT is 512 and the

size of the unique word is 16. Simulations are used to generate 107 realizations of the fading channel.

Figure 10 shows the performance for µ = 3 of the proposed method. We observe that, in this case, the

performance of the frequency-domain method is close to the performance of the time-domain method,

and that it is not too sensitive to the bias. A small improvement can be obtained by adding the

bias only at high SNR. At FEER=10−7 the performance of the proposed method with λ = 0.04 is

approximately 0.5 dB better than with λ = 0 and 0.9 dB worse than that of the time-domain method.

VII. Conclusion

The computation of the coefficients of the FIR FBF of the hybrid DFE by the spectral factorization

method due to Kolmogoroff is considered in the paper. The method works in the frequency-domain

by exploiting the efficiency of the FFT algorithm and, as such, it becomes attractive in case of a FBF

with a large number of taps. Together with this advantage, a potential drawback of the frequency-

domain method should also be considered. Specifically, the frequency-domain method can potentially

lead to performance degradation compared to the time-domain method when substantial truncation

of the spectral factor is necessary to fit the number of taps of an FIR FBF. A solution to mitigate the

performance loss due to truncation is described. This solution consists in adding a bias in the power

spectrum to be factored.

The experimental results presented in the paper show that, for the examined channels, which are

representative of a wide class of severe multipath channels characterized by impulse responses of

long duration, the performance of the frequency-domain method is virtually the same as that of the

time-domain method even in the case where important tails of the impulse response obtained from

the spectral factorization are truncated. Moreover, the effectiveness of the biasing used to make the

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performance of the truncated DFE close to the performance of the DFE with FBF obtained from the

Yule-Walker equations has been demonstrated considering severely frequency-selective channels with

short delay spread.

The presented results lead to the conclusion that the benefit induced by the efficiency of the FFT

algorithm comes free, at least for the examined channels.

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18

List of Figures

1 Block diagram of the feedback section of the DFE. (a) ISI cancellation. (b) noise prediction. 20

2 Block diagram of the hybrid noise predictive DFE. The sampled frequency response of the

FFF is given in equation (22). The sampled power spectral density of the error sequence

{un} is given in equation (21). The impulse response of the FBF is represented in the

Figure by the sequence {bn}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 FEER versus SNR for “channel ]1” of [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 FEER versus SNR for “channel ]2” of [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 FEER versus SNR for the IMT-2000 model B channel of the vehicular test environment

of [20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 MSE versus λ with length of the FBF truncated to µ = 4 for the AWGN channel of [21]

with ν = 6. Curves are parametric in the SNR that is reported next to them. The dashed

lines represent the MSE that is obtained when the coefficients of the FBF are computed

by the Yule-Walker equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7 Zeros of the impulse responses of the FBF with length truncated to µ = 4 obtained by

the application of the frequency and time-domain methods at SNR=20 dB for the AWGN

channel of [21] with ν = 6. Frequency-domain method with λ = 0: ’◦’; Frequency-domain

method with λ = 0.08: ’¦’; Time-domain method: ’∗’. . . . . . . . . . . . . . . . . . . . . 26

8 BER versus SNR for the AWGN channel of [21] with length of the FBF truncated to µ = 4. 27

9 FEER versus SNR for the AWGN channel of [21] with length of the FBF truncated to

µ = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

10 FEER versus SNR for the Hiperlan type A channel of [22] with length of the FBF truncated

to µ = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

19

List of Tables

I Complexity of the computation of the feedback filter. . . . . . . . . . . . . . . . . . . . . 30

20

Fig. 1. Block diagram of the feedback section of the DFE. (a) ISI cancellation. (b) noise prediction.

21

Fig. 2. Block diagram of the hybrid noise predictive DFE. The sampled frequency response of the FFF is given in

equation (22). The sampled power spectral density of the error sequence {un} is given in equation (21). The impulse

response of the FBF is represented in the Figure by the sequence {bn}.

22

8 10 12 14 16 1810

−7

10−6

10−5

10−4

10−3

10−2

10−1

SNR (dB)

FE

ER µ=128

µ=256

MMSE−LEFrequency−domain computationTime−domain computation

Fig. 3. FEER versus SNR for “channel ]1” of [19].

23

10 12 14 16 18 2010

−7

10−6

10−5

10−4

10−3

10−2

10−1

µ=128

µ=256

SNR (dB)

FE

ER

MMSE−LEFrequency−domain computationTime−domain computation

Fig. 4. FEER versus SNR for “channel ]2” of [19].

24

10 12 14 16 18 20 22 24 26 2810

−7

10−6

10−5

10−4

10−3

10−2

10−1

SNR (dB)

FE

ER µ=32

µ=64

µ=128

MMSE−LEFrequency−domain computationTime−domain computation

Fig. 5. FEER versus SNR for the IMT-2000 model B channel of the vehicular test environment of [20].

25

−0.1 0 0.1 0.2 0.3 0.4 0.5−14

−12

−10

−8

−6

−4

−2

0

2

4

30dB

25dB

20dB

15dB

10dB

λ

J (d

B)

Fig. 6. MSE versus λ with length of the FBF truncated to µ = 4 for the AWGN channel of [21] with ν = 6. Curves are

parametric in the SNR that is reported next to them. The dashed lines represent the MSE that is obtained when the

coefficients of the FBF are computed by the Yule-Walker equations.

26

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Real(z)

Imag

inar

y(z)

Fig. 7. Zeros of the impulse responses of the FBF with length truncated to µ = 4 obtained by the application of the

frequency and time-domain methods at SNR=20 dB for the AWGN channel of [21] with ν = 6. Frequency-domain

method with λ = 0: ’◦’; Frequency-domain method with λ = 0.08: ’¦’; Time-domain method: ’∗’.

27

10 15 20 25 3010

−4

10−3

10−2

10−1

100

SNR (dB)

BE

R

MMSE−LEFrequency−domain computation, λ=0Frequency−domain computation, λ=0.08Time−domain computation

Fig. 8. BER versus SNR for the AWGN channel of [21] with length of the FBF truncated to µ = 4.

28

10 15 20 25 3010

−4

10−3

10−2

10−1

100

SNR (dB)

FE

ER

MMSE−LEFrequency−domain computation, λ=0Frequency−domain computation, λ=0.08Time−domain computation

Fig. 9. FEER versus SNR for the AWGN channel of [21] with length of the FBF truncated to µ = 4.

29

10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

SNR (dB)

FE

ER

MMSE−LEFrequency−domain computation, λ=0Frequency−domain computation, λ=0.04Time−domain computation

Fig. 10. FEER versus SNR for the Hiperlan type A channel of [22] with length of the FBF truncated to µ = 3.

30

TABLE I

Complexity of the computation of the feedback filter.

Computation method Number of complex multiplications

Levinson µ2 + (N/2) log2 N

Kolmogoroff 3(N/2) log2 N