BER Modeling for Interference Canceling FIR Wiener Equalizer

Tamoghna Roy and A. A. (Louis) Beex

DSPRL – Wireless@VT – Department of Electrical and Computer Engineering Virginia Tech, Blacksburg, VA 24061-0111, USA

Abstract—The performance of a narrowband interference canceling FIR Wiener equalizer is analyzed. While mean squared error (MSE) relates to bit error rate (BER), their connection is not necessarily a direct one when the detector output noise is not Gaussian. We show that BER can be increasing for increasing signal power (or decreasing noise power) even though MSE is decreasing. A Gaussian BER model may not be accurate then.

For digital modulation schemes using FIR Wiener equalizers in a narrowband interference dominated environment, a Gaussian sum model is derived for the Wiener filter output. The analytical evaluation of the probability of bit error based on the Gaussian sum model produces a BER prediction that is shown to provide a close match with observed/estimated BER, in particular for lower order equalizers.

Index Terms—narrowband interference, FIR equalization,

BER, Gaussian sum model.

I. INTRODUCTION Wiener Filtering theory provides an analytical expression

for the Mean Square Error (MSE). However, for practical communication systems, Bit Error Rate (BER) is used as a component of the Quality of Service (QoS) metric. We propose a model to predict the BER of a communication system in a narrowband interference (dominated) environment, where a transversal (FIR) Wiener equalizer mitigates the interference.

Previous work in BER modeling was done for systems under various scenarios. In some of the earlier works [1-2] the BER model is a Gaussian one and shown to work reasonably well for larger size equalizers. The Gaussian BER approach was also adopted in [3]. A different model was proposed to predict the BER for OFDM under narrowband interference [4]. However, no model is given to predict the BER after interference mitigation. More recently there has been work in developing non-Gaussian models to estimate BER [5]. However, the latter effort pertains to a fading channel and the interference is caused by multiple access.

II. PROBLEM STATEMENT Figure 1 shows the block diagram of the system under consideration. The symbols transmitted at time instant n are denoted by nd , while ni and nn respectively denote the interference and the zero-mean AWGN (additive, white, Gaussian noise). Thus, the input process nx to the Wiener equalizer is given by:

n n n nx d i n (1)

which is seen to be a summation of three independent wide sense stationary processes.

Figure 1: Block diagram of the system under consideration.

Let 1 2T

Lw w ww be the vector of FIR Wiener

filter weights, and 1 1T

n n n n Lx x xx the vector input to the Wiener filter at time n; T denotes the transpose operator. The output of the Wiener filter at time n is then

n n

n n n

d i nn n n

y

y y y

H

H H H

w xw d + w i + w n (2)

where H denotes the Hermitian transpose operator, and the other definitions follow the vector convention above. The FIR Wiener filter is found by solving the Wiener-Hopf equation,

xR w p (3)

with the definitions: Hn nExR x x and

0n n nE dp x . The resulting minimum MSE is then given by

2 HdMMSE w p (4)

Note that in the narrowband interference canceling environment, the FIR Wiener weights will be dependent on the interference frequency if , which affects xR .

To evaluate BER performance, the PDF of ny conditioned on

0n nd (usually 0 1 2n L , the “center” of equalization, but this reference point can be located anywhere) is needed for each of the possible symbol values the latter can take on.

III. GAUSSIAN SUM MODEL The mean and variance of the right hand side components in

(2) can be evaluated separately because the inputs to the Wiener equalizer are independent and the equalizer is linear time invariant (LTI). We now look at each component, conditioned on a particular symbol.

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A. Conditional PDF of nny

The AWGN is independent of the symbol sequence and the interference, so that this output component does not depend on the conditioning symbol. A linear combination of independent Gaussian random variables produces a Gaussian random variable. So in order to completely describe the PDF of n

ny we need to evaluate its mean and variance. Note that, as we are working at baseband, the zero-mean AWGN is circularly symmetric, with real and imaginary parts that are independent and identically distributed.

The mean of nny is given in (5),

11

1

( )

0

nn n

L

i n ii

L

i ni

E y E

E w n

w E n

Hw n

(5)

and the variance of nny is given in (6),

1

1

2

1

2

nn n

L

i n ii

L

i ni

n

Var y Var

Var w n

w Var n

H

2

2

w n

w

(6)

where 2n is the input noise power and

2. denotes the

Euclidean norm. As a result, the PDF of nny is given by:

2~ ( , )nn ny CN 2

20 w (7)

where, 2~ ( , 2 )CN indicates that is a complex normal random variable, or a real vector random variable with mean

Re

Im

E

E and covariance

2

2

00

.

B. Conditional PDF of dny

Let the modulation scheme have M symbols denoted by

1

Mm m

, so that the conditional PDF of interest is

0

dn n n mf y d . Note that this fixes the “equalization-

point” component in nd , which is multiplied by the corresponding element of the weight vector, say jw , while the

L-1 remaining terms in Hnw d produce a sum of random

variables, i.e.

0

1

0,n n m

Ldn j m l kd

l j

y w w (8)

The first term on the right-hand-side is deterministic, while the symbols under the sum are random and independent. In addition assuming a modulation scheme such that the mean of all possibilities in the constellation is zero, we can then evaluate mean and variance and find:

0

12 2

0,~ ,

n n m

Ldn j m ld

l j

y w w (9)

where 2 is the symbol power. However, the PDF of the left-hand-side in (9) is not

Gaussian. Each of the terms under the sum in (8) contributes a PDF (actually a PMF, or probability mass function) corresponding to a symbol constellation that is rotated and scaled, by lw .

1

1~M

l k l mm

w x wM

(10)

The terms under the sum in (8) are independent, so that the overall PMF is given by the convolution of the various PMFs of the form in (10). The resulting conditional PDF is therefore:

0

1 1

1, , 1

1

1,

1

L

i

Mdn n n m mL

k k

L

m j m l kl j

f y d xM

w w

k

k

(11)

Figure 2 shows an example to illustrate the PDF described in (11). For our example we used the QPSK modulation scheme so that 4M . The filter length L is taken to be 3 and the interference frequency 1if e .

-0.2 0 0.2 0.4 0.6 0.8 1 1.2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Real

Imag

inar

y

Output ConstellationInput Symbol

Figure 2: Output Constellation for QPSK with 3L and 1if e .

Figure 3 shows the output constellation for 1 4if e . For the scenario under consideration, the change in interference frequency changes the Wiener weights; the latter then change the rotations of the contributing components. Under both configurations,

0

dn n n mf y d has 1 16LM discrete

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values which are equally likely. The results in Figs. 2 and 3 are consistent with (11).

-0.2 0 0.2 0.4 0.6 0.8 1 1.2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Real

Imag

inar

y

Output ConstellationInput Symbol

Figure 3: Output Constellation for QPSK with 3L and 1 4if e .

C. Conditional PDF of iny

The Wiener filter aims to minimize MSE. In an interference dominated environment this means that the interference must (pretty much) be canceled. In a narrowband interference case, the interference can be canceled (or predicted) equally well by each of the available weights. The magnitudes of the off-center weights are equal and together produce an estimate that cancels the interference at the center weight in a trade-off with distorting the desired signal (symbol) by minimizing MSE. The trade-off is such that in an interference dominant environment, the residual interference in the Wiener filter output, i

ny , is made negligible in comparison with the other two components contributing to the Wiener filter output.

D. Conditional PDF of ny

The noise component and the signal component of the LTI Wiener filter output are independent, so that the overall PDF is the convolution of the corresponding PDFs, i.e. the convolution of the results in (7) and (11), which is a Gaussian Sum Model (GSM).

0

1 1

2 21 2

, , 1

1

1,

1 ,L

i

M

n n n m m nLk k

L

m j m l kl j

f y d CNM

w w

k

k

w (12)

Based on the conditional PDF model in (12), we can calculate eP , the probability of bit-error, by evaluating the volume enclosed by the complex normal distribution in the regions corresponding to a bit error. The regions are determined by the modulation scheme used and the choice of decision boundaries.

For example, for the QPSK modulation scheme as used in the subsequent simulations, a received symbol is detected as a first quadrant symbol if both the in-phase and quadrature component are positive. If the transmitted symbol is in the first

quadrant the probability of bit error ( eP ) is evaluated by computing the volume enclosed by each of the complex normal distributions in (12) in the second, third, and fourth quadrant. The volume enclosed in the third quadrant is multiplied by 2 (as the third quadrant implies that both bits are in error) before being added to the contributions from the second and fourth quadrant. That final sum is then divided by the factor 1LM which is the probability associated with each of the terms in the GSM.

IV. GAUSSIAN MODEL To improve the BER performance of a communication

system, longer equalizers are generally used. With an increase in the length of an equalizer, the number of individual Gaussians in (12) increases and the GSM becomes computationally expensive.

For example, 3L for a QPSK modulation scheme gives rise to 16 Gaussian distribution terms in the GSM and the centers of example distributions were shown in Figs. 2 and 3.

Figure 4 shows the output constellation for an equalizer of length 10L , containing 262,144 Gaussian sum terms.

Figure 4: Output Constellation for QPSK with 10L and 1if e .

While there are clearly non-Gaussian features, it seems reasonable to approximate this fairly circular probability mass with its best Gaussian fit. So for larger values of L we propose an approximated version of the Gaussian Sum Model, consisting of a single Gaussian term.

A. Conditional PDF of ny

The Gaussian model assumes the PDF of the conditional output

0n n n mf y d to follow a Gaussian distribution. The mean of this distribution is the same as that of the PDF in (9) and the variance is given by the sum of the variances described in (7) and (9). Thus, the Gaussian model PDF is given by:

0

122 2

0,

~ ,L

n n n m j m n ll j

f y d CN w w2

2w (13)

We use the PDF characterized in (13) to then model the probability of bit error for the Gaussian model.

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V. SIMULATION RESULTS

A. Gaussian Sum Model In the simulation example, we look into a communication

system that employs QPSK modulation, i.e. 4M , and for which the average over all constellation values equals zero. The QPSK signal is corrupted by a narrowband interference (represented by a complex sinusoid) and zero mean additive white Gaussian noise. The signal, interference, and noise are all wide sense stationary and independent processes with known statistics, so that the Wiener filter is LTI and its MSE performance can be readily evaluated, according to (4).

Figure 5 shows the MSE performance of the communication system for a Wiener equalizer of length 3L for two different situations: when there is no narrowband interference at all and when there is strong narrowband interference (ISR = 20), and for two different interference frequencies of 1if e and 1 4if e .

0 2 4 6 8 10 12 14-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

Eb/N0 (dB)

MS

E (d

B)

Theoretical MSE: No InterferenceTheoretical MSE: ISR = 20, f = 1/eTheoretical MSE: ISR = 20, f = 1/4eObserved MSE: No InterferenceObserved MSE: ISR = 20, f = 1/eObserved MSE: ISR = 20, f = 1/4e

Figure 5: MSE performance for L = 3.

It is to be noted that for all cases the MSE performance is monotonically decreasing with a decrease in noise power; behavior one might reasonably expect. The observed MSE values are from simulation results, averaging the results from 1000 independent realizations of 10,000 QPSK symbols each. The observed MSE behave as predicted by theory.

The scenario when there is no interference is the same as an AWGN channel. For a QPSK system the theoretical expression for eP in the AWGN channel is given by [8]:

0

2 be

EP Q N (14)

The latter result provides a sanity check for a limiting case of the proposed model.

Figure 6 shows Pe, the probability of bit-error performance, for the same system when there is no interference and when ISR = 20 dB and 1if e .

As expected, the curves generated by (14) and GSM with ISR = - dB (no interference) in (12) are indistinguishable.

0 2 4 6 8 10 12 14-14

-12

-10

-8

-6

-4

-2

0

Eb/N0 (dB)

log 10

(Pe)

BER TheoreticalBER Gaussian Sum: No InterferenceBER Gaussian Sum: ISR = 20

Figure 6: eP performance for L = 3 under no interference and for ISR = 20 dB

and 1if e .

The curve for ISR = 20 dB and 1if e shows that performance in terms of probability of bit error has deteriorated substantially when ISR is dominant and – as observed from Fig. 5 – in terms of MSE performance also.

To take a closer look at the case for ISR = 20 dB, we next look at BER performance based on simulation results and its prediction using models.

Figure 7 shows the BER performance when there is a strong narrowband interference corrupting the desired signal.

0 5 10 15 20 25-1.3

-1.2

-1.1

-1

-0.9

-0.8

-0.7

Eb/N0 (dB)

log 10

(BE

R)

Gaussian sum model: f = 1/eGaussian model: f = 1/eObserved: f = 1/eGaussian sum model: f = 1/4eGaussian model: f = 1/4eObserved: f = 1/4e

Figure 7: BER performance for L = 3 and ISR = 20 dB.

For this simulation again ISR = 20 dB, with 1if e and 1 4if e . The observed BER values were generated by

Monte-Carlo simulation, in the same way the observed MSE results were generated for Fig. 5 earlier.

We see that unlike the MSE curve in Fig. 5 the BER curve is not necessarily monotonically decreasing as noise power is decreasing (or, equivalently, signal power is increasing). This result indicates that MSE and BER are not linked directly, not even in terms of overall behavior. Similar observations were

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made in [9] while dealing with interference mitigation and equalization in band limited channels. Note that the Gaussian sum model from (12) provides a good prediction of the BER of the system, for both interference frequencies.

The explanation for the increase in BER when SNR (signal-to-noise ratio) is increasing comes from the Gaussian Sum Model. The centers of gravity (COG) or means for each of the components in the GSM are shown in Fig. 2. SNR is reflected in the radius of the circular probability mass concentrated at each of these COG. When SNR is low these radii are large and a reduction in these radii corresponds to less probability mass spilling outside of the first quadrant. At some point, after increasing SNR, the radii are small enough that the probability mass associated with the COG inside the first quadrant spills outside less while more of the probability mass associated with the COG outside of the first quadrant (there are four of these) becomes concentrated outside of the first quadrant. The latter causes the BER to then increase and eventually saturate at the fraction of the number of COG outside the first quadrant relative to the total number of COG. For 3L and 1if e that saturation BER is (4/32=) 0.125 (or -0.9031 dB), while for 3L and 1 4if e that saturation BER is (2/32=) 0.0625 (or -1.204 dB). As seen in Fig. 7, SNR needs to be higher to reach the latter limiting BER, which is understandable from Fig. 3, as some of the constellation points are very close to a decision boundary (requiring tighter concentration of the probability mass at those COG).

B. Gaussian Model In Fig. 7 the performance predicted by the Gaussian model

was shown for the 3L case. Figure 8 shows the BER performance of the same system under the same strong narrowband interference (ISR = 20 dB), for 1if e , but now the equalizer length L is set to 10.

0 5 10 15 20 25-3.5

-3

-2.5

-2

-1.5

-1

-0.5

Eb/N0 (dB)

log 10

(BE

R)

Gaussian sum modelGaussian modelObserved

Figure 8: BER performance for L = 10 with ISR = 20 dB and 1if e .

While not shown in Fig. 8, the same experiment was performed for interference frequency 1 4if e ; the results were indistinguishable from those in Fig. 8.

For 3L we saw (in Fig. 7) that the Gaussian model provided less accurate results than the Gaussian sum model, especially at higher SNR. The actual BER performance was over-estimated by the Gaussian model over most of the SNR range. For 10L (in Fig. 8) we see that the prediction of the Gaussian model is now under-estimating actual BER performance but it is much closer to the observed values based on simulation. In fact, for the most practical range of SNR values, Eb/N0 below 15 dB, the Gaussian model is a reasonably good one for the narrowband interference dominated environment. Observe that the Gaussian sum model provides very accurate results; we note, however, that this accuracy comes at the expense of a considerably increased computational burden.

The BER saturation effect is still observed for 10L , it just happens at a lower level. For larger equalizers the estimation of the narrowband interference gets better, but the residual interference power starts to act as a limiting factor to performance as SNR becomes very large.

VI. CONCLUSION We have shown that the BER performance of a digital

communication system – when operating in a narrowband interference dominated environment – does not always follow along with the MSE performance. The BER performance of the system can be predicted accurately using the Gaussian sum model that was derived. For practical considerations, where larger equalizer filters are used, the Gaussian sum model can be approximated reasonably well by using a Gaussian model which then has the advantage of being much more efficient computationally.

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[3] M. E. Davis and L. B. Milstein, "Anti-jamming properties of a DS-CDMA equalization filter," MILCOM 1993, vol. 3, pp. 1008-1012, 11-14 Oct 1993.

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