Simulation of frequency-flat fading

channels

Anil KandangathEEE-558 Wireless Communications

Project 1

October 13, 2003

Contents

1 Introduction 2

2 Fading Channels 2

3 Simulation of Frequency-Flat Fading Channels 23.1 Filtered Gaussian Noise Model . . . . . . . . . . . . . . . . . . . 33.2 Jakes’ Model (Sum of Sinusoids method) . . . . . . . . . . . . . . 4

4 Implementation of Channel Model 4

5 Conclusions 8

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1 Introduction

We wish to study the effect of data transmission over frequency-flat fadingchannels with or without channel state estimates. We use two methods, namelyJakes model and the Filtered Gaussian Noise model to simulate a fading channelwith a given doppler power spectrum. For each such channel we transmit datausing uncoded BPSK and QPSK modulated symbols and perform Monte-Carlosimulations to estimate the bit error probability (BER) as a function of thesignal-to-noise ratio (SNR). We change the channel properties to study theeffect of the rapidity of channel variation over the performance of our channelestimation methods. We also change the period of the pilot signals to study it’seffect on our channel performance.

2 Fading Channels

The delays associated with different signal paths in a multipath fading channelchange in an unpredictable manner and can only be characterized statistically.When there are a large number of paths, the central limit theorem can be appliedto model the time-variant impulse response of the channel as a complex-valuedGaussian random process. When the impulse response is modeled as a zero-mean complex-valued Gaussian process, the channel is said to be a Rayleighfading channel. If the coherence bandwidth of channel is much greater than thebandwidth of the signal, the channel is said to be frequency-flat since it affectsall signal frequencies in almost the same manner. Our simulation concerns suchfrequency-flat fading channels. Channels can also be characterized as slow orfast fading.

3 Simulation of Frequency-Flat Fading Chan-nels

We simulate the frequency-flat fading channel using two methods, namely theFiltered Gaussian model and the Jakes’ model. We perform the simulation withand without the channel state information. For the latter case, we send periodicpilot symbols which are known to the receiver so that the channel characteristicscan be estimated to aid the detection process. Obviously a balance has to bestruck between the need to estimate the channel accurately and to send asfew pilot symbols as possible to improve transmission efficiency, we study theeffect of changing the pilot interval over the channel estimates. We perform the

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simulation over slow fading channels by choosing the doppler spectrum Bd ofthe channel so that it is smaller than the bandwidth W of the signal and thenvary the doppler spectrum so that the effect of rapidity of channel variation canbe understood.

3.1 Filtered Gaussian Noise Model

The Filtered Gaussian Noise model uses two independent white Gaussiannoise sources which are filtered using low pass filters. For discrete-time simu-lation the low-pass filter h(t) is usually implemented as a first-order low-passdigital filter which models the fading process as a Markov process. Stuber [1]notes that this method has the limitation that it can produce only rationalforms of the Doppler spectrum whereas the Doppler spectrum is typically non-rational. The advantage of using low-pass filtered WGN is the ease by whichwe can generate multiple uncorrelated fading waveforms by using uncorrelatednoise sources.

Figure 1: Channel output (Filtered Gaussian) when fmT = 0.001 (slowvarying).

We simulate the channel as a Markov process. An alternate method would beto use a low pass filter( such as a Butterworth or Chebyshev filter) to achievethe same effect.As suggested in [1], we may use the following parameters todefine the Gaussian random process in each step of the Markov process.

ζ = 2− cos(2πfmT )−√

(2− cos2πfmT )2 − 1 (1)

We choose the value of σ2 as

σ2 =(1 + ζ)(1− ζ)

Ωp

2(2)

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Figure 2: Channel output (Filtered Gaussian) when fmT = 0.1 (fastvarying).

The Markov model as used in this simulation implements the channel as

(gI,k+1, gQ,k+1) = ζ(gI,k, gQ,k) + (1− ζ)(w1,k, w2,k) (3)

3.2 Jakes’ Model (Sum of Sinusoids method)

The Jakes model implements the channel as a sum of sinusoids as defined bythe following equation

g(t) = gI(t) + jgQ(t)

=√

2[2∑M

n=1 cosβncos2πfnt +√

2cosαcos2πfmt]

+j[2∑M

n=1 cosβncos2πfnt +√

2sinαcos2πfmt] (4)

where the parameters are as defined in [1].In our simulation we use 8 sinusoids to generate the channel where the real

and imaginary parts of the complex channel are defined by the sinusoids andtheir phase shifted versions. The plots show the variation caused by the channelas fm is changed. On increasing fm the channel becomes fast varying and causesmore signal distortion which bears out our concepts.

4 Implementation of Channel Model

The channel simulation uses both uncoded BPSK and QPSK modulated sig-nals over the fading channels designed. We can either assume complete Channel

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Figure 3: Channel output (Jakes) when fmT = 0.001 (slow varying).

Figure 4: Channel output (Jakes) when fmT = 0.1 (fast varying).

State Information (CSI) or estimate the channel using pilot symbols. For thecomplete CSI case,

r(t) = h(t)s(t) + w(t)

orr(t)h(t)

= s(t) +w(t)h(t)

(5)

Here r(t) and h(t) are the complex valued received signals and channel fadingcoefficients respectively while w(t) is additive white gaussian noise. Since thechannel h(t) is known to us, detection is a simple affair of checking the signallevel. For QPSK, we check the quadrant that the received signal r(t)/h(t) liesin to make our decision.

If we do not have complete CSI, we can estimate the channel using a varietyof methods. The method used here involves transmission of pilot signals peri-

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Figure 5: BER vs. SNR using a BPSK signal for fmT = 0.001

Figure 6: BER vs. SNR using a QPSK signal for fmT = 0.001

odically to estimate the channel state. If the pilot symbols are y1, y2, ...yL, thenthe received signals are of the form

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r1 = α1y1 + w1

r2 = α2y2 + w2

...

rL = αLyL + wL (6)

If we choose the doppler spectrum such that the symbol time T 1/Bd

where Bd represents the doppler spread, we can take advantage of the slowfading properties of the channel to assume that α1 = α2 = ... = αL = α. Thus

L∑i=1

ri = αL∑

i=1

yi +L∑

i=1

wi (7)

Since w(t) is white gaussian noise, we can assume that∑

wi → 0 so that

α =∑

ri∑yi

(8)

By carefully choosing the number of pilot symbols and the ’period’ of the

Figure 7: Comparison of actual channel and estimated channel fordifferent pilot intervals.(The smooth curve represents the actual channel)

symbols, it is possible to get a very good estimate of the channel so that thedetection is almost as good as that in the complete CSI case. We use datatransmitted over an AWGN channel without fading as our benchmark. Thechannel coefficients are normalized in our implementation.

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Figure 8: BER vs. SNR using a BPSK signal for fmT = 0.001 usingchannel estimation

Figure 9: BER vs. SNR using a QPSK signal for fmT = 0.001 usingchannel estimation

5 Conclusions

We use a simple AWGN channel as our benchmark for all BER vs. SNR plots.For the AWGN case, the BER reduces with increasing SNR as expected andgoes to zero for SNRs above approximately 7dB. We plot the channel envelopesfor two different Doppler spectrums when a sinusoidal signal is transmitted. We

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notice that the fading and distortion increases as we increase the Doppler spreadfm, which is to be expected since an increase in fm causes more rapid variationand hence more fading in the channel.From the plots of BER vs. SNR for the channel where CSI is available, we findthat the BER decreases with increasing SNR as expected and both the Jakesmodel and the Filtered Gaussian Noise model perform in a similar way. TheMonte Carlo simulations are performed over 105 bits to get a good estimateof the BER. It is seen that the BER is much greater than the ones obtainedwhen only AWGN is present which is due to the random distortions producedby multipath fading.We also obtain the plots for the case where CSI is not available and the channel

Figure 10: BER vs. SNR using a QPSK signal for fmT = 0.0001 usingchannel estimation.(The channel is more slow varying than the that in Fig 9and shows a better BER)

has to be estimated using pilot signals. We notice that the channel estimationis better when the channel is not fast varying. Also, if the period of the pilotsignals is small, the estimation is more accurate. This is due to the fact thatthe channel estimates obtained by sending a few pilot signals gives the averagenature of the channel for the duration of the pilot signals.If the channel is slow varying, we can expect the estimates to give a good ap-proximation, else the estimates are not valid. Similarly, a smaller period forthe pilot signals signifies a more frequent estimation of the channel. Obviously,a more frequent estimation will give a better approximation of the channel asseen in the plots. As we change the Doppler spectrum to make the channel vary

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faster, we need to reduce the pilot period to get better estimates. We also seethat the BER is less when we have ideal channel state estimates.

References

[1] Gordon L. Stuber, ”Principles of Mobile Communication,” 2ed., KluwerAcademic Publishers, 2001.

[2] Theodore S. Rappaport, ”Wireless Communications: Principle and Prac-tice,” 2ed., Pearson Education, 2002.

[3] John G. Proakis, Digital Communications, 4ed., McGraw-Hill, 2001.

[4] R. Clarke, ”A statistical theory of mobile radio reception,” Bell SystemTech. J., Vol. 47, pp. 957-1000, 1968.

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