[American Institute of Aeronautics and Astronautics 26th International Communications Satellite...

1American Institute of Aeronautics and Astronautics

Evaluation of the Satellite Downlink Performance in thePresence of Phase Noise

Dr. Rajendra Kumar1The Aerospace Corporation, Los Angeles, California 90009-2957

This paper presents the performance evaluation of the satellite downlink by simulations,in the presence of phase noise caused by the reference oscillator in the mixer. The oscillatorphase noise power spectral density has an asymptotic behavior as f1 for the low-frequencyrange, where f denotes the frequency. Such a behavior is termed 1/f noise. Higher-frequencyranges have asymptotes of the form fn for n taking different integer values over differentfrequency ranges, although in some cases n may also take noninteger values. Thus thesimulation of the noise process with the specified power spectral density requires a filterwith its magnitude response with some segments having asymptotic behavior as fn/2 with awhite noise process at the filter input. Note that for any finite dimensional filter such as theButterworth filter, each segment of the magnitude response varies as fn, with n takinginteger values. Therefore, the given noise process cannot be simulated by such filters andrequires infinite dimensional filters. Using Barnes approach the paper first presents adesign methodology implemented on MATLAB to create an analog or digital filter so as toapproximate any arbitrary filter response with any desired accuracy. The method is thenapplied to generate a phase noise sequence according to the phase noise power spectraldensity specific to any specific satellite link. The phase noise sequence is then used toevaluate the performance of the digital modulation scheme, such as MPSK, used on thedownlink. A raised cosine filter characteristic is used in the simulation examples to bandlimit the spectrum of the signal. The performance is evaluated both in terms of theprobability of symbol error and the error vector magnitude (EVM).

Nomenclature = vector of the normalized slopes of the segments of the asymptotic frequency responsedI(t) = envelope of the inphase component of the modulated signaldQ(t) = envelope of the quadrature component of the modulated signalEVM = error vector magnitudeFs = sampling frequency(t) = phase noise processH(j) = frequency response of the jth section of the designed filterPv(f) = power spectral density of signal v(t)d(t) = phase modulation due to information symbolsPE = probability of symbol errorp = negative of the inverse of the pole locationz = negative of the inverse of the zero location

I. IntroductionHIS paper presents the performance evaluation of the satellite downlink by simulations, in the presence of phasenoise caused by the reference oscillator in the mixer. The present and future generations of satellites may

operate with different multiple accessing (MA) techniques on the uplinks and downlinks so as to optimize theperformance of both links. The feasibility of mixed-mode MA techniques requires efficient channelization

1 Senior Engineering Specialist, Communication Systems Engineering Department, P.O. Box 92957, Los Angeles,CA 90009-2957, M1/937, Member AIAA. Also Professor in the Department of Electrical Engineering at CaliforniaState University, Long Beach.

T

26th International Communications Satellite Systems Conference (ICSSC)10 - 12 June 2008, San Diego, CA

AIAA 2008-5418

Copyright 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


techniques. In addition, multibeam satellites require switching of channels among various uplink and downlinkbeams. More modern satellite architectures may even involve complete detection and routing of data packets. Due toinherent advantages of the digital techniques in terms of size, weight, cost, and flexibility, all of these functions,including the channelization and switching, are performed in the digital domain using modern digital signalprocessing (DSP) techniques. The uplink RF wideband signal, after downconversion to IF, is input into the analog-to-digital converter (ADC). The channelization, switching, and routing functions are then performed in the digitaldomain.18 Finally the digitally synthesized (multiplexed) signals at the DSP processor output are upconverted to theIF and RF frequencies for transmission over the downlinks.

In terms of broadband satellites (bandwidth in GHz), major limitations are seen in the analog-to-digitalconversion of the received broadband signals. Such an ADC needs to operate at a rate equal to at least two times thereceived signal bandwidth. This imposes serious limitations on the availability of an ADC operating at high speedsand providing the required number of quantization bits. The ADC forms a source of additional noise in the downlinkarising from the quantization and clipping processes in the ADC. References 1619 present mathematical analysisand simulation results for the ADC when the input to the ADC is a wideband noise, a number of sinusoidalwaveforms, or a number of digitally modulated signals. These references consider both the uniform and nonuniformquantizers and present the results on the power spectral density of the quantization noise. Such results are applicablewhen the clipping is negligible.

A very important source of noise in the downlink is the signal clipping that occurs in the ADC whenever theinstantaneous signal amplitude at the ADC input exceeds the maximum linear range of the quantizer. Since clippingcannot be avoided in most practical situations, the signal power to the quantization plus clipping noise power ratio isof most interest. In principle, clipping can be avoided by reducing the input signal average power to a level so thatthe peak signal amplitude never exceeds the quantizer input linear range. However, since in practice the signal ischaracterized by a random process that is not bounded by probability one, there will always be some nonzeroprobability of signal clipping irrespective of the input signal average power level. Reference 20 presents results onthe signal-to-total distortion power ratio, arising from quantization noise and clipping effects, when the input to theADC is a sum of a number of digitally modulated signals such as 16-QAM or 8-PSK signals including both theband-limiting filtering or no filtering cases. Reference 20 also considers nonuniform quantizers. Reference 21 alsopresents results on the probability of bit error when multiple digitally modulated signals such as 16-QAM or 8-PSKsignals are input to uniform or nonuniform quantizers showing that the effect of clipping and quantization noise isvery different in terms of the probability of bit error.

There is another important source of noise in the downlink arising in the analog-to-digital conversion process.This is the additional distortion caused due to jitter in the sampling clock. Both the analytical and simulation resultsin the published literature assume that the signal is sampled at a uniform interval, i.e., an ideal sampling clock isassumed. However, in practice the sampling clock has some timing jitter, the extent of which depends upon thedesign. The important parameters are the stability of the frequency standard of the master clock used in the system,the synthesis process such as indirect, direct, direct digital, or a hybrid of these procedures, and various otherparameters such as the order and the bandwidth of the phase-locked loop used in the design of the sampling clock.Reference 22 presents a detailed analysis of the impact of the sampling clock phase jitter on the signal-to-noisepower ratio at the output of the ADC. Reference 22 shows that the ADC output noise is the sum of the twoindependent noise processes, namely the quantization and the noise due to clock jitter referred to simply as the jitternoise, assuming that the ADC is loaded to avoid any significant clipping. The power spectral density (PSD) and thevariance of the jitter noise are presented in terms of the input signal power spectral density and the statistics of theclock jitter.22 Several special cases are presented for the signal at the ADC input, including baseband white noise,test tone sine wave and bandpass white noise for which explicit expressions are presented for the signal-to-jitternoise power ratio. It is shown22 that for the case of bandpass sampling, the jitter noise power increases as the squareof the carrier frequency.

There is another important source of noise in the satellite downlink performance. This is the phase noise arisingfrom various sources in the satellite payload, including the oscillator phase noise and mixers. The impact of thephase noise on the performance of the downlink is of a very different nature compared to other sources of noise,including the quantization, clipping, clock jitter, and receiver thermal noise as the effect of all these sources is of anadditive nature. Moreover, the noise generated by these sources is a wideband noise and except under some veryspecific conditions can be modeled as additional thermal noise power. In contrast to this, the phase noise is anonadditive process and is a narrowband process with bandwidth compared to the signal bandwidth. Thus the effectof phase noise cannot be modeled in terms of an equivalent additional additive noise power. This is true more sowhen the downlink signal is encoded with powerful error correction codes as is usually the case. Due to therelatively low bandwidth of the phase noise, the channel errors created by the phase noise cannot be modeled as


random errors as is normally assumed in a simplified analysis. This is so because the phase noise can easily causethe number of channel errors to exceed the error correction capability of the code, resulting in a code block error.For a large code block size this corresponds to a relatively large number of information bit errors, resulting in amuch poorer performance compared to that predicted on the basis of random error model. Such an effect may notalso be ameliorated by the interleaver due to the practical constraints on the length of the interleaver. Thus it is veryimportant to correctly evaluate the impact of the phase noise on the downlink performance in the presence of variousother sources of distortion, including those introduced in the ADC such as the quantization, clipping, and jitter noise,and due to the nonlinearity of the downlink power amplifier and any interchannel interference that may be present.Due to analytical difficulties involved in such a modeling, this paper evaluates the performance by simulation.

The modeling of the phase noise in digital computer simulations of modern communication systems is animportant problem. The oscillator phase noise power spectral density Pn(f) has an asymptotic behavior f 1 for thelow-frequency range. Such a behavior is termed 1/f noise. Higher-frequency ranges have asymptotes of the form f n for n taking different integer values over different frequency ranges although in some cases n may also takenoninteger values. Thus the simulation of the noise process with the specified power spectral density requires a filterwith its magnitude response 2/nf)f2j(H whose input is a white noise process. Note that for any finitedimensional filter such as the Butterworth filter, 2/nf)f2j(H for some integer n and thus the noise at the filteroutput in this case has asymptotes with Pn(f) f 2n. Therefore, the specified phase noise process cannot besimulated by such filters and requires infinite dimensional filters. However, an approximate technique has beenproposed in an earlier work by Barnes23 and is used in this paper to synthesize a filter with an arbitrary asymptoticresponse. The paper first reviews the Barnes approach, which is implemented on MATLAB to create an analog ordigital filter so as to approximate any arbitrary filter response with any desired accuracy. Of course, the higher theaccuracy of approximation, the higher the required filter order. In practice, though, as illustrated by examples in thepaper, there is no significant change in accuracy when the filter order exceeds a value. The paper presents severalexamples to illustrate the design of filter of any specified magnitude response.

The method is then applied to generate a phase noise sequence according to the phase noise power spectraldensity specific to any specific satellite link. The phase noise sequence is then used to evaluate the performance ofthe digital modulation scheme, such as MPSK, used on the downlink. A raised cosine filter characteristic is used toband limit the spectrum of the signal. The performance is evaluated both in terms of the probability of symbol errorand the error vector magnitude (EVM). In practice due to some of the above-mentioned difficulties encountered inthe simulations, a simplified approach is used in which first the EVM is estimated due to phase noise alone, which isthen used in a somewhat simplistic manner to evaluate its impact on the probability of symbol error. In contrast tothat approach, the approach of the paper provides a precise result on the probability of symbol error in the presenceof both the phase error and the receiver noise over any arbitrary range of these noise processes. Although the resultspresented in the paper focus on only the receiver noise and the phase noise, the method can be applied to includeother sources of distortion such as amplifier nonlinearity.

II. Filter Design for the Phase Noise SimulationFollowing the approach of Ref. 23, any segment of the specified asymptotic filter response with its magnitude

response 2/nf)f2j(H with noninteger n, and thus requiring an infinite dimensional filter, is approximated by afinite dimensional filter of order N. The filter order N is selected on the basis of the desired approximation accuracy.However, as shown by several design examples, in practice an excellent match between the specified response andthat achieved by the design can be obtained with only moderate values of N.

The desired approximate finite dimensional filter is obtained by a cascade of N filters of the type

N,,2,1i;)(j1

j1)j(H i2

i1

i2

i K=++

+= (1)

where = 2f, 1i and 2i are some appropriate time constants and N is selected according to the range offrequencies over which the desired f characteristics are required. The ith filter in the cascade has the property thatfor 1)( i2i1 , )/()j(H i2i1i2i += . Therefore if the filter time constants areselected according to


=

= =

i i 01 1i i 02 2 ;i 1,2, ,NK

(2)

for some positive constants


Figure 1. Designed filter response for the desiredresponse f1 over the interval (10, 100) Hz.

The range of frequencies over which the desired response is obtained is approximately equal to N, thus

N/1hl )/( = (11)

where h is the upper limit of the specified frequency range in rad/sec. From Eq. (9) it follows that

02

2/01 )1( = (12)

Note that in Eq. (12) is negative. Also, the higher the filter order N, the better the filter approximation. Tosummarize the design procedure, one computes , 20, and 10 from Eqs. (10) (12) for any specified values offrequencies l, h, slope, and the selected filter order N. The time constants of other filter sections of the designedfilter are then obtained from Eq. (2).

For example, with N = 4, =1, l = 10 rad/sec, h = 104 rad/sec, = 0.1 , 20 =0.1, and 10 =0.216. The firstfilter section has the frequency response + + ( 1 j / 10 ) / (1 j / 10 ) . The complete filter response is given by

)1010/j1()10/j1(

)1010/j1()10/j1(

)1010/j1()10/j1(

)10/j1()10/j1()j(H 3

4

2

32

+

+

+

+

+

+

+

+=

III. Filter Design ExamplesSeveral examples of the filter designs are presented in the following. These designs have been obtained by theMATLAB program developed for the design of filters to simulate any specified phase noise power spectral density.The program requires as its inputs only the slopes of various segments, corner frequencies, and the order N for thedistributed filter sections. The sections with slopes that are multiples of 20 dB/decade are implemented using theButterworth filter design approach.

A. Example 1A frequency range of (10, 100) Hz is specified and a 10 dB/decade slope is required over the specified range

with constant response outside the range. The value of N is selected equal to 10. Since only one section is specified,a better match is obtained in the specified range bymodifying the frequencies fl and fh to be (10/K) Hzand (100*K) Hz for some appropriate value of theconstant K. Figure 1 shows the filter response withK=3. As may be inferred from the figure, a goodmatch is obtained with the specifications.

The values of the time constants zi $ 2i and pi $1i + 2i for i=1,2,N obtained from the design areshown below.

p = [5.98 10-2, 3.8110-2, 2.4310-2, 1.5510-2,9.8810-3, 6.3010-3, 4.0210-3, 2.5610-3, 1.6310-3 1.0410-3];

z = [4.77 10-2, 3.0410-2, 1.9410-2, 1.2410-2,7.8910-3, 5.0310-3, 3.2110-3, 2.0510-3,1.3010-3, 8.3210-4];

Figure 2 compares the filter response with thespecified filter response over the specified frequencyinterval. As may be inferred from the figure, the difference between the two is small over the specified range.


Figure 3. 10 dB/dec slope over (100, 1000) Hz,30 dB/dec slope for frequency > 1000 Hz.

Figure 4. Designed filter response with specified = [1 3 5] and F = [10 100 1000 10000] Hz.

Figure 2. Comparison of the desired and designedresponse over the specified frequency interval.

For the multiple section response, the distributedfilter sections may be combined with Butterworth filtersections. For example, to obtain a 30 dB/decade slopeover some interval, a distributed filter with a 10dB/decade slope is cascaded with a first-order low-passfilter (LPF) of first order. Note, however, that the first-order LPF has 20 dB/decade slope for all frequenciesgreater than the 3 dB cut-off frequency. This must beincluded in the design of subsequent filter sections, asillustrated by the following example.

B. Example 2It is required to design a filter with a 10 dB/decade

slope over the frequency range (100, 1000) Hz rangeand a slope of 30 dB/decade for frequencies higher than1000 Hz. Figure 3 shows the response obtained andcompares with the specified asymptotic response. Thetwo are again in good agreement.

In general the filter response may be specified as avector of slopes and the set of corner frequencies F.The MATLAB program then generates the desired filterdesign as shown in the following example.

C. Example 3The specified slope vector and the set of corner

frequencies F are given by

= [1 3 5];F = [10 100 1000 10000] Hz;

Figure 4 plots the specified asymptotic response andthe response of the designed filter with N=10 for eachsection requiring infinite dimensional filterimplementation. As may be inferred from the figure thetwo are in good agreement.

D. Digital FilterFor the purpose of digital simulations, each of the

filter sections (1) may be replaced by its digital filterversion obtained by the bilinear transformation. Thuswith

p

z

s1s1)s(H++

= (13)

The corresponding transfer function for the digital filteris given by

)1z/()1z(F2spz

s

s1s1)z(H

==

++

= (14)

and may be evaluated as


104 105 106 107 108 109-70

-60

-50

-40

-30

-20

-10

0

frequency in Hz

mag

nitu

dere

spo

nse

indB

number of filter sections N =10

r: Specified asymptotic response

b: Response of the designed filter

Figure 5. Phase noise filter response of the designedanalog filter (N=10).

104 105 106 107 108 109-70

-60

-50

-40

-30

-20

-10

0

frequency in Hz

mag

nitu

dere

spo

nse

indB




Figure 6. Phase noise filter response of thedesigned analog filter (N=20).

+= = + = +

+ = + + = =

1

pn pn zn zn1

zn pn zn z s pn p s

1 bzH( z ) r ; a (1 ) / (1 ); b (1 ) / (1 );1 az

r (1 ) / (1 ); ( 2 / F ); ( 2 / F )(15)

where Fs denotes the sampling frequency of the digital filter. The digital filter is implemented in the simulations bythe following difference equation

y(k+1) = -a y(k) + r x(k+1) + r b x(k) (16)

where x(k) and y(k) are the filter input and output, respectively.

IV. Design of Digital Filter for Phase Noise SimulationFigure 5 below shows the desired asymptotic response of the filter required to generate the requisite phase noise

sequence for some specified phase noise power spectral density. The corner frequencies of the filter are given by[104 105 106 107 108 109] Hz with the coefficients over consecutive segments in the specifiedfrequency range given by [2 1.5 0.5 1.5 1.0].Recall that 20 is the slope of the magnituderesponse in dB of the corresponding filter segment.Thus the first segment can be designed by a first-order Butterworth filter with the remainingsegments designed using the approach of the paper.The frequency response of the filter obtained by theapproach of this paper is also presented in Figure 5.Since the true system is an infinite dimensional, theapproximation accuracy depends upon the order Nof the filter selected for each segment, and thus a

trial and error approach may be necessary to synthesizethe filter. Figures 5 to 7 below plot the frequency responseof the analog filter thus designed with N equal to 10, 20,and 30, respectively. These figures also include thedesired asymptotic response for comparison. As may beinferred from these figures, the difference from thespecified asymptotic response is relatively small for allthree cases. The significant difference at the cornerfrequencies is expected since the specified response isonly an asymptotic response, not an actual response. Interms of the filter order N, the difference between thefilters for N=20 and N=30 is relatively small but issignificant when compared to the selection of N=10. Thus

a value of N=20 has been selected for simulations of the paper. As the specified response depicts four segments withslopes that are noninteger multiples of 20 dB/decade and one segment with 20 dB/decade slope, the overall filterhas 81 poles and 81 zeros.

As explained in the previous section, for the purpose of digital simulations, the corresponding digital filter isobtained by using the standard bilinear transform

)1z/()1z(F2s s)s(H)z(H +==


104 105 106 107 108 109-70

-60

-50

-40

-30

-20

-10

0

frequency in Hz

mag

nitu

dere

spon

se

indB




Figure 7. Phase noise filter response of thedesigned analog filter (N=30).

104 105 106 107 108 109-70

-60

-50

-40

-30

-20

-10

0

frequency in Hz

mag

nitu

dere

spon

se

indB




Figure 8. Phase noise filter response of thedesigned digital filter (N=20, Fs = 10 GHz).

104 105 106 107 108 109-70

-60

-50

-40

-30

-20

-10

0

frequency in Hz

mag

nitu

dere

spon

sein

dB




Figure 9. Phase noise filter response of thedesigned digital filter (N=20, Fs = 2 GHz).

where H(s) is the transfer function of the analog filter, H(z) isthe transfer function of the corresponding digital filter, and Fsis the sampling frequency. Figure 8 plots the frequencyresponse of the resulting digital filter with the samplingfrequency Fs equal to 10 GHz. As may be inferred from thefigure, the frequency response of the designed filter matchesvery well with the specified response over the entirefrequency range of interest. Figure 9 shows the correspondingresponse with Fs = 2 GHz, with a very similar responseshowing that the frequency response of the filter thusdesigned is not too sensitive to the sampling frequency aslong as it is higher than the Nyquist rate.

A. Power Spectral DensityTo further verify the performance of the designed filter, the

filter is simulated with a white Gaussian noise input sequenceof variance equal to 1. Figure 10 plots the power spectraldensity (PSD) estimated at the filter output. In theory theoutput power spectral density Pn0(f) is given by

)f(P)f(H)f(Pi0 n

2n = , where )f(P in , is the input noise PSD

and is equal to s2n

F/2i

% , where 2ni

% is the input noise

variance. In dB scale the output the relationship is given by

[ ] ( ) )(log10)2/F(log10)f(Hlog20)f(P 2n10s1010dBn i0

%+=

( ) Hz/dBW90)f(Hlog20 10 = (17)

and is in good agreement with the simulation result of Figure10. It may be noted that the simulation result of Figure 10 issomewhat different than the usual plots in the literature due toa different method applied in the estimation. In the literature,the PSD is obtained by using the Welch method, whichsimply averages out the absolute value square of the fastFourier transform (FFT) of the signal over several segments.Here the averaging is performed not only over the segmentbut also over the frequency intervals around any desiredfrequency. There is, of course, a theoretical basis forperforming such an averaging. The PSD estimation methodused in the paper is described below.

For any signal v(t), let {vi(k), k=0,1,,N-1} denote asegment of the sampled version of v(t) of length N fori=1,2, M, and let Vi(j) denote the FFT of vi(k). The PSD ofv(t) denoted by Pv(f) at any frequency f = j *f , *f = 1/ Nts, isestimated as

+++

=*==

2

1

n

nk

2i

M

1i 12v )k(V)1nn(

1M1)fj(P (18)


104 105 106 107 108 109-160

-150

-140

-130

-120

-110

-100

-90

-80

frequency in Hz

Pow

ersp

ectra

lden

sity

(dBW

/Hz)


sampling frequency 2e9 Hz

b: estimated from the designed filter simulations

r: specified asymptotic power spectral density plot

Figure 10. Filter output noise power spectraldensity (dBW/Hz), input noise variance = 1 W.

104 105 106 107 108 109-140

-130

-120

-110

-100

-90

-80

-70

-60

frequency in HzPo

wer

spec

trald

ensi

ty(dB

W/H

z)





Figure 11. Filter output noise power spectraldensity (dBW/Hz), input noise variance = 25 dBW.

104 105 106 107 108 109-140

-130

-120

-110

-100

-90

-80

-70

-60

frequency in Hz

Pow

ersp

ectra

lden

sity

(dBW

/Hz)





Figure 12. Filter output noise PSD with25 dBW input noise power and a second-order high-pass filter with fc = 10 kHz.

104 105 106 107 108 109-130

-120

-110

-100

-90

-80

-70

-60

frequency in Hz

pow

ersp

ectra

lden

sity

(dBW

/Hz)

Figure 13. Filter output noise PSD with 18 dBWinput noise power, a second-order high-passfilter with fc = 10 kHz , ,f = 4.2 kHz.

where n1 and n2 are selected so that

*,

=f2fjn1 ;

*,

+=f2fjn 2 for any specified frequency resolution

bandwidth ,f, where x denotes the highest integer lowerthan x for any real number x with a similar definition for x . The usual PSD estimation methods select ,f =*f,

resulting in large fluctuations in the estimation of thePSD (high estimation error). By increasing the resolutionbandwidth such fluctuations are greatly reduced. In fact,f can be made a function of f for further improvement inthe estimation. Such is the case with the result in Figure 6wherein ,f = [1 10 102 103 104] 2 kHz overconsecutive frequency interval decades starting with thefirst frequency interval decade of 10 to 90 kHz. Figure 11

shows the estimated power spectral density from thesimulations when the input noise power is equal to 25 dBWand compares it with the specified power spectral density,which is equal to the filter magnitude response multiplied bythe filter input noise spectral density. The simulation results arebased on averaging over M = 10 runs of N = 106 points eachaccording to Eq. (18) above. As may be inferred from thefigure the two graphs match very closely, as expected, exceptfor some deviation in the low-frequency range. The result canbe easily improved by reducing the frequency resolution in the10 kHz to 90 kHz frequency range.

B. Phase Noise VarianceThe phase noise variance in rad2 is simply the integral

of P (f) over the interval [0,-), i.e.,

.=-

0

2rms df)f(P (19)

When the PSD in Figure 11 is integrated over thecomplete range, the result is equal to 5.5103 rad2, whichcorresponds to rms = 4.25 deg. With the low-pass filterwith cut-off frequency at 250 MHz and a high-pass filter


0 100 200 300 400 500 600 700 800 900 1000-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

time index

phas

enoi

sein

rad

Figure 14. Phase noise sequence.

with cut-off frequency at 10 kHz as in Figure 12, the power is reduced to 3.3 103 rad2, corresponding torms = 3.29 deg. The result obtained by time averaging of the actual phase noise sequence generated is very close tothat obtained by integration of the power spectral density.

V. Simulation of 8 PSK Modulation with Phase NoiseFor the PSK modulation, the signal before raised cosine filtering is given by

))t(tf2sin(A)t(v dc /++= (20)

In Eq. (20) A, fc, and / represent the amplitude, frequency, and phase of the signal, while d(t) represents thedata phase modulation. The MPSK signal may equivalently be expressed in the following I & Q (inphase andquadrature) form:

)tf2cos()t(dA)tf2sin()t(dA)t(v cQcI /++/+= (21)

))t(sin()t(d));t(cos()t(d dQdI =$ (22)

The signals are band limited by a filter with a square root raised cosine filter characteristic with filter roll-offfactor r equal to 0.20, as in

)tf2cos()t(dA)tf2sin()t(dA)t(v cfQcfI /++/+= (23)

where )t(d fI and )t(d fQ represent the impulse sampled and square root raised cosine filtered versions of )t(dI and

)t(dQ , respectively. In the implementation of the square root raised cosine filter, an FIR filter of length equal to 65is used for the approximation with the sampling rate equal to four times the symbol rate. An additive Gaussian noisen(t) and a phase noise (t) are added to the signal to yield the received signal r(t), which is input into the PSKdemodulator.

)t(n))t(tf2cos()t(dA))t(tf2sin()t(dA)t(r cfQcfI ++/+++/+= (24)

The demodulator is composed of a complex mixer with its local reference frequency and phase equal to fc and /,respectively. The inphase and quadrature components of Eq. (22) are separately filtered by a square root raisedcosine filter that is identical to the one used in the modulator. The square root raised cosine filter output may beexpressed in the form of the following complex baseband signal:

)t(jn)t(n))t(cos()t(dA))t(cos()t(dA)t(r QIrcQb rcI +++= (25)

where dIrc(t) denotes the raised cosine filtered version of dI(t),obtained by filtering dIf(t) with the square root raised cosinefilter in the receiver, with a similar definition for dQrc(t). Thesignal rb(t) is then sampled at the symbol intervals. The sampledvalues of the filters outputs are input into a decision device toprovide the detected symbols. The detected symbols arecompared with the transmitted symbols to arrive at theprobability of symbol error. In the simulations the symbol rateis selected equal to 250 Msps.

Figure 14 depicts a sample plot of the phase noise sequencegenerated by the digital filter designed in the previous sectionof the paper.


0 2 4 6 8 10 12 14 16 1810-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N0 (dB)

Prob

abilit

yof

sym

bole

rror

b: ideal case

r: With phase noise

Figure 15. Performance of 8-PSK signals inthe presence of phase noise (3.29 deg rms,r = 0.20).

0 2 4 6 8 10 1210-3

10-2

10-1

100

Eb/N0 (dB)

Pro

babi

lity

ofsy

mbo

lerr

or

b: ideal case

r: With phase noise

Figure 16. Performance of 8-PSK signals inthe presence of phase noise (3.29 deg rms,r = 0.20).

0 2 4 6 8 10 12 14 16 1810-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N0 (dB)

Prob

abi

lity

ofs

ymbo

lerr

or

b: ideal case

r: reference phase noise

m: reference + 1 dB phase noise

c: reference + 2 dB phase noise

g: reference + 3 dB phase noise

k: reference + 4 dB phase noise

Figure 17. Performance of 8-PSK signals inthe presence of phase.

0 2 4 6 8 10 12 1410-4

10-3

10-2

10-1

100

Eb/N0 (dB)

Pro

babi

lity

ofsy

mbo

lerr

or

b: ideal case

r: reference phase noise

m: reference + 1 dB phase noise

c: reference + 2 dB phase noise

g: reference + 3 dB phase noise

k: reference + 4 dB phase noise

Figure 18. Performance of 8-PSK signals in thepresence of phase.

Figure 15 plots the probability of symbol error with andwithout the phase noise. As may be inferred from the figure,there is a very significant degradation due to the phase noise,especially at the low probability of symbol error PE. Figure 16shows the result at relatively high values of PE. There is adegradation of about 1 dB in terms of Eb/N0 at PE equal to 103.Another measure of performance measure is the rms EVM(Error Vector Magnitude) defined as

++ ===

Ns

1k

20b

2Ns

1k0bbrms )k(r/)k(r)k(rEVM (26)

where rb(k) denotes the sampled version of rb(t) in Eq. (25) andrb0(k) denotes the corresponding value in the ideal case. For thesimulation example, the EVMrms is equal to 0.054.

C. Sensitivity of PE to the Phase Noise VarianceIn order to assess the sensitivity of the probability of symbol

error, the simulations have also been performed with a numberof different phase noise variances. Figures 17 and 18 plot the PEwhen the phase noise variance is increased above the referencevalue used in Figure 15. As expected, the figure shows strongdependence of the PE on the phase noise variance.

ConclusionsThis paper has presented a design methodology and a

MATLAB implementation for designing an arbitrary responsefilter to simulate any specified power spectral density functionfor the phase noise. Several design examples using the approachhave also been presented to illustrate the approach. The methodhas been applied to generate a phase noise sequence accordingto the phase noise power spectral density specific to any specific

satellite link. The phase noise sequence is then used to evaluate the performance of the digital modulation scheme,such as MPSK, used on the downlink. A raised cosine filter characteristic is used in the simulation examples to bandlimit the spectrum of the signal. The performance has been evaluated both in terms of the probability of symbolerror and the error vector magnitude (EVM). In practice due to the difficulty of simulating the phase noise, oftensomewhat simplistic methods are used, which do not provide accurate results. The discrepancy can be morepronounced when the signal is encoded with powerful error correction codes. Although the results in the paper arefor the case of no coding, the methodology can be applied to the coded case as well. A future paper will present theresults for the coded case and also present the differences in performance between the precise results and thoseobtained by a simplified approach that ignores the power spectral characteristics of the phase noise.


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