Modeling of the Simultaneous Influence of the Thermal Noise and the Phase Noise in Space Communication Systems

Transcription

1 618 O. BARAN, M. KASAL, MODELING OF THE SIMULTANEOUS INFLUENCE OF THE THERMAL NOISE AND THE HASE NOISE Modeling o the Siultaneous Inluence o the Theral Noise and the hase Noise in Space Counication Systes Ondřej BARAN, Miroslav KASAL Dept. o Radio Electronics, Brno University o Technology, urkyňova 118, Brno, Czech Republic xbaran03@stud.eec.vutbr.cz, kasal@eec.vutbr.cz Abstract. Our work deals with studies o a noise behavior in space counication systes. Two ost iportant noise types, the additive theral noise and the ultiplicative phase noise, respectively, are included. A siple odel o the narrowband counication syste is created and siulated in the Ansot Designer syste siulator. The additive theral noise is odeled as AWGN in a counication channel. The phase noise is produced in transitter and receiver oscillators. The ain intention is to investigate the receiver ilter bandwidth decrease eect on powers o both noise types. Results proposed in this paper show that or deined syste conditions and or a certain ilter bandwidth value, the power o the ultiplicative phase noise equals to the additive theral noise power. Another decrease o the ilter bandwidth causes the phase noise power exceeding. To deonstrate the noise behavior transparently, input syste paraeters are properly selected. All siulation results are docuented by theoretical calculations. Siulation outcoes express a good coincidence with presuptions and calculations. during the design is an additive theral noise projected ro the counication channel and ro the receiver part. The narrowband low rate counication [1], e. g. with deep space probes, is in addition aected by the ultiplicative phase noise, which arises ro all oscillators in the whole counication syste. The long distance causes huge signal attenuation during the ree space transission and syste noise paraeters are then very critical to the syste quality. On the other hand, in space counications, a ground receiver antenna is pointed to the cold sky, thus the appropriate noise teperature o the counication channel is quite low. This leads to a act that the inluence o the phase noise increases and it can t be neglected. The siultaneous incidence o both noise types on the syste signal to noise ratio is an object o the investigation. The decreasing bandwidth o the receiver ilter restricts the inluence o the additive theral noise and SNR is increasing. The sae procedure is ade with the ultiplicative phase noise and the eect is observed. Keywords Ansot Designer, theral noise, AWGN, phase noise, FIR ilter, noise bandwidth. 1. Introduction Generally, every counication syste consists o three basic parts, a transitter, a receiver and the useul inoration is transerred through a transission channel. Each part has its negative eatures which inluence and/or disturb the transitted useul signal in a certain way. These negative eatures are very oten in a contrary with high user deands or such a counication syste. In the area o space counication systes, conditions are even ore critical than in terrestrial systes. Very long counication distances, oving objects connected with the Doppler requency shit eect and a noise belong to the ost severe disturbing actors in a counication with space objects. In the wideband systes, the ain noise type considered. Theoretical Description o the Counication Syste Model In spite o a act that transitters and receivers are very coplex subsystes, their sipler ors are used or the odeling and or the investigation o the noise inluence. Obtained results have general validity. The proposed odel can be extended and the previous results can be applied or that extension..1 Basic araeters Declaration The odel is created in Ansot Designer syste siulator and Fig. 1 depicts its look. All signals used in the siulated syste have to be properly sapled according to the sapling theore []. The sapling requency is denoted as sa. For a sipliication, the useul baseband signal, denoted as x(t), is a haronic cosine wave. Its tie doain or is described by the equation

2 RADIOENGINEERING, VOL. 19, NO. 4, DECEMBER x A jxt jxt t e e (1) where A is an aplitude and ω x is an instantaneous angular requency. To coply with the sapling theore [], the inequality ω x (π sa ) needs to be ulilled. For a deodulation, the analytic signal o the local oscillator has a or j t jo jo t O e e e o (4) where O is the signal aplitude, ω O is the angular requency, Ф O is the initial phase and ψ (t) represents tie doain phase luctuations. For the proper deodulation, requencies and initial phases o transitter and receiver oscillator signals have to be the sae (ω O = ω O and Ф O = Ф O ). This condition is supposed to be accoplished in the whole text. For the sipliication in the ollowing text, the initial phase o transitter and receiver oscillators is considered to be zero radians (Ф O = Ф O = 0 rad). Ansot Designer siulator works with analytic signal representations, thus, derivations, used urther in the text, also utilize this representation. Analytic signals are always denoted with sybol + in a lower index. Fig. 1. A block diagra o the odeled counication syste. For a useul signal odulation, a carrier ro a local oscillator is used. Its signal is described in the tie doain as o t Ocos t t () O O where O is the oscillator aplitude, ω O is the instantaneous angular requency, Ф O is the initial phase and paraeter ψ(t) denotes tie doain phase luctuations (it is the expression o the requency doain oscillator phase noise). It is supposed that oscillators have the aplitude liitation ability. Thus in a coparison with phase luctuations, oscillator aplitude luctuations can be neglected [3]. The requency ω O can reach about several orders higher values then the requency o the useul odulating signal ω x. To ulill the sapling theore or the oscillator signal o(t), the sapling requency sa has to be at least twice as high as the requency O ( O is the carrier instantaneous requency, ω O = π O ). In addition, to cover the statistical behavior o the noise, one needs a huge nuber o saples to process. This puts very severe clais or the coputer power and the eory. To overcoe this proble, the sapling o a passband signal as a baseband signal is used. The oscillator passband signal o(t) is expressed as an analytic signal [], [4] o + (t) with a one-sided spectru according to j jo jot t Oe e e o. (3) The irst three ters o (3) reer to a coplex envelope [4] that, in act, represents the baseband signal which can be sapled by the sapling requency sa. The coplex sinusoid exp(jω o t) causes the requency transposition o the coplex envelope to the vicinity around the carrier requency ω O.. Additive Theral Noise Modeling The additive theral noise is coposed o the theral noise o all blocks on the receiver side o the syste (transission channel as well as the single receiver). In this text, it is supposed that the overall additive theral noise is recalculated to the input o the receiver antenna [5], [6]. Thus, it is expressed as the transission channel additive theral noise. The transission channel is odeled as a su-block which perors the saple-by-saple suation o the odulated signal + (t) and the noise signal n(t). The noise signal n(t) has the noral Gaussian distribution with a zero ean and a variance σ AWGN, that is directly equal to a noise power N. The channel odeled in such a way is denoted as AWGN (Additive White Gaussian Noise) channel [7]. The whole syste odel works with coplex signals, thus, even the noise signal has to be coplex n t n t j n t. (5) I Both, the real part noise and the iaginary part noise, are independent non-correlated processes. Originally, the signal n(t) is represented in a baseband and is sapled by sa. Ater the noise signal n(t) transposition to the passband (to the vicinity around the carrier O ), a theral noise power spectral density N 0 can be expressed as N AWGN N0. (6) sa Q sa Fig.. Noise power spectral density derivation.

3 60 O. BARAN, M. KASAL, MODELING OF THE SIMULTANEOUS INFLUENCE OF THE THERMAL NOISE AND THE HASE NOISE The noise power N is obtained as a su o real and iaginary parts powers. One supposes a one-sided spectru representation [8] and the situation is deonstrated by Fig...3 Multiplicative hase Noise Modeling Carrier waves are produced in oscillator odels which can also siulate a phase noise behavior. The single sideband (SSB) phase noise denotes the requency doain representation o tie doain phase luctuations (e.g. ψ(t) ro (3)). Ansot Designer odels the SSB phase noise L( ) according to the Leeson orula [3] L c 10 log 1 3 load load LO c LO FkT q q where is the oset requency ro the carrier LO, c denotes the licker noise corner requency [3]. The oscillator power is LO, q load reers to the loaded quality actor o the oscillator resonator and F denotes a noise igure o the oscillator active eleent. k is the Boltzan constant, T is the therodynaic teperature o the oscillator (k = JK -1, usually T = 90 K). The Leeson orula divides the requency doain SSB phase noise course into 4 areas (see equation (7)). Each area corresponds to the individual phase noise type (see Tab. 1) [6], [9]. Fro Tab. 1, β denotes the slope o the individual phase noise type ater a linear approxiation o L( ) in a log-log plot. It also expresses the power o a L( ) decoposition into a power series. phase noise type white M noise 0 licker M noise 1 white FM noise licker FM noise 3 Tab. 1. hase noise types covered by the Leeson orula [6]..4 Theoretical Derivations The ollowing subsections provide theoretical derivations o tie doain signals and their powers in signiicant points o the syste odel (designations agree with Fig. 1). Transitter point 1 According to Fig. 1, the transitter contains only a DSB-SC odulator. The odulator is odeled as a coplex ultiplier that ultiplies the odulating signal x(t) with the carrier o + (t). The result is a odulated signal + (t) and its tie doain representation ollows the equation t x t o t AO. jo x jo x 1 j t e e β LO (7) (8) Ater subtracting the signal + (t) (8) ro the ideal odulated signal (with a zero noise ter ψ(t)), only the noise signal with the power N (1) is obtained N AO 1 t. (9) This result iplies ro the act that powers o a coplex sine wave or a coplex cosine wave are equal to the unity. Transission channel point The lossless transission channel is odeled according to the earlier description. The additive theral noise (siulated as AWGN) is added to the odulated signal + (t) (8) and the result is n(t) + + (t). The noise power N () observed in the point is equal to AO N n t. (10) Receiver passband part point 3 A received signal passes through the band-pass ilter. It is considered that the ilter doesn t aect the useul signal, but it inluences the noise o both types. Let one denote noise signals passed through the ilter as n(t) n (t) and ψ(t) ψ (t). With these considerations, equation (10) can be rewritten to the or AO N 3 n t t. (11) Now, the question is, how exactly the band-pass ilter aects the additive noise and the phase noise. For each ilter, its noise bandwidth BN can be calculated with a help o a ilter s power transer unction [5], [10]. When the ilter is high order, the steepness o its transer unction is very high and the ilter noise bandwidth can be considered to be equal to the ilter 3 db bandwidth. The usage o analytic signals and coplex envelopes brings a certain sipliication into the iltering. Instead o a band-pass ilter, the low-pass iltering is used [4]. In a classical conception, the band-pass ilter inluence on the additive noise can be explained according to the Fig. 3. Fig. 3. A derivation o the additive noise power ater band-pass iltration. A coplex analytic signals conception enables shiting the signal to the baseband, where the noise bandwidth o the equivalent low-pass ilter is BN /. I the additive noise power spectral density N 0 is known (see (6)), then the additive noise power ater iltration equals to [5]

4 RADIOENGINEERING, VOL. 19, NO. 4, DECEMBER BN. (1) n t N0 N0 BN The actor denotes the reality that, in act, one works with both lower and upper requency coponents ater the odulation (the situation is in accordance with Fig. 3). In a case o the ultiplicative phase noise, the iltration proble is ore coplicated. The situation is deonstrated in Fig. 4. At the beginning, one expects a high steepness o the ilter transer unction, thus, the concept o the noise bandwidth can be used. Behind the ilter passband, noise sidebands are considered to be suppressed so well that they can be neglected. Fig. 4 deonstrates the real situation, while in siulations, the concept o analytic signals and the passband signal shiting to the baseband is utilized (indeed, with sae results). The low-pass ilter is used instead o the band-pass one (in the sae way as in the previous case with the additive noise). Fig. 4. The odulation and iltration inluence on the phase noise, a) carrier phase noise sidebands, b) a transoration o the phase noise ater the odulation, c) a principle o the iltered phase noise calculation ater the odulation. hase noise sidebands in the vicinity o the carrier (e.g. o + (t)) are deonstrated in Fig. 4 a). The phase noise power in one sideband (upper or lower) can be calculated as a nuerical integration o the SSB phase noise course L( ) Ncarr L d (13) where 1 and are integral liits. In a case o the carrier phase noise, the liits are ollowing 1 = Δ and = sa /. The low liit requency 1 is not allowed being zero (the phase noise doesn t cover the carrier), thus, the paraeter Δ represents the oset step ro the carrier. The total carrier phase noise power is two ties the power calculated 1 according to (13) (the upper sideband power plus the lower sideband power). Ater the odulation, the situation with phase noise changes as can be seen in Fig. 4 b). The upper coponent (vicinity o O + x ) and the lower coponent (vicinity o O - x ) are created. The phase noise power o each coponent independently is a hal copared to the carrier phase noise power (integrated according to thin doted lines around each coponent in Fig. 4 b)). The total odulated phase noise power equals to a suation o upper noise coponent and lower noise coponent contributors (in other words, the proper integration according to the thick solid line in Fig. 4 b) has to be done). The result directly corresponds to the carrier phase noise power. To calculate the phase noise power ater the iltration, the ollowing solution is proposed. One ocuses only to the upper coponent o the odulated signal, where the proble is divided in to the lower part A and the upper part B (see Fig. 4. c)). In each part, the integration N 1 od L d (14) is utilized. A lower index od is in turn replaced by a particular part (A U, A L, B U, B L ; see Fig. 4 c)) in which liits 1 and or integration o L( ) dier. The upper part B is created by the upper phase noise sideband B U o the upper coponent (a thick dashed curve in Fig. 4 c)) and by the upper phase noise sideband B L o the lower coponent (a thick doted curve in Fig. 4 c)). Then or the B U part, the liits are 1 =Δ and =BN / - x. The liits or the B L part are 1 = x + Δ and = BN / + x. The lower part A is shaped by the lower phase noise sideband A U o the upper coponent (a thick dashed curve in Fig. 4 c)) and by the upper phase noise sideband A L o the lower coponent (a thick doted curve in Fig. 4 c)). Then, the liits or A U part are 1 = Δ and = x (it iplies ro the lipped course o L( )). The liits or the A L part are 1 = x and = x - Δ. According to the previous procedure, ater integral (14) calculations, one obtains our partial noise powers. The total noise power o one coponent is a su o these our partial noise powers 1 Nod_1 cop NB NB U L N N 1 x BN L AU x x d L x x d L d. L x AL BN d (15) Doubling this total noise power, the overall phase noise power {ψ (t)} in the odulated signal is gained (upper

5 6 O. BARAN, M. KASAL, MODELING OF THE SIMULTANEOUS INFLUENCE OF THE THERMAL NOISE AND THE HASE NOISE and lower coponents have identical power and are syetrical around the carrier requency, {ψ (t)}={n od_1cop }). With a lower ilter order, the ilter transer unction steepness decreases and the above procedure gives less accurate results. Again, one has our parts (A U, A L, B U, B L ) to be integrated and their su ors the total noise power o each coponent. But in this case, or parts B U and B L, in the requency doain, the SSB phase noise course L( ) needs to be ultiplied by a real ilter transer unction shited to the right position on the requency axis. For both parts, the integration liits changes to the ollowing: or B U 1 = Δ and = sa /, or B L 1 = x and = sa /. The process or the calculation o powers o parts A U and A L stays the sae as above. The overall odulated phase noise power is also the double o the coponent noise power. Receiver baseband part point 6 A deodulation is ipleented by a help o a quadrature syste [8]. The quadrature deodulator consists o two coplex ultipliers that are ed by the received iltered signal on the irst input and by the local oscillator signal on the second input. The in-phase ultiplier N I is driven directly by the oscillator signal o +I (t) = o + (t). The quadrature ultiplier N Q is controlled by the oscillator signal that is phase shited about 90 o j t o t e. (16) Q The su o in-phase and quadrature products gives the coplex analytic deodulated signal d + (t). For the deodulation back to the baseband, requencies o odulator and deodulator oscillators are the sae, ω O = ω O. When this condition is ulilled, Ansot Designer takes only the real part o the quadrature deodulator output signal d t n Red t t O I cos Ot sin Ot cos Ot sin Ot n O Q cos Ot sin Ot cos Ot sin Ot 1 t t AOOcos t. x (17) In the deodulated signal, both parts (in-phase and quadrature) o the coplex additive noise are presented. Ater the sae presuptions as in other points o the syste, subtracting the distorted deodulated signal (17) ro the ideal one (zero ters n (t) = 0, ψ (t) = 0 and ψ (t) = 0), the single noise signal is obtained. Its power equals to N 6 O n t O n AOO t. (18) The power o the deodulator oscillator phase noise {ψ (t)} is gained as a double o the integration according to (13) with the ollowing liits 1 = Δ and = sa /. Receiver inal signal point 7 The inal step to get the deodulated signal x (t) in the baseband is the low-pass L iltration o the signal d(t) (17). The ain assuption is that the L ilter doesn t aect the useul signal. The L inluences only the aount o the noise power, thus, tie doain noise signals can be rewritten as n (t) n L (t), ψ (t) ψ L (t) and ψ (t) ψ L (t). With these conditions, the noise power can be obtained ater a siple odiication o equation (18) to the or N 7 O n O n AOO t. L L L L L (19) The sae question here is, how exactly the low-pass ilter aect the additive noise and the ultiplicative phase noise and how the power o such a iltered noise can be calculated. Siilarly as above, the low-pass ilter noise bandwidth BN L can be calculated [5], [10]. The high ilter order secures a suicient steepness o the ilter transer unction which also allows using the noise bandwidth concept or the phase noise. Let one ake another assuption, the low-pass ilter bandwidth is always equal to or less than the hal o the band-pass ilter bandwidth. The reason will eerge ro the ollowing paragraphs and pictures. In a case o the additive noise, the procedure depicted in Fig. 5 can be utilized (a siilar way as or the band-pass ilter earlier [8]). When the noise power spectral density N 0 is known (7), then, ater substituting BN L instead o BN into (1), the ollowing orula can be used n L N0 BNL. (0) Factor denotes the two-sided spectru division o the power between positive and negative requencies. Fig. 5. An expression o the additive noise power ater a low-pass iltration. For the phase noise, the inluence o the low-pass iltration can be also described siilarly as or the band-pass ilter case. The whole procedure stays the sae, just BN L is substituted or BN /. Calculations are ade twice, once or the transored odulator oscillator phase noise ψ (t), and the second tie or the deodulator oscillator

6 RADIOENGINEERING, VOL. 19, NO. 4, DECEMBER phase noise ψ (t). The sae rules as or the ilter holds or the decreasing steepness o the L ilter. The results are used in calculations according to (19). In a case o high order ilters with a high steepness o their transer unction, i BN / < BN L, noise contributors n (t) and ψ (t) are considered not to be aected by the L ilter. Only the phase noise o the deodulator oscillator is inluenced. On the other hand, when the ilter steepness decreases, the real ilters transer unctions have to be used or the iltration o the phase noise courses. 3. Model Settings in Ansot Designer The syste odel according to Fig. 1 can be created in dierent siulating progras. Each o the has its speciic deands that need to be kept to obtain correct results. The ollowing paragraphs su up iportant eatures and paraeters that have to be coped with in Ansot Designer siulator. The inial oset requency o the generated phase noise in Ansot Designer is 100 Hz ro the carrier. This restriction is established in the paraeter Δ, that is used in the previous equations. The space between the carrier and the liit 100 Hz is created by the distortion iplying ro the iperect signal sapling. This act has to be considered in theoretical calculations to a proper coparison with siulation results. Upper sideband requency spectru details o the odulator oscillator with and without the phase noise are deonstrated in Fig. 6. Both ilters and a L ilter are realized by the FIR low-pass ilters. The received signal beore the ilter (point, see Fig. 1) is coplex. To peror the FIR iltration, this coplex signal is divided into its real and iaginary parts that are iltered separately. The resulting parts are cobined back to the coplex signal (point 3). The deodulated signal (point 6) is just real, thus, this odiication isn t necessary. Fig. 6. A coparison o requency spectra o the oscillator signal o + (t) with and without the phase noise. In reality, the odeled quadrature deodulator contains two separate oscillators. Their signals are set to be 90 phase shited. This solution has an advantage in a contrary to a realization with one oscillator and a phase shiter. The phase shiter can be realized by a delay block that delays the incoing signal about a certain nuber o saples. The proble is that the integer nuber o saples can t cover exactly the phase shit 90 and eerging deviations can increase errors. Every noise generating block has a paraeter seed, which is used or the rando nuber generator. For each block, the seed paraeter has to be set dierently to obtain the uncorrelated independent noise signals. The exception is in a case o the quadrature deodulator, where both oscillators ust have equal seed paraeters (in act, they express one oscillator with two phase shited output signals with the sae phase noise distortion). 4. Siulation Results and Coparison The basic counication syste odel according to Fig. 1 is created in Ansot Designer siulator. The chain odel is ade in two parallel branches. One represents a reerence ideal syste without any noise source. The second branch is, in turn, degraded by an additive noise, then by a ultiplicative phase noise and inally, by both noise types siultaneously. This organization allows one to easure signal to noise ratios between ideal and distorted branches in the siulator (in points and 3 CNR (1) and CNR (), in points 6 and 7 SNR (1) and SNR (), see Fig. 1). Signal to noise ratios are used or noise powers derivations. Siulation results are copared with theoretical calculations processed in Matlab (prograed according to derivations in chapter ). araeters used during siulations are chosen properly to provide transparent results. Concrete settings o the syste odel Basic paraeters that have to be set in signal sources odels are suarized in Tab.. The sapling requency in the whole siulation is chosen to sa = 4096 Hz. quantity nae value unit Modulating signal x(t) aplitude A V power {x(t)} * 1 W requency x 150 Hz Modulator carrier o(t) aplitude O 1 V power {o + (t)} * 1 W requency O 1 MHz N power {ψ(t)} * -3,678 dbw Deodulator carrier o (t) aplitude O 1 V power {o + (t)} * 1 W requency O 1 MHz N power {ψ (t)} * -36,735 dbw * powers are calculated over 1 Ω resistance Tab.. Settings o signal sources paraeters.

7 64 O. BARAN, M. KASAL, MODELING OF THE SIMULTANEOUS INFLUENCE OF THE THERMAL NOISE AND THE HASE NOISE The requency doain phase noise courses corresponding to odulator oscillator phase luctuations ψ(t) and to deodulator oscillator phase luctuations ψ (t) are shown in Fig. 7. Siulation results are copared with calculations according to (7). Fig. 7. Siulated and calculated requency doain SSB phase noise courses. Ansot Designer siulator enables the direct connection o a coplex AWGN channel. The ain paraeter characterizing the aount o added noise is the signal to noise ratio SNR. The AWGN channel odel easures the power o the input useul signal and, according to SNR, it generates the noise signal n(t) with the power N (equations (5) and (6)). The AWGN block output is the signal degraded by an additive noise. The paraeter SNR is set to SNR = 7 db. ilter designation 3 db bandwidth noise bandwidth BN noise bandwidth BN / * Hz Hz Hz band-pass ilters ,19 160, ,15 180, ,18 10, ,18 50, ,19 300, ,17 400, ,0 500, ,18 650, ,17 750, ,17 850, ,0 1000,60 low-pass ilter BN L BN L L ,59 160,59 * the noise bandwidth o an equivalent low-pass ilter expression used or a band-pass ilters siulation Tab. 3. A characterization o FIR ilters used in siulations. In a passband part (between points and 3, in Fig. 1), or a iltration, a bank o band-pass ilters is created. As was written in section 3, they are coposed as low-pass FIR ilters. Filters are characterized by a vector o ilter coeicients []. The advantage o FIR ilters usage is a constant group delay and a aster siulation processing. Their disadvantage is a need o high ilter order to reach or the sharp steepness o the ilter transer unction. For a inal signal iltration (between points 6 and 7, in Fig. 1), a low-pass FIR ilter is used. The order o all ilters in the syste odel is set to Bandwidths o individual ilters calculated according to [5], [10] are suarized in Tab. 3. The ollowing paragraphs suarize siulation results copared with theoretical presuptions and calculations. The ain concern is devoted to the investigation o the ilter bandwidth reduction inluence on the noise behavior. Results are expressed graphically. X axis shows the noise bandwidth change and y axis represents the noise power (in dbw). Transission channel point In this point o the syste odel, no ilters are used so ar. There is no noise dependency on the ilter bandwidth. Ater substituting o all quantities ro Tab. 3 into equation (10), theoretical noise powers are obtained (see Tab. 4). (An abbreviation N eans the phase noise.) Tab. 4. Theoretical versus siulation results in point. Receiver passband part point 3 The irst part o the receiver odel (see Fig. 1) is the band-pass ilter peroring the noise reduction. Ater calculating the ilter inluence on both noise types (equations (11) - (15)), the noise power dependency on the noise bandwidth change is depicted in Fig. 8 (both theoretical and siulation results are covered). A nuerical exaple or the ilter 1 is suarized in Tab. 5. point 3 BN [Hz] N(3) [dbw] noise power N () noise type AWGN only N only AWGN & N unit dbw dbw dbw theory -6,993-3,649-5,949 siulations -6,993-3,65-5,950 conditions ψ(t) = 0 n(t) = 0 ψ(t) 0, n(t) 0 N_siul AWGN_siul AWGN&N_siul N_theory AWGN_theory AWGN&N_theory Fig. 8. The noise bandwidth inluence on the noise power in the passband part o the syste point 3. noise power N (3) noise type AWGN only N only AWGN & N unit dbw dbw dbw theory -38,049-36,180-34,004 siulations -38,056-36,17-34,033 conditions ψ(t) = 0 n(t) = 0 ψ(t) 0, n(t) 0 Tab. 5. Theoretical versus siulation results in point 3, or a ilter 1.

8 RADIOENGINEERING, VOL. 19, NO. 4, DECEMBER Receiver baseband part point 6 The passband signal is deodulated to the baseband with a help o the deodulator oscillator that is distorted by the phase noise (proportional to phase luctuations ψ (t)). Using the description in section (equations (1) - (15) and (18)), the noise power dependency on the noise bandwidth change is calculated. Coparisons between both theoretical and siulation results are shown in Fig. 9. point 6 N(6) [dbw] BN [Hz] N_siul AWGN_siul AWGN&N_siul N_theory AWGN_theory AWGN&N_theory Fig. 9. The noise bandwidth inluence on the noise power in the baseband part o the syste point 6. Tab. 6 deonstrates dierences between theoretical and siulation values on an exaple o the ilter 1. noise power N (6) noise type AWGN only N only AWGN & N unit dbw dbw dbw theory -38,049-33,394-3,115 siulations -38,089-33,480-3,13 conditions ψ(t) = 0, ψ (t) = 0 n(t) = 0 ψ(t) 0, ψ (t) 0, n(t) 0 Tab. 6. Theoretical versus siulation results in point 6, or a ilter 1. Receiver inal signal point 7 The inal deodulated signal in the baseband is created ater the L iltration. The results calculated according to (19) and the siulation results iplying ro the lowpass iltration are depicted in Fig. 10. point 7 BN [Hz] N(7) [dbw] N_siul AWGN_siul AWGN&N_siul N_theory AWGN_theory AWGN&N_theory Fig. 10. The noise bandwidth inluence on the noise power o the inal deodulated signal point 7. When the low-pass ilter noise bandwidth is equal to or is less than a hal o the noise bandwidth o the bandpass ilter, then the inal noise curves are constant. This presuption is conired by siulations (see Fig. 10). The aount o the inal noise is ostly given by a requency band speciied by the low-pass ilter, no atter how the band-pass ilter bandwidth is. Again, as an exaple, theoretical and siulation nuerical values or the ilter 1 case are suarized in Tab. 7. noise power N (7) noise type AWGN only N only AWGN & N unit dbw dbw dbw theory -38,049-34,739-33,076 siulations -38,095-34,757-33,131 conditions ψ(t) = 0, ψ (t) = 0 n(t) = 0 ψ(t) 0, ψ (t) 0, n(t) 0 Tab. 7. Theoretical versus siulation results in point 7, or a ilter 1. Fro previous graphical results, the ilter bandwidth inluence on the aount o the additive theral noise and the ultiplicative phase noise is obvious. Fig. 8 and Fig. 9 show the boundary o the ilter noise bandwidth, when the power o the additive noise equals to the power o the phase noise. The aount o the additive noise power decreases linearly with the descending ilter bandwidth (in Fig. 8 and Fig. 9, courses are not linear because o a logarithic expression o power levels on y axis). The phase noise power decreases very slowly with the reducing ilter bandwidth. In Fig. 8, or the received passband signal, the boundary ilter bandwidth is approxiately 1000 Hz. In the baseband, ater the deodulation, this bandwidth boundary even increases to approxiately 1500 Hz. The ilter bandwidth reduction below these boundary bandwidths causes an increase o the phase noise power portion in the inal noise power very expressively (over the additive noise power portion). I the useul haronic signal requency is x = 150 Hz, a hal o the band-pass ilter bandwidth can be e.g. 1.5 x. According to siulation results, in this band, the ultiplicative phase noise has uch higher portion than the additive noise. In the inal baseband signal, ater the low-pass iltration, the phase noise inluence predoinates. Even i powers o the individual oscillators phase noise are uch lower than the power o the channel additive noise, the inal noise ratios are exactly opposite. The solution leading to iproveents o syste noise paraeters lies in the reduction o the syste oscillators phase noise. A reark on Ansot Designer phase noise results As can be seen in Fig. 8 and Fig. 9, courses o the phase noise power show a quite sharp step down when decreasing the band-pass ilter noise bandwidth. The sharp all-down ends on the requency approxiately 500 Hz. This value iplies ro the ollowing consideration. The Ansot Designer phase noise generation is restricted by the inial start oset requency Δ = 100 Hz. Another act is the odulating signal requency x = 150 Hz. Suing this two values (on the one side o the odulated signal spectru) and doubling the result (lower and upper requency coponents), one obtains the desired value 500 Hz. This consideration is suggested in Fig. 11.

9 66 O. BARAN, M. KASAL, MODELING OF THE SIMULTANEOUS INFLUENCE OF THE THERMAL NOISE AND THE HASE NOISE The research described in this paper shows the iportance o the phase noise especially in space counication systes, where the narrowband counication with space probes is exploited. The siple counication syste odel is designed and the siultaneous eect o an additive theral noise (siulated as AWGN in a counication channel) and a ultiplicative phase noise o oscillators is observed, while the receiver ilter bandwidth is being decreased. The ain results give noise powers in iportant syste points. The siulation is run three ties, irstly only with the additive noise, then only with the phase noise and inally with an inluence o both noise types. Fro the obtained noise powers, the ilter bandwidth boundary, where powers o the additive noise and the phase noise are equal, can be ound. Coparing the useul signal requency with the ilter bandwidth boundary, one can ind out both the powers o both noise types and which noise type is predoinating. The deep ocus is devoted to descriptions o the phase noise and the additive noise ater the iltration and the repeated iltration. The obtained siulation results are in a very good coincidence with presuptions and theoretical calculations that are ade or all siulation cases. The presented siple siulation procedure provides general conclusions that can be used in extended and uch coplicated odels. In our uture work, the current odel will be extended. The input haronic signal will be replaced by the rando bit generator and the siultaneous inluence o both noise types on the deodulated signal BER will be observed. This odel will siulate the SK narrowband low rate counication. Other extensions can include a ixing o odulated signals to icrowave requency bands, where the phase noise o other oscillators has to be counted in. Acknowledgeent Fig. 11. An explanation o steps in phase noise power courses. I a hal o the ilter noise bandwidth BN / is less than the su Δ + x, the phase noise power is alost the sae (see Fig. 11, BN A, only the noise part N A around odulated coponents is included). But ater increasing BN / over the su Δ + x (see Fig. 11, BN B ), the phase noise power abruptly increases because reaining phase noise sidebands (see Fig. 11, denoted as N B parts) start to be included. Theoretically, i Ansot Designer hadn t the restriction, the course (in Fig. 8 and Fig. 9) o the phase noise power wouldn t decrease so rapidly, no sharp steps would be presented. 5. Conclusion The research leading to these results has received unding ro the European Counity's Seventh Fraework rograe (F7/ ) under grant agreeent no The research described in the paper was also supported by the Czech Grant Agency under the grant no. 10/10/1853 Advanced Microwave Coponents or Satellite Counication Systes, the grant no. 10/08/H07 "Advanced Methods, Structures and Coponents o Electronic Wireless Counication" and by the research progra MSM "Advanced Electronic Counication Systes and Technologies (ELCOM)". Reerences [1] ŠAČEK, J., KASAL, M. The low rate teleetry transission siulator. Radioegineering, 007, vol. 16, no. 4, p [] ROAKIS, J. G. Digital Counications (4 th Edition). New York: McGraw-Hill, 001. p [3] LEE, T. H., HAJIMIRI, A. Oscillator phase noise: A tutorial. IEEE Journal o Solid-State Circuits, 000, vol. 35, no. 3, p [4] MEYR, H., MOENECLAEY, M., FECHTEL, S. A. Digital Counication Receivers: Synchronization, Channel Estiation and Signal rocessing. New Jersey: John Willey & Sons, [5] KASAL, M. Radio Relay and Satellite Counication. Lecture notes. Brno: MJ Servis, 003. ISBN (In Czech.) [6] BARAN, O., KASAL, M. Modeling o the phase noise in space counication systes. Radioengineering, 010, vol. 19, no. 1, p [7] VASILESCU, G. Electronic Noise and Interering Signals: rincipals and Applications. Berlin: Springer, pages. [8] ROBINS, W.. hase Noise in Signal Sources. London: eter eregrinus Ltd, 007. ISBN: [9] RILEY, J. Handbook o Frequency Stability Analysis. Beauort: Hailton Technical Services, 007. [Online] Cited Available at: [10] CHANG, K. Encyclopedia o RF and Microwave Engineering. New Jersey: John Willey & Sons, 005. About Authors... Ondřej BARAN, Miroslav KASAL or biography see Radioengineering, April 010, vol. 19, no. 1, p. 148.