Probability Density Functions of Envelope and Phase of the Sum of a PSK Modulated Carrier and Narrowband Gaussian Noise

Transcription

1 robabiliy Desiy Fuios of Eveloe a hase of he Su of a SK oulae Carrier a Narrowba Gaussia Noise Auhor: Lohar Frieerihs AUDENS Teleouiaios Cosulig Dae: 6.6. wih orreios a eesios of Coes age. Soe. Uoulae Carrier lus Narrowba Gaussia Noise DF of Aliue Eveloe DF of he hase 5 Coleeary Cuulaive robabiliy Disribuio Fuio 6 CCDF of hase Differee ary SK Carrier lus Narrowba Gaussia Noise ary SK Carrier lus Narrowba Gaussia Noise wih Ia of Frequey Offse 5. Uoulae Carrier lus Narrowba Gaussia Noise wih Ia of Frequey Offse 5 6. Referees 6 The orreios oer oly soe eails i he rasiio fro equaio -8 o equaio -9 o he e resul. The eesios iroue Coleeary Cuulaive Disribuio Fuios as useful aliaios of he hase FD. SKEvhDF_rev age

2 . Soe I ay aliaios i is of ieres o have a aalyial eressio for he robabiliy Desiy Fuios DF of he eveloe a he hase of he su of a -ary SK oulae or a uoulae e wave e.g. 4 8 a aiive arrowba Gaussia oise uer he oiios of ohere or o-ohere reeio i.e. iluig he ia of arrier frequey off-se. The wae DF eressios are erive here alog he lies show e.g. i ref. [] or [] for he ase of a uoulae arrier lus arrowba Gaussia oise. As a ase of ariular ieres he hase esiy fuio of a uoulae arrier lus NB Gaussia oise has bee alie o rese Coleeary Cuulaive Disribuio Fuios of he hase a hase ifferee. Graeful akowlegee is ue o Wolfgag Seier who siulae he ivesigaios erfore a ofire he orreess of he aalyial resuls by rigorous siulaios ru uer ATLAB.. Uoulae Carrier lus Narrowba Gaussia Noise [ ] DF of Aliue Eveloe The arrier shall have osa bu uiforly isribue hase ieee of he oise roess reresee by a sale fuio os. Here a s are ieee Gaussia variables wih zero ea a variae ². The su of arrier a oise is y os[ ] os s - or y os[ ] - os[ ] os [ ] wih a s [ ] os os -3a [ ]. -3b s Sie s a are sasially ieee he joi robabiliy esiy fuio of rereseig he ossible values of s rereseig he ossible values of s a a be faorise io he gle DFs of s a. The rile of rao variables s a is rasfore by equaio -3 io he rile of rao variables a where a rerese he ossible values of a. The rasforaio eeria "Jaobia" is s s s [ ] [ ] [ ] os[ ] [ ] os[ ] J. os The joi robabiliy esiy fuio of a is ow s SKEvhDF_rev age

3 SKEvhDF_rev age 3 [ ] s s os os os e 4 ; os e e 4-4 By iegraio of equ. -4 over he joi robabiliy esiy fuio of isaaeous aliue or eveloe a hase is obaie. ; os e e -5 The DF of he eveloe of he su of a uoulae arrier a arrowba Gaussia oise is ow eerie by iegraig equ. -5 wih rese o : { } os e e os e -6 Ug he Jaobi-Ager relaio > for for where os os e z I z ε θ ε θ where I eoes he oifie Bessel fuio of firs ki he iegral i equ. -6 beoes os os e I I. I is obvious ha he iegrals o he righ-ha sie for will vaish. Thus he DF of he eveloe of he su of a uoulae arrier a arrowba Gaussia oise is e I -7

4 Equ. -7 esribes a Rie robabiliy esiy fuio. I is illusrae for various CN raios i Fig. - wih oralisaio o arrier aliue a i Fig. - wih oralisaio o he oise saar eviaio. For i.e. ure oise he Rie DF aroahes he Rayleigh DF. 6 5 CN DF of Nor. Eveloe Carr. Al. 4 3 B 5 B B 5 B B -5 B Noralize Aliue Fig. -: DF of he Eveloe of a Uoulae Carrier lus Gaussia Noise Noralisaio o Carrier Aliue.7.6 CN - B DF of Nor. Eveloe siga siga siga B B 5 B B 5 B Eveloe Noralize o Noise Saar Deviaio siga Fig. -: DF of he Eveloe of a Uoulae Carrier lus Gaussia Noise Noralisaio o Noise Saar Deviaio SKEvhDF_rev age 4

5 DF of he hase Siilarly he DF of he hase of he su of a uoulae arrier a arrowba Gaussia oise is eerie by iegraig equ. -5 wih rese o : os The iegral o he righ-ha sie i equ. -8 is solve by subsiuig os u. The iegral he beoes: e e -8 os e os os [ u os ] e [ u ] u os os erf e The FD of he hase of a uoulae arrier a arrowba Gaussia oise for a give raio a is ow: a { e a a os e[ a ] erf[ a os ]} -9 Fig. -3 gives a illusraio of a for various raios of C N log a. 6 DF of Uo. Carrier lus NB Gaussia Noise CN rob. Desiy Fuio B 5 B B 5 B B B - B ra Fig. -3: DF of he hase of a Uoulae Carrier lus Narrowba Gaussia Noise SKEvhDF_rev age 5

6 Two liiig ases are of ieres: Gaussia oise oly: li ; - is equivale o C N B a he robabiliy esiy ges reagular. Carrier wihou oise: li for δ elsewhere The robabiliy esiy reues o a gle lie. wih li Coleeary Cuulaive robabiliy Disribuio Fuio I is of soe ieres o look a he Coleeary Cuulaive robabiliy Disribuio Fuio CCDF of he hase of a uoulae arrier lus arrowba Gaussia oise. The CCDF is obaie fro ϕ CCDF ϕ a a wih a fro equ. -9. The iegral ao be solve i lose for. Therefore i has bee evaluae uerially for a se of CN values. Fig. -4a shows he resulig CCDF urves. While he rereseaio i liear sale for he robabiliy sees o ake a quie oral shae for ϕ values lose o he seifi haraerisis beoe evie fro he logarihi rereseaio i Fig. -4b. Thus he CCDF is well suie o reveal he ourree of eree values lose o CN B 5 B B B 7 B 3 B B - B CCDF ra Fig. -4a: CCDF of he hase of a Uoulae Carrier lus Narrowba Gaussia Noise Liear Sale SKEvhDF_rev age 6

7 CN - -4 CCDF -6-8 B 5 B B B 7 B 3 B B - B ra Fig. -4b: CCDF of he hase of a Uoulae Carrier lus Narrowba Gaussia Noise Logarihi Sale CCDF of hase Differee The relaio for he hase of a uoulae arrier lus Gaussia oise is ariularly useful i he aalysis of ilo assise frequey syhroisaio ehiques base o hase esiaio. Here he isribuio of he absolue value of he hase ifferee Φ of wo hasors is of iorae. Sie he wo hasors are saisially ieee he DF of he ifferee of heir hases resuls fro he ovoluio of he robabiliy esiy fuio aorig o equ. -9 wih iself: Φ. a Φ a Φ a where Φ. - D The wo-sie CCDF of he hase ifferee Ψ CCDFD Ψ a D Φ a Φ where Ψ. - The oe-sie CCDF of he absolue value of he hase ifferee is ow Ψ Ψ a D Φ a CCDFD Φ wih Ψ. - The resuls of uerial evaluaios of equ. - are show i Fig. -5a i liear sale a i Fig. -5b i logarihi sale for he sae se of CN values as above. Agai he logarihi sale i Fig. -5b reveals he ourree of eree values lose o wih raher high robabiliy. SKEvhDF_rev age 7

8 CN B 5 B B B 7 B 3 B B - B.6 CCDF D ra Fig. -5a: CCDF of he hase Differee of wo hasors Rereseig a Uoulae Carrier lus Narrowba Gaussia Noise Liear Sale ^ CN - B ^- B 3 B 7 B ^-4 CCDF D B ^-6 B ^-8 5 B B ß^ ra Fig -5b: CCDF of he hase Differee of wo hasors Rereseig a Uoulae Carrier lus Narrowba Gaussia Noise Logarihi Sale SKEvhDF_rev age 8

9 SKEvhDF_rev age ary SK Carrier lus Narrowba Gaussia Noise I eesio of seio he oulae arrier a be wrie as [ ] os y wih he wo ieee rao variables [uiforly isribue i ] a he oulaio hase whih raoly akes ay of he values -... a µ wih µ N. δ. The -ary SK arrier lus arrowba Gaussia oise is reresee by [ ] y s os os 3- or [ ] [ ] [ ] y os os os 3- wih [ ] os a [ ] s s. The rao variables a s are all saisially ieee. Therefore he joi DF is he rou of he gle-variable DFs: s s e δ 3-3 Wih he relaios [ ] [ ] os os [ ] [ ] s he joi DF s is rasfore io he joi DF. The rasforaio eeria is s s s s s s s s D [ ] [ ] [ ] [ ] s s D os os The we have for :

10 SKEvhDF_rev age [ ] [ ] os os e δ [ ] os e δ 3-4 Iegraig equ. 3-4 wih rese o a yiels he joi DF of a : 3 [ ] δ os e 3 os e e 3-5 v os e e where. By oariso of he las for of 3 wih equ. -5 i is see ha eah of he righha ers orresos ealy o ies he joi DF of aliue a hase of a uoulae arrier lus arrowba Gaussia oise bu shife i hase by -. Thus he joi DF of aliue a hase of a -ary SK oulae arrier lus arrowba Gaussia oise is obaie as he -fol suerosiio of he joi DF of aliue a hase of a uoulae arrier lus Gaussia oise he hase beig saggere i ses of a he aliue weighe by. Beause of he relaio os e I for all... he DF of he aliue of a -ary SK oulae arrier lus arrowba Gaussia oise is ieial o ha of a uoulae arrier lus Gaussia oise equ. -7.

11 The DF of he hase follows fro iegraig equ. 3-5 wih rese o : e e v os Ug he oaio a as i equ. -9 his beoes a e a a os e a erf a os 3-6 wih. The DF of a 4-SK oulae arrier lus arrowba Gaussia oise is illusrae i Fig DF of he hase 4-SK Carrier lu Noise CN B hi ra Fig. 3-: DF of he hase of a 4-SK oulae Carrier lus Gaussia Noise SKEvhDF_rev age

12 SKEvhDF_rev age 4. -ary SK Carrier lus Narrowba Gaussia Noise wih Ia of Frequey Offse Whe a -ary SK arrier is reeive by a o-ohere reeiver he eoulaor will aly a esiae arrier frequey o he sigal whih i geeral oes o oiie wih he ea arrier frequey bu has soe offse f. The -ary SK arrier a he aiive Gaussia oise a he be reresee by [ ] y s os os 4- where f. All oher eries i equ. 4- have he sae eaig as i equ. 3- above. Followig he sae roeure as i seio 3 we obai ow [ ] [ ] [ ] os os os a [ ] [ ] [ ] s os os. The joi DF of he variables a s is agai give by equ. 3-3 bu ow we have s os os a hus s δ [ ] os os e. The joi DF of he variables he is [ ] os e δ 4- The joi aliue a hase DF is obaie fro iegraio wih rese o a : 3 os e e

13 SKEvhDF_rev age 3 3 os e e 4-3 Iegraio of equ. 4-3 wih rese o leas o he aliue DF wih osieraio of Frequey offse. The resul is he sae as give i equ. -7 for a uoulae arrier lus arrowba Gaussia oise. The hase DF is obaie by iegraig equ. 4-3 wih rese o. The iegraio follows he sae eho as use i equ. -8 a equ Due o he frequey offse he hase DF is ow also eee o ie. The resul is: os e os erf e 4-4 Soe rereseaive ases are illusrae i Figures 4- a hrough. Fig. 4-: robabiliy Desiy Fuio of he hase of a 4-SK Carrier lus NB Gaussia Noise wih Ia of Frequey Offse a CN B a f Hz b CN B a f Hz CN B a f Hz

14 Fig. 4- a Fig. 4- b SKEvhDF_rev age 4

15 Fig Uoulae Carrier lus Narrowba Gaussia Noise wih Ia of Frequey Offse The hase DF forula aorig o equ alhough erive for -ary SK arriers wih aiive arrowba Gaussia oise - a be use also o oue he DF of he hase of a uoulae arrier lus arrowba Gaussia oise by forally seig a resriig o. This ase is of ieres whe osierig he oi of arrier syhroisaio i ouiaio reeivers. A eale is show i Fig. 5-. SKEvhDF_rev age 5

16 Fig. 5-: robabiliy Desiy Fuio of he hase of a Uoulae Carrier lus Narrowba Gaussia Noise wih Ia of Frequey Offse 6. Referees [] W.B. Daveor W.L. Roo: A Irouio o he Theory of Rao Sigals a Noise; Graw-Hill New York Toroo Loo 958 [] J.G. roakis: Digial Couiaios; Graw-Hill Iera. Eiios 995 SKEvhDF_rev age 6