Performance Analysis for Multichannel Reception of OOFSK Signaling

Performance Analysis for Multic hannel Reception of OOFSK Signaling Qingyun W ang Mustafa Cenk Gursoy Departmen t of E lectrical En gineerin g University of Neb raska-Linco ln, Lincoln, NE 68588 Email: q wang4@bigred .unl.edu , gursoy@unl.ed u Abstract — 1 In this paper , th e erro r performance of on- off frequency shift ke ying (OOFSK) modulation ov er fadi ng channels is analyzed when t he recei ve r is equipped with multiple antennas. The analysis is conducted in two cases: the c oherent scenario where the fading is perfectly known at the receiv er , and the noncoherent scenario where neither the receiv er nor the transmitter knows the fading coefficients. For both cases, the maximum a p osteriori probability (M AP) detection rule is derived and an alytical probability of error expressions are obtained. T he effect of fading correlation among the recei ver antennas is also studied . Simulation re sults indi cate that f or sufficiently low duty cycle values, lower probability of erro r values with r espect to FSK signaling ar e achiev ed. Equivalently , when compar ed to FS K modulation , OOFS K with low duty cycle requires less energy to achieve the same p robability of error , which renders th is modulation a more energy efficien t transmission techni que. I . I N T RO D U C T I O N Frequency-sh ift k eying (FSK) is a modu lation forma t that is well-known a nd well-stud ied in the co mmunic ations liter ature [15]. FSK is an attractive transmission scheme d ue to its hig h energy efficiency and suitability for non coheren t commu nica- tions. In un known channel conditions, energy detection can be emp loyed to detec t the FSK signals. Indeed, the analysis of FSK mod ulation da tes back to 1960 s (see e.g., [1], and [2]). Recen tly , it has been shown in [5] that unless th e c hannel condition s are perfectly known at the recei ver , signals that ha ve very high peak-to-average power ratio is r equired to achieve the capacity in the lo w SNR regime. This has initiated work on p eaky sign aling. Lu o an d M ´ edard [6] h av e shown th at FSK with small duty cycle can achieve rates of the order of capacity in ultrawideba nd systems with limits on bandwidth and p eak power . In [8], the au thors have studied the error perf ormanc e of peaky FSK signalin g over multipath fading channe ls by obtaining upper and lower bo unds o n th e e rror proba bility . In [9 ], on -off frequency-shif t keying (OOFSK) is defin ed as FSK overlaid on on -off keying, and its c apacity and energy efficiency is an alyzed. Note that OOFSK c an be seen a s joint pulse po sition m odulation (PPM) and FSK. In th is signaling , peakedness is introduced in both time and frequ ency . The erro r perfor mance of O OFSK signaling when the tran smitter and receiver are each equipped with a single anten na is recently studied in [10]. 1 This work was supported in part by the NSF CAREER Grant CCF- 0546384. One of the important techniques to improve the perf ormanc e in wireless co mmun ications is to use multiple antenn as to achieve diversity gain . Considerable amo unt o f work h as been done on multiple reception channels. In [2], it is shown fo r binary and M -ar y signaling over Rician fading chan nels that increasing the numb er of reception chan nels can im prove the error perfor mance significantly . By finding the pr obability distribution function of the instantaneous SNR in flat fading multi-recep tion chan nels and s ubstituting it into the p robability of er ror expressions of P AM, PSK and QAM over A WGN channel, the author s in [12] obtained expr essions for the av erage probability of error of multi-reception fading channels. In [14], th e probability of erro r of BPSK over Rician fadin g multi-recep tion channels is given and extensions to o ther modulatio n techn iques are discussed. In [17], average symbo l error rate of selection diversity of M -ary FSK modulated signal tra nsmitted over fading channels is stu died. In this paper, the error perfo rmance of OOFSK over multiple reception Rician fadin g ch annels is studied . I n Section I I, the system m odel is p resented. In Section III, the erro r perfor mance in co herent fading chan nels is stud ied. In Sectio n IV, we invest igate th e error per forman ce in noncoher ent Rician fading chan nels. I I . S Y S T E M M O D E L W e assume that OOFSK m odulatio n is emp loyed at the transmitter to send the inform ation. In OOFSK mod ulation, the transm itted signal du ring the symbol interval 0 ≤ t ≤ T s can b e expressed as s m ( t ) = ( q P v e j ( w m t + θ m ) m = 1 , 2 , 3 , . . . M 0 m = 0 (1) where w m and θ m are the f requen cy in radian s per second and p hase, respectively , o f the signal s m ( t ) when m 6 = 0 . Note th at we h av e M FSK signals and a zero signal denoted by s 0 ( t ) . The fr equencies of the FSK signa ls are cho sen so that the signals are orthog onal. It is assum ed that an FSK signal s m ( t ) , m 6 = 0 , is transmitted with a prob ability of v M while s 0 ( t ) is tran smitted with a proba bility of 1 − v where v is the d uty cycle of transmission. With these defin itions, it is easily seen that P and P v are the a verage an d peak powers, respectively , of the modulatio n technique. The receiver is equip ped with L a ntennas that enab le th e multiple reception of the tra nsmitted signal. If, with out loss of generality , we assume that s k ( t ) is the tra nsmitted sign al, the received signa l a t th e l th antenna is r l ( t ) = h l s k ( t ) + n l ( t ) l = 1 , 2 , . . . , L (2) where h l is the fading coefficient at the l th reception channel and n l ( t ) is a wh ite Gaussian noise with sing le-sided spectral density of N 0 . It is assumed th at the additive Gaussian noise compon ents at different anten nas are in depend ent. Fur ther- more, the received signal mode l (2) p resumes that the fading is freq uency-flat and slo w enoug h so that the fading coefficient stays c onstant over one sym bol dur ation. Follo wing each anten na, there is a bank of M c orrelator s, each co rrelating the received signal with one of the o rthogo nal frequen cies. The output o f the m th correlator employed after the l th antenna is giv en by Y l,m = 1 √ N 0 T s Z T s 0 r l ( t ) e − j w m t dt = ( q P T s vN 0 h l e j θ m + n l,m m = k n l,m m 6 = k =  Ah l e j θ m + n l,m m = k n l,m m 6 = k , (3) where n l,m is a circula rly symm etric complex G aussian ran - dom variable with zero-m ean and a variance of 1 a nd fo r notational conv enience, we have defined A = q P T s vN 0 . Since the f requen cies are o rthog onal and the additive Gaussian no ise is indepen dent at e ach an tenna, { n l,m } for l ∈ { 1 , . . . , L } and m ∈ { 1 , . . . , M } form s an indepen dent and identically distributed (i.i.d. ) seque nce. Note also that R l,m = | Y l,m | 2 giv es th e energy p resent in the m th frequen cy at the l th antenna. I I I . O O F S K O V E R C O H E R E N T F AD I N G C H A N N E L S A. Detection Rule In this section, we assume that tran smission takes place ov er coheren t fading channels and h ence h l for all l is known to th e receiver while the tran smitter d oes not h ave such knowledge. Conditioned on h l and the transmitted sign al s k ( t ) , Y l,m is a pro per complex Gaussian ran dom variable with mean value and variance giv en by E { Y l,m | h l , s k } =  Ah l e j θ k m = k 0 m 6 = k (4) v ar { Y l,m | h l , s k } = 1 . (5) Therefo re, R l,m = | Y l,m | 2 is chi- square d istributed with the following condition al prob ability density function (pdf): f R l,m | | h l | ,s k ( R l,m ) =  e − ( R l,m + A 2 | h l | 2 ) I 0 ` 2 A | h l | p R l,m ´ m = k e − R l,m m 6 = k It is assumed that the receiver , using equ al gain comb ining (EGC), combines the en ergies of the m th frequen cy compo- nents at each anten na, i.e., co mputes the total energy R m = L X l =1 R l,m . (6) Since the no ise compon ents a re indepe ndent, R m is a su m of indepen dent chi-sq uare random v ariables, and is its elf also chi- square distrib uted with 2 L de grees o f fr eedom. The condition al pdf is g iv en by f R m | h ,s k ( R m ) = 8 > < > : “ R m ξ ” L − 1 2 e − ( R m + ξ ) I L − 1 (2 √ R m ξ ) m = k R L − 1 m Γ( L ) e − R m m 6 = k where ξ = P L l =1 A 2 | h l | 2 , h = [ h 1 , . . . , h L ] , I L − 1 ( · ) is the ( L − 1) th order modified Bessel function o f th e first k ind, and Γ( · ) is th e ga mma fun ction. The receiver employs maximu m a posteriori pro bability (MAP) cr iterion to de tect the tran smitted signals. Let R = [ R 1 , R 2 , . . . , R M ] be th e vector of energy values correspo nding to ea ch fre- quency . Sin ce the noise comp onents n l,m are indep endent for different m ∈ { 1 , . . . , M } , compo nents of R are mu tually indepen dent. Hence, the co nditiona l pdf of R is f R | h ,s k ( R ) = 8 < : “ R k ξ ” L − 1 2 e − ( R k + ξ ) I L − 1 (2 √ R k ξ ) Q M n =1 n 6 = k R L − 1 n e − R n Γ( L ) k 6 = 0 1 [Γ( L )] M Q M n =1 R L − 1 n e − R n k = 0 (7) Then, th e MAP r ule th at detects s k for k 6 = 0 is  f R | h ,s k > f R | h ,s m ∀ m 6 = 0 , k f R | h ,s k > M (1 − v ) v f R | h ,s 0 (8) where w e have used th e fact that the p rior p robab ilities o f the tr ansmitted signa ls are p ( s m ) = v M for m 6 = 0 , and p ( s 0 ) = (1 − v ) . Substituting (7) in to to ( 8), the decision rule is simplified to : ( g 1 ( R k ) > g 1 ( R m ) ∀ m 6 = k g 1 ( R k ) > M (1 − v ) e ξ ξ L − 1 2 v ( L − 1)! (9) where g 1 ( R k ) = R − L − 1 2 k I L − 1 (2 p R k ξ ) , ξ > 0 . (10) The following Le mma enab les us to further simplif y the detection ru le. Lemma 1 : T he f unction g 1 ( x ) = x − L − 1 2 I L − 1 (2 p xξ ) for x > 0 , ξ > 0 (11) is a mo noton ically increasing fu nction o f x . Pr oof : The der iv ative of the n th order modified Bessel function is dI n ( x ) dx = I n +1 ( x ) + n x I n ( x ) . (12) Hence, dI L − 1 (2 √ xξ ) dx = r ξ x I L (2 p xξ ) + L − 1 2 x I L − 1 (2 p xξ ) > L − 1 2 x I L − 1 (2 p xξ ) (13) where we use the fact th at q ξ x I L (2 √ xξ ) > 0 f or x > 0 . Then, th e deriv ati ve of g 1 ( · ) satisfies dg 1 ( x ) dx = − L − 1 2 x − L +1 2 I L − 1 (2 p xξ ) + x − L − 1 2 dI L − 1 (2 √ xξ ) dx > − L − 1 2 x − L +1 2 I L − 1 (2 p xξ ) + x − L − 1 2 L − 1 2 x I L − 1 (2 p xξ ) = 0 , proving that g 1 ( x ) is a m onoton ically increasing fun ction of x > 0 .  By the above result, the detection rule (9) further simplifies to  R k > R m ∀ m 6 = k R k > g − 1 1 ( T ) (14) where T = M (1 − v ) e ξ ξ L − 1 2 v ( L − 1)! . Since g 1 ( · ) is a monoto nically increasing function , the in verse g − 1 1 ( · ) is well-defined . Note that (14) is the rule that d etects th e sig nal s k ( t ) for k 6 = 0 . The z ero sig nal s 0 ( t ) is d etected if R k < g − 1 1 ( T ) ∀ k. (15) B. Pr oba bility of Err or In this section, we ana lyze the error proba bility of O OFSK modulatio n when MAP d etection is u sed at the re ceiv er . Suppose with out loss o f g enerality that s 1 ( t ) is the tran smitted signal. Let τ = g − 1 1 ( T ) . Then the correc t detection p robab ility is P c, 1 = P ( R 1 > R 2 , R 1 > R 3 , . . . , R 1 > R M , R 1 > τ | s 1 ) = Z ∞ τ  Z x 0 f R 2 | h ,s 1 ( t ) dt  M − 1 f R 1 | h ,s 1 ( x ) dx = Z ∞ τ  Z x 0 t L − 1 Γ( L ) e − t dt  M − 1 f R 1 | h ,s 1 ( x ) dx. From [15], we have Z x 0 1 Γ( L ) t L − 1 e − t dt = 1 − e − x L − 1 X l =0 x l l ! (16) Therefo re, the correct detection pro bability can now b e ex- pressed as P c, 1 = Z ∞ τ " 1 − e − x L − 1 X l =0 x l l ! # M − 1 f R 1 | h ,s 1 ( x ) dx Using th e binomial the orem, P c, 1 becomes P c, 1 = Z ∞ τ M − 1 X n =0 ( − 1) n  M − 1 n  " L − 1 X l =0 x l l ! e − x # n f R 1 | h ,s 1 ( x ) dx Using the m ultinomial theorem , w e have the f ollowing expan- sion " L − 1 X l =0 x l l ! e − x # n = e − nx n ( L − 1) X i =0 c in x i (17) where c in is th e coefficient of x i in the expansion. c in can be ev alu ated from the recursive equation [18] c in = i X q = i − L +1 c q ( n − 1) ( i − q )! I [0 , ( n − 1)( L − 1)] ( q ) (18) where I [ a,b ] ( q ) =  1 , a ≤ q ≤ b 0 , otherwise . (19) Using th e multinomial expa nsion, P c, 1 becomes P c, 1 = M − 1 X n =0 ( − 1) n  M − 1 n  n ( L − 1) X i =0 c in Z ∞ τ x i  x ξ  L − 1 2 × e − [( n +1) x + ξ ] I L − 1 (2 p xξ ) dx (20) Let ξ = a 2 and x = t 2 , then, P c, 1 can be written as P c, 1 = M − 1 X n =0 ( − 1) n  M − 1 n  n ( L − 1) X i =0 c in × Z ∞ √ τ t 2 i e − nt 2  t a  L − 1 e − ( t 2 + a 2 ) I L − 1 (2 at )2 tdt = M − 1 X n =0 ( − 1) n  M − 1 n  n ( L − 1) X i =0 2 c in e a 2 a − ( L − 1) ×  a L − 1 Γ( i + L ) 2( n + 1) i + L Γ( L ) e a 2 n +1 F  − i, L ; − a 2 n + 1  − Z √ τ 0 t 2 i + L e − ( n +1) t 2 I L − 1 (2 at ) dt # (21) where F ( a, c ; x ) is the con fluent hy pergeometr ic function [ 16, Chap 1 0]. The prob ability o f co rrect detection when sig nal s 0 ( t ) is tr ansmitted is: P c, 0 = P ( R 1 < τ , . . . , R M < τ | h , s 0 ) = 1 − e − τ L − 1 X l =0 τ l l ! ! M . (22) Hence, the probab ility of error as a fun ction o f the instanta- neous signal-to-n oise ratio is P e = 1 − ( v P c, 1 + (1 − v ) P c, 0 ) . (23) Since the chann el is assum ed to be known, er ror probab ility in ( 23) is a functio n of the fading coefficients through χ = P L l =1 | h l | 2 . Hen ce, th e average probability of e rror is o btained by c omputin g ¯ P e = Z ∞ 0 P e f χ ( χ ) dχ. (24) If h l is a com plex Gau ssian random variable with mean v alue d l and variance σ 2 and { h l } are mutually independen t, χ is a chi-square rand om variable with 2 L d egrees of f reedom and has a pd f given by f χ ( χ ) = 1 σ 2  χ s 2  L − 1 2 e − χ + s 2 σ 2 I L − 1 2 p χs 2 σ 2 ! (25) 0 1 2 3 4 5 6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR(dB) Probability of error Two Rician channel with known fading coefficients dashed:v=0.2 v=0.5 v=0.8 Conventional FSK Fig. 1. Error probability vs. SNR for 4-OOFSK signaling over two indepen dent coherent Rician fading channels with equal Rician factor K = 1 8 . Duty factor value s are v = 1 , 0 . 8 , 0 . 5 , and 0 . 2 . 0 1 2 3 4 5 6 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 SNR (dB) Probability of error Three Rician channel with known fading coefficients dashed:v=0.2 v=0.5 v=0.8 conventional FSK Fig. 2. Error probabili ty vs. SNR for 4-OOFSK signaling over three indepen dent coherent Rician fading channels with equal Rician factor K = 1 8 . Duty factor value s are v = 1 , 0 . 8 , 0 . 5 , and 0 . 2 . where, s 2 = P L l =1 | d l | 2 . When the fading coefficients { h l } are corr elated, th e av- erage error proba bility ¯ P e can be obtain ed by evaluating the expected value of P e with r espect to the join t d istribution o f ( | h 1 | , . . . , | h L | ) , wh ich inv olves L -f old integration. However , if {| h l |} are Nakagami- m distributed, closed-fo rm expressions for f χ ( χ ) are p rovided in [ 19], which lead to a single in tegra- tion. Next, we present the simulation results. W e define the Rician factor a s K = | d | 2 σ 2 and correlatio n coefficient as ρ = cov ( h i ,h j ) √ var ( h i ) var ( h j ) . Figur es 1, 2, and 3 plot the proba bility of er ror curves as a function of SNR f or 4-OOFSK signalin g over Rician fading ch annels with d ifferent numb er of receiver antennas and d ifferent duty factor s. T wo indep endent ch annels are considered in Fig . 1. No te th at co n ventional FSK c orre- 0 1 2 3 4 5 6 10 −3 10 −2 10 −1 10 0 SNR (dB) Probability of error Two correlated Rician channel with known fading coefficients v=0.2 v=0.5 v=0.8 Conventional FSK Fig. 3. E rror probabilit y vs. S NR for 4-OOFSK signali ng ov er two correlat ed coheren t Ricia n fading channels with equal Rician fact or K = 1 8 and correla tion coef ficient ρ = 1 4 . Duty factor value s are v = 1 , 0 . 8 , 0 . 5 , and 0 . 2 . sponds to OOFSK with d uty factor v = 1 . In Fig. 2, three indepen dent channels are assumed. In both figur es, we observe an impr ovement in the error pr obability cu rves if the du ty factor v of OOFSK signaling is less th an 0 .5. When v = 0 . 2 , we see appr oximately an order o f magn itude im provement in the error perfo rmance. This results in substantial ene rgy gains fo r fixed value of error prob ability , renderin g OOFSK signaling a very energy efficient transmission techniq ue. It can also immediately be noted th at due to increased d iv ersity , having more an tennas decreases th e err or rates. I n Fig. 3, the case of two correlated c hannels is in vestigated . Altho ugh the perfor mance is deterio rated due to correlation , OOFSK with sufficiently small duty factor still con siderably impr oves th e error per forman ce. It should b e no ted that ha ving small duty factor means that FSK signals ha ve high peak p ower but they are tran smitted less f requen tly to satisfy the average p ower constraint. Consequen tly , having high p eak power signals decrease the err or rates. I V . O O F S K O V E R N O N C O H E R E N T F A D I N G C H A N N E L S A. Detection Rule In the non coheren t channe l case, we assume that the realiza- tions of th e fading c oefficients { h l } are unkn own at bo th th e receiver and transmitter . The r eceiver is only equipp ed with the kn owledge of th e statistics of { h l } . W e fur ther assume that { h l } are i.i.d. complex Gaussian r andom variables with E { h l } = d l and v ar { h l } = σ 2 . Therefo re, condition ed on s k ( t ) bein g the transmitted sign al, Y l,m is a com plex Gaussian random variable with E { Y l,m | s k } =  Ad l e j θ k m = k 0 m 6 = k , v ar { Y l,m | s k } =  A 2 σ 2 + 1 m = k 1 m 6 = k . Similarly as in the coheren t case, we combine the energies of the m th frequen cy componen ts across the antennas, and obtain R m = P L l =1 R m,l . Condition ed on transmitted signal s k ( t ) , R m is a chi-squar e random variable with the fo llowing p df: f R m | s k ( R m ) =    1 σ 2 y  R m ξ  L − 1 2 e − R m + ξ σ 2 y I L − 1  2 √ R m ξ σ 2 y  m = k R L − 1 m Γ( L ) e − R m m 6 = k (26) where ξ = A 2 P L l =1 | d l | 2 and σ 2 y = A 2 σ 2 + 1 . Th e vector R = [ R 1 , . . . , R M ] h as the fo llowing conditiona l joint p df f R | s k ( R ) = 8 > < > : 1 σ 2 y “ R k ξ ” L − 1 2 e − R k + ξ σ 2 y I L − 1 „ 2 √ R k ξ σ 2 y « Q M n =1 n 6 = k R L − 1 n e − R n Γ( L ) k 6 = 0 1 [Γ( L )] M Q M n =1 R L − 1 n e − R n k = 0 The M AP d ecision rule th at detects s k for k 6 = 0 is  f R | s k > f R | s m ∀ m 6 = 0 , k f R | s k > M (1 − v ) v f R | s 0 (27) Similarly a s in Sectio n II I, it can be easily shown that g 2 ( R k ) = R − L − 1 2 k e R k A 2 σ 2 σ 2 y I L − 1  2 √ R k ξ σ 2 y  , ξ > 0 (28) is a mon otonically increasing function . With this observation, the decision ru le in (2 7) simplifies to  R k > R m ∀ m 6 = k R k > g − 1 2 ( T 2 ) (29) where T 2 = M (1 − v ) σ 2 y ξ L − 1 2 e ξ σ 2 y v Γ( L ) . Note that s 0 is the dete cted signal if R k < g − 1 2 ( T 2 ) f or a ll k . B. Pr oba bility of Err or W e first assum e tha t s 1 ( t ) is tr ansmitted. Let τ 2 = g − 1 2 ( T 2 ) . Then, th e probab ility of correct dete ction is P c, 1 = P ( R 2 > R 1 , R 3 > R 1 , . . . , R M > R 1 , R 1 > τ 2 | s 1 ) (30) Follo wing an ap proach similar to th at in Section III, we have P c, 1 = M − 1 X n =0 ( − 1) n  M − 1 n  n ( L − 1) X i =0 c in Z ∞ τ 2 x i e − nx f R 1 | s 1 ( x ) dx = M − 1 X n =0 ( − 1) n  M − 1 n  n ( L − 1) X i =0 c in Z ∞ τ 2 x i e − nx × 1 σ 2 y  x ξ  L − 1 2 e − x + ξ σ 2 y I L − 1  2 √ xξ σ 2 y  dx = M − 1 X n =0 ( − 1) n  M − 1 n  n ( L − 1) X i =0 c in ξ − L − 1 2 e − ξ σ 2 y σ 2 y × " ξ L − 1 2 ( i + L )! 2(1 + nσ 2 y ) i + L 2 σ L − 2 − i y L ! F  − i, L, ξ σ 2 y (1 + nσ 2 y )  × e ξ σ 2 y (1+ nσ 2 y ) − Z τ 2 0 x 2 i + L − 1 2 e − 1+ nσ 2 y σ 2 y x I L − 1  2 √ xξ σ 2 y  dx # 0 1 2 3 4 5 6 10 −4 10 −3 10 −2 10 −1 10 0 SNR(dB) Probability of error Rician channel with unknown fading coefficients v=0.2 v=0.5 v=0.8 conventional FSK Fig. 4. Error probability vs. SNR for 4-OOFSK signaling over two indepen dent noncoherent Rician fadi ng channels with equal Rician factor K = 1 8 . If s 0 ( t ) is th e tr ansmitted signal, the pro bability of co rrect detection is P c, 0 = P ( R 1 < τ 2 , . . . , R M < τ 2 | s 0 ) = 1 − e − τ L − 1 X l =0 τ l l ! ! M . (31) Finally , the average proba bility of error is P e = 1 − ( v P c, 1 + (1 − v ) P c, 0 ) . (32) Figures 4 , 5, and 6 provide the simulatio n re sults of er- ror probab ility when 4 -OOFSK signa ls ar e transm itted over nonco herent Rician fading ch annels. In Fig s. 4 a nd 5, the channels are assum ed to b e in depend ent. In these figu res, it is seen that OOFSK sign aling with v = 0 . 8 and v = 0 . 5 have worse error perfo rmance when co mpared to that of conv entional FSK (OOFSK with v = 1 ). As evidenced in the graph of v = 0 . 2 , if the d uty factor is sufficiently decreased, and hence c onsequen tly th e peak p ower is increased, we start seeing impr ovements. Since fading is not kn own in the nonco herent case, the advantage of using OOFSK signaling is twofold. Ha ving low d uty cycle allo ws the FSK signals to have high pe ak p ower which is especially beneficial when channel characteristics are unkn own. In addition, when the zero signal s 0 ( t ) is sent, th e r eceived signal is co mposed o f ad ditiv e no ise and is free of fading coefficients. Finally , Fig. 6 plots the error prob abilities when 4- OOFSK sign als are sent over two correlated n oncoh erent channels. These curves a re obtained when the detection ru le (29) d eriv ed for in depend ent chan nels are employed at the re ceiv er . W e n ote that the perfo rmance degrades du e to cor relation in no ncohere nt channels as well. Similar co nclusions ab out OO FSK m odulation are d rawn. R E F E R E N C E S [1] J. R. Pierc e, “Ultimate performance of M -ary transmissions on fad ing channe ls, ” IEEE T rans. Inform. Theory , vol. IT -12, pp. 2-5, Jan. 1966. 0 1 2 3 4 5 6 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR(dB) Probability of error Rician channel with unknown fading coefficients v=0.2 v=0.5 v=0.8 conventional FSK Fig. 5. Error probabili ty vs. SNR for 4-OOFSK signaling over three indeped ent noncoherent Rician fadi ng channe ls with equal Rician factor K = 1 8 . 0 1 2 3 4 5 6 10 −3 10 −2 10 −1 10 0 SNR(dB) Probability of error Rician channel with unknown fading coefficients v=0.2 v=0.5 v=0.8 conventional FSK Fig. 6. Error probabilit y vs. SN R for 4-OOFSK s ignali ng ove r two correlated noncoher ent Ric ian fadi ng channels with equal Rician fac tor K = 1 8 . [2] W .C.L indsey , “Error probabil itie s for Rician fading multicha nnel recep- tion of binary and N-ary signals, ” IE EE T ransact ion On Information Theory , pp.330-350, October 1964. [3] J.C.Hancock and W .C.Lindsey , “Optimum performance of self-adapt i ve communicat ion system operating through a Raylei gh-fa ding medium, ” IEEE Tr ansacti on O n Communication Systems. , vol.CS-11 , pp.443-454, December 1963 [4] C. W . Helstrom, Statistical Theory of Signal Detecti on, New Y ork, Perga mon Press, 1968. [5] S. V erd ´ u, “Spectral ef ficienc y in the wideband re gime, ” IEEE T rans. Inform. Theory , vol. 48, pp. 1319-1343, June 2002 [6] C. Luo and M. M ´ edard, “Frequenc y-shift ke ying for ultrawide band - Achie ving rates of the order of capac ity , ” Pro c. 40th A nnual Allerton Confer ence on Communicati on, Contr ol, and Computing , Monti cello , IL, 2002. [7] D. S. Lun, M. M ´ edard, and I. C. Abou-Fayc al, “On the performance of peak y capaci ty-achi e ving signaling on multipat h fading channels, ” IEEE T rans. Commun. , v ol. 52, pp. 931-938, June 2004. [8] D. S. L un, M. M ´ edard and I. C.Abou-Fayca l, “On performance of peak y capaci ty-achi e ving signaling on multipat h fading channels, ” IEEE T ransactions On Communications , pp.931-938, vol.52, no.6, Jun 2004. [9] M. C. Gursoy , H. V . Poor and S. V erdu, “On-Off frequenc y-shift ke ying for wideba nd fadin g channel s, ” EU RASIP J ournal on W ire less Communicat ions and Networking , 2006. [10] Q. W ang and M. C. Gursoy , “Error performanc e of OOFSK signaling ov er fadi ng channels, ” Confer ence on Information Sciences and Sys- tems (CISS), Princeton Univ ersity , Princet on, NJ, 2006. [11] N. L. Johnson and S. Kot z, Conti nuous univariate distrib ution s, Ne w Y ork, Wile y , 1970. [12] H. Zhang and T .A. Gulli ver , “Error probability for maximum ratio combing multi channel reception for M-ary coherent system ov er flat Ricea n fadi ng channels, ” IEE E W ireless Communicat ions and Network- ing Confere nce , vol. 1, 21-25 Mar 2004, pp.306-310. [13] D. D.N Be v an, V .T Ermolaye v and A.G Flaksman, “Coherent multiple recept ion of binary modulated signals with dependent Rician fad ing channe l, ” IEE Proc eedin gs-Communica tions , V ol.48, Issue 2, April 2001, pp.105 - 111. [14] V .V . V eera v alli and A. Mantrav adi, “Performance analysis for dive rsity recept ion of digital signals over correlat ed fading channels, ” IEE E V ehicular T echnolo gy Confer ence , vol. 2, 16-20 May 1999, pp.1291 - 1295. [15] J . G .Proakis, Digital Communications , New Y ork, McGraw-Hil l, 1989. [16] L . C. Andrews, Special Functions of Mathemat ics for Engineers, Oxford Uni ve rsity Press, 1998. [17] Y . Chen and N.C.Beaulie u, “SER of select ion di ve rsity MFSK with channe l estimation errors, ” IEEE T ransactions on W ireless Communica- tions , V ol.5, No.7, July 2006, pp.1920-1928. [18] M. K.Simon and M.S . Alouini, Digital Communication over F ading Channel s , N e w Y ork, John Wi le y & Sons, 2000. [19] G. K. Karagiannindi s, N. C. Sagias and T .A. Tsiftsis, “Close d-form statisti cs for the sum of squared Nakagami- m vari ates and its appli- catio ns, ” IEEE T ransaction On Communicat ions , vol.54, no.8, pp.1353- 1359, August 2006. [20] P .Gupta, N,Bansal and R.K. Mallik , “ Analysis of minimum s elect ion H- S/MRC in Rayleig h channel , ” IEEE T ransaction On Communicati ons , vol.53, no.5, pp.780-784, May 2005.