Simple and Exact Closed-Form Expressions for the Expectation of the Linn-Peleg M-PSK Lock Detector Yair Linn, Member IEEE University of British Columbia, Vancouver, BC, Canada e-mail: [email protected] Abstract – Exact closed-form expressions for the expectation of the Linn-Peleg M-PSK lock detector [1] are derived through usage of Fourier series analysis. It is shown that the expectation can be expressed as a summation of a finite number of terms which are composed of exponentials and polynomials. These results are then used to model the behavior of the lock detector under conditions of frequency-flat slow fading, and closed-form results are derived for Nakagami-m fading. The results derived in this paper thus allow for lock detector threshold values, lock probabilities, and false alarm rates to be determined easily and accurately without the need to perform numerical integrations or any other complicated mathematical operation. I. INTRODUCTION In [1] a new lock detector for M-PSK (M-ary Phase Shift Keying) receivers was presented. The expectation of the lock detector was given through an integral formula [1 eq. (20)]. Closed-form expressions for the expectation seemed unattainable due to the complicated nature of the integrand, so an approximate closed-form formula [1 eq. (23)] was derived. However, that approximation proved somewhat inaccurate at low SNRs, especially for BPSK and QPSK (see [1 Fig. 3]). In this paper we show that through usage of Fourier series analysis we can in fact arrive at exact closed-form expressions for the lock detector expectation. We then demonstrate a further application of these results, which is the derivation of closed-form expressions for the lock detector expectation in the presence of Nakagami-m fading. The signal and receiver models used in this paper are identical to those of [1 Sec. II]. Moreover, since many derivations depend upon [1], the reader is encouraged to review that reference before continuing. II. EXPRESSIONS FOR THE DETECTOR EXPECTATION

explicitly denoting the dependence on some minor notational changes: f M ( χ )  E[lˆM , N | ES / N0 = χ ] π

Rician phase distribution, given below (from [1 eq. (19)], but here we explicitly spell out the conditioning of the distribution upon the SNR, denoted by the variable χ ):

pC (∆φ | χ )  p ( ∆φn = ∆φ ES / N0 = χ )

cos( ∆φ ) e− χ ª 2 = «1+ 2χ cos ( ∆φ ) exp χ ⋅ cos ( ∆φ ) ⋅ ³ 2𠬫 −∞

(

)

2χ 2

º (2)

e− y / 2dy»

¼»

For convenience, formula [1 eq. (20)] for the computation of the lock detector expectation is now repeated here, while 1-4244-1190-4/07/$25.00 ©2007 IEEE.

(3)

−π

In [1 pp. 1663] it was stated that due to the complicated nature of the Rician phase distribution (given in (2)) computation of (3) “yields results which elude closed-form representations”. In fact, here we show that via Fourier series analysis we can reach exact expressions for (3) which are given entirely as finite sums of elementary functions. We begin by noting that the domain of pC (∆φ | χ ) is [ −π , π ] and hence we can represent it as a Fourier series, i.e. pC (∆φ | χ ) = 1 2π

¦

∞

y k ( χ ) exp( − jk ∆ φ ) where yk ( χ ) are the Fourier

k = −∞

series coefficients, given by (from [2 eq. (4.A.9)]):

yk ( χ )  ³

π

pC (τ | χ ) ⋅ e jkτ dτ

−π

=³

π

−π

pC (τ | χ ) cos( kτ ) dτ + j ³

π

−π

(4)

pC (τ | χ ) sin( kτ ) dτ

Since pC ( ∆φ | χ ) is even, the imaginary part of (4) vanishes, so that:

yk ( χ ) =

π

³π −

p C (τ | χ ) cos( k τ ) d τ

(5)

Comparing (3) to (5), we find that fM (χ) = yM (χ) . Hence, if we can find a closed-form formula for yk ( χ ) then we can find a closed-form expression for fM ( χ ) . Fortunately, the coefficients yk ( χ ) have been investigated in the literature, and they are given by [2 eq. (4.A.11)], which leads to:

f M (χ) = yM (χ) =

where N is a user-defined positive integer, and I(n), Q(n) are the in-phase and quadrature received signals (respectively) whose sample rate is 1/T, which is also the symbol rate. It was found in [1 eq. (20)] that the lock detector expectation is E[lˆM , N ] = E[cos( M ∆φn )] , where ∆φn ∈ [ −π , π ] and has the

(the SNR) and with

= E[cos(M ∆φn ) | ES / N0 = χ ] = ³ cos ( Mτ ) pC (τ χ ) ⋅ dτ

In [1] a lock detector for M-PSK was defined as [1 eq. (2)]: N lˆM , N  21N ¦n=− N +1 Re[(I (n) + jQ(n))M ]/(I 2 (n) + Q2 (n))M / 2 (1)

χ

π ⋅χ 2

⋅ exp

( ) ª¬I −χ 2

(M −1)/ 2

( ) +I χ 2

(M +1)/2

( )º¼ (6) χ

2

where I n (•) is the n-th order modified Bessel function of the first kind (see [3 Chap. 24]). Moreover, since M is an even number and using [8 eq. (9)] we can simplify (6) to:

fM (χ ) = 1 +

M 2

( − 1) ( M ⋅ ¦ k ! (M k =1

M /2

k

/ 2 + k − 1) !

/ 2 − k )! χ k

M /2 ª ( M / 2 + k − 1) ! º 1 M / 2 +1 + exp ( − χ ) « ( − 1) ⋅¦ ⋅ k » k =1 ( k − 1) ! ( M / 2 − k ) ! χ ¼ ¬

(7)

For example, for M = 2 : f2 (χ ) = 1 − χ −1 + χ −1 exp(−χ ) (8) It is emphasized that (7) is a summation of a finite number of terms which are composed entirely of exponentials and polynomials (as exemplified in (8) for M = 2 ). Hence, eq. (7) is easy to compute (even manually for small M ). As an immediate application, this allows for lock thresholds, lock

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PACRIM'07

detection probabilities and false alarm rates to be computed with ease via [1 eq. (35)-(38)]. Another application is in the generation of lookup tables for the estimation of the SNR from the value of lˆM ,N (see [4 Sec. III-B]). In the following section, we discuss yet another application, namely the investigation of the detector’s behavior in the presence of fading. III. APPLICATION: EXACT EXPRESSIONS FOR DETECTOR EXPECTATION IN THE PRESENCE OF FADING

A. Discussion: The Detector in the Presence of Fading In [1] and [4] fading effects were not treated and it was assumed that the SNR is constant. In many cases, however, the signal experiences fading ([5 Chap. 14], [6]) which must be taken into account when modeling the lock detector’s behavior. In this section we assume1 that the fading is frequency-flat and slow, i.e. that 1/ T  ∆fC where ∆fC is the channel’s coherence bandwidth, and that TCOH  T , where TCOH is the channel coherence time [5 Secs. 14.1.1, 14.2]. Using χ to denote the average SNR ratio (defined as

χ  E[ ES / N0 ] ), the conditional probability density function of the SNR due to fading is denoted as: p F ( χ | χ )  p ( E S / N 0 = χ | E[ ES / N 0 ] = χ ) (9) The channel coherence time TCOH is a measure of how fast the channel’s characteristics are changing (see [5 Chap. 14]). In general, we can assume that the SNR remains constant during time intervals significantly shorter than TCOH . Hence, to evaluate the lock detector’s behavior during fading, we must differentiate between two cases according to TCOH and the lock detector calculation period 2N ⋅ T : (a) 2 N ⋅ T  TCOH : The calculation interval is much shorter than the coherence time. Thus, the SNR can be considered constant during the lock detection computation process. Hence, the analysis undertaken in [1] and [4] is applicable, and the results of Sec. II can be used assuming a constant χ . (b) 2N ⋅ T  TCOH : The calculation interval is much longer than the coherence time. Thus, the effects of fading must be taken into account, and we can assume that the SNR values encountered during the calculation interval are distributed according to p F ( χ | χ ) .

As noted above, case (a) has been analyzed in [1], [4], and in Sec. II. In the following subsections, we analyze case (b) ( 2N ⋅ T  TCOH ). Before doing so, we comment that the case 2 N ⋅ T  TCOH (where “  ” means the same order of magnitude) is undesirable since in that case the SNR distribution during the lock detector computation period cannot be accurately predicted. This is because since 2N ⋅ T  TCOH we are not statistically guaranteed that during the lock detector 1

This assumption is a very good one for the system under discussion (coherent suppressed-carrier M-PSK). When the channel is frequency selective and/or has fast fading, then other modulation techniques (e.g. differential, noncoherent, pilot-aided, OFDM) are appropriate (see [9]).

computation period of 2N ⋅ T seconds the distribution of the SNR values encountered will be a sufficiently accurate approximation of p F ( χ | χ ) , nor are we guaranteed that the SNR remains constant. Thus, for the case of 2 N ⋅ T  TCOH , the lock detector’s value cannot be predicted and, ipso facto, the lock detection algorithm is rendered useless. Fortunately, 2 2 N ⋅ T  TCOH can always be avoided by choosing a large enough N , which ensures that 2N ⋅ T  TCOH (case (b)). We thus assume for the remainder of this section that 2 N ⋅ T  TCOH . B. Detector Expectation for Fading with 2 N ⋅ T  TCOH

Since 2 N ⋅ T  TCOH , we have that the detector expectation is weighted by p F ( χ | χ ) , i.e.: ∞

f M ( χ )  E[lˆM , N | E[ ES / N0 ] = χ ] = ³ f M ( χ ) pF ( χ | χ ) d χ (10) 0

Sec. II has already afforded us the advantage of reducing (10) to a single integral since we now do not have to compute fM ( χ ) via the integration formula of (3) (rather, we can use (7)). However, further simplifications can also be attained. Depending upon the fading probability distribution, we can sometimes use the results of Sec. II in order to obtain closedform expressions for the lock detector expectation in the presence of fading. A case in point is the Nakagami-m distribution, which is one of the most important distributions in use today [7 Sec. I]. From [6 Table 2] we have that the Nakagami-m fading distribution is: mm χ m −1 pNak − m ( χ χ )  m exp( −mχ / χ ) , with m ≥ 0.5 (11) χ Γ(m) To facilitate computation of f M ( χ ) we use the formula

³

∞

0

τ n exp(− aτ ) dτ = Γ ( n + 1) a n +1 [3 eq. (15.76)] to define:

ϒ k ,A,m ( χ ) 

³

∞

0

τ − k e − Aτ ⋅ p Nak − m (τ χ ) dτ ∞

= (m m /( χ m Γ ( m))) ³ τ m − k −1e −( A + m / χ )τ dτ 0

=

χ

−k

(12)

m ⋅ Γ (m − k ) m

(Aχ + m )m − k Γ (m ) where k , A ≥ 0 and m ≥ 0.5 . A minor difficulty arises when using (12): when m − k is a non-positive integer, then Γ( m − k ) tends to positive or negative infinity (see [3 Fig.

16.1]). This problem is elegantly solved by substituting m = m + ε instead of m in (12), where 0 < ε  m , such as ε = 0.001 ⋅ m . This has a negligible effect upon the results while avoiding the singularity of Γ( m − k ) . Hence we define the following function: 2

Ideally we prefer 2 N ⋅ T  TCOH (case (a)), so that we can rapidly generate

lock detector values. However if TCOH is small, it might not be possible to find N that simultaneously satisfies the detector accuracy requirements (given in [1 Sec. V]) and 2 N ⋅ T  TCOH . Then, we must chose N so that

2 N ⋅ T  TCOH and accept the longer lock detector calculation period.

103

­ϒ ° k ,A ,m ( χ ) m-k ∉ {0,-1,-2,-3,...} ϒ k ,A , m ( χ ) = ® (13) °¯ ϒ k ,A ,m ( χ ) m-k ∈ {0,-1,-2,-3,...} where k , A ≥ 0 , m ≥ 0.5 , and m  m + m /1000 . We can now use (7), (8) and (10)-(13) in order to arrive at closed-form expressions for f M ( χ ) . For example, for M = 2 and m  1 we have from (8), (10), (11), (12) and (13): f 2 ( χ ) = ϒ 0 ,0 , m ( χ ) − ϒ 1,0 , m ( χ ) + ϒ 1,1, m ( χ )

probabilities, and false alarm rates. They are also useful in the generation of lookup tables for estimation of the SNR from the lock detector value. Additionally, we analyzed the detector’s operation during frequency-flat slow fading and showed that in this case, depending upon the fading distribution, it is sometimes possible to arrive at closed-form expressions for the detector expectation.

(14) ⋅ Γ ( m − 1) χ − 1 m m ⋅ Γ ( m − 1) + m −1 ⋅ Γ (m ) ( χ + m ) m −1 ⋅ Γ ( m ) and if m = 1 then eq. (14) will work if in the last two terms m is replaced by m = m + m /1000 = 1.001 , as per (13). Generally, for even M we have from (7), (10), (11), (12) and (13) that:

=1−

χ

−1

f M (χ )

( −1) ⋅ ( M / 2 + k −1)! ⋅ ϒ (χ) ¦ k! ( M / 2 − k )! k ,0,m k =1 M /2 ª º ( M / 2 + k −1) !  1 +1 + «( −1) ⋅ ¦ ⋅ ϒk ,1,m ( χ ) » ( M / 2 − k )! k =1 ( k −1) ! ¬ ¼ = ϒ 0,0,m ( χ ) + M 2

M /2

k

(15) Fig. 1. Theoretical expectation of lock detector vs. simulations, for cases of (i) no fading (or, equivalently, fading with 2N ⋅ T  TCOH ), and (ii) for frequency-

M 2

We emphasize that (15) is a summation of a finite number of terms containing ratios of polynomials and gamma functions. This is easily calculated using numerical computation packages. Furthermore, we note that Rayleigh and one-sided Gaussian fading are particular cases of Nakagami-m fading (with m=1 and m=0.5 respectively (see [6 Table 2])), so closed-form expressions for these fading distributions can be obtained from (15) as well. C. Detector Distribution for Fading with 2 N ⋅ T  TCOH

Regarding the detector’s variance and distribution when 2 N ⋅ T  TCOH , we note that the assumptions in [1 Sec. III-D] that led to the bound var(lˆM , N ) ≤ 1 (2 N ) and to the conclusion is Gaussian are still valid for frequency-flat slow that lˆ M ,N

fading. Thus, if 2 N ⋅ T  TCOH then: lˆM , N locked ~ N ( f M ( χ ) , 1/(2N ) ) lˆM , N unlocked or noise only input ~ N ( 0 , 1 /(2 N ) )

(16)

where 1 (2 N ) is an upper bound. IV. SIMULATION RESULTS To validate the closed-form expressions that were derived, we compare results computed via (7) and (15) to results obtained via simulations (i.e., by computing (1) with simulated inputs). This is shown in Fig. 1. As we see there, the theoretical results agree completely with the simulations. V. CONCLUSIONS Using Fourier analysis, we have found exact closed-form expressions for the expectation of the Linn-Peleg M-PSK lock detector. These expressions are finite sums whose terms are composed of polynomials and exponentials, and hence facilitate easy computation of lock thresholds, detection

flat slow Nakagami-m fading with 2 N ⋅ T  TCOH for various values of m . For case (i) the graph shows fM ( χ ) vs. χ , while for case (ii) the graph is of

fM ( χ ) vs. χ . The simulations used N=1024 and 100 lock metrics were averaged to compute each data point. For computation of fM ( χ ) we used

TCOH = 50 ⋅ T , while for computation of fM ( χ ) the SNR was assumed constant (i.e. no fading, or equivalently TCOH = ∞ ).

REFERENCES [1] Y. Linn and N. Peleg, "A family of self-normalizing carrier lock detectors and Es/N0 estimators for M-PSK and other phase modulation schemes," IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 1659-1668, Sep. 2004. [2] K.-P. Ho, Phase-modulated optical communication systems. NY: Springer, 2005. [3] M. R. Spiegel, Mathematical handbook of formulas and tables. NY: McGraw-Hill, 1968. [4] Y. Linn, "Quantitative analysis of a new method for real-time generation of SNR estimates for digital phase modulation signals," IEEE Trans. Wireless Commun., vol. 3, no. 6, pp. 19841988, Nov. 2004. [5] J. G. Proakis, Digital communications, 4th ed. Boston: McGrawHill, 2001. [6] M. K. Simon and M. Alouini, "A unified approach to the performance analysis of digital communication over generalized fading channels," Proc. IEEE, vol. 86, no. 9, pp. 1860-1877, Sep. 1998. [7] A. Annamalai and C. Tellambura, "Error rates for Nakagami-m fading multichannel reception of binary and M-ary signals," IEEE Trans. Commun., vol. 49, no. 1, pp. 58-68, Jan. 2001. [8] S. A. Butman and J. R. Lesh, "The effects of bandpass limiters on n-phase tracking systems," IEEE Trans. Commun., vol. 25, no. 6, pp. 569-576, Jun. 1977. [9] B. Sklar, "Rayleigh Fading Channels in Mobile Digital Communication Systems Part II: Mitigation," IEEE Comm. Mag., vol. 35, no. 9, pp. 148-155, Sep. 1997.

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