- Email: [email protected]
Performance analysis of the adaptive line enhancer with multiple sinusoids in noisy environment
Signal Processing 82 (2002) 93 – 101 www.elsevier.com/locate/sigpro
Performance analysis of the adaptive line enhancer with multiple sinusoids in noisy environment R.L. Campbell Jr.a , N.H. Younanb; ∗ , J. Gub;1 a Information
b Department
Technology Laboratory, U.S. Army Corps of Engineers, Vicksburg, MS 39180, USA of Electrical and Computer Engineering, Mississippi State University, Box 9571, Mississippi State, MS 39762-9571, USA
Abstract In this paper, the performance analysis of the adaptive line enhancer when the input signal consists of multiple sinusoids embedded in noise is investigated. The performance is evaluated in terms of the signal-to-noise ratio gain at the /lter’s output. It is shown that, for multiple sinusoids, this gain is not only a function of the /lter length, but also of three additional factors — the number of sinusoids, the noise power, and the amplitude of each sinusoid. Simulation results for a dual noisy sinusoidal input are presented to illustrate the validity of this analysis. ? 2002 Elsevier Science B.V. All rights reserved.
1. Introduction In general, the physical telecommunications infrastructure in the United States is primarily copper between local telecommunication companies and their users. Although the late-90’s ideologues forecasted a timely transition from old copper telephone infrastructure to /ber, where there was talk of /ber-to-the-curb or /ber to each person’s home, it appears now that there is little to no push by the local telephone exchanges to change the distribution medium to their customers. It is copper and will remain copper for sometime to come. This places the local exchange in a precarious situation in terms of competition with its natural nemesis; the cable television provider [1]. Aside from power cables, telephone wires and coaxial cable television are commonly distributed to the ∗ Corresponding author. Tel.: +1-601-325-3912; fax: +1-601325-2298. E-mail address: [email protected] (N.H. Younan). 1 J. Gu is with Ansoft Corporation, 669 River Drive, Suite 200, Elmwood park, NJ 07407, USA.
majority of homes. However, a coaxial cable has a distinct advantage over a two-wire telephone connection, since a coaxial cable is naturally shielded [8,9]. Therefore, the standard telephone connection faces greater noise and will pose a particular design problem to those telephone engineers that are working to bring Internet access to densely populated areas [1]. In competition with this service, local telephone companies provide a service known as Asynchronous Digital Subscriber Line (ADSL) over their non-coaxial infrastructure [7]. ADSL has been touted as the solution for an end-user that plans to use telephone wires to gain access, since it provides a high speed uplink and a much higher speed downlink, which is ideal for web browsing [1]. Since ADSL occupies the 0 –500 kHz band, a broadband noise management approach is in order [8]. To provide this service reliably, they must combat a number of problems including line attenuation, group delay, and noise, with the noise being the unique obstacle. An interesting application for performing noise reduction on a telephone wire connection is the use of
0165-1684/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 0 1 ) 0 0 1 6 0 - 8
94
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
adaptive /ltering, where the adaptive line enhancer (ALE) is chosen in this investigation due to its simplicity and ease of implementation [11]. Although the ALE is known to be applicable to narrowband signals in broadband noise [3,5], it is not clear as to why the ALE does not extend well to a problem like this one, since the ALE provides enhancement on a sinusoid by sinusoid basis. In other terms, it is not evident as to why its performance does not extrapolate well to a host of sinusoids (i.e. a band). Adaptive line enhancement techniques for tracking sinusoids in noise have been widely used and their performance for improving the signal-to-noise ratio (SNR) has been also studied [4,10,2,12]. The main goal of this paper is to explore the limitations of the adaptive line enhancer by analytically describing its performance to predict the algorithm’s SNR gain. For multiple sinusoids, it is shown that this gain is not only a function of the /lter length L; but also of three other additional factors — the number of sinusoids, the noise power, and the amplitude of each sinusoid. Furthermore, it is shown that the gain can be exactly predicted for all L corresponding to any integer multiple of the noise-free signal’s fundamental frequency. At other values of L; the gain is approximated. Simulation results for a noisy sinusoidal input are presented to illustrate the validity of this analysis. Note that the model of a fundamental and its harmonics are used for signaling in this simulation. Furthermore, since a set of harmonic sinusoids is a subset of the entire domain of individual sinusoids, the extrapolation of individual sinusoids into a broadband is somehow similar to the extrapolation of groups of harmonic sinusoids into a broadband. 2. Analysis 2.1. General concept The ALE is one of the most useful applications of adaptive /ltering. It is a method of optimal /ltering that can be applied to enhance noise corrupted signals. It is designed to suppress the broadband noise component of an input signal, while passing the narrowband component, such as a sinusoid, with little distortion [11]. It is also capable of automatically turning itself oI when no SNR improvement is achieved. In contrast
d(n)
+ delay
x(n)
f
y(n) e(n)
Fig. 1. Model of the adaptive line enhancer.
to other techniques, the ALE does not require a priori knowledge of the signal or noise. The block diagram of the ALE is shown in Fig. 1. Assuming that the input signal d(n) consists of a deterministic or noise-free component plus an additive noise component, i.e., a narrowband signal plus broadband noise, and denote x(n) as a delayed version of d(n); then x(n) decorrolates the broadband noise while leaving the narrowband component correlated. Ideally, the output of the adaptive /lter, y(n); is an estimate of the noise-free input signal. The computational algorithm for the ALE is [6] y(n) =
L−1
fi (n)x(n − i);
x(n) = d(n − );
i=0
fi (n + 1) = fi (n) + 2 e(n)x(n − i);
(1)
e(n) = d(n) − y(n); where is the adaptation parameter that controls the speed of convergence and the stability of the /lter, is the decorrelation parameter which depends on the correlation lag of the input signal components, L is the /lter length, fi (n) is the ith set of /lter coeLcients, and n ∈ [0; N − 1] with N being the data length. Note that the adaptation parameter plays a major role and its value results in a compromise or a tradeoI between stability and convergence, thus eventually aIecting the steady-state error. 2.2. SNR derivation It is of interest to explore the limitations of the ALE algorithm, described by Eq. (1), for multiple
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
sinusoids in noisy environment. Accordingly, a quantitative analysis in terms of the SNR gain at the /lter output is performed. Note that throughout the derivation of the SNR gain, the following assumptions are considered: 1. For M sinusoids with AWGN, the input signal is expressed as x(n) =
M −1
Am cos(!m nT + m ) + w(n);
(2)
m=0
where Am is the mth amplitude of the sinusoidal input, !m is the mth sinusoidal frequency, m is the mth phase angle, T is the sampling period, and w(n) is an additive white–Gaussian noise with zero mean and a variance w2 , i.e., w(n) ∼ N(0; w2 ). Note that the phase angle m is chosen to be zero since the phase has no impact on the SNR derivation [3]. 2. The desired signal d(n) is d(n) = x(n + ):
(3)
3. In order to decorrelate the noise, the delay, , is chosen such that E{w(n)w(n − )} ≈ 0;
L∗ = k
!sampling ; !fundamental
(5)
where k is an arbitrary integer, !fundamental is the fundamental frequency of the sinusoidal signal satisfying {!m = nm !fundamental ; m = 0; 1; : : : ; M − 1; nm is an integer} and !sampling is the sampling frequency with !sampling ¿ 2!h according to the Nyquist theorem to avoid aliasing, with !h being the highest frequency of the sinusoid, i.e., !h = max{!m ; m = 0; 1; : : : ; M − 1}. Note that Eq. (5) holds regardless of the number of sinusoids M . Under these assumptions [11], the steady-state /lter coeLcients are f∗ (i) =
M −1 m=0
Bm cos(!m iT + m );
where i ∈ [0; L∗ − 1] and Bm =
2=L∗ m2 ; w2 2=L∗ + m2 =w2
2=L∗ : m = N!m T = 1 + (2=L∗ )(w2 =m2 )
(6)
(7)
Note that m2 is the signal power of the mth sinusoid, i.e., m2 = A2m =2, and w2 is the overall noise power. Furthermore, it is clearly seen that as w2 → 0, Bm → 2=L∗ . Accordingly, the /lter output y(n), i.e., y(n) = x(n) ∗ f(n), with ∗ denotes convolution, can be separated into two distinct terms y(n) = ysignal (n) + ynoise (n):
(8)
Assuming complete coherence, the signal and noise portions of the output signal can be expressed as ysignal (n) =
∗ L −1 M −1 M −1
i=0
[Bj cos(!j iT )]
j=0 k=0
×[Ak cos(!k (n − i)T )]
(9)
and
(4)
with E{· · ·} being the expected value operator. 4. The ideal /lter length is an integer multiple of the sampling rate, i.e.,
95
ynoise (n) =
∗ L −1 M −1
i=0
Bj cos(!j iT )w(n − i):
(10)
j=0
Due to the constraint placed on L∗ and assuming that orthogonality holds for any set of arbitrary fundamental frequencies, the signal component of the output becomes (see Appendix A) ysignal (n) =
M −1 L∗ Bj Aj cos(!j nT ): 2
(11)
j=0
For a /nite signal duration, N , the signal power can be approximated by Psignal =
N −1 1 [ysignal (n)]2 : N
(12)
n=0
If, in addition, N is an integer multiple of L∗ , the signal power is Psignal =
M −1 (L∗ )2 2 2 Bj A j : 8 j=0
(13)
96
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
In a similar way, the noise power can be obtained from Pnoise = E{[ynoise (n)]2 };
(14)
yielding, Pnoise =
M −1 L∗ 2 2 w Bj : 2
(15)
j=0
Accordingly, the output signal-to-noise ratio, SNR out , is SNR out =
Psignal Pnoise
(16)
or
M −1 2 2 L∗ j=0 Bj Aj : SNR out = 2 M −1 2 4w j=0 Bj
It is also worthy to mention that if L is not an integer multiple of the fundamental frequency of the sinusoidal input signal, Eq. (20) is no longer valid to use for the computation of the performance or SNR gain. Instead, the SNR gain can be obtained directly from Eq. (19) with SNR out being expressed by Eq. (16). Note that the theoretical derivation of SNR out for multiple sinusoids, i.e., M ¿ 2, is somewhat complicated. However, for illustrative purposes, it can be shown that for a dual sinusoid, M = 2, the theoretical output signal and noise powers are: Psignal =
B02 A20 L2 8 L−1
+
(17)
i=0
Since the input SNR can be expressed as M −1 2 j=0 Aj SNR in = ; 2w2
+
or
SNR out SNR in
L−1 L−1 B12 A20 cos(!1 iT ) 2 i=0 j=0
(18)
×cos(!1 jT ) cos(!0 (j − i)T ) +
the performance or SNR gain is, SNR gain =
B0 B1 A20 L cos(!1 iT )cos(!0 iT ) 2
L−1
(19)
B0 B1 A21 L cos(!0 iT ) cos(!1 iT ) 2
+
i=0
M −1
SNR gain =
2 2 L∗ j=0 Bj Aj M −1 2 M −1 2 2 j=0 Bj j=0 Aj
(20)
=
∗
M −1
+ (4=L )(w2 =A2j )]−2 M −1 (4=L∗ )(w2 =A2j )]−2 j=0
A2j [1
L j=0 M −1 2 j=0 [1 +
L−1 L−1 B02 A21 cos(!0 iT ) cos(!0 jT ) 2
+
i=0 j=0
and in terms of the /lter coeLcients, Bm , Eq. (20) can be written as: SNR gain
B12 A21 L2 8
∗
A2j
:
(21)
Note that Eq. (20) represents the theoretical gain or performance of the ALE. It is clearly seen that this gain is a function of the /lter length, the number of sinusoids, as well as their corresponding amplitudes. Furthermore, if the amplitudes of the sinusoidal signal are equal, Eq. (20) reduces simply to SNR gain = L∗ =2M , an indication that the SNR gain increases linearly with L∗ and inversely with M . Note also that for M = 1, Eq. (20) reduces to the conventional ALE gain of L∗ =2.
×cos(!1 (j − i)T )
(22)
N −1 1 ≈ [y(n) − ysignal (n)]2 N
(23)
and Pnoise
n=0
with ysignal (n) =
B0 A0 L cos(!0 nT ) 2 + B1 A0
L−1
cos(!1 iT )cos(!0 (n − i)T )
i=0
+ B 0 A1
L−1
cos(!0 iT )cos(!1 (n − i)T )
i=0
+
B 1 A1 L cos(!1 nT ): 2
(24)
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
Note that the noise power approximation is valid as long as the noise is covariance-ergodic.
3. Simulation results To verify the performance of the ALE in terms of the SNR gain, a dual sinusoidal signal of length N = 8192 is considered. In this simulation, N is chosen suLciently long to ensure the convergence of the ALE algorithm. DiIerent cases are simulated to investigate the eIects of the sinusoidal frequencies, their amplitudes, as well as the noise content on the performance. For illustrative purposes, the following parameters associated with the ALE implementation are used in this simulation: = 100; L ∈ [1; 2; : : : ; 50], and ∈ [0:0001; 0:0002; : : : ; 0:0075]. Note that the
value used corresponds to the optimal value. In each case considered, the experimental SNR gain values are obtained via averaging 40 runs, which results in what is known as the statistical sample mean. Furthermore, the eIect of the delay parameter on the overall performance of the ALE, thus the SNR gain,
97
is not investigated in this study. The only consideration taken is that should be large enough to decorrelate the noise component. Figs. 2 and 3 are plots of the SNR gain versus the /lter length L for a dual sinusoidal input, with different noise levels, for L∗ = k · 20, with k being an arbitrary integer. These /gures represent the resulting SNR gain for the dual sinusoidal signal with separate frequencies for two cases: equal and diIerent amplitudes. For an ideal /lter length L∗ satisfying Eq. (5), the theoretical SNR gain values are obtained directly from Eq. (20). These values are denoted by the symbol “∗ ”. Accordingly, only the discrete point values at L∗ = k · 20 correspond to the theoretical values. Everywhere else, there is no basis for comparison since the theoretical values are not determined over the interval of interest. Then, the SNR gain values obtained from the simulation results via Eqs. (22) – (24) for L values within the speci/ed range, L ∈ [1; 2; : : : ; 50], are directly compared with their corresponding theoretical values to verify the performance of the ALE method. Theoretically, both should match exactly at L = L∗ . These /gures illustrate clearly this attribute, except in few cases where the degradation in the performance
SNR Gain
15
σ w2 = 1
10 5 0
0
5
10
15
20
25
30
35
40
45
50
10
15
20
25
30
35
40
45
50
SNR Gain
15 10
σ w2 = 3
5 0
0
5
SNR Gain
15 10
σ w2 = 5
5 0
Theoretical simulation 0
5
10
15
20 25 30 Filter Length: L
Fig. 2. SNR gain comparison of a dual sinusoid with equal x(n) = cos((2 =5)n) + cos((9 =10)n) + w(n); L∗ = k · 20; k = 1; 2; : : :.
35
amplitudes
40
and
45
separate
50
individual
frequencies,
i.e.,
98
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
SNR gain
15
σ w2 = 1
10 5 0
0
5
10
15
20
25
30
35
40
45
50
10
15
20
25
30
35
40
45
50
SNR gain
15
σ w2 = 3
10 5 0
0
5
SNR gain
20
σ w2 = 5
10 0
Theoretical Simulation 0
5
10
15
20 25 30 Filter Length: L
35
40
45
50
Fig. 3. SNR gain comparison of a dual sinusoid with diIerent amplitudes and separate individual frequencies, i.e., x(n) = cos((2 =5)n) + 4 cos((9 =10)n) + w(n); L∗ = k · 20; k = 1; 2; : : :
Fig. 4. The eIect of the amplitude on the SNR gain for a dual sinusoidal signal at various noise levels, L∗ = 20.
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
99
Fig. 5. The eIect of the amplitude on the SNR gain for a multiple sinusoidal signal (M = 4) at various noise levels, L∗ = 20.
can be attributed to the algorithm’s instability induced by a substantial increase in the input power since 0 ¡ stability ¡
2 ; LPinput
(25)
where Pinput is the corresponding power of the dual sinusoidal input signal and=or higher noise levels, i.e., lower SNR values. Note that the L = 1 case is a degenerative case and is of no importance to this comparison. It is also of interest to further investigate the effect of the noise level as well as the amplitudes of the sinusoidal signal on the SNR gain. Figs. 4 and 5 illustrate this concept for M = 2 and 4, respectively. It is apparent from these /gures that higher amplitudes result in higher SNR gain values at any given noise variance. Furthermore, the noise level, from which the input SNR can be obtained via Eq. (18), does clearly aIect the performance or SNR gain. Note that, for illustrative purposes, only one sinusoidal amplitude is varied while the rest is kept the same. It is worthy to mention that for sinusoidal signals with equal amplitudes, a broadband eIect is obtained. Accordingly, as one sinusoidal term increases in amplitude, i.e., A4 ¿ A1 ; A2 ; A3 for the M =4 case, the broadband case starts looking like a narrowband case.
Since the general equation derived for the SNR gain (Eq. (20)) has proven to be a good indicator of the performance of the ALE algorithm, the performance trend as a function of the number of sinusoids, for a /xed noise power and sinusoidal amplitudes, is examined. Fig. 6 illustrates this trend for a sinusoidal signal with unity equal amplitudes and a unity noise variance. This /gure reveal that the ALE performance is degraded as the input signal’s bandwidth is increased (M increases) and careful consideration has to be taken in the implementation of the ALE algorithm. 4. Conclusion A quantitative analysis of the ALE method has been performed in details. For multiple sinusoids in noisy environment, it is shown that the output SNR gain is a function of not only the /lter length, but also is a function of three additional factors — the number of sinusoids, the noise power, and the amplitude of each sinusoid. Simulation results reveal the performance trends in the ALE implementation and the eIect of these factors. Accordingly, careful consideration has to be taken before using the ALE in telecommunication type applications.
100
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
Fig. 6. Theoretical SNR gain vs. the number of sinusoids for the signal model, x(n) = Am = 1; m = 0; 1; : : : ; M − 1, w2 = 1; and L∗ = 20.
Appendix A. It is desired to derive the signal and noise components of the /lter output and subsequently the SNR gain of the adaptive line enhancer under the assumptions made (please refer to Eqs. 2–5). Accordingly, the /lter output y(n) can be expressed: y(n) =
∗ L −1
∗
f (i)x(n − i);
i=0
(A.1)
i=0
×
Am cos(!m nT ) + N (0; w2 );
×
j=0
M −1
Ak cos(!k (n − i)T )
k=0
where f∗ (n) and x(n) denote the /lter coeLcients and sinusoidal input signal, respectively. A direct substitution of Eqs. (2) and (6) into (A.1) yields, y(n) =
m=0
Assume that the phase angles have the same coherence and separate y(n) into two distinct components, the signal and noise components can be obtained as follows: ∗ L −1 M −1 Bj cos(!j iT ) ysignal (n) =
i=0
∗ L −1
M −1
M −1 k=0
M −1
=
i=0
=
j=0
∗ L −1 M −1 M −1
i=0
Ak cos(!k (n − i)T + k ) + w(n − i) : (A.2)
Bj Ak cos(!j iT )
j=0 k=0
×cos(!k (n − i)T )
Bj cos(!j iT + j )
∗ L −1 M −1 M −1
Bj Ak
j=0 k=0
1 cos{(!j + !k )i−!k n}T × 2 +cos{(!j −!k )i+!k n}T (A.3)
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
and
101
References
ynoise (n) =
∗ L −1
i=0
M −1
Bj cos(!j iT ) w(n − i):
j=0
(A.4) Eq. (A.3) can be further reduced using the following assumptions: Let !j = m · !fundamental and !k = n · !fundamental , where m and n are arbitrary integers, i.e., m and n ∈ I . Accordingly, the frequency sum and diIerence becomes !j + !k = (m + n) · !fundamental and !j − !k = (m − n) · !fundamental , where (m + n) and (m − n) ∈ I . Since the optimal /lter length L∗ satis/es Eq. (5), all sinusoidal components summed over the range {0; 1; : : : ; L∗ − 1} with a spacing of T and a frequency of ‘ · !fundamental , where ‘ ∈ I and ‘ = 0 will add to zero. Under these assumptions, Eq. (A.3) reduces to, ysignal (n) =
∗ L −1 M −1
i=0
=
Bj Aj [1=2 cos(!j nT )]
j=0
M −1 L∗ Bj Aj cos(!j nT ); 2
(A.5)
j=0
from which the signal power can be derived. Note that the orthogonality property stated above holds for any arbitrary fundamental frequency and for the M = 2 case, it is suLcient to integrate over a period of Nf = (f2 − f1 )=2 = 1=L∗ for the sinusoidal frequencies to be orthogonal.
[1] G. Abe, Residential Broadband, Cisco Press, Indianapolis, IN, 1997. [2] T.R. Albert, H. Abusalem, M.D. Juniper, Experimental results: Detection and tracking of low SNR sinusoids using real-time LMS and RLS lattice adaptive line enhancers, Proc. IEEE Int. Conf. Acoust. Speech Signal Process 3 (May 1991) 1857–1860. [3] P.M. Clarkson, Optimal and Adaptive Signal Processing, CRC Press Inc., Boca Raton, FL, 1993. [4] M. Ghogho, M. Ibnkahla, J.N. Bershad, Analytic behavior of the LMS adaptive line enhancer for sinusoids corrupted by multiplicative and additive noise, IEEE Trans. Signal Process. 46 (9) (September 1998) 2386–2393. [5] S. Haykin, Adaptive Filter Theory, 3rd Edition, Prentice-Hall Inc., Englewood CliIs, NJ, 1996. [6] S.J. Orfanidis, Optimum Signal Processing, McGraw-Hill Publishing Company, New York, 1988. [7] S.V. Raghavan, S.K. Tripathi, Networked Multimedia Systems: Concepts, Architecture, and Design, Prentice-Hall, Inc., Englewood CliIs, NJ, 1998. [8] W.D. Reeve, Subscriber Loop Signaling and Transmission Handbook: Digital, IEEE Press, New York, 1992. [9] J.D. Spragins, J.L. Hammond, K. Pawlikowski, Telecommunications: Protocols and Design, Addison-Wesley Publishing Company, Reading, MA, 1994. [10] E. Watanabe, Y. Okamura, A. Nishihara, Adaptive line enhancers with discriminated structures, Electron. Commun. Jpn, Part III: Fundam. Electron. Sci. 79 (11) (November 1996) 57–68. [11] B. Widrow, S. Stearns, Adaptive Signal Processing, Prentice-Hall Inc., Englewood CliIs, NJ, 1985. [12] J.R. Zeidler, E.H. Satorius, D.M. Chabries, H.T. Wexler, Adaptive enhancement of multiple sinusoids in uncorrelated noise, IEEE Trans. Acoust. Speech Signal Process. ASSP-26 (1978) 240–254.
Performance analysis of the adaptive line enhancer with multiple sinusoids in noisy environment R.L. Campbell Jr.a , N.H. Younanb; ∗ , J. Gub;1 a Information
b Department
Technology Laboratory, U.S. Army Corps of Engineers, Vicksburg, MS 39180, USA of Electrical and Computer Engineering, Mississippi State University, Box 9571, Mississippi State, MS 39762-9571, USA
Abstract In this paper, the performance analysis of the adaptive line enhancer when the input signal consists of multiple sinusoids embedded in noise is investigated. The performance is evaluated in terms of the signal-to-noise ratio gain at the /lter’s output. It is shown that, for multiple sinusoids, this gain is not only a function of the /lter length, but also of three additional factors — the number of sinusoids, the noise power, and the amplitude of each sinusoid. Simulation results for a dual noisy sinusoidal input are presented to illustrate the validity of this analysis. ? 2002 Elsevier Science B.V. All rights reserved.
1. Introduction In general, the physical telecommunications infrastructure in the United States is primarily copper between local telecommunication companies and their users. Although the late-90’s ideologues forecasted a timely transition from old copper telephone infrastructure to /ber, where there was talk of /ber-to-the-curb or /ber to each person’s home, it appears now that there is little to no push by the local telephone exchanges to change the distribution medium to their customers. It is copper and will remain copper for sometime to come. This places the local exchange in a precarious situation in terms of competition with its natural nemesis; the cable television provider [1]. Aside from power cables, telephone wires and coaxial cable television are commonly distributed to the ∗ Corresponding author. Tel.: +1-601-325-3912; fax: +1-601325-2298. E-mail address: [email protected] (N.H. Younan). 1 J. Gu is with Ansoft Corporation, 669 River Drive, Suite 200, Elmwood park, NJ 07407, USA.
majority of homes. However, a coaxial cable has a distinct advantage over a two-wire telephone connection, since a coaxial cable is naturally shielded [8,9]. Therefore, the standard telephone connection faces greater noise and will pose a particular design problem to those telephone engineers that are working to bring Internet access to densely populated areas [1]. In competition with this service, local telephone companies provide a service known as Asynchronous Digital Subscriber Line (ADSL) over their non-coaxial infrastructure [7]. ADSL has been touted as the solution for an end-user that plans to use telephone wires to gain access, since it provides a high speed uplink and a much higher speed downlink, which is ideal for web browsing [1]. Since ADSL occupies the 0 –500 kHz band, a broadband noise management approach is in order [8]. To provide this service reliably, they must combat a number of problems including line attenuation, group delay, and noise, with the noise being the unique obstacle. An interesting application for performing noise reduction on a telephone wire connection is the use of
0165-1684/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 0 1 ) 0 0 1 6 0 - 8
94
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
adaptive /ltering, where the adaptive line enhancer (ALE) is chosen in this investigation due to its simplicity and ease of implementation [11]. Although the ALE is known to be applicable to narrowband signals in broadband noise [3,5], it is not clear as to why the ALE does not extend well to a problem like this one, since the ALE provides enhancement on a sinusoid by sinusoid basis. In other terms, it is not evident as to why its performance does not extrapolate well to a host of sinusoids (i.e. a band). Adaptive line enhancement techniques for tracking sinusoids in noise have been widely used and their performance for improving the signal-to-noise ratio (SNR) has been also studied [4,10,2,12]. The main goal of this paper is to explore the limitations of the adaptive line enhancer by analytically describing its performance to predict the algorithm’s SNR gain. For multiple sinusoids, it is shown that this gain is not only a function of the /lter length L; but also of three other additional factors — the number of sinusoids, the noise power, and the amplitude of each sinusoid. Furthermore, it is shown that the gain can be exactly predicted for all L corresponding to any integer multiple of the noise-free signal’s fundamental frequency. At other values of L; the gain is approximated. Simulation results for a noisy sinusoidal input are presented to illustrate the validity of this analysis. Note that the model of a fundamental and its harmonics are used for signaling in this simulation. Furthermore, since a set of harmonic sinusoids is a subset of the entire domain of individual sinusoids, the extrapolation of individual sinusoids into a broadband is somehow similar to the extrapolation of groups of harmonic sinusoids into a broadband. 2. Analysis 2.1. General concept The ALE is one of the most useful applications of adaptive /ltering. It is a method of optimal /ltering that can be applied to enhance noise corrupted signals. It is designed to suppress the broadband noise component of an input signal, while passing the narrowband component, such as a sinusoid, with little distortion [11]. It is also capable of automatically turning itself oI when no SNR improvement is achieved. In contrast
d(n)
+ delay
x(n)
f
y(n) e(n)
Fig. 1. Model of the adaptive line enhancer.
to other techniques, the ALE does not require a priori knowledge of the signal or noise. The block diagram of the ALE is shown in Fig. 1. Assuming that the input signal d(n) consists of a deterministic or noise-free component plus an additive noise component, i.e., a narrowband signal plus broadband noise, and denote x(n) as a delayed version of d(n); then x(n) decorrolates the broadband noise while leaving the narrowband component correlated. Ideally, the output of the adaptive /lter, y(n); is an estimate of the noise-free input signal. The computational algorithm for the ALE is [6] y(n) =
L−1
fi (n)x(n − i);
x(n) = d(n − );
i=0
fi (n + 1) = fi (n) + 2 e(n)x(n − i);
(1)
e(n) = d(n) − y(n); where is the adaptation parameter that controls the speed of convergence and the stability of the /lter, is the decorrelation parameter which depends on the correlation lag of the input signal components, L is the /lter length, fi (n) is the ith set of /lter coeLcients, and n ∈ [0; N − 1] with N being the data length. Note that the adaptation parameter plays a major role and its value results in a compromise or a tradeoI between stability and convergence, thus eventually aIecting the steady-state error. 2.2. SNR derivation It is of interest to explore the limitations of the ALE algorithm, described by Eq. (1), for multiple
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
sinusoids in noisy environment. Accordingly, a quantitative analysis in terms of the SNR gain at the /lter output is performed. Note that throughout the derivation of the SNR gain, the following assumptions are considered: 1. For M sinusoids with AWGN, the input signal is expressed as x(n) =
M −1
Am cos(!m nT + m ) + w(n);
(2)
m=0
where Am is the mth amplitude of the sinusoidal input, !m is the mth sinusoidal frequency, m is the mth phase angle, T is the sampling period, and w(n) is an additive white–Gaussian noise with zero mean and a variance w2 , i.e., w(n) ∼ N(0; w2 ). Note that the phase angle m is chosen to be zero since the phase has no impact on the SNR derivation [3]. 2. The desired signal d(n) is d(n) = x(n + ):
(3)
3. In order to decorrelate the noise, the delay, , is chosen such that E{w(n)w(n − )} ≈ 0;
L∗ = k
!sampling ; !fundamental
(5)
where k is an arbitrary integer, !fundamental is the fundamental frequency of the sinusoidal signal satisfying {!m = nm !fundamental ; m = 0; 1; : : : ; M − 1; nm is an integer} and !sampling is the sampling frequency with !sampling ¿ 2!h according to the Nyquist theorem to avoid aliasing, with !h being the highest frequency of the sinusoid, i.e., !h = max{!m ; m = 0; 1; : : : ; M − 1}. Note that Eq. (5) holds regardless of the number of sinusoids M . Under these assumptions [11], the steady-state /lter coeLcients are f∗ (i) =
M −1 m=0
Bm cos(!m iT + m );
where i ∈ [0; L∗ − 1] and Bm =
2=L∗ m2 ; w2 2=L∗ + m2 =w2
2=L∗ : m = N!m T = 1 + (2=L∗ )(w2 =m2 )
(6)
(7)
Note that m2 is the signal power of the mth sinusoid, i.e., m2 = A2m =2, and w2 is the overall noise power. Furthermore, it is clearly seen that as w2 → 0, Bm → 2=L∗ . Accordingly, the /lter output y(n), i.e., y(n) = x(n) ∗ f(n), with ∗ denotes convolution, can be separated into two distinct terms y(n) = ysignal (n) + ynoise (n):
(8)
Assuming complete coherence, the signal and noise portions of the output signal can be expressed as ysignal (n) =
∗ L −1 M −1 M −1
i=0
[Bj cos(!j iT )]
j=0 k=0
×[Ak cos(!k (n − i)T )]
(9)
and
(4)
with E{· · ·} being the expected value operator. 4. The ideal /lter length is an integer multiple of the sampling rate, i.e.,
95
ynoise (n) =
∗ L −1 M −1
i=0
Bj cos(!j iT )w(n − i):
(10)
j=0
Due to the constraint placed on L∗ and assuming that orthogonality holds for any set of arbitrary fundamental frequencies, the signal component of the output becomes (see Appendix A) ysignal (n) =
M −1 L∗ Bj Aj cos(!j nT ): 2
(11)
j=0
For a /nite signal duration, N , the signal power can be approximated by Psignal =
N −1 1 [ysignal (n)]2 : N
(12)
n=0
If, in addition, N is an integer multiple of L∗ , the signal power is Psignal =
M −1 (L∗ )2 2 2 Bj A j : 8 j=0
(13)
96
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
In a similar way, the noise power can be obtained from Pnoise = E{[ynoise (n)]2 };
(14)
yielding, Pnoise =
M −1 L∗ 2 2 w Bj : 2
(15)
j=0
Accordingly, the output signal-to-noise ratio, SNR out , is SNR out =
Psignal Pnoise
(16)
or
M −1 2 2 L∗ j=0 Bj Aj : SNR out = 2 M −1 2 4w j=0 Bj
It is also worthy to mention that if L is not an integer multiple of the fundamental frequency of the sinusoidal input signal, Eq. (20) is no longer valid to use for the computation of the performance or SNR gain. Instead, the SNR gain can be obtained directly from Eq. (19) with SNR out being expressed by Eq. (16). Note that the theoretical derivation of SNR out for multiple sinusoids, i.e., M ¿ 2, is somewhat complicated. However, for illustrative purposes, it can be shown that for a dual sinusoid, M = 2, the theoretical output signal and noise powers are: Psignal =
B02 A20 L2 8 L−1
+
(17)
i=0
Since the input SNR can be expressed as M −1 2 j=0 Aj SNR in = ; 2w2
+
or
SNR out SNR in
L−1 L−1 B12 A20 cos(!1 iT ) 2 i=0 j=0
(18)
×cos(!1 jT ) cos(!0 (j − i)T ) +
the performance or SNR gain is, SNR gain =
B0 B1 A20 L cos(!1 iT )cos(!0 iT ) 2
L−1
(19)
B0 B1 A21 L cos(!0 iT ) cos(!1 iT ) 2
+
i=0
M −1
SNR gain =
2 2 L∗ j=0 Bj Aj M −1 2 M −1 2 2 j=0 Bj j=0 Aj
(20)
=
∗
M −1
+ (4=L )(w2 =A2j )]−2 M −1 (4=L∗ )(w2 =A2j )]−2 j=0
A2j [1
L j=0 M −1 2 j=0 [1 +
L−1 L−1 B02 A21 cos(!0 iT ) cos(!0 jT ) 2
+
i=0 j=0
and in terms of the /lter coeLcients, Bm , Eq. (20) can be written as: SNR gain
B12 A21 L2 8
∗
A2j
:
(21)
Note that Eq. (20) represents the theoretical gain or performance of the ALE. It is clearly seen that this gain is a function of the /lter length, the number of sinusoids, as well as their corresponding amplitudes. Furthermore, if the amplitudes of the sinusoidal signal are equal, Eq. (20) reduces simply to SNR gain = L∗ =2M , an indication that the SNR gain increases linearly with L∗ and inversely with M . Note also that for M = 1, Eq. (20) reduces to the conventional ALE gain of L∗ =2.
×cos(!1 (j − i)T )
(22)
N −1 1 ≈ [y(n) − ysignal (n)]2 N
(23)
and Pnoise
n=0
with ysignal (n) =
B0 A0 L cos(!0 nT ) 2 + B1 A0
L−1
cos(!1 iT )cos(!0 (n − i)T )
i=0
+ B 0 A1
L−1
cos(!0 iT )cos(!1 (n − i)T )
i=0
+
B 1 A1 L cos(!1 nT ): 2
(24)
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
Note that the noise power approximation is valid as long as the noise is covariance-ergodic.
3. Simulation results To verify the performance of the ALE in terms of the SNR gain, a dual sinusoidal signal of length N = 8192 is considered. In this simulation, N is chosen suLciently long to ensure the convergence of the ALE algorithm. DiIerent cases are simulated to investigate the eIects of the sinusoidal frequencies, their amplitudes, as well as the noise content on the performance. For illustrative purposes, the following parameters associated with the ALE implementation are used in this simulation: = 100; L ∈ [1; 2; : : : ; 50], and ∈ [0:0001; 0:0002; : : : ; 0:0075]. Note that the
value used corresponds to the optimal value. In each case considered, the experimental SNR gain values are obtained via averaging 40 runs, which results in what is known as the statistical sample mean. Furthermore, the eIect of the delay parameter on the overall performance of the ALE, thus the SNR gain,
97
is not investigated in this study. The only consideration taken is that should be large enough to decorrelate the noise component. Figs. 2 and 3 are plots of the SNR gain versus the /lter length L for a dual sinusoidal input, with different noise levels, for L∗ = k · 20, with k being an arbitrary integer. These /gures represent the resulting SNR gain for the dual sinusoidal signal with separate frequencies for two cases: equal and diIerent amplitudes. For an ideal /lter length L∗ satisfying Eq. (5), the theoretical SNR gain values are obtained directly from Eq. (20). These values are denoted by the symbol “∗ ”. Accordingly, only the discrete point values at L∗ = k · 20 correspond to the theoretical values. Everywhere else, there is no basis for comparison since the theoretical values are not determined over the interval of interest. Then, the SNR gain values obtained from the simulation results via Eqs. (22) – (24) for L values within the speci/ed range, L ∈ [1; 2; : : : ; 50], are directly compared with their corresponding theoretical values to verify the performance of the ALE method. Theoretically, both should match exactly at L = L∗ . These /gures illustrate clearly this attribute, except in few cases where the degradation in the performance
SNR Gain
15
σ w2 = 1
10 5 0
0
5
10
15
20
25
30
35
40
45
50
10
15
20
25
30
35
40
45
50
SNR Gain
15 10
σ w2 = 3
5 0
0
5
SNR Gain
15 10
σ w2 = 5
5 0
Theoretical simulation 0
5
10
15
20 25 30 Filter Length: L
Fig. 2. SNR gain comparison of a dual sinusoid with equal x(n) = cos((2 =5)n) + cos((9 =10)n) + w(n); L∗ = k · 20; k = 1; 2; : : :.
35
amplitudes
40
and
45
separate
50
individual
frequencies,
i.e.,
98
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
SNR gain
15
σ w2 = 1
10 5 0
0
5
10
15
20
25
30
35
40
45
50
10
15
20
25
30
35
40
45
50
SNR gain
15
σ w2 = 3
10 5 0
0
5
SNR gain
20
σ w2 = 5
10 0
Theoretical Simulation 0
5
10
15
20 25 30 Filter Length: L
35
40
45
50
Fig. 3. SNR gain comparison of a dual sinusoid with diIerent amplitudes and separate individual frequencies, i.e., x(n) = cos((2 =5)n) + 4 cos((9 =10)n) + w(n); L∗ = k · 20; k = 1; 2; : : :
Fig. 4. The eIect of the amplitude on the SNR gain for a dual sinusoidal signal at various noise levels, L∗ = 20.
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
99
Fig. 5. The eIect of the amplitude on the SNR gain for a multiple sinusoidal signal (M = 4) at various noise levels, L∗ = 20.
can be attributed to the algorithm’s instability induced by a substantial increase in the input power since 0 ¡ stability ¡
2 ; LPinput
(25)
where Pinput is the corresponding power of the dual sinusoidal input signal and=or higher noise levels, i.e., lower SNR values. Note that the L = 1 case is a degenerative case and is of no importance to this comparison. It is also of interest to further investigate the effect of the noise level as well as the amplitudes of the sinusoidal signal on the SNR gain. Figs. 4 and 5 illustrate this concept for M = 2 and 4, respectively. It is apparent from these /gures that higher amplitudes result in higher SNR gain values at any given noise variance. Furthermore, the noise level, from which the input SNR can be obtained via Eq. (18), does clearly aIect the performance or SNR gain. Note that, for illustrative purposes, only one sinusoidal amplitude is varied while the rest is kept the same. It is worthy to mention that for sinusoidal signals with equal amplitudes, a broadband eIect is obtained. Accordingly, as one sinusoidal term increases in amplitude, i.e., A4 ¿ A1 ; A2 ; A3 for the M =4 case, the broadband case starts looking like a narrowband case.
Since the general equation derived for the SNR gain (Eq. (20)) has proven to be a good indicator of the performance of the ALE algorithm, the performance trend as a function of the number of sinusoids, for a /xed noise power and sinusoidal amplitudes, is examined. Fig. 6 illustrates this trend for a sinusoidal signal with unity equal amplitudes and a unity noise variance. This /gure reveal that the ALE performance is degraded as the input signal’s bandwidth is increased (M increases) and careful consideration has to be taken in the implementation of the ALE algorithm. 4. Conclusion A quantitative analysis of the ALE method has been performed in details. For multiple sinusoids in noisy environment, it is shown that the output SNR gain is a function of not only the /lter length, but also is a function of three additional factors — the number of sinusoids, the noise power, and the amplitude of each sinusoid. Simulation results reveal the performance trends in the ALE implementation and the eIect of these factors. Accordingly, careful consideration has to be taken before using the ALE in telecommunication type applications.
100
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
Fig. 6. Theoretical SNR gain vs. the number of sinusoids for the signal model, x(n) = Am = 1; m = 0; 1; : : : ; M − 1, w2 = 1; and L∗ = 20.
Appendix A. It is desired to derive the signal and noise components of the /lter output and subsequently the SNR gain of the adaptive line enhancer under the assumptions made (please refer to Eqs. 2–5). Accordingly, the /lter output y(n) can be expressed: y(n) =
∗ L −1
∗
f (i)x(n − i);
i=0
(A.1)
i=0
×
Am cos(!m nT ) + N (0; w2 );
×
j=0
M −1
Ak cos(!k (n − i)T )
k=0
where f∗ (n) and x(n) denote the /lter coeLcients and sinusoidal input signal, respectively. A direct substitution of Eqs. (2) and (6) into (A.1) yields, y(n) =
m=0
Assume that the phase angles have the same coherence and separate y(n) into two distinct components, the signal and noise components can be obtained as follows: ∗ L −1 M −1 Bj cos(!j iT ) ysignal (n) =
i=0
∗ L −1
M −1
M −1 k=0
M −1
=
i=0
=
j=0
∗ L −1 M −1 M −1
i=0
Ak cos(!k (n − i)T + k ) + w(n − i) : (A.2)
Bj Ak cos(!j iT )
j=0 k=0
×cos(!k (n − i)T )
Bj cos(!j iT + j )
∗ L −1 M −1 M −1
Bj Ak
j=0 k=0
1 cos{(!j + !k )i−!k n}T × 2 +cos{(!j −!k )i+!k n}T (A.3)
R.L. Campbell Jr. et al. / Signal Processing 82 (2002) 93 – 101
and
101
References
ynoise (n) =
∗ L −1
i=0
M −1
Bj cos(!j iT ) w(n − i):
j=0
(A.4) Eq. (A.3) can be further reduced using the following assumptions: Let !j = m · !fundamental and !k = n · !fundamental , where m and n are arbitrary integers, i.e., m and n ∈ I . Accordingly, the frequency sum and diIerence becomes !j + !k = (m + n) · !fundamental and !j − !k = (m − n) · !fundamental , where (m + n) and (m − n) ∈ I . Since the optimal /lter length L∗ satis/es Eq. (5), all sinusoidal components summed over the range {0; 1; : : : ; L∗ − 1} with a spacing of T and a frequency of ‘ · !fundamental , where ‘ ∈ I and ‘ = 0 will add to zero. Under these assumptions, Eq. (A.3) reduces to, ysignal (n) =
∗ L −1 M −1
i=0
=
Bj Aj [1=2 cos(!j nT )]
j=0
M −1 L∗ Bj Aj cos(!j nT ); 2
(A.5)
j=0
from which the signal power can be derived. Note that the orthogonality property stated above holds for any arbitrary fundamental frequency and for the M = 2 case, it is suLcient to integrate over a period of Nf = (f2 − f1 )=2 = 1=L∗ for the sinusoidal frequencies to be orthogonal.
[1] G. Abe, Residential Broadband, Cisco Press, Indianapolis, IN, 1997. [2] T.R. Albert, H. Abusalem, M.D. Juniper, Experimental results: Detection and tracking of low SNR sinusoids using real-time LMS and RLS lattice adaptive line enhancers, Proc. IEEE Int. Conf. Acoust. Speech Signal Process 3 (May 1991) 1857–1860. [3] P.M. Clarkson, Optimal and Adaptive Signal Processing, CRC Press Inc., Boca Raton, FL, 1993. [4] M. Ghogho, M. Ibnkahla, J.N. Bershad, Analytic behavior of the LMS adaptive line enhancer for sinusoids corrupted by multiplicative and additive noise, IEEE Trans. Signal Process. 46 (9) (September 1998) 2386–2393. [5] S. Haykin, Adaptive Filter Theory, 3rd Edition, Prentice-Hall Inc., Englewood CliIs, NJ, 1996. [6] S.J. Orfanidis, Optimum Signal Processing, McGraw-Hill Publishing Company, New York, 1988. [7] S.V. Raghavan, S.K. Tripathi, Networked Multimedia Systems: Concepts, Architecture, and Design, Prentice-Hall, Inc., Englewood CliIs, NJ, 1998. [8] W.D. Reeve, Subscriber Loop Signaling and Transmission Handbook: Digital, IEEE Press, New York, 1992. [9] J.D. Spragins, J.L. Hammond, K. Pawlikowski, Telecommunications: Protocols and Design, Addison-Wesley Publishing Company, Reading, MA, 1994. [10] E. Watanabe, Y. Okamura, A. Nishihara, Adaptive line enhancers with discriminated structures, Electron. Commun. Jpn, Part III: Fundam. Electron. Sci. 79 (11) (November 1996) 57–68. [11] B. Widrow, S. Stearns, Adaptive Signal Processing, Prentice-Hall Inc., Englewood CliIs, NJ, 1985. [12] J.R. Zeidler, E.H. Satorius, D.M. Chabries, H.T. Wexler, Adaptive enhancement of multiple sinusoids in uncorrelated noise, IEEE Trans. Acoust. Speech Signal Process. ASSP-26 (1978) 240–254.