Orthogonal Multi-carrier Bandwidth Modulation ... - Semantic Scholar
ORTHOGONAL MULTICARRIER BANDWIDTH MODULATION SCHEME FOR WIRELESS COMMUNICATIONS Yi Ma, Yi Huang, Waleed Al-Nuaimy and Yiyuan Xiong Dept. of Electrical Engineering & Electronics, University of Liverpool, L69 3GJ, Liverpool, U.K. [email protected], [email protected]
Abstract - A new orthogonal multicarrier bandwidth modulation (OMBM) scheme is proposed in this paper. The distinct feature of this scheme is that the frequency domain data is used to decide the number of orthogonal subcarriers for wireless transmissions. Different bandwidth of the OMBM symbol in the time domain reflects on the different data in the frequency domain. A Discrete Fourier Transform (DFT) based Minimum Mean Square Error (MMSE) estimator is employed to conduct the bandwidth detection. Analysis and simulation results demonstrate that the OMBM scheme is a flexible modulation structure and not sensitive to noise interference. Some properties and possible application of the OMBM code are also introduced. Keywords - Orthogonal multicarrier scheme, carrier frequency offset, DFT, MMSE
all OFDM systems, the signal bandwidth and the number of subcarriers are fixed. In this paper, we propose an orthogonal multicarrier bandwidth modulation scheme, where the subchannel bandwidth varies according to the data in the frequency domain. In other words, the OMBM system gets the sum of I m subcarriers without modulation of the complex symbol, and I m < the total subcarriers N m of the subchannel, where m denotes the mth subchannel of the whole signal bandwidth. Therefore different frequency domain data are represented by the different bandwidth of the subchannel. In order to accurately detect the subchannel bandwidth from a single OMBM symbol, the DFT-MMSE estimator is introduced in this paper. II. OMBM BASEBAND MODEL
I. INTRODUCTION Orthogonal multicarrier modulation techniques were first introduced in military communications in the 1960s [1]. In recent years, there has been considerable interest in applying orthogonal frequency division multiplex (OFDM) in the physical layer of broadband wireless access systems [2], as well as mobile multimedia communications because of its advantages in lessening the severe effects of frequencyselective fading [3]. However, the high data-rate and spectrum-efficient OFDM systems are sensitive to carrier frequency offset (CFO) errors and phase noise interference [4]-[9]. Some complex algorithms are required to estimate and track the fading channel parameters, and then to adjust timing and frequency errors. The spectrum of the OFDM signal is derived from its time domain representation. A single OFDM symbol without inter-symbol interference (ISI) in the time domain can be represented as
N −1 t SOFDM (t ) = ∑ g n e jϖ nt × rect N ⋅ Ts n =0 which is the sum of N subcarriers e a complex symbol
g n = Gn e
jφ n
jω n ⋅t
subcarriers and modulated by a K-bit data. The mathematical model for one OMBM subchannel is represented as:
I m − 1 j ω ⋅t S m ( t ) = ∑ e m , n × rect n=0
t N m ⋅ Ts
(2)
where the I m th subcarrier is called characteristic subcarrier (1)
, each modulated by
[6], and windowed by a
rectangular window of the OFDM symbol duration N Ts . In
0-7803-7589-0/02/$17.00 ©2002 IEEE
We consider an orthogonal multi-carrier wireless communication system that consists of N subcarriers. K-bit data is encoded by the channel encoder, where K is a variable for the corresponding modulation method. For example, K = 2, 4, and 6 for BPSK, QPSK and 8-QAM respectively. The encoded data goes through a serial-toparallel converter, and is divided into l streams of data. In OFDM systems, l = N , and each subcarrier is modulated by the encoded data. Unlike in OFDM systems, OMBM splits the whole bandwidth of the orthogonal multicarrier system into l subchannels; each subchannel is of N m
(CS) of the mth subchannel, and it is decided by the value of k
the K-bit data, with 2 < max( I m ) < N m ; Ts is the
ω m,n is the frequency of the nth subcarrier in subchannel; and Bm = I m ⋅ ∆f is called the
sample time; the mth
effective bandwidth of the mth subchannel. The OMBM transmitters employ an inverse fast Fourier transform (IFFT)
PIMRC 2002
of size N for modulation, which require the frequency difference of consecutive subcarriers ∆f = 1 NTs . Hence,
When min(G n ,k < I ) > max(Q n ,k > I ) , we can find a threshold m m
A = (min(Gn,k < I m ) + max(Qn,k > I m )) 2
a single OMBM symbol in time domain is
s(k ) =
1 N
N
∑g n=0
n
⋅ e j ( 2 π n ⋅k / N )
(3)
1 , n < I m , m = 1,2,L, l gn = otherwise 0 ,
and
so that the efficient bandwidth can be easily detected by using a DFT based estimator. According to the central limitation theorem, n q , k obeys to the same Gaussian distribution as n( k ) , and the cumulative distribution function (CDF) is
After transmission over a multipath channel, the samples of the transmitted baseband OMBM signal at the receiver are Lh
r (k ) = ∑ s (k − i ) ⋅ hq (i ) + n(k )
(4)
where hq (i ) is the sampled complex channel impulse response for the ith propagation path at the qth time slot and n( k ) is complex white Gaussian noise. It is assumed that the channel does not vary within an OMBM symbol duration, and the synchronization is perfect, the DFT demodulated result of the received signal at time slot q and subcarrier n can therefore be expressed as (5)
where nq ,n is also the complex Gaussian white noise and the transfer function:
H q ,n =
∑h i =0
q
(k ) ⋅ e j 2πn⋅i / N
m = 1,2,L, l (7)
and,
Gn , k < I m = g q , k < I m H q , k < I m + n q , k < I m
(8)
Qn , k > I m = g q , k > I m H q , k > I m + n q , k > I m
(9)
Qn,k>Im is the redundant bandwidth within the
subchannel. Substituting Equation (3) for g q ,k in Equation (8) and (9) gives
Qn , k > I m = n q , k > I m
e −( x − µ )
2
2σ n2
dx
(13)
σ n and µ are the standard deviation and mean of the m
Hence the error probability in the direct DFT detection algorithm is 1 − Q r ( A) , with a signal to noise ratio (SNR) 2
of E b
2
σn.
To improve the demodulation performance, we employed a DFT-MMSE based bandwidth estimator, which uses the adaptive filter [10], as illustrated in Fig.1. x (n ) = d (n ) + v(n ) e(n )
dˆ (n ) Wn(z)
y(ϖ ) DFT
x( n ) is the input signal, d ( n ) the OMBM signal, v ( n ) the wide-sense stationary noise, dˆ ( n ) the Where
estimated signal, Wn (z ) the adaptive weighting function and
y (ϖ ) the frequency domain representation of the estimated
signal. As the noise is cancelled by the adaptive filter, we can employ differential operation in the frequency domain to find the CS of each subchannel. IV. OMBM CODE AND ITS PROPERTIES
where Gn,k < I m is the effective bandwidth of the mth
Gn ,k < I m = H q ,k < I m + nq ,k < I m
x
−∞
Fig. 1 DFT-MMSE based bandwidth estimator
From Equation (5), the demodulation result at the receiver can be represented as
Yq ,n = Gn ,k < I m + Qn ,k > I m
2π σ n
∫
Gn,k I cannot be separated by the threshold.
(6)
III. DFT-MMSE BASED DEMODULATION
subchannel;
where
Z − k0
Lh −1
1
Qr ( x) =
noise n( k ) respectively. Errors will only occur if x > A , as
i =0
Yq ,n = g q ,n H q ,n + nq ,n
(12)
(10) (11)
It is well known that the amplitude, frequency and phase of transmitted signals are usually affected by channel noise, CFO and phase noise, but the signal bandwidth is usually not sensitive to these interferences. OMBM receives the benefits from the bandwidth estimate technique and produces relatively low BER. In our simulation, we notice that the CS suffers from noise interference if the SNR is lower than 10dB, so a dynamic range should be employed for the CS to reduce the BER. It requires enough subcarriers
within one subchannel, the more the subcarriers , and the smaller the BER . Performance is best when N m = N . Although the OMBM code is not bandwidth efficient, it offers some good properties in interference cancellation, ID recognition, etc. It can be combined with the OFDM to enhance the physical layer of broadband wireless access systems. A. Phase noise cancelling Phase noise is commonly caused by the timing synchronization error and local oscillator. Assuming that the channel is flat and the signal is only affected by phase noise φ N at the receiver, then the received signal can be represented by
r (n) = s(n) ⋅ e
jφ N
(14)
The received signal is processed by DFT. In order to separate the signal and noise terms, Equation (14) can be expressed as a Taylor series,
( jφ N ) m m! m =1 ∞
r (n) = s (n) + s(n) ⋅ ∑
(15)
The signal in the frequency domain is, Yk = g k +
1 2π
( jφ N ) m g k ∗ F m! m =1 ∞
∑
where g k is the DFT result of
(16)
s ( n) .
Let Ψ (k ) = F ( jφ N ) m , which according to the central m
m!
limitation theorem, Ψ m (k ) is also white Gaussian noise. Obviously, the differential of the DFT result is ∞
∆Yk ≈ ∑Ψ m (k ) 10dB. Fig. 3 shows that different dynamic ranges also result in different BER performance, and the more subcarriers are employed, the lower the BER can be achieved. Our simulation also indicates if the dynamic range of the CS has 3 redundant subcarriers, then the demodulation result is not sensitive to the CFO.
C. ID recognition and the application in TDMA/OFDM systems The symbol timing property offers a useful application of the OMBM code in TDMA/OFDM wireless communication systems. Assume that the OFDM code employs 512 subcarriers in the frequency domain with a guard interval T g in the time domain, and one downlink frame of the TDMA system includes N u time slots, with each slot for a specified user. Obviously, this kind of frame structure is fixed and the transmission efficiency is low as all slots within the frame are usually not fully taken up. We generate an OMBM code with 512 subcarriers, and add the head of the time domain code with duration Th < T g to the guard interval of the OFDM code. When the OFDM code passes
through the DFT-MMSE estimator, we find the CS of the OMBM code in the frequency domain. If the TDMA/OFDM system allocates N u different OMBM codes to each user as the user’s ID, then the DFT-MMSE algorithm of the receiver can detect the user’s ID from the OFDM codes. In this case, the TDMA downlink frame structure can be dynamically configured. The base-station can allocates the slots within one frame based on the required data rate of the terminals, and all slots are used even if there is only one user. Thus the OMBM code offers a dynamic frame structure to the TDMA/OFDM downlink. VI. CONCLUSION In this paper, we have presented and analysed the OMBM modulation scheme, code and the DFT-MMSE demodulation algorithm in noise fading channels. Analysis and simulation results have shown that the OMBM code has advantages in CFO, symbol timing, phase noise cancelling, and ID recognition. Moreover, it can be employed in TDMA/OFDM systems to construct a dynamic downlink model to improve the transmission efficiency REFERENCES [1] R. V. Nee, R. Prasad, OFDM for wireless multimedia communications, pp.1-6 Artech House, Hardcover, January 2000 [2] IEEE802.16 unapproved working document, “Standard air interface for fixed broadband wireless access systems-media access control modifications and additional physical layer for 2-11GHz”. Sep. 2001 [3] B. Yang, Z. Cao, K.B. Letaief, “Analysis of lowcomplexity windowed DFT-based MMSE channel estimation for OFDM systems”, IEEE Trans. On Comm., Vol. 49, No. 11, pp. 1977-1985, November 2001. [4] P. Shelswell, “The COFDM modulation system: the heart of digital audio broadcasting,” BBC Research and Development Report, Aug 1996. [5] T. Keller and L. Hanzo, “Adaptive multicarrier modulation: a convenient framework for time-frequency in wireless communications,” Proceedings of the IEEE, vol. 88, no. 5, pp. 611-640, May 2000. [6] T. Keller, L. Piazzo, P. Mandarini, and L. Hanzo, “Orthogonal frequency division multiplex synchronization techniques for frequency-selective fading channels,” IEEE Journal On Selected Areas in Comm, vol. 19, no. 6, pp. 999-1008, June 2001 [7] S. B. Weinstein and P. M. Ebert, “Data transmission by frequency division multiplex using the discrete Fourier transform,” IEEE Trans. Comm., vol. 19, pp.628-634, Oct. 1971 [8] A. G. Armada, “Understanding the effects of phase noise in orthogonal Frequency division multiplexing
(OFDM),” IEEE Trans. Broadcasting, vol. 47, no. 2 pp.153-159, June 2001. [9] H. Steendam, M. Moeneclaey, "Sensitivity of OFDM and MC-CDMA to Carrier Phase Errors," 6th Symposium on Vehicular Technology and Communications, Oct 1213, 1998 Brussels, Belgium [10] M. H. Hayes, Statistical digital signal processing and modeling, pp. 493-554, John Wiley & Sons, Inc, 1996.
Abstract - A new orthogonal multicarrier bandwidth modulation (OMBM) scheme is proposed in this paper. The distinct feature of this scheme is that the frequency domain data is used to decide the number of orthogonal subcarriers for wireless transmissions. Different bandwidth of the OMBM symbol in the time domain reflects on the different data in the frequency domain. A Discrete Fourier Transform (DFT) based Minimum Mean Square Error (MMSE) estimator is employed to conduct the bandwidth detection. Analysis and simulation results demonstrate that the OMBM scheme is a flexible modulation structure and not sensitive to noise interference. Some properties and possible application of the OMBM code are also introduced. Keywords - Orthogonal multicarrier scheme, carrier frequency offset, DFT, MMSE
all OFDM systems, the signal bandwidth and the number of subcarriers are fixed. In this paper, we propose an orthogonal multicarrier bandwidth modulation scheme, where the subchannel bandwidth varies according to the data in the frequency domain. In other words, the OMBM system gets the sum of I m subcarriers without modulation of the complex symbol, and I m < the total subcarriers N m of the subchannel, where m denotes the mth subchannel of the whole signal bandwidth. Therefore different frequency domain data are represented by the different bandwidth of the subchannel. In order to accurately detect the subchannel bandwidth from a single OMBM symbol, the DFT-MMSE estimator is introduced in this paper. II. OMBM BASEBAND MODEL
I. INTRODUCTION Orthogonal multicarrier modulation techniques were first introduced in military communications in the 1960s [1]. In recent years, there has been considerable interest in applying orthogonal frequency division multiplex (OFDM) in the physical layer of broadband wireless access systems [2], as well as mobile multimedia communications because of its advantages in lessening the severe effects of frequencyselective fading [3]. However, the high data-rate and spectrum-efficient OFDM systems are sensitive to carrier frequency offset (CFO) errors and phase noise interference [4]-[9]. Some complex algorithms are required to estimate and track the fading channel parameters, and then to adjust timing and frequency errors. The spectrum of the OFDM signal is derived from its time domain representation. A single OFDM symbol without inter-symbol interference (ISI) in the time domain can be represented as
N −1 t SOFDM (t ) = ∑ g n e jϖ nt × rect N ⋅ Ts n =0 which is the sum of N subcarriers e a complex symbol
g n = Gn e
jφ n
jω n ⋅t
subcarriers and modulated by a K-bit data. The mathematical model for one OMBM subchannel is represented as:
I m − 1 j ω ⋅t S m ( t ) = ∑ e m , n × rect n=0
t N m ⋅ Ts
(2)
where the I m th subcarrier is called characteristic subcarrier (1)
, each modulated by
[6], and windowed by a
rectangular window of the OFDM symbol duration N Ts . In
0-7803-7589-0/02/$17.00 ©2002 IEEE
We consider an orthogonal multi-carrier wireless communication system that consists of N subcarriers. K-bit data is encoded by the channel encoder, where K is a variable for the corresponding modulation method. For example, K = 2, 4, and 6 for BPSK, QPSK and 8-QAM respectively. The encoded data goes through a serial-toparallel converter, and is divided into l streams of data. In OFDM systems, l = N , and each subcarrier is modulated by the encoded data. Unlike in OFDM systems, OMBM splits the whole bandwidth of the orthogonal multicarrier system into l subchannels; each subchannel is of N m
(CS) of the mth subchannel, and it is decided by the value of k
the K-bit data, with 2 < max( I m ) < N m ; Ts is the
ω m,n is the frequency of the nth subcarrier in subchannel; and Bm = I m ⋅ ∆f is called the
sample time; the mth
effective bandwidth of the mth subchannel. The OMBM transmitters employ an inverse fast Fourier transform (IFFT)
PIMRC 2002
of size N for modulation, which require the frequency difference of consecutive subcarriers ∆f = 1 NTs . Hence,
When min(G n ,k < I ) > max(Q n ,k > I ) , we can find a threshold m m
A = (min(Gn,k < I m ) + max(Qn,k > I m )) 2
a single OMBM symbol in time domain is
s(k ) =
1 N
N
∑g n=0
n
⋅ e j ( 2 π n ⋅k / N )
(3)
1 , n < I m , m = 1,2,L, l gn = otherwise 0 ,
and
so that the efficient bandwidth can be easily detected by using a DFT based estimator. According to the central limitation theorem, n q , k obeys to the same Gaussian distribution as n( k ) , and the cumulative distribution function (CDF) is
After transmission over a multipath channel, the samples of the transmitted baseband OMBM signal at the receiver are Lh
r (k ) = ∑ s (k − i ) ⋅ hq (i ) + n(k )
(4)
where hq (i ) is the sampled complex channel impulse response for the ith propagation path at the qth time slot and n( k ) is complex white Gaussian noise. It is assumed that the channel does not vary within an OMBM symbol duration, and the synchronization is perfect, the DFT demodulated result of the received signal at time slot q and subcarrier n can therefore be expressed as (5)
where nq ,n is also the complex Gaussian white noise and the transfer function:
H q ,n =
∑h i =0
q
(k ) ⋅ e j 2πn⋅i / N
m = 1,2,L, l (7)
and,
Gn , k < I m = g q , k < I m H q , k < I m + n q , k < I m
(8)
Qn , k > I m = g q , k > I m H q , k > I m + n q , k > I m
(9)
Qn,k>Im is the redundant bandwidth within the
subchannel. Substituting Equation (3) for g q ,k in Equation (8) and (9) gives
Qn , k > I m = n q , k > I m
e −( x − µ )
2
2σ n2
dx
(13)
σ n and µ are the standard deviation and mean of the m
Hence the error probability in the direct DFT detection algorithm is 1 − Q r ( A) , with a signal to noise ratio (SNR) 2
of E b
2
σn.
To improve the demodulation performance, we employed a DFT-MMSE based bandwidth estimator, which uses the adaptive filter [10], as illustrated in Fig.1. x (n ) = d (n ) + v(n ) e(n )
dˆ (n ) Wn(z)
y(ϖ ) DFT
x( n ) is the input signal, d ( n ) the OMBM signal, v ( n ) the wide-sense stationary noise, dˆ ( n ) the Where
estimated signal, Wn (z ) the adaptive weighting function and
y (ϖ ) the frequency domain representation of the estimated
signal. As the noise is cancelled by the adaptive filter, we can employ differential operation in the frequency domain to find the CS of each subchannel. IV. OMBM CODE AND ITS PROPERTIES
where Gn,k < I m is the effective bandwidth of the mth
Gn ,k < I m = H q ,k < I m + nq ,k < I m
x
−∞
Fig. 1 DFT-MMSE based bandwidth estimator
From Equation (5), the demodulation result at the receiver can be represented as
Yq ,n = Gn ,k < I m + Qn ,k > I m
2π σ n
∫
Gn,k I cannot be separated by the threshold.
(6)
III. DFT-MMSE BASED DEMODULATION
subchannel;
where
Z − k0
Lh −1
1
Qr ( x) =
noise n( k ) respectively. Errors will only occur if x > A , as
i =0
Yq ,n = g q ,n H q ,n + nq ,n
(12)
(10) (11)
It is well known that the amplitude, frequency and phase of transmitted signals are usually affected by channel noise, CFO and phase noise, but the signal bandwidth is usually not sensitive to these interferences. OMBM receives the benefits from the bandwidth estimate technique and produces relatively low BER. In our simulation, we notice that the CS suffers from noise interference if the SNR is lower than 10dB, so a dynamic range should be employed for the CS to reduce the BER. It requires enough subcarriers
within one subchannel, the more the subcarriers , and the smaller the BER . Performance is best when N m = N . Although the OMBM code is not bandwidth efficient, it offers some good properties in interference cancellation, ID recognition, etc. It can be combined with the OFDM to enhance the physical layer of broadband wireless access systems. A. Phase noise cancelling Phase noise is commonly caused by the timing synchronization error and local oscillator. Assuming that the channel is flat and the signal is only affected by phase noise φ N at the receiver, then the received signal can be represented by
r (n) = s(n) ⋅ e
jφ N
(14)
The received signal is processed by DFT. In order to separate the signal and noise terms, Equation (14) can be expressed as a Taylor series,
( jφ N ) m m! m =1 ∞
r (n) = s (n) + s(n) ⋅ ∑
(15)
The signal in the frequency domain is, Yk = g k +
1 2π
( jφ N ) m g k ∗ F m! m =1 ∞
∑
where g k is the DFT result of
(16)
s ( n) .
Let Ψ (k ) = F ( jφ N ) m , which according to the central m
m!
limitation theorem, Ψ m (k ) is also white Gaussian noise. Obviously, the differential of the DFT result is ∞
∆Yk ≈ ∑Ψ m (k ) 10dB. Fig. 3 shows that different dynamic ranges also result in different BER performance, and the more subcarriers are employed, the lower the BER can be achieved. Our simulation also indicates if the dynamic range of the CS has 3 redundant subcarriers, then the demodulation result is not sensitive to the CFO.
C. ID recognition and the application in TDMA/OFDM systems The symbol timing property offers a useful application of the OMBM code in TDMA/OFDM wireless communication systems. Assume that the OFDM code employs 512 subcarriers in the frequency domain with a guard interval T g in the time domain, and one downlink frame of the TDMA system includes N u time slots, with each slot for a specified user. Obviously, this kind of frame structure is fixed and the transmission efficiency is low as all slots within the frame are usually not fully taken up. We generate an OMBM code with 512 subcarriers, and add the head of the time domain code with duration Th < T g to the guard interval of the OFDM code. When the OFDM code passes
through the DFT-MMSE estimator, we find the CS of the OMBM code in the frequency domain. If the TDMA/OFDM system allocates N u different OMBM codes to each user as the user’s ID, then the DFT-MMSE algorithm of the receiver can detect the user’s ID from the OFDM codes. In this case, the TDMA downlink frame structure can be dynamically configured. The base-station can allocates the slots within one frame based on the required data rate of the terminals, and all slots are used even if there is only one user. Thus the OMBM code offers a dynamic frame structure to the TDMA/OFDM downlink. VI. CONCLUSION In this paper, we have presented and analysed the OMBM modulation scheme, code and the DFT-MMSE demodulation algorithm in noise fading channels. Analysis and simulation results have shown that the OMBM code has advantages in CFO, symbol timing, phase noise cancelling, and ID recognition. Moreover, it can be employed in TDMA/OFDM systems to construct a dynamic downlink model to improve the transmission efficiency REFERENCES [1] R. V. Nee, R. Prasad, OFDM for wireless multimedia communications, pp.1-6 Artech House, Hardcover, January 2000 [2] IEEE802.16 unapproved working document, “Standard air interface for fixed broadband wireless access systems-media access control modifications and additional physical layer for 2-11GHz”. Sep. 2001 [3] B. Yang, Z. Cao, K.B. Letaief, “Analysis of lowcomplexity windowed DFT-based MMSE channel estimation for OFDM systems”, IEEE Trans. On Comm., Vol. 49, No. 11, pp. 1977-1985, November 2001. [4] P. Shelswell, “The COFDM modulation system: the heart of digital audio broadcasting,” BBC Research and Development Report, Aug 1996. [5] T. Keller and L. Hanzo, “Adaptive multicarrier modulation: a convenient framework for time-frequency in wireless communications,” Proceedings of the IEEE, vol. 88, no. 5, pp. 611-640, May 2000. [6] T. Keller, L. Piazzo, P. Mandarini, and L. Hanzo, “Orthogonal frequency division multiplex synchronization techniques for frequency-selective fading channels,” IEEE Journal On Selected Areas in Comm, vol. 19, no. 6, pp. 999-1008, June 2001 [7] S. B. Weinstein and P. M. Ebert, “Data transmission by frequency division multiplex using the discrete Fourier transform,” IEEE Trans. Comm., vol. 19, pp.628-634, Oct. 1971 [8] A. G. Armada, “Understanding the effects of phase noise in orthogonal Frequency division multiplexing
(OFDM),” IEEE Trans. Broadcasting, vol. 47, no. 2 pp.153-159, June 2001. [9] H. Steendam, M. Moeneclaey, "Sensitivity of OFDM and MC-CDMA to Carrier Phase Errors," 6th Symposium on Vehicular Technology and Communications, Oct 1213, 1998 Brussels, Belgium [10] M. H. Hayes, Statistical digital signal processing and modeling, pp. 493-554, John Wiley & Sons, Inc, 1996.
