Optimal Wavelet Design for Multicarrier Modulation ... - Semantic Scholar
Optimal Wavelet Design for Multicarrier Modulation with Time Synchronization Error D. Karamehmedović, M. K. Lakshmanan, H. Nikookar International Research Center for Telecommunications and Radar (IRCTR) Department of Electrical Engineering, Mathematics and Computer Science Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands [email protected], [email protected]
Abstract— Wavelet Packet based Multi-Carrier Modulation (WPMCM) is an efficient transmission technique which has the advantage of being a generic scheme whose characteristics can be customized to fulfill a design specification. However, WPMCM is sensitive and vulnerable to time synchronization errors because its symbols overlap. In this paper, we design new wavelets to alleviate WPMCM’s vulnerability to timing errors. First, a filter design framework that facilitates the development of new wavelet bases is built. Then the expressions for errors due to time offset in WPMCM transmission are derived and stated as a convex optimization problem. Finally, an optimal filter that best handles these deleterious effects is designed by means of Semi Definite Programming (SDP). Through computer simulations the performance advantages of the newly designed filter over standard wavelet filters are proven.
I. INTRODUCTION Wavelet Packet based Multi Carrier Modulation is a multiplexing method that uses orthogonal waveforms derived from a wavelet packet transform to combine a collection of parallel signals into a single composite signal. WPMCM is similar to the traditionally popular Fast Fourier Transform (FFT) based Orthogonal Frequency Division Multiplexing (OFDM) in the sense that both use orthogonal waveforms as subcarriers and achieve high spectral efficiency by allowing subcarriers’ spectra to overlap. The adjacent subcarriers do not interfere with each other as long as the orthogonality between the subcarriers is preserved. The difference between OFDM and WPMCM is in the shape of the subcarriers and the way they are generated. OFDM makes use of Fourier bases while WPMCM uses wavelet packet bases which are generated from a class of FIR filters called paraunitary filters. WPMCM was first proposed by Lindsey who laid out the theoretical foundations and propounded its use as an alternative to OFDM [1]. His idea has since been carried forward by many researchers. Maximum likelihood decoding for wavelet packet modulation has been addressed by Suzuki [2]. The study of an equalization scheme suited for WPMCM has been conducted by Gracias [3]. In [4]–[5] an investigation on the performance of WPMCM systems in the presence of time offset is performed. The greatest motivation for pursuing WPMCM systems is in the flexibility and adaptability that they offer. Unlike OFDM where the carriers are static sine/cosine bases, WPMCM uses wavelets whose features can be tailored to satisfy an engineering demand. Different wavelets result in different subcarriers leading to different transmission characteristics. By careful selection of proper wavelets it is possible to optimize WPMCM performance in terms of bandwidth efficiency, frequency selectivity of subcarriers, sensitivity to synchronization errors, PAPR, etc. The WPMCM is a developmental system and a lot of key research questions still remain to be addressed before it can become practically viable. One of them is their high sensitivity and vulnerability to time synchronization errors because of the overlap in WPMCM symbols.
The goals of this paper are two-fold. First to utilize wavelet theory to establish a framework that facilitates the design of new wavelet bases. Then to focus on the reduction of timesynchronization errors in WPMCM by developing a family of new wavelets that better cope with infarctions caused by time offset. To this end the expressions for Inter Carrier Interference (ICI) and Inter Symbol Interference (ISI) in WPMCM transmission caused by time offset are first derived and stated as a convex optimization problem. Then an optimal filter that best handles these deleterious effects is derived using optimization algorithms. Through computer simulations the performance advantages of the newly designed filter over standard wavelet filters is demonstrated. The rest of the paper is organized as follows. Section II outlines the basics of WPMCM transceiver while section III discusses the criteria for two-channel filter bank. Time offset in WPMCM is discussed in Section IV. In Section V the optimization problem is formulated and finally numerical results are presented in Section VI followed by conclusions which appear in Section VII. II. WAVELET PACKET BASED MULTI-CARRIER MODULATION The wavelet packet theory can be viewed as an extension of Fourier analysis. The basic idea of both transformations is the same: projecting an unknown signal on a set of known basis functions to obtain insights on the nature of the signal. Any function S(n) in L2 (ℜ) can be expressed as the sum of weighted wavelet packets. In communication systems, this means that a signal can be seen as the sum of modulated wavelet packets leading to the idea of WPMCM. The WPMCM signal is composed of symbols obtained from a sum of modulated and weighted wavelet packet waveforms ξ . In the discrete time domain this signal S(n) can be expressed as: N −1
S (n) = ∑∑ au , k ξ logk 2 ( N ) (n − uN ) u
(1)
k =0
In (1) N denotes the number of subcarriers while u and k are the symbol and subcarrier indices, respectively. The constellation symbol modulating kth subcarrier in uth symbol is represented as au,k. The sub-index log2(N) denotes the levels of decomposition required to generate N subcarriers. It is well known from the theory of wavelets that compactly supported orthonormal wavelet bases can be obtained from two-band paraunitary filter banks [6]–[8]. Time and frequency limited wavelet packet bases ξ(t) can be derived by iterating discrete half-band high g[n] and low-pass h[n] filters, recursively defined by: ξl2+p1 (t ) = 2 ∑ h[n]ξl p (t − 2l n) n
ξl2+p1 +1 (t ) = 2 ∑ g[ n]ξl p (t − 2l n)
(2)
n
In (2) the subscript l denotes the level in the tree structure and superscript p indicates the tree depth. The number of channels
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
in WPMCM is determined by the number of iterations of the two-channel filter bank. In wavelet jargon (2) is called the 2scale equation and the low and high-pass filters are referred as scaling and wavelet filters, respectively. WPMCM systems are realized by a set of four paraunitary filters called Quadrature Mirror Filters (QMF): a pair halfband high g[n] and low pass filters h[n] called synthesis filters and another pair of half-band high g’[n] and low pass filters h’[n] called analysis filters. These four filters share a strict and tight relation and hence it is enough if the specifications of one of these filters are available. The wavelet packet sub-carriers (used at the transmitter end) are generated from the synthesis filters. And the wavelet packet duals (used at the receiver end) are obtained from the analysis filters. The exact procedure to derive these carriers can be found in [9]. III. WPMCM, WAVELETS AND FILTER CRITERIA The attributes of a WPMCM system greatly depend on the set of waveforms it uses which in turn is determined by the underlying filters used. This means that by adapting the filters one can conceive a WPMCM transceiver that best handles a system specification. The design of wavelets is bounded by multiple constraints. Besides the design objectives there are other budgets that should be considered in order to guarantee that the wavelet designed is valid. The constraints include properties such as orthogonality, compact support, symmetry, and smoothness and are usually stated in terms of the scaling filter h[n]. Here we shall touch upon a few of them. 1) Wavelet Existence and Compact Support: This property ensures that the wavelet has a finite number of non-vanishing coefficients and the filters are of finite length. Wavelet existence imposes a single linear constraint on h[n] [6]-[7]: L −1
∑ h[n] =
(3)
2
n =0
2) Paraunitary Condition: The paraunitary condition is essential for many reasons. First, it is a prerequisite for generating orthonormal wavelets [7]–[8]. Second, it automatically ensures perfect reconstruction of the decomposed signal i.e., the original signal can be reconstructed without amplitude or phase or aliasing distortion. To satisfy the paraunitary constraint the scaling filter coefficients have to be orthogonal at even shift [6]-[7]: ⎧1 if m = 0 h[n]h[n − 2m] = δ [m] = ⎨ ∑ n =0 ⎩0 otherwise L −1
(4)
For a filter of length L the orthogonality condition (4) imposes L /2 non-linear constraints on h[n]. 3) Flatness/K-Regularity: This property is a rough measure of smoothness of the wavelet. The regularity condition requires that the wavelet be locally smooth and concentrated in both time and frequency domains. It is normally quantified by the number of times a wavelet is continuously differentiable. The simplest regularity condition is the flatness constraint which is stated on the low pass filter. A LPF is said to satisfy Kth order flatness if its transfer function H(ω) contains K zeroes located at the Nyquist frequency (ω = π). For any function Q(ω) with no poles or zeros at (ω = π) this can be written as [6]-[7]: K
⎛ 1 + e jω ⎞ H (ω ) = ⎜ ⎟ Q(ω ) ⎝ 2 ⎠
(5)
Parameter K is called the regularity order and for a filter of length L its degree is limited by 1 ≤ K ≤ L/2. In the time domain we can impose regularity condition (5) as [6]-[7]: L −1
∑n
k
(−1) n h[n] = 0
(6)
for k = 0,1,… K − 1
n =0
K-Regularity condition enforces K constraints on h[n]. IV. TIME OFFSET EFFECTS IN WPMCM One of the major concerns of multicarrier systems is their vulnerability to time synchronization errors which occur when the symbols are not perfectly aligned at the receiver. Because of the time offset, samples outside a WPMCM symbol get erroneously selected, while useful samples at the beginning or at the end of the symbol get discarded. Time offset degrades the performances of multicarrier transceivers by introducing ISI and ICI. WPMCM and OFDM share many similarities as both are orthogonal multicarrier systems but with regard to performance under timing error there are major differences. First, the length of the WPMCM symbols (which is determined by the wavelet used) is significantly longer than the OFDM symbol. Under a loss in time synchronization, the overlap of the symbols in WPMCM causes it to interfere with several other symbols while in OFDM each symbol interferes only with its neighbors. The second important difference is in the usage of guard intervals. OFDM uses cyclic prefix that significantly improves its performance under timing errors. WPMCM cannot benefit from guard interval since the WPMCM symbols overlap. Under ideal conditions when the WPMCM transmitter and receiver are perfectly synchronized and the channel is benign, the data estimate aˆu ', k ' is the same as the transmitted data au,k . The demodulation process is elucidated below: k' aˆu ', k ' = ∑ R(n)ξlog n
2
(N)
(u ' N − n )
N −1
= ∑∑∑ au , k ξ logk 2 ( N ) (n − uN )ξ logk ' 2 ( N ) (u ' N − n) n
u k =0
⎛ ⎞ = ∑∑ au , k ⎜ ∑ ξ logk 2 ( N ) (n − uN )ξ logk ' 2 ( N ) (u ' N − n) ⎟ u k =0 ⎝ n ⎠ N −1
(7)
N −1
= ∑ au ', k δ (k − k ') = au ', k ' k =0
Here R(n) stands for the received sequence and ξ k ' are receiver carriers which are the duals of the transmitter carriers ξ k . The apostrophe is used to indicate receiver side. The time synchronization error is modeled by shifting the received data samples by a time offset tε to the left or right, depending on the sign of the tε. The demodulated data with a time offset tε can be written as: aˆu ', k ' = ∑ R (n)ξ k ' (u ' N − n + tε ) log ( N ) 2
n
N −1
k = ∑∑∑ au , k ξ log n
u
k =0
2
(N )
(n − uN )ξ logk ' 2 ( N ) (u ' N − n + tε ) + w(n)
(8)
N −1 ⎛ ⎞ = ∑∑ au , k ⎜ ∑ ξ logk 2 ( N ) (n − uN )ξ logk ' 2 ( N ) (u ' N − n + tε ) ⎟ + w(n) u k =0 ⎝ n ⎠
As before the apostrophe is used to indicate the receiver side. To shorten the derivation we define a new notation:
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
Ω uk ,,uk '' (tε ) = ∑ ξ logk 2 ( N ) ( n − uN )ξ logk '2 ( N ) (u ' N − n + tε )
(9)
n
This equation represents the autocorrelation and the crosscorrelation of the WPMCM waveforms referred by the subcarrier index k. When k = k’ the two subcarrier waveforms are time-reversed images of each other and (10) gives the autocorrelation sequence of the waveform k. In the other cases when k ≠ k’ the two waveforms correspond to different subcarriers and (10) stands for the cross-correlation between waveforms k and k’. Using (8) and (9) we can express the output of the receiver for kth subcarrier and uth symbol as: aˆu ',k ' = au ', k ' Ωuk ',' ku'' + Useful Signal
N −1
∑
au ', k Ωuk ,',ku'' +
k = 0; k ≠ k '
N −1
∑ ∑a
u,k
u ;u ≠ u ' k = 0 ICI
Ωuk ,,uk '' + wu ', k ' (10) Noise
ISI
In (11) the first term stands for the attenuated useful signal, second term denotes ICI, third term gives ISI and the last term stands for Gaussian noise. V. DESIGNING BEST BASES TO TACKLE TIME OFFSET ERRORS A. The Design Procedure The objective of the design process is to construct WPMCM carriers that better handle time offset. This is done by generating WPMCM subcarriers with large distances between each other, i.e. the cross-correlation between the waveforms is made as small as possible. Since the wavelet bases used as the WPMCM carriers implicitly share a relationship with the underlying filter banks, the design process can be converted into a filter design problem. Indeed in addition to the objective of reduction of time-error, the filters also have to satisfy additional constraints such as: • the paraunitary condition • the flatness/regularity condition • the filters must allow orthonormal expansion and perfect reconstruction of discrete-time signals • have finite impulse response For a filter of length L this is essentially solving L unknown filter variables from L linear equations. Of these L linear equations, one equation is required to satisfy the wavelet existence condition, L/2 equations come from the paraunitaryness constraint, K-1 equations come from the regularity constraint and the remaining L/2-K conditions offer the possibility for establishing the design objective. Larger the value of L/2-K, greater the degree of freedom for design and greater is the loss in regularity. There is therefore a trade-off on offer. The L/2-K degrees of freedom that remain after satisfying wavelet existence, orthogonality and K-regularity condition can be used to design a scaling filter which minimizes the interference energy caused by the timing offset. The design procedure consists of 3 major steps: 1. Formulation of design problem, i.e. stating the design objectives and the constraints necessary for wavelet generation 2. Making the problem tractable by converting it into a convex optimization problem 3. Using suitable optimization tools to solve the problem and obtain the required filter coefficients. In the following segments we will elaborate on each of the points stated above.
B. Problem Formulation The information carried by the waveforms that overlap one another can only be correctly decoded if the waveforms used have large distances between one another. In WPMCM this is achieved through the orthogonality of the waveforms. Therefore, in disturbance-free environments the crosscorrelations of WPMCM waveforms equal zero and perfect reconstruction is possible despite time and frequency overlap. A timing error tε, however, leads to the loss of the orthogonality between the waveforms and they begin to interfere one with another leading to ICI and ISI, stated as: k k' (11) Ωuk ,,uk '' (tε ) = ∑ ξ log ( N ) ( n − uN )ξ log ( N ) (u ' N − n + tε ) k ≠k '
2
n
2
The design objective would therefore be to generate wavelet bases ξ and their duals ξ ' that minimize interference energy in the presence of timing error, i.e. Ωuk ,,uk '' (tε )
MINIMIZE:
2 u , k ;k ≠ k '
with respect to {ξ , ξ '}
(12)
C. Wavelet Domain to Filter Bank Domain The waveforms in WPMCM are created by the multilayered tree structure filter bank. Using Parseval’s theorem of energy conservation it can be easily proved that the total energy at each level is equal regardless of the tree’s depth. Therefore, minimizing the interfering energy at the roots of the tree will automatically lead to the decrease of total interfering energy at the higher tree branches. Furthermore, the two-channel filter banks through the 2-scale equation are unambiguously related to the WPMCM waveforms. Therefore the design process can be easily converted into a tractable filter design problem. We should hence be able to minimize deleterious effects of time synchronization errors in WPMCM by minimizing the following cross-correlation function: 2 2 2 rhg [m] = ∑ h[n]g[n − m] = ∑ h[n]((−1) n h[ L − n + m]) (13) n
n
The design problem of minimizing the interference energy due to timing offset can now be formally stated as an optimization problem with objective function (13) and constraints (3), (4) and (6), i.e. 2
MINIMIZE:
rhg [m] with respect to h[n] L −1
∑ h[n] =
(14)
2
n =0
SUBJECT TO:
L −1
∑ h[n]h[n − 2m] = δ (m) n =0
L −1
∑n
k
n =0
(−1)n h[n] = 0
⎛L ⎞ for m = 0,1,… , ⎜ − 1⎟ ⎝2 ⎠
(15)
for k = 0,1,… , ( K − 1)
D. Conversion of Non-Convex Problem to Convex Problem The objective function and the paraunitary condition are non-convex. Although there exist general purpose solvers which can solve non-convex problems, these algorithms are susceptible to being trapped in a local minima. In order to overcome this difficulty, some authors have suggested multiple starting point technique or branch-and-bound method to approach real minimum. However, these algorithms can be unstable, inefficient and do not guarantee a global minimum. Therefore, we will rewrite the original problem into a convex
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
form by expressing the objectives and constraints as convex functions of the scaling filter’s autocorrelation rh[n]. ⎧ L − n −1 ⎪ ∑ h[m]h[m + n] n ≥ 0 rh [n] = ⎨ m = 0 ⎪ r [ − n] n
Abstract— Wavelet Packet based Multi-Carrier Modulation (WPMCM) is an efficient transmission technique which has the advantage of being a generic scheme whose characteristics can be customized to fulfill a design specification. However, WPMCM is sensitive and vulnerable to time synchronization errors because its symbols overlap. In this paper, we design new wavelets to alleviate WPMCM’s vulnerability to timing errors. First, a filter design framework that facilitates the development of new wavelet bases is built. Then the expressions for errors due to time offset in WPMCM transmission are derived and stated as a convex optimization problem. Finally, an optimal filter that best handles these deleterious effects is designed by means of Semi Definite Programming (SDP). Through computer simulations the performance advantages of the newly designed filter over standard wavelet filters are proven.
I. INTRODUCTION Wavelet Packet based Multi Carrier Modulation is a multiplexing method that uses orthogonal waveforms derived from a wavelet packet transform to combine a collection of parallel signals into a single composite signal. WPMCM is similar to the traditionally popular Fast Fourier Transform (FFT) based Orthogonal Frequency Division Multiplexing (OFDM) in the sense that both use orthogonal waveforms as subcarriers and achieve high spectral efficiency by allowing subcarriers’ spectra to overlap. The adjacent subcarriers do not interfere with each other as long as the orthogonality between the subcarriers is preserved. The difference between OFDM and WPMCM is in the shape of the subcarriers and the way they are generated. OFDM makes use of Fourier bases while WPMCM uses wavelet packet bases which are generated from a class of FIR filters called paraunitary filters. WPMCM was first proposed by Lindsey who laid out the theoretical foundations and propounded its use as an alternative to OFDM [1]. His idea has since been carried forward by many researchers. Maximum likelihood decoding for wavelet packet modulation has been addressed by Suzuki [2]. The study of an equalization scheme suited for WPMCM has been conducted by Gracias [3]. In [4]–[5] an investigation on the performance of WPMCM systems in the presence of time offset is performed. The greatest motivation for pursuing WPMCM systems is in the flexibility and adaptability that they offer. Unlike OFDM where the carriers are static sine/cosine bases, WPMCM uses wavelets whose features can be tailored to satisfy an engineering demand. Different wavelets result in different subcarriers leading to different transmission characteristics. By careful selection of proper wavelets it is possible to optimize WPMCM performance in terms of bandwidth efficiency, frequency selectivity of subcarriers, sensitivity to synchronization errors, PAPR, etc. The WPMCM is a developmental system and a lot of key research questions still remain to be addressed before it can become practically viable. One of them is their high sensitivity and vulnerability to time synchronization errors because of the overlap in WPMCM symbols.
The goals of this paper are two-fold. First to utilize wavelet theory to establish a framework that facilitates the design of new wavelet bases. Then to focus on the reduction of timesynchronization errors in WPMCM by developing a family of new wavelets that better cope with infarctions caused by time offset. To this end the expressions for Inter Carrier Interference (ICI) and Inter Symbol Interference (ISI) in WPMCM transmission caused by time offset are first derived and stated as a convex optimization problem. Then an optimal filter that best handles these deleterious effects is derived using optimization algorithms. Through computer simulations the performance advantages of the newly designed filter over standard wavelet filters is demonstrated. The rest of the paper is organized as follows. Section II outlines the basics of WPMCM transceiver while section III discusses the criteria for two-channel filter bank. Time offset in WPMCM is discussed in Section IV. In Section V the optimization problem is formulated and finally numerical results are presented in Section VI followed by conclusions which appear in Section VII. II. WAVELET PACKET BASED MULTI-CARRIER MODULATION The wavelet packet theory can be viewed as an extension of Fourier analysis. The basic idea of both transformations is the same: projecting an unknown signal on a set of known basis functions to obtain insights on the nature of the signal. Any function S(n) in L2 (ℜ) can be expressed as the sum of weighted wavelet packets. In communication systems, this means that a signal can be seen as the sum of modulated wavelet packets leading to the idea of WPMCM. The WPMCM signal is composed of symbols obtained from a sum of modulated and weighted wavelet packet waveforms ξ . In the discrete time domain this signal S(n) can be expressed as: N −1
S (n) = ∑∑ au , k ξ logk 2 ( N ) (n − uN ) u
(1)
k =0
In (1) N denotes the number of subcarriers while u and k are the symbol and subcarrier indices, respectively. The constellation symbol modulating kth subcarrier in uth symbol is represented as au,k. The sub-index log2(N) denotes the levels of decomposition required to generate N subcarriers. It is well known from the theory of wavelets that compactly supported orthonormal wavelet bases can be obtained from two-band paraunitary filter banks [6]–[8]. Time and frequency limited wavelet packet bases ξ(t) can be derived by iterating discrete half-band high g[n] and low-pass h[n] filters, recursively defined by: ξl2+p1 (t ) = 2 ∑ h[n]ξl p (t − 2l n) n
ξl2+p1 +1 (t ) = 2 ∑ g[ n]ξl p (t − 2l n)
(2)
n
In (2) the subscript l denotes the level in the tree structure and superscript p indicates the tree depth. The number of channels
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
in WPMCM is determined by the number of iterations of the two-channel filter bank. In wavelet jargon (2) is called the 2scale equation and the low and high-pass filters are referred as scaling and wavelet filters, respectively. WPMCM systems are realized by a set of four paraunitary filters called Quadrature Mirror Filters (QMF): a pair halfband high g[n] and low pass filters h[n] called synthesis filters and another pair of half-band high g’[n] and low pass filters h’[n] called analysis filters. These four filters share a strict and tight relation and hence it is enough if the specifications of one of these filters are available. The wavelet packet sub-carriers (used at the transmitter end) are generated from the synthesis filters. And the wavelet packet duals (used at the receiver end) are obtained from the analysis filters. The exact procedure to derive these carriers can be found in [9]. III. WPMCM, WAVELETS AND FILTER CRITERIA The attributes of a WPMCM system greatly depend on the set of waveforms it uses which in turn is determined by the underlying filters used. This means that by adapting the filters one can conceive a WPMCM transceiver that best handles a system specification. The design of wavelets is bounded by multiple constraints. Besides the design objectives there are other budgets that should be considered in order to guarantee that the wavelet designed is valid. The constraints include properties such as orthogonality, compact support, symmetry, and smoothness and are usually stated in terms of the scaling filter h[n]. Here we shall touch upon a few of them. 1) Wavelet Existence and Compact Support: This property ensures that the wavelet has a finite number of non-vanishing coefficients and the filters are of finite length. Wavelet existence imposes a single linear constraint on h[n] [6]-[7]: L −1
∑ h[n] =
(3)
2
n =0
2) Paraunitary Condition: The paraunitary condition is essential for many reasons. First, it is a prerequisite for generating orthonormal wavelets [7]–[8]. Second, it automatically ensures perfect reconstruction of the decomposed signal i.e., the original signal can be reconstructed without amplitude or phase or aliasing distortion. To satisfy the paraunitary constraint the scaling filter coefficients have to be orthogonal at even shift [6]-[7]: ⎧1 if m = 0 h[n]h[n − 2m] = δ [m] = ⎨ ∑ n =0 ⎩0 otherwise L −1
(4)
For a filter of length L the orthogonality condition (4) imposes L /2 non-linear constraints on h[n]. 3) Flatness/K-Regularity: This property is a rough measure of smoothness of the wavelet. The regularity condition requires that the wavelet be locally smooth and concentrated in both time and frequency domains. It is normally quantified by the number of times a wavelet is continuously differentiable. The simplest regularity condition is the flatness constraint which is stated on the low pass filter. A LPF is said to satisfy Kth order flatness if its transfer function H(ω) contains K zeroes located at the Nyquist frequency (ω = π). For any function Q(ω) with no poles or zeros at (ω = π) this can be written as [6]-[7]: K
⎛ 1 + e jω ⎞ H (ω ) = ⎜ ⎟ Q(ω ) ⎝ 2 ⎠
(5)
Parameter K is called the regularity order and for a filter of length L its degree is limited by 1 ≤ K ≤ L/2. In the time domain we can impose regularity condition (5) as [6]-[7]: L −1
∑n
k
(−1) n h[n] = 0
(6)
for k = 0,1,… K − 1
n =0
K-Regularity condition enforces K constraints on h[n]. IV. TIME OFFSET EFFECTS IN WPMCM One of the major concerns of multicarrier systems is their vulnerability to time synchronization errors which occur when the symbols are not perfectly aligned at the receiver. Because of the time offset, samples outside a WPMCM symbol get erroneously selected, while useful samples at the beginning or at the end of the symbol get discarded. Time offset degrades the performances of multicarrier transceivers by introducing ISI and ICI. WPMCM and OFDM share many similarities as both are orthogonal multicarrier systems but with regard to performance under timing error there are major differences. First, the length of the WPMCM symbols (which is determined by the wavelet used) is significantly longer than the OFDM symbol. Under a loss in time synchronization, the overlap of the symbols in WPMCM causes it to interfere with several other symbols while in OFDM each symbol interferes only with its neighbors. The second important difference is in the usage of guard intervals. OFDM uses cyclic prefix that significantly improves its performance under timing errors. WPMCM cannot benefit from guard interval since the WPMCM symbols overlap. Under ideal conditions when the WPMCM transmitter and receiver are perfectly synchronized and the channel is benign, the data estimate aˆu ', k ' is the same as the transmitted data au,k . The demodulation process is elucidated below: k' aˆu ', k ' = ∑ R(n)ξlog n
2
(N)
(u ' N − n )
N −1
= ∑∑∑ au , k ξ logk 2 ( N ) (n − uN )ξ logk ' 2 ( N ) (u ' N − n) n
u k =0
⎛ ⎞ = ∑∑ au , k ⎜ ∑ ξ logk 2 ( N ) (n − uN )ξ logk ' 2 ( N ) (u ' N − n) ⎟ u k =0 ⎝ n ⎠ N −1
(7)
N −1
= ∑ au ', k δ (k − k ') = au ', k ' k =0
Here R(n) stands for the received sequence and ξ k ' are receiver carriers which are the duals of the transmitter carriers ξ k . The apostrophe is used to indicate receiver side. The time synchronization error is modeled by shifting the received data samples by a time offset tε to the left or right, depending on the sign of the tε. The demodulated data with a time offset tε can be written as: aˆu ', k ' = ∑ R (n)ξ k ' (u ' N − n + tε ) log ( N ) 2
n
N −1
k = ∑∑∑ au , k ξ log n
u
k =0
2
(N )
(n − uN )ξ logk ' 2 ( N ) (u ' N − n + tε ) + w(n)
(8)
N −1 ⎛ ⎞ = ∑∑ au , k ⎜ ∑ ξ logk 2 ( N ) (n − uN )ξ logk ' 2 ( N ) (u ' N − n + tε ) ⎟ + w(n) u k =0 ⎝ n ⎠
As before the apostrophe is used to indicate the receiver side. To shorten the derivation we define a new notation:
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
Ω uk ,,uk '' (tε ) = ∑ ξ logk 2 ( N ) ( n − uN )ξ logk '2 ( N ) (u ' N − n + tε )
(9)
n
This equation represents the autocorrelation and the crosscorrelation of the WPMCM waveforms referred by the subcarrier index k. When k = k’ the two subcarrier waveforms are time-reversed images of each other and (10) gives the autocorrelation sequence of the waveform k. In the other cases when k ≠ k’ the two waveforms correspond to different subcarriers and (10) stands for the cross-correlation between waveforms k and k’. Using (8) and (9) we can express the output of the receiver for kth subcarrier and uth symbol as: aˆu ',k ' = au ', k ' Ωuk ',' ku'' + Useful Signal
N −1
∑
au ', k Ωuk ,',ku'' +
k = 0; k ≠ k '
N −1
∑ ∑a
u,k
u ;u ≠ u ' k = 0 ICI
Ωuk ,,uk '' + wu ', k ' (10) Noise
ISI
In (11) the first term stands for the attenuated useful signal, second term denotes ICI, third term gives ISI and the last term stands for Gaussian noise. V. DESIGNING BEST BASES TO TACKLE TIME OFFSET ERRORS A. The Design Procedure The objective of the design process is to construct WPMCM carriers that better handle time offset. This is done by generating WPMCM subcarriers with large distances between each other, i.e. the cross-correlation between the waveforms is made as small as possible. Since the wavelet bases used as the WPMCM carriers implicitly share a relationship with the underlying filter banks, the design process can be converted into a filter design problem. Indeed in addition to the objective of reduction of time-error, the filters also have to satisfy additional constraints such as: • the paraunitary condition • the flatness/regularity condition • the filters must allow orthonormal expansion and perfect reconstruction of discrete-time signals • have finite impulse response For a filter of length L this is essentially solving L unknown filter variables from L linear equations. Of these L linear equations, one equation is required to satisfy the wavelet existence condition, L/2 equations come from the paraunitaryness constraint, K-1 equations come from the regularity constraint and the remaining L/2-K conditions offer the possibility for establishing the design objective. Larger the value of L/2-K, greater the degree of freedom for design and greater is the loss in regularity. There is therefore a trade-off on offer. The L/2-K degrees of freedom that remain after satisfying wavelet existence, orthogonality and K-regularity condition can be used to design a scaling filter which minimizes the interference energy caused by the timing offset. The design procedure consists of 3 major steps: 1. Formulation of design problem, i.e. stating the design objectives and the constraints necessary for wavelet generation 2. Making the problem tractable by converting it into a convex optimization problem 3. Using suitable optimization tools to solve the problem and obtain the required filter coefficients. In the following segments we will elaborate on each of the points stated above.
B. Problem Formulation The information carried by the waveforms that overlap one another can only be correctly decoded if the waveforms used have large distances between one another. In WPMCM this is achieved through the orthogonality of the waveforms. Therefore, in disturbance-free environments the crosscorrelations of WPMCM waveforms equal zero and perfect reconstruction is possible despite time and frequency overlap. A timing error tε, however, leads to the loss of the orthogonality between the waveforms and they begin to interfere one with another leading to ICI and ISI, stated as: k k' (11) Ωuk ,,uk '' (tε ) = ∑ ξ log ( N ) ( n − uN )ξ log ( N ) (u ' N − n + tε ) k ≠k '
2
n
2
The design objective would therefore be to generate wavelet bases ξ and their duals ξ ' that minimize interference energy in the presence of timing error, i.e. Ωuk ,,uk '' (tε )
MINIMIZE:
2 u , k ;k ≠ k '
with respect to {ξ , ξ '}
(12)
C. Wavelet Domain to Filter Bank Domain The waveforms in WPMCM are created by the multilayered tree structure filter bank. Using Parseval’s theorem of energy conservation it can be easily proved that the total energy at each level is equal regardless of the tree’s depth. Therefore, minimizing the interfering energy at the roots of the tree will automatically lead to the decrease of total interfering energy at the higher tree branches. Furthermore, the two-channel filter banks through the 2-scale equation are unambiguously related to the WPMCM waveforms. Therefore the design process can be easily converted into a tractable filter design problem. We should hence be able to minimize deleterious effects of time synchronization errors in WPMCM by minimizing the following cross-correlation function: 2 2 2 rhg [m] = ∑ h[n]g[n − m] = ∑ h[n]((−1) n h[ L − n + m]) (13) n
n
The design problem of minimizing the interference energy due to timing offset can now be formally stated as an optimization problem with objective function (13) and constraints (3), (4) and (6), i.e. 2
MINIMIZE:
rhg [m] with respect to h[n] L −1
∑ h[n] =
(14)
2
n =0
SUBJECT TO:
L −1
∑ h[n]h[n − 2m] = δ (m) n =0
L −1
∑n
k
n =0
(−1)n h[n] = 0
⎛L ⎞ for m = 0,1,… , ⎜ − 1⎟ ⎝2 ⎠
(15)
for k = 0,1,… , ( K − 1)
D. Conversion of Non-Convex Problem to Convex Problem The objective function and the paraunitary condition are non-convex. Although there exist general purpose solvers which can solve non-convex problems, these algorithms are susceptible to being trapped in a local minima. In order to overcome this difficulty, some authors have suggested multiple starting point technique or branch-and-bound method to approach real minimum. However, these algorithms can be unstable, inefficient and do not guarantee a global minimum. Therefore, we will rewrite the original problem into a convex
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
form by expressing the objectives and constraints as convex functions of the scaling filter’s autocorrelation rh[n]. ⎧ L − n −1 ⎪ ∑ h[m]h[m + n] n ≥ 0 rh [n] = ⎨ m = 0 ⎪ r [ − n] n
