Teletrac modeling and estimation

Transcription

1 Teletrac modeling and estimation File 3 José Roberto Amazonas jra@lcs.poli.usp.br Telecommunications and Control Engineering Dept. - PTC Escola Politécnica University of São Paulo - USP São Paulo 11/2008 Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

2 Outline 1 Long Memory Teletrac Modeling Introduction Non-parametric modeling Modelo MWM Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

3 Introduction Outline 1 Long Memory Teletrac Modeling Introduction Non-parametric modeling Modelo MWM Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

4 Introduction Classes of modeling Trac models may be classied as heterogenous or homogeneous. Heterogeneous models simulate the aggregate trac (trac generated by several users, protocols and applications) over a network link. Homogeneous models refer to a specic kind of trac, as the MPEG video trac. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

5 Introduction Classes of modeling Heterogeneous models may be subdivided in two classes: behavioral or structural. Behavioral models model the trac statistics, as correlation, marginal distribution or even higher order statistics (third and fourth orders, for example) without taking into account the physical mechanism of trac generation (i. e., behavioral models' parameters are not directly related to the communications network's parameters). Structural models are related to the packets generation mechanisms and their parameters may be mapped to network's parameters, as number of users and bandwidth. FGN, ARFIMA and MWM are behavioral models of aggregate trac. On/O processes may be used as trac structural models. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

6 Non-parametric modeling Outline 1 Long Memory Teletrac Modeling Introduction Non-parametric modeling Modelo MWM Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

7 Non-parametric modeling FGN process The FGN process, proposed by Mandelbrot and Van Ness in 1968 for modeling LRD hydrological series, is the rst important long memory model that appears in the literature. If {x t } t Z is a FGN, then x t is a stationary process with autocovariance given by C x (τ) = σ2 x 2 [ τ + 1 2H 2 τ 2H + τ 1 2H ], τ =..., 1, 0, 1,...). The FGN corresponds to the rst dierence of a continuous time stochastic process known as Fractionary Brownian Motion, FBM {B H (t) : 0 t } with Hurst parameter 0 < H < 1, i. e., x t = B H (t) = B H (t + 1) B H (t), t = 0, 1, 2,.... (1) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

8 Non-parametric modeling FBM realizations 1.5 Realizações de processos FBM H=0,6 H=0,5 H=0,7 H=0,8 H=0, t Figure: Realizations of FBM processes for several Hurst parameters values. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

9 Non-parametric modeling FBM Process The FBM has a special name when H = 1/2: (Brownian motion) and it is designated by B 1/2 (t). In this case, x 1, x 2,... are independent Gaussian random variables. We can create a discrete time FBM (DFBM), denoted by B t, by means of the cumulative sum of the FGN {x t } samples: t 1 B t B H (t) = x u, t = 1, 2,.... (2) u=0 Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

10 Non-parametric modeling FBM Process - cont. The DFBM SDF is given by P Bt (f ) = σ 2 x C H j= 1 f + j 2H+1, 1 2 f 1 2, (3) in which σx 2 is the power of a zero mean FGN, C H = and 0 < H < 1. Γ(2H+1) sin (πh) 2π 2H+1 According to (3), the DFBM SDF has a f α, 0 < α < 1, singularity, in the origin, as P Bt f 1 2H, f 0. (4) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

11 Non-parametric modeling FGN's SDF The FGN and the DFBM are related by the transfer function H(z) = X (z) B(z) = 1 z 1, (5) in which X (z) e B(z) denote the z-transforms of x t and B t, respectively. The frequency response associated to (5) is H(f ) = H(z) z=e j2πf = 1 e j2πf. (6) As the input/output relation in terms of the SDFs is P x (f ) = H(f ) 2 P Bt (f ), (7) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

12 Non-parametric modeling FGN's SDF - cont. H(f ) 2 is given by, then the FGN's SDF is equal to H(f ) 2 = G(f ) = 4 sin 2 (πf ), (8) P x (f ) = 4 sin 2 (πf )P Bt (f ). (9) (3) e (9) show that the FGN's SDF is characterized by only two parameters:σx 2 e H (responsible for the spectrum shape). The FGN is completely specied by its mean and by its SDF, as it is Gaussian. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

13 Non-parametric modeling FGN's SDF - cont. It may be shown that (9) may be rewritten as: P x (f ) = A(f, H)[ 2πf 2H 1 + B(f, H)], (10) in which A(f, H) = 2 sin (πh)γ(2h + 1)(1 cos (2πf )) e B(f, H) = j=1 [(2πj + 2πf ) 2H 1 + (2πj 2πf ) 2H 1 ]. For small values of f, P x (f ) f 1 2H. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

14 Outline 1 Long Memory Teletrac Modeling Introduction Non-parametric modeling Modelo MWM Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

15 FT e WFT The Fourier transform of a signal x(t), if exists, is dened as X (ν) = TF{x(t)} = x(t)e j2πνt dt, (11) in which ν denotes the frequency in cycles/second [Hz]. Gabor has shown that it is possible to represent the local spectral content of a signal x(t) around an instant of time τ by the windowed Fourier transform (WFT) X T (ν, τ) = x(t)g T (t τ)e j2πνt dt, (12) in which g T (t) is a window of nite duration support T and ν denotes frequency. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

16 FT and WFT - cont. The WFT is a bi-dimensional representation dened on the time-frequency plane or domain as it depends on the ν and τ parameters. The WFT would be equivalent to a kind of continuous sheet music description of x(t). According to the Heisenberg's uncertainty principle, a signal whose energy content is quite well localized in time has this energy quite spread out in the frequency domain. As the window of (12) has a xed size T, we may conclude that the WFT is not good to analyze (or identify) behaviors of x(t) occurring in time intervals much smaller or much larger than T, as, for example, transient phenomena of duration t << T or cycles that exist in periods larger than T. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

17 CWT A wavelet ψ 0 (t) (sometimes also called mother wavelet), t R, is a function that satises three conditions. 1 Its Fourier transform Ψ(ν), < ν <, is such that exists a nite constant C ψ that obeys the admissibility condition 0 < C ψ = 2 The integral of ψ 0 (t) is null: 3 Its energy is unitary: 0 Ψ(ν) 2 dν <. (13) ν ψ 0 (t) dt = 0. (14) ψ 0 (t) 2 dt = 1. (15) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

18 Examples of wavelet functions Haar Wavelet D4 Wavelet C3 Coiflet 0.2 S8 Symmlet Figure: Four examples of wavelet functions. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

19 Examples of wavelet functions - cont Figure: Meyer's wavelet. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

20 Examples of wavelet functions - cont wavelet Gaussiana de ordem 1 wavelet chapéu mexicano t Figure: Gaussian wavelet (related to the rst derivative of a Gaussian PDF) t Figure: Mexican hat wavelet (related to the second derivative of a Gaussian PDF). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

21 CWT The wavelet transform has been originally developed as an analysis and synthesis tool of continuous time energy signals. An energy signal x(t), t R (t denotes time), obeys the constraint x 2 = x, x x(t) 2 dt <, (16) i. e., x(t) that obeys the constraint (16) belongs to the squared summable functions space L 2 (R). Presently, the wavelet transform has also been used as an analysis tool of discrete time signals. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

22 CWT - cont. There are continuous time (CWT) and discrete time wavelet decompositions. The CWT of a signal x(t) consists of a set C = {W ψ (s, τ), s R +, τ R}, in which τ is the time localization parameter, s represents scale and ψ denotes a wavelet function of wavelet coecients on the continuous time-scale plane (also known as time-frequency plane)given by ( λ τ W ψ (s, τ) = ψ 0(s,τ), x = 1 s ψ 0 s ) x(λ)dλ. (17) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

23 CWT - cont. ( ψ 0(s,τ) (t) = s 1/2 ψ t τ ) 0 s denotes a dilated and shifted version of the mother wavelet ψ 0 (t). The factor 1/ s in (17) provides all functions of the class { ( ) } 1 s t τ W = ψ 0 R s have the same energy (norm). (18) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

24 1 Measure the likelihood. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67 CWMT - cont. The basic idea of the CWT dened by (17) is to correlate 1 a signal x(t) with shifted (by τ) and dilated (by s) versions of a mother wavelet (that has a pass-band spectrum). The CWT is a two parameters function. So, it is a redundant transform, because it consists on mapping an one-dimension signal on the time-scale plane.

25 ICWT Dierently from the WFT, where the reconstruction is made from the same family of functions as that used in the analysis, in the CWT the synthesis is made with functions ψ s,τ that have to satisfy ψ s,τ (t) = 1 C ψ 1 s 2 ψ s,τ (t). (19) So, x(t) is completely recovered by the inverse continuous wavelet transform (ICWT): [ ( t τ x(t) = 1 C ψ 0 W ψ (s, τ) 1 s ψ s ) ] ds dτ s. (20) 2 Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

26 CWT vs. WFT The fundamental dierence between the CWT and the WFT consists of the fact that the functions ψ s,τ undergo dilations and compressions. The analysis on rened scales of time (small values of s) requires fast ψ s,τ functions, i. e., of a small support, while the analysis on aggregate scales of time (large values of s) requires slower ψ s,τ functions, i. e., of a wider support. As already mentioned, the internal product dened by (17) is a likelihood measure between the wavelet ψ ( ) t τ s and the signal x(t) on a certain instant of time τ and on a determined scale s. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

27 CWT vs WFT - cont. For a xed τ, large values of s correspond to a low-frequency analysis, while small values of s are associated to a high-frequency analysis. Therefore, the wavelet transform has a variable time resolution (i. e., the capacity of analyzing a signal from close - zoom in - or from far - zoom out), being adequate to analyze phenomena that occur in dierent time scales. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

28 Example of CWT 3 2 sinal x(t) CWT log 2 (s) tempo Figure: The image on the bottom part of the gure is the CWT W ψ (s, τ) of the signal on the top part, evaluated with a wavelet that is the rst derivative of the Gaussian PDF. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

29 Multiresolution analysis and the discrete wavelet transform There are two kinds of DWT: the DWT for discrete time signals and the DWT for continuous time signals. The DWT may be formulated for discrete time signals (as it is done, for example, by Percival and Walden) without establishing any explicit connection with the CWT. On the other hand, we should not understand the term discrete of the DWT for continuous time signals as meaning that this transform is dened over a discrete time signal. But only that the coecients produced by this transform belong to a subset D = {w j,k = W ψ (2 j, 2 j k), j Z, k Z} of the set C. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

30 Multiresolution analysis and the discrete wavelet transform - cont. In fact, the DWT coecients for continuous time signals can also be directly obtained by means of the integral w j,k = ψ, x = 0(2j,2j k) 2 j/2 ψ0(2 j λ k)x(λ) dλ, (21) in which the indices j and k are called scale and localization, respectively, does not involve any discrete time signal, but the continuous time signal x(t). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

31 Multiresolution analysis and the discrete wavelet transform - cont. Eq. (21) shows that the continuous time DWT corresponds to a critically sampled version of the CWT dened by (17) in the dyadic scales s = 2 j, j =..., 1, 0, 1, 2,..., in which the instants of time in the dyadic scale s = 2 j are separated by multiples of 2 j. The function ψ 0 of (21) must be dened from a multiresolution analysis (MRA) of the signal x(t). Observe that the continuous time MRA theory is similar to that of discrete time. Although the teletrac signals are discrete time, we decided to present the continuous time MRA version because the Hurst parameter estimator based on wavelets proposed by Abry and Veitch is based on the spectral analysis of a ctitious process { x t, t R} that is associated to the discrete time process {x n, n Z}. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

32 CWT sampling j = 0 j = 1 j = 2 j = 3 j = 4 Figure: Critical sampling of the time-scale plane by means of the CWT parameters (s = 2 j e τ = 2 j k) discretization. s Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

33 MRA A MRA is, by denition, a sequence of closed subspaces {V j } j Z de L 2 (R) such that: 1... V 2 V 1 V 0 V 1 V 2...; 2 j Z V j = {}; 3 j Z V j = L 2 (R); 4 x(t) V j x(2 j t) V 0, j > 0 (in which t denotes time and x(t) is an energy signal); 5 There is a function φ j (t) = 2 j/2 φ 0 (2 j t) in V j, called scale function, such that the set {φ j,k, k Z} is an orthonormal basis of V j, with φ j,k (t) = 2 j/2 φ 0 (2 j t k) j, k Z. The subspace V j is known as the approximation space associated to the time scale s j = 2 j (assuming that V 0 is the approximation space with unit scale). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

34 MRA - cont. If the x(t) projection on V j de x(t) is represented by the scale coecients u j,k = φ j,k, x = 2 j/2 φ 0(2 j t k)x(t) dt, (22) then the properties 1 and 3 assure that lim φ k j,k(t)u j,k = x(t), j x L 2 (R). Property 4 implies that the subspace V j is a scaled version of subspace V 0 (multiresolution). The orthonormal basis mentioned in property 5 is obtained by time shifts of the low-pass function φ j. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

35 MRA - cont. Consider the successive approximations sequence (also known in the literature as wavelet smooths) of x(t) S j (t) = k φ j,k (t)u j,k j =..., 1, 0, 1,.... (23) As V j+1 V j, we have S j+1 (t) is a coarser approximation of x(t) than S j (t). This fact illustrates the MRA's fundamental idea, that consists in examining the loss of information when one goes from S j (t) to S j+1 (t): S j (t) = S j+1 (t) + x j+1 (t). (24) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

36 2 Besides, Wj+1 is contained in the subspace V j. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67 MRA - cont. x j+1 (t) (called detail of x j (t)) belongs to the subspace W j+1, named detail space that is associated to the uctuations (or variations) of the signal in the more rened time scale s j = 2 j and that corresponds to the orthogonal complement of V j+1 in V j 2. The MRA shows that the detail signals x j+1 (t) = D j+1 (t) may be directly obtained by successive projections of the original signal x(t) on wavelet subspaces W j. Besides, the MRA theory shows that exists a function ψ 0 (t), called mother wavelet, that is obtained from φ 0 (t), in which ψ j,k (t) = 2 j/2 φ 0 (2 j t k) k Z is an orthonormal basis of W j.

37 MRA - cont. The detail D j+1 (t) is obtained by the equation D j+1 (t) = k ψ j+1,k (t) ψ j+1,k (t), x(t). (25) The internal product ψ j+1,k (t), x(t) = w j+1,k denotes the wavelet coecient associated to scale j + 1 and discrete time k and {ψ j+1,k (t)} is a family of wavelet functions that generates the subspace W j+1, orthogonal to subspace V j+1 (W j+1 V j+1 ), i. e., ψ j+1,n, φ j+1,p = 0, n, p. (26) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

38 MRA - cont. Therefore, the detail signal D j+1 (t) belongs to the complementary subspace W j+1 de V j, because V j = V j+1 W j+1. (27) That is, V j is given by the direct addition of V j+1 and W j+1, and this means that any element in V j may be determined from the addition of two orthogonal elements belonging to V j+1 and W j+1. Iterating (27), we have V j = W j+1 W j (28) Eq. (28) says that the approximation S j (t) is given by S j (t) = i=j+1 w i,k ψ i,k (t). (29) k Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

39 3 The sequence u0(k) is obtained sampling the lter's output whose impulse response is φ ( t) (matched lter with a function φ 0(t) = φ(t)) at instants k = 0, 1, 2,..., i. e., u 0(k) = x(t) φ ( t) for k = 0, 1, 2,..., in which denotes convolution. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67 MRA - cont. The MRA of a continuous time signal x(t) is initiated by determining the coecients 3 u 0 (k) = φ 0,k (t), x(t), in which k = 0, 1,..., N 1, that are associated to the projection of x(t) on the approximation subspace V 0.

40 MRA - cont. Following, the sequence {u 0 (k)} is decomposed by ltering and sub-sampling by a factor of 2 (downsampling) in two sequences: {u 1 (k)} and {w 1 (k)}, each one with N/2 points. This ltering and sub-sampling process is repeated several times, producing the sequences and {{u 0 (k)} N, {u 1 (k)} N 2, {u 2 (k)} N 4,..., {u j (k)} N 2 j,..., {u J (k)} N 2 J } (30) {{w 1 (k)} N 2, {w 2 (k)} N 4,..., {w j (k)} N 2 j, {w J (k)} N 2 J }. (31) The literature calls the set of coecients { } {w 1 (k)} N, {w 2 (k)} N,..., {w J (k)} N, {u J (k)} N J 2 J as the DWT of the x(t) signal. (32) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

41 MRA - cont. Fig. 8 illustrates a 3-levels DWT (decomposition in scales j = 1, 2, 3) associated to 1024 samples of the discrete time x(k) = sin (3k) + sin (0.3k) + sin (0.03k), that corresponds to the superposition of 3 sinusoids in frequencies f , f and f Fig. 9 shows the SDF of this signal. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

42 DWT example 4 sinal original DWT Figure: An illustration of the 3-levels DWT of the discrete time signal x(k) = sin (3k) + sin (0.3k) + sin (0.03k). The graph concatenates the sequences of the scale coecients {u 3 (k)} 128 and of the wavelet coecients {w 3 (k)} 128, {w 2 (k)} 256 e {w 1 (k)} 512 from left to right, i. e., the rst 128 points correspond to the sequence {u 3 (k)} 128 ; then follow the 128 points of the sequence {w 3 (k)} 128, the 256 points of the sequence {w 2 (k)} 256 and 512 points of the sequence {w 1 (k)} 512. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

43 20 DEP 10 Power/frequency (db/hz) Frequency (Hz) Figure: SDF of the signal x(k) = sin (3k) + sin (0.3k) + sin (0.03k). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

44 MRA - cont. The reconstruction of x(t) is implemented by ltering and oversampling by a factor of 2 (upsampling) of the sequences (30) and (31), obtaining an approximation of x(t) in the subspace V 0 or S 0 (t) = S J (t) + D 1 (t) + D 2 (t) + + D J (t) (33) x(t) k u(j, k)φ J,k (t) + J w j,k ψ j,k (t). (34) Eq. (34) denes the Inverse Discrete - IDWT. j=1 k Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

45 Figure: Synthesis of the signal x(k) = sin (3k) + sin (0.3k) + sin (0.03k) in terms of the sum S 3 (t) + D 1 (t) + D 2 (t) + D 3 (t). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

46 We say that the function φ 0 (t) = φ(t) determines a MRA of x(t) according to (33), if it obeys the following conditions: 1 intra-scale orthonormality (property 5) φ(t m), φ(t n) = δ m,n, (35) in which δ m,n is the Kronecker's delta (δ m,n = 1 if m = n, δ m,n = 0 for m n). Eq. (35) imposes an orthonormality condition at scale j = 0. 2 unit mean φ(t) dt = 1. (36) φ( t 2 ) = n g n φ(t n), (37) as several φ(t k) t in φ( t ) (is a consequence of property (1) of the 2 MRA). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

47 φ(t) = n Eq. 37 may be rewritten as 2gn φ(2t n), (38) known as Dilation Equation. Eqs. 37 and 38 may be written, respectively, in the frequency domain as 2Φ(2ν) = G(ν)Φ(ν), (39) and Φ(ν) = 1 2 G(ν)Φ( ν 2 ), (40) in which Φ(ν) is the Fourier transform of φ(t) and G(ν) = n g ne j2πνn, known as scale lter (low-pass), represents a periodic lter in ν. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

48 ψ(t) = n As the subspace W j+1 is orthogonal to V j+1 and is in V j, we have 1 2 ψ( t 2 ) = n h n φ(t n), (41) or 2hn φ(2t n), (42) that is the Wavelet Equation. Applying the Fourier transform to (41) and (42) we get, respectively, (2)Ψ(2ν) = H(ν)Φ(ν), (43) and Ψ(ν) = 1 2 H(ν)Φ( ν 2 ). (44) in which H(ν) is the wavelet lter (high-pass). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

49 Rewriting (26) in terms of the frequency domain and using (39) and (43) results the orthogonality condition G(ν)H (ν) Φ(ν) 2 dν = 0, (45) that the lter H has to obey so the family {ψ 1,k (t)} is orthogonal to the family {φ 1,k (t)}. We may show that the condition h n = ( 1) n g L 1 n, H(z) = z L+1 G( z 1 ), (46) in which L denotes the length of a FIR lter g n, is sucient to (45) to hold. We say that g n e h n are quadrature mirrored lters (or QMF) when they are related by (46). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

50 G(f) QMF H(f) f G(f) Filtros brickwall H(f) Figure: QMF lters frequency response (graphs on the upper part) vs brickwall-type lters frequency response. f Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

51 According to (38), the MRA departs from a denition (from several possible) of the scale function φ(t), that is related to the scale lter g n by (37). Eq. (46) says that the choice of a FIR-type lter {g n } implies a {h n } that is also FIR. At last, the wavelet function is determined by (41). The scale φ(t) and wavelet ψ(t) functions associated to the FIR lters {g n } e {h n } have compact support, thus oering the time resolution functionality. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

52 The simplest scale function that satises (35) is the characteristic function of the interval I = [0, 1), that corresponds to the Haar's scale function: { φ (H) 1 se 0 t < 1 (t) = χ [0,1) (t) = (47) 0 otherwise. In this case (Haar MRA), the associated Haar scale lter is given by the Haar wavelet lter by g n = {..., 0, g 0 = 1/ 2, g 1 = 1/ 2, 0,...}, (48) h n = {..., 0, h 0 = g 1 = 1/ 2, h 1 = g 0 = 1/ 2, 0,...} (49) and the Haar wavelet function by ψ (H) (t) = χ [0,1/2) (t) χ [1/2,1) (t). (50) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

53 Fig. 12 shows the Daubechies' scale and wavelet functions with N = 2, 3, 4 vanishing moments t m ψ(t) dt = 0, m = 0, 1,..., N 1. (51) Ingrid Daubechies was the rst one to propose a method for building sequences of transfer functions {G (N) (z)} N=1,2,3,... and {H (N) (z)} N=1,2,3,..., in which G (N) (z) is associated to the low-pass FIR lter g (N) n and H (N) (z) to the high-pass lter h (N) n. The corresponding scale and wavelet functions have support in [0, 2N 1]. The rst member of the sequence is the Haar system φ (1) = φ (H), ψ (1) = ψ (H). The Daubechies' lters are generalizations of the Haar system for N 2. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

54 Figure: The graphs in the lower part show the Daubechies' wavelets with N = 2, 3, 4 vanishing moments, from left to right, respectively. The corresponding scale functions are in the upper part. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

55 We can demonstrate that: u j (n) = k g(k 2n)u j 1 (k) (52) and that w j (n) = k h(k 2n)u j 1 (k). (53) According to (52) and (53), we can obtain the coecients u j (n) and w j (n) from the scale coecients u j 1 (m) by means of decimation operation of the sequence {u j 1 (m)} by a factor of 2. The decimation consists in cascading a low-pass lter g( m) (with a transfer function Ḡ(z) = G(1/z) and frequency response G (f )) or a high-pass h( m) (with transfer function H(z) = H(1/z) and frequency response H (f )) with a compressor (or decimator) by a factor of 2. Decimate a signal by a factor D is the same as to reduce its sampling rate by D times. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

56 The MRA is implemented by a low-pass and high-pass analysis lter banks G (f ) and H (f ) adequately positioned for separating the scale and wavelet coecients sequences. Later, it is possible to rebuild the original signal using dual QMF reconstruction lter banks, low-pass G(f ) and high-pass H(f ). It is important to emphasize that the pyramid algorithm's complexity is O(N) (assuming we want to evaluate the DWT of N samples), while the direct evaluation of the DWT (that involves matrices multiplication) is O(N 2 ). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

57 u j-1(m) QMF g(-m) G*(f) h(-m) H*(f) 2 2 u j(n) w j(n) Figure: QMF analysis lter banks G (f ) (low-pass) and H (f ) (high-pass) with decimation (downsampling) by a factor of 2 uj(m) wj(m) 2 2 u up j(n) w up j(n) QMF dual g(n) G(f) h(n) H(f) + uj-1(n) Figure: QMF reconstruction lter banks with interpolation (upsampling) by a factor of 2. Observe that are used dual low-pass and high-pass lters, G(f ) and H(f ). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

58 N/4 amostras N/2 amostras G* k/(n/2)] 2 V 2 u 2 (n) G* k/n] 2 V 1 u 1 (n) x(t) t V 0 u 0 (n) N amostras H* k/n] 2 W 1 w 1 (n) H* k/(n/2)] 2 W 2 u 2 (n) N/4 amostras N/2 amostras Figure: Flow diagram that shows the initial projection of a signal x(t) on V 0 followed by the decomposition in W 1, W 2 and V 2. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

59 u 2 (n) 2 V 2 w 2 (n) 2 W 2 N/4 amostras G k/(n/2)] H k/(n/2)] + u 1 (n) w 1 (n) 2 G k/n] N amostras 2 H k/n] u 0 (n) N/2 amostras D/A t + x(t) Figure: Flow diagram that illustrates the approximate synthesis of x(t) from W 1, W 2 and V 2. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

60 U 2 W 2 W 1 f Figure: Block diagram that shows that the DWT works as a sub-bands codication scheme. The spectrum U 0 (f ) of the signal u 0 (n) is subdivided in three frequency bands (that cover two octaves): 0 f < 1/8, 1/8 f < 1/4 and 1/4 f 1. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

61 Modelo MWM Outline 1 Long Memory Teletrac Modeling Introduction Non-parametric modeling Modelo MWM Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

62 Modelo MWM The MWM uses the Haar's MRA and is based on a multiplicative binomial cascade in the wavelet domain, that ensures that the simulated series are positive. The binomial cascade is a random binomial tree whose root is the N coecient u J 1,0 (the MWM considers that 2 J 1 = 1, where N is the number of samples) J 1 x t = x 0,k = u J 1,0 φ J 1,0 (t) + w j,k ψ j,k (t), (54) in which φ J 1,0 (t) denotes a Haar's scale function in the slowest scale (ordem J 1) and w j,k are the wavelet coecients. j=1 k Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

63 Modelo MWM The Haar's scale and wavelet coecients may be recursively computed by the following set of synthesis equations, u j 1,2k = 2 1/2 (u j,k + w j,k ) (55) u j 1,2k+1 = 2 1/2 (u j,k w j,k ). (56) So, strictly positive signals may be modeled if u j,k 0 and w j,k u j,k. (57) It is possible to choose a statistical model for w j,k that incorporates the condition (57). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

64 Modelo MWM 4 Riedi et al have also investigated other distributions for the multipliers. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67 The MWM species a multiplicative model, w j,k = M j,k u j,k, (58) in which the multiplier M j,k may be modeled as a random variable with symmetric β distribution with shape parameter p j, i. e., M j β(p j, p j ). In this case, the MWM is known as β-mwm and we assume that the M j,k 's are mutually independent and independent from u j,k 4.

65 Modelo MWM The variance of M j is given by Var[M j ] = 1 2p j + 1. (59) In this way, equations (55) and (56) may be rewritten as ( ) 1 + Mj,k u j 1,2k = u j,k (60) 2 ( ) 1 Mj,k u j 1,2k+1 = u j,k. (61) 2 These equations show that the MWM is, in fact, a binomial cascade. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

66 Modelo MWM The MWM may approximate the SDF of a training sequence by modeling the variance decay of the wavelet coecients η j = Var[w j,k] Var[w j 1,k ] = 2p (j 1) + 1, (62) p (j) + 1 that leads to and p (j 1) = η j 2 (p (j) + 1) 1/2 (63) p (j) = 2p (j 1) + 1 η j 1. (64) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67

67 Modelo MWM The MWM assumes that u J 1,0 (the root scale coecient) is approximately Gaussian. We can show that p (j) converges to p = lim p (j) = 2α 1 j 2 2 α. (65) Table: Asymptotic values of the shape parameter p of the β multipliers as function of α (or H). α p H The MWM model has multifractal properties and the marginal probability density is lognormal. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 67