MATLAB/SIMULINK Implementation of Phi-Transforms A New Toolbox Only or the Rival of Wavelet Toolbox for the Next Decade?

Transcription

1 MALAB/SIMULINK Imlementation of Phi-ransforms A New oolbox Only or the Rival of Wavelet oolbox for the Next Decade? Peteris Misans (Professo Riga echnical University, Riga, Latvia) Maris erauds (Assistant, Riga echnical University, Riga, Latvia) Arturs Aboltins (Doctoral student, Riga echnical University, Riga, Latvia) Gatis Valters (Doctoral student, Riga echnical University, Riga, Latvia) Abstract he aer gives an introductory descrition of novel classes of fast orthogonal transforms (Phi-ransforms) which are defined, described and exloited by the team of authors during the recent three years. he main goal of the aer is the resentation of oen collection of MALAB functions and the library of SIMULINK blocks (Phi-ransforms oolbox/blockset). Presently, the toolbox contains almost 50 units and sulements existing Signal Processing, Communication, Image Processing and other toolboxes/blocksets of MALAB/SIMULINK. Here are several categories of the units basis functions generators, sectrum analyzers, signal synthesizers, arametrical orthogonal filters, arametrical communication engines, GUI based interactive tools, demo tools, and others. he units are develoed for both the -D and -D case. he reliminary comarison with orthogonal wavelets is rovided. Advantages of Phi-ransforms are demonstrated by simle examles from the signal synthesis, comression and filtering. Authors treat the toolbox as a rich base for teaching of students and the develoment of novel DSP algorithms and devices. Keywords Orthogonal ransforms, Phi-ransforms oolbox, Parametrical Orthogonal ransforms. Introduction he term "Phi-transforms" (Ps) (also hi-functions) was introduced in []. his term originates from the rotation of lanes in Euclidian (also Hilbert) sace. Howeve discussions are necessary about the usefulness of this term and conflicts with similar recognizable terms on the web []. In the IEEE Member Digital Library [3] we can find out about a dozen of aers concerning signal rocessing based on rotation angles. We suose that the rotation angles of lanes in Euclidian sace are cornerstones of the descrition of discrete orthogonal (orthonormal) transforms [4]. he angular aroach is very well known in linear algebra (e.g. QR-algorithms, Jacobi rotation, etc.) but it has not been used very often by signal rocessing eoles. It seems that H. C. Andrews [5] is a ioneer in the area of arameterization of fast orthogonal (orthonormal) transforms by rotation angles of lanes. P. Rieder with colleagues (for examle, [6]) uses rotation angles in the context of CORDIC-based imlementation of orthogonal wavelet transforms and DC. he ioneer in introducing and using of rotation angle based orthogonal filters is P. P. Vaidyanathan [7]. We avoid here a detailed overview of all available works on rotation angle aroach because of the limited sace of aer. We

2 exect that such overview will be available later this year in [8]. It seems that the famous work of B. Fino and R. Algazi [9] is the basic aer dealing with generalization of discrete unitary transforms. Previously we focused mainly on the real discrete orthonormal transforms. In [4] we introduced several classes of real discrete Rotation Angle Based Orthogonal ransforms (RABO). he latter can be interreted also as a single arametrical transform with infinite number of shaes of basis functions (BFs). On the other hand, we can use rotation angles as the base for different classification schemes of orthogonal transforms [4]. We exect that the use of RABO transforms could be very romising in the signal analysis and synthesis. We work on that. Particularly, reliminary results show that one of the subclasses of RABO (in [4] called as CRAIMO) is very useful for comression of seech. We need a relatively small number of orthogonal functions (sometimes only a few BFs) to reresent, for examle, the vowels of Latvian seech [7]. Our team works on the develoment of FPGA-based "angular" devices. Exerimental generators [0], [], analyzers [] and orthogonal filters [3], [4] are our results in the revious and this year. We exect an aearance of devices for more wide range alications in the nearest future. Simultaneously with the ractical rogramming of FPGAs we work on the "theory" of angle-based functions (Phi-functions). One of the recent aers deals with a ossible generalization of Haar functions [5]. Our team has roduced also several aers on the accuracy of Ps in the FPGA imlementation context (e.g. [6]). his aer deals with our efforts aimed towards the building of MALAB/SIMULINK functions/blocks using Ps.. Basics of Orthogonal ransforms We will talk here about the discrete transforms only. It is very well known that we can reresent the direct/inverse fast real/comlex -D transform in the matrix form Y = H X, X = H Y () where ( ) means the transosition/hermitian transosition of matrix, X inut signal vecto Y sectrum vecto and = l H = B( ϕ ), H = B ( ϕ ) () l = where for l n = log ( N ) (3) B(ϕ ) (further B ) can be sarse orthonormal/unitary matrix (the Stairs-like Orthonormal Generalized Rotation Matrix (SOGRM) [4]) resented by (5). N is the size of matrix H and the length of vectors X, Y. In the case when l ( N / ) ( N ) (4) (also in the case of (3)) the matrix B has a structure that may differ from the stairs-like structure. he "nonstairs-like" structures can be derived from SOGRM by an easy ermutation of rows. In our work we use mainly SOGRM because of the mathematical convenience (see, for examle, Good's factorization theorem). he SOGRM can be exressed as τ, τ, O... = τ N /, B (5) τ, τ, O τ N /, where τ and τ are -element row-vectors which originate from an elementary four-element unitary rotation matrix that can be defined in many ways, for examle τ τ cosr, e = ± sin e jγ jψ msin e r = jψ jγ cosr, e. (6) In the case when ψ,=γ =0 and in the right-uer corner has been used sin(), (6) is the well-known Given s matrix. For γ =0 we obtain the well-known Jacobi rotation matrix. he elementary rotation matrix erforms the rotation (or other maniulations) of a lane or a two-element vector in N-dimensional Euclidian/Hilbert sace. here are sixty four alternatives (for the maniulations with lanes (vectors)) how to build u a SOGRM [8]. In such a way we can create an infinite number of orthonormal (also orthogonal) transforms using the array (matrix with size N/ l 3) of rotation angles (angle matrix N/ l in [4]) [ ϕ, K,,, ] Φ = ϕ K ϕ l. (7) In formula () and (7) ϕ reresents the -th column of angle array (matrix) Φ, ϕ = ψ, ψ γ, γ, K,, K, ψ, K, γ N / N / N / We can also reresent the direct/inverse -D transform in the matrix form taking into account linear searability (8)

3 Y = ( H ( H X ) ), X = ( H ( H Y ) ) (9) where X -D matrix (image), Y -D sectrum, H -D orthonormal (unitary) matrix. For the orthogonal transforms, the roducts in (), (), and (9) may contain scaling terms (scalars).. Classification of RABOs We can define some relations between elements of angle matrix and get classes of orthogonal transforms (O). Particularly, we defined four different classes of real orthogonal transforms in [4]. A brief summary of these transforms is shown in next table. Reader can also choose their "own angles" and define own classes of Phi-transforms. For examle, [5] defines one of the ossible generalization of real Haar transform Rotation Angles Based Haar ransform (RA-H). For this transform each next column of angle matrix has twice less nonzero angles as the revious one. For examle, for N = 8 (l = 3) we have 3 Φ = [ ] = (0) In the secial case, when all nonzero angles are equal to 45, we obtain classical normalized Haar transform. It is necessary to mention that the structures of matrices B for RA-H generally differ from SOGRM (see [5]). Very briefly the rows of SOGRM for zero angles are ermuted in such a way that all ones are laced on the main diagonal. A natural extension and generalization of RABO is a Comlex RABO (CRABO or the comlex orthogonal (unitary) Phi-transform CP). We will review CPs in deth in our subsequent aers (a coule of them are in a bluerint). Briefly we define subclasses of CRABO similarly transforms rovided in able, but the limits for angles are extended and classification scheme is more sohisticated. able Summary of real RABO and RA-H (ψ,=γ =0) ransform Limits for angles Comments to (7) and (0) RABO - Rotation Angle Based O = angles in (7) are different CRAO Constant Rotation Angle O CRAIMO Constant Rotation Angle Inside Matrix O CRMO Constant Rotation Matrix O r r r r = all angles in (7) are equal r = r all rows in (7) are equal = all columns in (7) are equal RA-H Rotation Angles Based Haar ransform 0 angles in (0) are different r = r = CRA-H - Constant Rotation Angle H r 0 all nonzero angles in (0) are equal CRAIM-H - Constant Rotation Angle Inside Matrix H r = r 0 all nonzero angles within each angle matrix (0) column are equal = 0 r the first column of (0) contains RSA-H - H with Reduced Sequences of Rotation r N/ different angles, but each next Angles column contains twice reduced angle sequence.3 Shaes of Phi Basis Functions We can obtain an infinite diversity of shaes of BFs by the change of angles. he generalized H (CRA-H) in Fig. lets us see that BFs look like corruted Haar functions. A very good examle is the reresentation of the iece of the Latvian vowel "a" in Fig.. We see that functions look like random signals. In total we can observe very rich diversity of shaes from the simle ulse-like BFs to the arbitrary random signals [4]. Such a diversity offers almost infinite ossibilities for signal rocessing. Fig.. he iece of vowel (at bottom) "A" resented by the weighted sum of four CRAIMO BFs (4 uer waves)

4 oor coverage of US because of one angle change only) for the CRAO (similarly for CRA-H) BFs on the other side. For known orthogonal wavelets the trace is very simle and similar to the Hadamard or Haar (corresonding figure in [4]) functions trace only one single oint on US er BF.. Signal Processing and Phi-transforms. Signal Comression Fig.. Shaes of the first four discrete CRA-H BFs for certain values of rotation angles and N=8.4 Basis Functions and Unit Shere he big diversity of BFs shaes makes difficult to choose an aroriate transform for selected alication. he N-dimensional unit shere (US) is an alternative for the characterization of some general features of transforms. Geometrically, the rojections of ends of BF vectors of orthonormal transforms create a certain icture on the surface of N-D US if we change rotation angles for selected orthonormal transform in formula () [0]. he density of the icture deends on the number of angles used for the transform. he degree of coverage of US is very significant for signal comressing. For examle, if signal vector is coincident with some vector of BFs we have a ossibility to reresent this signal losslessly using a single weight coefficient, the index of BF and the identifier of set of angles. In such a way, the choice of otimal transform is a comromise between the number of angles, the degree of filling (coverage) of shere and the class of signals used. Our exerience confirms that the CRAIMO and CRAIM-H are a trade-off for signal comressing. Fig. 3 shows the trace of CRAIMO BF vector end rojection on the sread of one of the 3-D rojection of 4-D unit shere. We see an almost uniformly distributed coverage of the oints-ofends-rojections for the CRAIMO (similarly for CRAIM-H) BFs on one side and a simle lines (a very Elevation - [deg] "White hole" Axes,, 3 Hadamard Azimuth - [deg] CRAIMO CRAO Hadamard Fig. 3. race of CRAIMO (and CRAO) BFs vectors ends on the sread of surface of one of the unit shere rojections In [0] we demonstrated that the CRAIMO functions can be very useful for seech analysis, synthesis and comression. For seech it is ossible to achieve the comression degree -3 times (at least) higher than in the case of using of wavelets. We exect that RA-H functions are very ersective for the rocessing of ulse-like signals [], [3], [4]. For a simle demonstration we icked out the sum of two CRAIM-H BFs [7]. A erfect reconstruction of this signal is also ossible using the weighted sum of 5 Haar functions, but the average result can be achieved using - comonents. he reconstruction result is shown in Fig. 4. We see that the limited number of classical Haar functions does not allow to reconstruct the details of chosen signal even for 9% energy. his simle examle shows also that the comression degree can be imroved times by the roer choice of transform for large number of shaes of ulse-like signals (because the diversity of RA-H BFs shaes is infinite!). Fig. 4. Examle of reconstruction of ulse-like signal by the Haar BFs [7]. Orthogonal Wavelets and Phi-ransforms We can observe several differences and a few similarities between orthogonal wavelets and Ps. Generally, Ps lacks comactness in oosite to wavelets. But the RA-Hs shows that such a roerty is ossible also for Ps. We can observe the aroximate comactness of RABO BFs for many combinations of angles. Is this roerty so imortant? We suggest that it is not so in many cases. A very imortant roerty of wavelets is an

5 orthogonality to simle olynomials (vanishing moments). Generally, the hi-functions are not orthogonal to the olynomials but we can find out ractically an infinite number of more sohisticated functions (signals) orthogonal (or aroximate orthogonal) to hi-functions. Usually, O has a DC (constant) BF. yically for Ps is the lack of ure DC (it is sread between BFs). his means that the mean value of BF is nonzero in oosite to wavelets. Is it accetable or not, deends on alication..3 Phi-Filters In total, the architecture of filters based on Ps are similar to wavelet Decomosition-Reconstruction Filter(s) (DeReF) or other orthogonal filters. filters where the length can be > also. he IR of the elementary filters is controllable using the arameters (angles). he rincial difference is the existence of shelves in magnitudes for angles other than 45. hese shelves are located near the high/low frequencies for the PAx/PDx filters. We should mention that in general PDeReFs are more close to wavelet ackets filters than wavelet filters..3. Examle Rejection of Pulse We rovide a very simle examle for CRA-H filter. A simlified test scheme is shown in Fig. 7. In Fig. 5. Block diagrams of elementary PDe and PRe filters with Pxx blocks he rincial difference is in the arametrical nature of filters. We call those filters as Parametrical or hi- DeReF(s) (PDeReF). Similarly, we use the refix "P" for the names of units. For examle, the resective Aroximation Decomosition block is called PAD. he abbreviations of other blocks are PDD, PAR, and PDR, corresondingly (see Wavelet oolbox). We use also a Fig. 7. Simle filter test scheme (SIMULINK) the first examle we use the sine wave corruted by the additive ulse (b)). he ulse (a)) can be an arbitrary CRA-H BF or a wavelet. he corruted signal has been Fig. 6. Magnitudes of wavelet (Db8) and Pxx filters for different rotation angles generalized abbreviation Pxx. If we comare PDeReFs with wavelet filters we see the following relevant differences: he length of imulse resonse (IR) of elementary filters Pxx is always equal to in oosite to wavelet Fig. 8. Residuals after the filtering of corruted (by 4-th CRA-H BF) sine wave (rejection mode, N=64) rocessed by the PDeReF. he shae of outut signal (c)) deends on the angle(s) tuning (d)). he residual (e)) and the otal Harmonic Distortion (HD) characterize the quality of rejection (in the case of sine). We see that the best refinement is in the case of roer tuning (by angle) to the shae of chosen ulse. he corresonding wavelet filtering ('db', 'db4', etc.) gives very bad results (HD hundreds of ercents). he best that we can achieve is with Haar wavelet filter (3000 times worst than in the case of CRA-H filter).

6 But, if the sine is corruted, for examle, by the ulse with 'db' shae (or any other wavelet) and we erform the corresonding wavelet filtering, the HD is with the same order ( 0.0%) as reviously (Fig. 8). distortion ulses (CRA-H BFs) to sine wave. Fig. 0. HD of filtered sine wave for different distortion ulses (6 samles, A ulse =0*A sin ) Fig. 9. HD for the filtered sine wave in deendence on the tuning angle and the amlitude of the ulse In the case of RABO and RA-H filters we can observe, ossibly, a novel henomenon "shae resonance". In some sense the curves shown in Fig. 9 can be treated as the "resonant curves". here we change the angle instead of frequency. In the generalized case, the SNR (or some other measure) can be used instead of secific HD (defined only for the sine). When we oerate with a set of angles the "resonant curve" is multidimensional. he degree of nonorthogonality (NOD) (zero when signals are orthogonal and one when they are coherent) between signals which must be searated is the quantity for the estimation of otential searation quality. We exect that the NOD is more ersective than the classical vanishing moment tied to a simle olynomial. Fig. 0 shows the changes of HD in deendence on the nonorthogonality of different.3. Examle Extraction of Pulse he next examle is taken from [3]. he ulse (a)) (CRA-H BF) is masked by additive noise (b)). he selected signals are orthogonal, therefore NOD = 0. If so, we can erform the erfect extraction of masked ulse (c)) by the PDeReF for roer angle (d)). In reality NOD differs from zero and the extracted ulse is slightly corruted by some residual signal. Authors are sure that both rovided simle examles demonstrate a large otential of Ps and PDeReFs. he RA-H PDeReF and test scheme are already imlemented into FPGA chis. 3. Phi-ransforms oolbox/blockset We give a brief summary information about the oolbox/blockset in able. he names of units are Fig.. Extraction of single -nd CRA-H BF from additive noise [3]

7 given for MALAB only. If the unit has the -D version (column "D") the name has been aended by "". he names marked by the "(*)" have the modifications given by bolded abbreviations in the second column of table, for examle, hicrafot CRAFO transform. A "+" in the last column "M/S" (MALAB/SIMULINK) indicates that the unit has a similar block in SIMULINK library. At this moment we adot and imlement comlex transforms. he column "C" shows the research state at the aer submission moment. Most of the rimitives run also in symbolic mode. his means that the use instead of the numerical values of angles, may use their symbolic values or the shortcuts to sine and cosine. he total number of units is close to 50. A detailed descrition of functions and blocks will be available during MALAB Conference after the comletion of full testing of the collection. For beginners in Ps area we recommend the interactive GUI tools. In Fig. we can see the main window of -D BFs generator. his tool can generate any of the RABO/RA-H functions and has advanced exort facilities. Our recent efforts target on the develoment of hi-object constructor and corresonding methods (for examle, like wtree in Wavelet oolbox) to work with P. But, the sense of the object is different because of the different nature of P if we comare those with wavelets. Fig.. Main window of -D Phi-BFs generator (higen) showing the interolated -th CRAIMO BF for N=64 able Brief descrition of oolbox/blockset Unit Name Functionality S C D M/S Interactive GUI tools himain he main (to) GUI tool (something like wavmenu) /- hintro4d ool for the demonstration of angular reresentation of four-samle signal in the 4D Euclidian sace /- hidemo Live examles signal analysis/synthesis/comression/filtering/etc /+ higen ool for the generation and visualization of BFs /+ hiansyn ool for the sectrum analysis and synthesis of signals /+ hiofdtool ool for the design of orthogonal filters /+ hirun ool for the running of orthogonal filters (something like dstool) /- Primitives/Blocks elrotmat Elementary rotation matrix /- genkron Generalized Kronecker roduct (defined in [9]) /- hisogrm Stairs like Orthogonal Generalized Rotation Matrix /+ hi(*) Elementary PAD/PDD/PAR/PDR/PdeF/PreF filters /+ hi(*)fot Fast RABO/CRAO/CRAIMO/CRMO transform /+ hi(*)gen Generator of single RABO/CRAO/CRAIMO/CRMO BF /+ hi(*)fht Fast RA-H/CRA-H/CRAIM-H/RSA-H transform /+ hi(*)htgen Generator of single RA-H/CRA-H/CRAIM-H/RSA-H BF /+ hiofdm Generalized P-based OFDM module /+

8 4. Conclusions he resented toolbox is the first successful trial of virtual imlementation of rotation angle based arametrical transforms in the MALAB/SIMULINK environment. he imlemented library contains almost 50 MALAB functions and SIMULINK blocks. he main categories of the units are -D and -D BFs generators, arametrical signal analyzers/synthesizers, arametrical orthogonal filters, generalized arametrical OFDM module, GUI based interactive tools, demo tools, and others. Imlemented rimitives/blocks of arametrical DeRe Phi-filters esentially extend ossibilities for the building of novel arametrical orthogonal filters which allow to filter out the -D signals and images with almost any arbitrarily chosen shae. Our recent exeriments with generalized OFDM based data transmission systems show a very romissing ersective of Ps for the Software Defined Radio. In total, we see the P oolbox as very ersective. Will or will not be this collection a serious layer in the DSP area it deends on our common efforts. 5. Acknowledgments his work has been artly suorted by the roject (the grant from the Council of Science of Latvia), the grant of Ministry of Education and Science, and by the National Program "Modern echnologies in elecommunications". We are also thankful to Andra Martinsone for collaboration and hel. 6. References [] P. Misans, U. Derums, "Introduction into the novel two-dimensional discrete orthogonal transforms based on rotation angles," resented at the th International Conference Electronics, May 8-0, Kaunas, Lithuania, aer (6.) in ress - in Proc. Electronics and Electrical Engineering, Kaunas: echnologija, 008. [] htt:\\ [3] htt:\\ [4] P. Misans, M. erauds, "Introduction into the fast orthogonal transforms based on rotation angles: A new methodical aroach only or a gateway to novel DSP algorithms?," resented at the 5t Electronic Circuits and Systems Conference ECS'05, Bratislava, Slovakia on Set. 8-9, 005, in Conf. Proc., [5] H. C. Andrews, "Multidimensional Rotations in Feature Selection," IEEE ransactions On Comuters, Vol. 8, No. 5, , Set., 97. [6] P. Riede J. Goetze, J.A. Nossek, and C. S. Burrus, "Parameterization of Orthogonal Wavelet ransforms and heir Imlementation," IEEE ransactions On Circuits And Systems II: Analog And Digital Signal Processing, Vol. 45, no., February 998, [7] P. P. Vaidyanathan, "A unified aroach to orthogonal digital filters and wave digital filters, based on LBR twoair extraction," IEEE ransactions On Circuits And Systems, Vol. CAS-3, No. 7, , July, 985. [8] M. erauds, "Synthesis of discrete fast orthogonal transforms," the Ph.D. dissertation, Faculty of Electronics and elecommunications, Riga echnical University, to be ublished. [9] B. J. Fino, R. V. Algazi, "A unified treatment of discrete fast unitary transforms," SIAM J. Comut., 977, 6, No. 4, [0] G. Valters, P. Misans, "Initial version of FPGA-based CRAIMO basis functions generato" resented at the 4th International Conference Mixed Design of Integrated Circuits and Systems MIXDES 007, Ciechocinek, Poland, June -3, 007, in Conf. Proc., [] P. Misans, M. erauds, G. Valters, U. Derums, N. Vasilevskis, "FPGA-based CRAIMO basis function generato" resented at the 5th IEEE Norchi Conference, Aalborg, Denmark, November 9-0, 007, in Conf. Proc. (CD version), 6 ages, ISBN /07/ IEEE, IEEE Catalog Number 07EX896C. [] P. Misans, G. Valters, "Initial version of FPGA-based CRAIMO sectrum analyze" resented at the 6th Electronic Circuits and Systems Conference ECS'07, Bratislava, Slovakia, Set. 6-7, 007, in Conf. Proc., [3] P. Misans, G. Valters, "Introduction into the arametrical decomosition-reconstruction filters based on Haar-like orthonormal transforms," resented at the 6th Electronic Circuits and Systems Conference ECS'07, Bratislava, Slovakia, Set. 6-7, 007, in Conf. Proc., [4] P. Misans, G. Valters, M. erauds, N. Vasilevskis, "Initial imlementation of generalized Haar-like transforms into FPGA-based devices Part II: Parametrical decomosition-reconstruction filters," submitted to the 6th IEEE Norchi Conference, allinn, Estonia, November 0-, 008, 5 ages. [5] P. Misans, "Introduction into the Haar-like transforms based on rotation angles," Scientific Proc. of Riga echnical University, elecommunications and Electronics, Riga, RU, vol. 7, Dec., 007, [6] P. Misans, M. erauds, "Errors of constant rotation angle fast orthogonal transforms used for fixed-oint arithmetic DSP alications: Preliminary results," resented at the 9th International Conference Electronics, May 7-9, Kaunas, Lithuania, in Proc. Electronics and Electrical Engineering, Kaunas: echnologija, No. 4(60),. 7-, 005 (htt:// [7] P. Misans, G. Valters, M. erauds, A. Aboltins, "Initial imlementation of generalized Haar-like transforms into FPGA-based devices Part I: Signal sectrum analyzer-synthesizer module," submitted to the 6th IEEE Norchi 008 Conference, allinn, Estonia, November 0-, 008, 6 ages.