Integrated Wavelet Packet Modulation and Signal Analysis Using Analytic Wavelet Packets

Transcription

1 Integrate Wavelet Pacet Moulation an Signal Analysis Using Analytic Wavelet Pacets Michael auer, René Anselment, Klaus Dostert Institute of Inustrial Information Technology, Universität Karlsruhe TH Hertzstr Karlsruhe, Germany Astract Transmit signals for low-spee PLC may e seriously ostructe y impulsive noise, resulting in a significant egraation of system performance an reliaility In this paper we provie a possile solution to this prolem y using Wavelet Pacets for integrate signal moulation an analysis Moreover we show that Analytic Wavelet Pacets which are nown for superior signal analysis properties are also suitale for moulation Keywors moulation, power line communication, wavelet pacets I INTRODUCTION S is generally nown the PLC transmission channel Aposes a challenging environment for communication Orthogonal Frequency Division Multiplexing OFDM has so far prove to e a goo means to counter the prolems arising from a PLC channel, especially concerning frequency selective channels [] However, OFDM is ase on the Fast Fourier Transform FFT that elivers information on the spectral properties of a winowe signal, ut lacs the aility of resolving its ehaviour over time It is therefore extremely sensitive to the most challenging class of noises on a PLC channel, namely impulsive noise One alternative moulation scheme that coul prove to e promising in this context is the Wavelet Pacet Transform, ecause it provies time resolution in aition to frequency resolution The approach to exploit time resolution within the Wavelet Pacet Transform contrasts the using of Wavelet Pacets for Wavelet-OFDM as escrie in [] In this paper, we present the Wavelet Pacet Transform an its extension, the Complex Wavelet Pacet Transform, for general signal moulation as well as signal analysis at the receiver sie In Section II we provie a short overview of the PLC transmission channel an relate simplifying assumptions Section III summarizes the conventional Wavelet Pacet Moulation WPM first propose in [], which is extene to the Complex Wavelet Pacet Moulation CWPM in Section IV Section V compares CWPM to WPM an shows the enefits of using CWPM for integrate signal analysis Finally, Section VI provies a short summary of the results an the conclusion II PLC CHANNEL MODEL A general moel for the PLC transmission channel has een propose for example in [4], see Fig The transmit signal st is influence y the time-variant channel impulse response ht,τ, that represents a time-variant, frequency-selective transfer function In aition, the receive signal rt contains up to four ifferent ins of aitive noise, where impulsive noise, either perioic or aperioic, is to e consiere the most challenging class of noises ue to its generally wie-an characteristics For the following sections, we mae several simplifying assumptions on the general PLC channel that are vali for the frequency range etween 9 Hz an 9 Hz, the CENELEC A an as specifie in [] We assume that the channel transfer function is ieal, ie ht,τ=δt Moreover, narrowan noise an coloure acgroun noise are assume to e out of the frequency range that can actually e utilize We focus our iscussion on impulsive noise Fig isplays one example measurement of aperioic impulsive noise that has een recore at the usar of a transformer station Although it is ifficult to generalize the waveform of impulsive noise, we assume that impulsive noise preominantly exhiits transient ehaviour an can est e approximate y t nt = A sin πft e Here, A is the initial amplitue of the pulse, f is the frequency of the harmonic oscillation, an is a parameter specifying the ecay of the exponential function Perioic Impulsive Noise Aperioic Impulsive Noise Noise Scenario Narrowan Noise acgroun Noise ntτ, st htτ, i H f, τ rt Channel Transfer Function Fig : Moel of the PLC transmission channel /9/$ 9 IEEE 4 Authorize license use limite to: Karlsruhe Institute of Technology Downloae on April, at 7::6 UTC from IEEE Xplore Restrictions apply

2 Voltage V Time ms Fig : Waveform of impulsive noise For the sae of simplicity, we assume to e large enough so that the waveform ecays to values near zero quicly, which guarantees a finite uration of this waveform III CONVENTIONAL WAVELET PACKET MODULATION The Wavelet Pacet Moulation WPM has een introuce in [] Data symols consisting of parallel ata su-streams with iniviual symol rates / T with =,, are transmitte on frequency ans using an orthogonal set of wavelet pacet functions p, nt Here, /T enotes the average symol rate of all su-streams Each wavelet pacet function p n, t can e erive from an original function p t through time shifting y an integer multiple n of T an through scaling y a factor of - : n, = p t p t n T = p tn T The superscript consecutively numers the original unscale wavelet pacet functions in the orer of their spectral appearance eginning with p t = φ t eing the only lowpass function calle the scaling function Although the functions p n, t significantly overlap in time an frequency, their orthogonality properties ensure reception without intersymol interference uner ieal conitions Let the set α contain all cominations, which escrie the wavelet pacet functions suppose to e use for transmission The transmit signal can then e state as WPM =, n, α n= s t p, t where is the ata symol transmitte y means of the function p, t n A criterion for α to form a vali wavelet pacet asis was given y Coifman, Meyer an Wicerhauser an can e foun in [] Each function p, t can e expresse in terms of the n scaling function φ t filtere y the FIR filter f l,, n = φ l= p t f l t nt lt 4 Equation can e rewritten y using 4 to give an alternative representation of the transmitte signal: WPM = m=, α n= φ s t f m n t mt = ym φ tmt m= In the sequence ym simply is the iscrete Inverse Wavelet Pacet Transform IWPT of the ata symols, which can e efficiently implemente using the wellnown filter an structure introuce y Mallat [6] This interpretation of leas to a moulation scheme as epicte in Fig The ata symols are transforme into a time omain signal using a synthesis filter an with scaling filter hn an wavelet filter gn The transmitte continuous signal is finally generate y pulse shaping with φ t Mutual relations etween φ t, hn, an gn can e foun in [6] The FIR filter f l merges all filters corresponing to one specific frequency an inexe with In this way, together with one upsampling operation y a factor of, a compact notation is achieve, as can e seen in Nevertheless, implementation using consecutive stages of upsampling operations y a factor of followe y scaling filter an wavelet filter, respectively, was prove to lea to ientical results while eing the more efficient way to calculate the time omain signal [6] Demoulation is epicte in Fig 4 The receive signal is matche filtere y φ t an sample with rate /T to get an estimation ŷm of the sequence ym After that, the Wavelet Pacet Transform WPT, represente y an analysis filter an structure, recovers the ata symols As there are many possile choices for a vali set α, areas in the time-frequency plane can e allocate very flexily to the ata symols Fig isplays the resulting structure of the analysis filter an as well as the corresponing partitioning of the time-frequency plane if the set α={,,,4,,,,} is chosen for moulation an emoulation, respectively Authorize license use limite to: Karlsruhe Institute of Technology Downloae on April, at 7::6 UTC from IEEE Xplore Restrictions apply

3 φ t g g h h g g h h IWPT g g h h ym φ t Fig : loc iagram of the WPM moulation tree T y m g n h n WPT Fig 4: loc iagram of the WPM emoulation tree s WPM t As can e seen from Fig, WPM offers time as well as frequency resolution within one WPM symol, therey contrasting conventional multi-carrier moulation schemes lie OFDM Moreover, this time an frequency resolution can e auste to current requirements On the one han this flexiility proves avantageous for the moulation scheme itself, as the partitioning of the time-frequency plane can e chosen with respect to current characteristics of the channel an the noise scenario [] On the other han, emoulation in a WPM scheme is nothing ut time-frequency analysis of the receive signal Useful information aout channel an noise scenario can therefore e erive from the wavelet pacet coefficients The potential to aust resolution in time an frequency gives the WPT an avantage over Fourier ase methos also for this application, especially in a scenario with time variant noises such as impulsive noise 4 Frequency f Time t Fig : Possile WPM analysis tree structure an resulting partitioning of the time-frequency plane IV COMPLEX WAVELET PACKET MODULATION Two maor prolems occur when WPT is use for signal analysis The transform proves highly shift variant, which is the result of aliasing within a filter an ue to the ownsampling operations Furthermore, the resulting wavelet pacet coefficients show oscillations even if a signal with a constant energy istriution is analyze ue to oscillating wavelet pacet functions [7] A possile solution to these prolems was propose in [8]: Instea of real-value asis functions, approximately analytical an therefore necessarily complex-value functions are use for calculating the transform As this cannot e accomplishe with one single filter an, the application of two parallel filter ans for analysis as well as for synthesis is propose in [8] One filter an represents the real part of the transform, the other one represents the imaginary part Therefore this transform is calle Dual-tree complex wavelet transform For synthesis the average is calculate to fuse oth signals elivere inepenently y the synthesis filter ans [7] In orer for the asis functions to e analytical, the scaling filters of the two filter ans have to fulfil the so-calle half-sample elay HSD conition: the imaginary filter an s scaling filter h i n has to e ientical to the real filter an s scaling filter h r n, ut shifte y half a sample [7] The scaling filters in the first stage of a filter an, the socalle first stage filters, have to e shifte y one sample [7] As the HSD conition cannot e perfectly fulfille y FIR filters, the resulting asis functions can only e approximately analytical Several methos for esigning scaling filters which approximately fulfil the HSD conition were propose eg in [9] an [] The extension from the approximately analytical Complex Wavelet Transform CWT to the approximately analytical Complex Wavelet Pacet Transform CWPT was carrie out in [] The well-nown tree-lie structure of the filter ans remains unchange ut specific filters have to e interchange etween the real an the imaginary filter an In [] a etaile explanation is provie When a signal is analyse y means of the CWPT, two sets of coefficients are generate y the ual-tree filter an The coefficients at the output of the real filter an are laelle r, the coefficients at the output of the imaginary filter an are laelle i accoringly These coefficients result from the analysis of an input signal y means of the r i real-value wavelet pacet functions p n, t an p n, t, respectively, which together form an approximate Hilert pair In orer to leverage the avantages offere y the CWPT, the complex coefficients nee to e calculate This can e accomplishe in two ways: + n = n + n 6 c r i c r i n = n n 7 6 Authorize license use limite to: Karlsruhe Institute of Technology Downloae on April, at 7::6 UTC from IEEE Xplore Restrictions apply

4 These two sets of coefficients correspon to an analysis with the approximately analytical functions p t = p t + p t 8 c+ r i n, n, n, c r i n, n, n, p t = p t p t 9 If the analyze signal is real-value, c+ is the c complex conugate of an therefore oes not nee to e calculate Otherwise, the coefficients c+ an c oth nee to e nown in orer to uniquely otain r the coefficients an i As the CWPT is a linear transform, it can also e written using vector matrix notation [7] Let the orthogonal matrix F r represent the real filter an an F i the imaginary one Then signal analysis can e escrie as w r = -/ F r x an w i = -/ F i x, where the vector x consists of the samples of the time signal which is analyse an the vectors w r an w i contain the coefficients r an i respectively Inversely, synthesis can e - - r - r r i c+ c an state as = + x F w F w The coefficients are collecte in the vectors u an v respectively, where u= -/ w r + w i an v= -/ w r - w i If CWPT is suppose to e use in a moulation scheme, two approaches seem possile at first glance On the one han, c+ ata symols coul e interprete as coefficients or c approach On the other han, they coul e r interprete as coefficients or i approach For oth approaches it has to e verifie that the ual-tree filter an in transmultiplexer structure fulfils the perfect reconstruction PR conition, ie the coefficients can e exactly recovere after a synthesis operation followe y its inverse analysis operation Approach : Without loss of generality, the ata symols c+ are interprete as coefficients, ie the vector u contains the ata symols In orer to fulfil the PR conition, the vector û, resulting from executing ual-tree synthesis of u followe y its ual-tree analysis, must satisfy the conition û=u For this conition to e satisfie, oth, the real an the imaginary synthesis filter ans must each generate ientical outputs so that the averaging operation oes not cause any errors It can e shown that this is satisfie iff r i i r r i i r F F + F F v = F F F F I u to choose a vector v so that is satisfie, ecause the matrix M = F r F i + F i F r is very close to eing singular for orthogonal scaling filters an wavelet filters that fulfil the HSD conition This was verifie eg with so-calle q-shift filters from [9] an with filters esigne with the common factor metho introuce in [] All these filters were teste in comination with first stage filters of the Dauechies, the Coiflet, an the Symmlet family In all cases consiere, the eterminant of M was shown to e of the orer of - an elow Therefore, we postulate that cannot e resolve for aritrarily chosen ata vectors u The ata woul e require to fulfil certain conitions which reners this moulation scheme generally unfeasile for communications r Approach : Here, the coefficients or i are suppose to carry the ata symols Without loss of generality r we consier the coefficients as ata symols in the following The PR conition for the ata vector w r is satisfie if ŵ r =w r Again, this is fulfille if oth synthesis filter ans r i lea to ientical output signals, ie if F r w = F i w Therefore, w i has to e chosen accoring to w = F F w i r i r As F r an F i oth are orthogonal matrices, their inverses always exist an w i can always e chosen to satisfy Moreover, as oth filter ans lea to an ientical output signal, the result of the imaginary filter an oes not even nee to e calculate As only real-value wavelet pacet functions are actually use for transmission, moulation as well as emoulation can e accomplishe using a single synthesis an analysis filter an respectively Therefore the propose moulation scheme is very similar to conventional WPM Nevertheless, as all requirements for analytical wavelet pacet functions, such as the HSD conition, were consiere in the filter esign process an therefore are met y the filters, a ual-tree filter an can e optionally use in the receiver for the purpose of signal analysis The resulting moulation an emoulation scheme is enote in Fig 6 The real an imaginary analysis filter ans are separate y a ashe line V ANALYSIS OF NOISY RECEIVE SIGNALS In orer to show the avantages of the propose CWPM scheme over conventional WPM, we compare the analysis of several receive signals in this section For this purpose, oth systems were simulate in a MATLA environment For the WPM system, Dauechies filters of length were chosen for moulation an signal analysis respectively is fulfille, where I is the unity matrix Numerical tests have shown that it is in general not possile 7 Authorize license use limite to: Karlsruhe Institute of Technology Downloae on April, at 7::6 UTC from IEEE Xplore Restrictions apply

5 w r r w i w u = v = r i w + w r i w w Fig 6: Diagram of the propose CWPM scheme A single-tree synthesis filter an is use for moulation A ual-tree analysis filter an is use for emoulation an signal analysis at the same time The CWPM system generates the transmitte signal using a single-tree synthesis filter an structure as epicte in Fig 6 y means of q-shift filters of length 4 from [9] for moulation as well as for signal analysis Dauechies filters of length were chosen as first stage filters in orer to ensure comparaility etween oth systems Whenever a ata signal is consiere, inary real-value coing of the ata symols is use, ie each symol is chosen from the set {-,} This correspons to inary phase shift eying PSK each ata symol oth systems use ecomposition level, ie ==, for moulating the ata symols The uration of one WPM or CWPM supersymol is enote y T S, where N ata symols are comine to one supersymol so that T S =N T First, we present an analysis of a pure narrowan noise signal Narrowan noise is not a typical source of isturance for the channel uner consieration, the results are rather presente for the sae of completeness In aition, this example shows very clearly the isavantages regaring signal analysis ue to oscillating wavelet coefficients of conventional WPM / WPT The noise signal is simulate as a sine-wave of constant power at 6 % of the system s Nyquist frequency f N Fig 7 shows the energies of WPT coefficients on the left c+ an the energies of CWPT coefficients on the right Dar areas represent coefficients with high energy, right areas represent coefficients with low energy Signal analysis results are shown for ecomposition level = û v The WPT analysis oes not reveal that the power of the noise signal is constant in the relevant time interval, whereas this is clearly visile in the CWPT analysis This avantage ecomes particularly apparent in the analysis of a signal carrying ranom ata an eing istorte y the same sinewave noise signal Fig 8 shows the analysis of such a receive ata signal with a signal-to-noise ratio SNR of Here the narrowan noise signal can harly e etecte y WPT analysis ue to the unerlying ata signal The little numer of coefficients with increase energy cannot e allocate to a specific in of noise In the CWPT analysis, however, the noise signal can clearly e istinguishe from the ata signal an can e ientifie as narrowan noise In the following we present the analysis of impulsive noise which is much more typical for PLC transmission channels uner consieration Fig 9 shows the energies of WPT an CWPT coefficients at ecomposition level = when a signal consisting of two noise pulses is analyse No ata signal is consiere here, which leas to a more istinct representation of the results oth pulses are stochastic waveforms of high power uring a short perio of time In this case, each pulse has a uration of T S The pulses occur at times T S an 4 T S, therey enclosing a time span of T S where noise energy is asent Analysis y means of the CWPT clearly shows oth pulses an the time span free of noise energy etween them WPT analysis oes not allow for istinguishing etween the pulses ue to oscillations in the coefficients energies 7 n WPM 7 7 c+ n CWPM 7 Fig 8: Analysis of a ata signal istorte y narrowan noise n WPM c+ n CWPM n WPM c+ n CWPM Fig 7: Signal analysis of narrowan noise 7 7 Fig 9: Analysis of two intermittent noise pulses stochastic waveforms Finally, we present the analysis of noise pulses which are of 8 Authorize license use limite to: Karlsruhe Institute of Technology Downloae on April, at 7::6 UTC from IEEE Xplore Restrictions apply

6 the type introuce in section II accoring to A noise signal containing two pulses is superimpose with the ata signal Parameters of the pulses were chosen as follows: The first pulse has initial amplitue of A =7, a ecay of = -4 an a frequency of f =64 f N It appears after T S The secon pulse which appears after 6 T S has the same parameters, ut its initial amplitue is A =9 This results in an SNR at the receiver of - The receive signal is epicte in Fig Energies of WPT an CWPT coefficients at ecomposition level = are isplaye in Fig The exponential ecay of the noise energy can clearly e oserve in the CWPT coefficients The WPT coefficients, however, show oscillations that mae the characteristics of the original noise signal har to estimate This can e emonstrate even more explicitly when only the coefficients of the frequency an which contains the maor part of the noise energy is regare Fig shows the coefficients energies of the frequency an with inex = at ecomposition level = WPT coefficients show sharp notches for inices that correspon to time perios in which the receive signal contains large amounts of noise energy Voltage V - Analyse receive signal - 7 Fig : Data signal istorte y two transient noise pulses 7 n WPM 7 c+ n CWPM 7 7 Fig : Analysis of a ata signal istorte y transient noise pulses Energy n WPM Inex of coefficients n n Energy c CWPM Inex of coefficients n Fig : Coefficients energy istriution on one specific frequency an Improve shift invariance properties of CWPT can clearly e seen in Fig as well As oth noise pulses are similar apart from their ifferent initial amplitues, the shapes of their respective energy istriution over the CWPT coefficients o not significantly iffer Even the energy ratio of 7 /9 6 can e estimate very precisely from the CWPT coefficients energies This however is not possile regaring energy istriution of WPT coefficients ecause they show significantly ifferent shapes for the two noise pulses VI CONCLUSION In general, it woul e esirale to mitigate the influences of channel-relate noise on low-spee PLC transmission signals We propose an efficient metho for integrate ata moulation / emoulation an receive signal analysis as a first step towars this goal The simulation results encourage utilizing Analytic Wavelet Pacets for oth purposes Signal analysis at the receiver sie can asically e performe y CWPT in any in of moulation scheme However, if CWPM is use for moulation, CWPT analysis can e implemente in an efficient way using the alreay existing analysis filter an structure as well as the results provie y the emoulation Our propose moulation scheme therefore allows for leveraging the synergies of the preferale properties of Analytic Wavelet Pacets concerning signal analysis along with their applicaility for ata moulation REFERENCES [] K Dostert, Power line communications, Prentice Hall, [] S Galli, H Koga, N Koama, Avance signal processing for PLCs: Wavelet-OFDM, Proceeings of the 8 IEEE ISPLC, pp 87-9, Jeu City, Korea, April 8 [] AR Linsey, Orthogonally Multiplexe Communication via Wavelet Pacet ases, PhD Dissertation, Ohio University, Athens, Ohio, June 99 [4] M Goetz, K Dostert, A Universal High Spee Powerline Channel Emulator, International Zurich Seminar on roaan Communications, [] CENELEC EN 6, Signalling on low voltage electrical installations in the frequency range Hz to 48 Hz, russels, 99- [6] SG Mallat, a wavelet tour of signal processing, n e, Acaemic Press, San Diego, California, 999 [7] IW Selesnic, RG araniu an NG Kingsury, The Dual-Tree Complex Wavelet Transform, IEEE Signal Processing Magazine, Vol, Issue 6, pp -, Novemer [8] NG Kingsury, The Dual-Tree Complex Wavelet Transform: A New Technique for Shift Invariance an Directional Filters, Proceeings of the 8th IEEE DSP worshop, paper no 86, Utah, August 998 [9] NG Kingsury, A Dual-Tree Complex Wavelet Transform With Improve Orthogonality an Symmetry Properties, Proceeings of the IEEE International Conference on Image Processing, Vol, pp 7-78, Vancouver, Septemer [] IW Selesnic, The Design of Approximate Hilert Transform Pairs of Wavelet ases, IEEE Trans on Signal Processing, Vol, Issue, pp 44-, May [] T Weicert, C enaminsen, U Kience, Analytic Wavelet Pacets Comining the Dual-Tree Approach with Wavelet Pacets for Signal Analysis an Filtering, IEEE Trans on Signal Processing, January 9, in press 9 Authorize license use limite to: Karlsruhe Institute of Technology Downloae on April, at 7::6 UTC from IEEE Xplore Restrictions apply