Lv's Distribution for Time-Frequency Analysis

Transcription

1 Recent Researches in ircuits, Systems, ontrol and Signals Lv's Distribution for Time-Frequency Analysis SHAN LUO, XIAOLEI LV and GUOAN BI School of Electrical and Electronic Engineering Nanyang Technological University 50 Nanyang Avenue, Singapore, and Abstract: - A frequency-chirp rate (FR) distribution, nown as Lv's distribution (LVD), is introduced in this paper. The LVD can directly represents the information of centroid frequency and chirp rate of a linear frequency modulated (LFM) signal without using any searching process. It is able to suppress interference to represent clearly the signal pea in the FR plane. omparing to polynomial Fourier transform (PFT) and fractional Fourier transform (FrFT), the LVD has better performance on FR resolution and energy concentration. Based on LVD, a new method of time-frequency transform named inverse LVD (ILVD) is also introduced to obtain a time-frequency representation (TFR). Simulation results show that the ILVD has better energy distribution concentration than the Wigner-Ville distribution (WVD) and local polynomial timefrequency transform (LPTFT). ey-words: - Lv's distribution, inverse Lv's distribution, time-frequency transform Introduction Many signals in music, speech, geology, wireless communication and radar are non-stationary and belong to frequency modulated (FM) signals. The instantaneous frequency (IF) of the signal, defined as the derivative of the phase, usually varies with time. The Fourier transform (FT) is not suitable to process this ind of signals because it cannot tell when the frequencies occur or disappear. It is more useful to characterize the signals with methods of time-frequency analysis, such as short-time Fourier transform and Wigner-Ville distribution (WVD) []. Recently, a frequency-chirp rate (FR) distribution, nown as Lv's distribution (LVD), is proposed for linear FM (LFM) signals []. It can directly provide accurate centroid frequency-chirp rate representation on the FR plane by using a scaling operation and a -dimension (D) FT, without using any searching steps required by the polynomial Fourier transform (PFT) [, 4, 5] and fractional FT (FrFT) [6]. The LVD is simple on computation and asymptotic linear to be applied on multi-component signal. Based on the LVD, inverse LVD (ILVD) is proposed in [] as a time-frequency transform method to generate the time-frequency representation (TFR) of an LFM signal. It does not suffer from noise seriously, and can present a clear TFR in premise of the LVD provides a good estimation of centroid frequency-chirp rate. This paper presents the principles of the LVD and ILVD and their FR and TFR planes of a multicomponent LFM signal. Moreover, we give some performance analysis on energy distribution concentration in terms of distribution concentration rate (DR). omparing to PFT and FrFT, the LVD has better performance on FR resolution and higher value of DR. Experiments based on Monte arlo simulation show that the ILVD also provides higher DR than the WVD and local polynomial timefrequency transform (LPTFT) [7, 8]. Review of the LVD The LVD is a frequency-chirp rate distribution for LFM signals proposed in []. An LFM signal x( t ) can be expressed as () = () = exp( π + πγ ), () x t x t A j f t j t = = where is the number of signal components, is the constant amplitude, f and γ denote the centroid frequency and chirp rate, or the first derivatives of the IF, respectively. Its parametric symmetric instantaneous autocorrelation function (PSIAF) is defined as (, τ ) τ + a τ + a = + * Rx t x t x t = A ( π ( τ ) πγ ( τ ) ) = A exp j f + a + j + a t ISBN:

2 Recent Researches in ircuits, Systems, ontrol and Signals i= j= i+ ( R xx ( t, τ ) R ( )) i j xjx t τ i + + +, () where a denotes a constant time-delay and R xix j means the cross terms. We define a scaling operator t, τ as Γ of a phase function G with respect to ( ) t (, ) n Γ G t τ G, τ, () h ( τ + a ) where h is a scaling factor and t n is called the = + a ht. Let perform a scaling scaled time as t ( τ ) n operation Γ on the PSIAF in () and obtain γ Γ Rx ( t, τ) = A exp jπ f ( τ + a) + jπ tn = h R xx ( t, τ ) R (, ) i j xjx t τ. (4) i + Γ + i= j= i+ Then we tae the -D FT of the previous equation with respect to τ and t n as (, L ) { (, x f γ = Fτ Ft Γ R ) } n x t τ { } Lx ( f, γ ) (, ) Lx ix f γ, (5) j = + = i= j= i+ where F {} means the FT and the auto-terms L x is calculated by jπ af γ Lx ( f, γ) = A ( ) e δ f f δ γ h. (6) L f, γ is an Equation (5) is called the LVD, and ( ) FR plane. According to [], the optimal parameters are a = h=. For a multi-component LFM signal inputs, the LVD has a property of asymptotic linearity because the cross-terms of components are very small comparing to their auto-terms. We therefore have. (7) () = () (, γ ) (, γ ) x t x t L f L f x x = = From the LVD plane, all the parameters of an LFM signal, i.e., centroid frequencies and chirp rates, are f, γ. obtained by the locations of the peas ( ) Review of the ILVD The ILVD is a time-frequency transform generating signal's TFR based on LVD [], and defined as ( ) { { { ( ) }} }, X I t f = Fτ Γ Ff Fγ M Lx f, γ,(8) x Fig. The FR of a multi-component LFM signal based on the LVD. where the F {}, Γ [ ] and M [] represent the inverse FT, inverse scaling transform and masing operation, respectively. The masing operator is defined as Lx ( f, γ), for ( f, γ) X, M Lx ( f, γ ) = (9) 0, for ( f, γ ) X, where X denotes the support of the auto-term on the LVD plane. A TFR plane of a three-component LFM signal by using the ILVD is shown in Fig. 4(c). It is seen that the ILVD has good timefrequency energy concentration. In particular, it does not suffer from noise because the masing operator M [ ] in (9) only selects the signal terms for the transform. Therefore the ILVD can always generate a clear TFR in premise of the LVD f, γ. provides a good estimation of ( ) 4 Experimental Results This section presents some simulation results based on the LVD and ILVD compared with some other time-frequency methods. 4. The LVD An FR plane of a three-component LFM signal obtained by using the LVD is presented in Fig. in which the three peas located at ( f, γ ) are clearly seen. In this section, we will discuss some performance of the LVD, the FR resolution and DR, and compare with the FrFT and PFT. The PFT is defined as [, 4] ISBN:

3 Recent Researches in ircuits, Systems, ontrol and Signals frequency frequency centroid frequency PFT chirp rate (a) PFT FrFT angle (b) FrFT LVD chirp rate (c) LVD Fig. The FRs of the multi-component LFM signal in Example. M m XPFT ( f ) = x() t exp jπ fmt dt m= = F x f, f,..., f, (0) t { ( M )} m x f,..., fm = x t exp( j π ( fmt )). where ( ) ( ) M m= Because we are interested in dealing with the LFM signal in this paper, let us set M = in (0) and f denotes the chirp rate γ. As same as the FrFT, the PFT method requires to searching the chirp rate of the signal. In this paper, the searching steps of order of both PFT and FrFT are / ( N /), where N is input sample length. This maes the computational resolutions of PFT, FrFT and LVD are the same and we can compare them wisely. 4.. FR Resolution According to [], the resolutions of the centroid frequency and chirp rate achieved by the LVD are Δ f = / T and Δ γ = / ( ht ), where T is the time interval. The signal used in the simulation is denoted as Example and given by s () t = A exp( jπ ft+ jπγt ), () = where the number of LFM components =, amplitudes A = for all, centroid frequencies f = f = 5. Hz and f = 5. Hz, the chirp rates γ = γ =.8 Hz/s and γ = 4.8 Hz/s, the sampling frequency f s = 56 Hz and signal length N = 04. This signal has three LFM components with differences in centroid frequencies by Hz and in chirp rates by Hz/s. Fig. shows the FRs of the signal specified in Example. In this figure, these LFM components are clearly represented with three peas only by the LVD since the differences in frequencies and chirp rates are too small to be separated by the resolutions of PFT and FrFT. It is also observed in Fig. (a) that the interferences around the three peas mae the detection difficult. These interferences are mainly due to the finite length of the signal samples so that the large main-lobe width of the window effects may result in interferences between pea values of the signal components. 4.. DR Next let us evaluate the energy concentration performance of the frequency-chirp rate. Energy distribution concentration is an important performa- ISBN:

4 Recent Researches in ircuits, Systems, ontrol and Signals nce in time-frequency analysis. Similar to the concept of the distribution concentration used in [9], the DR is defined to measure the signal-term energy concentration in the FR domain as follows, DR ave FR f ave FR f (, γ ) ( ) f, γ S (, ) = 0log f γ 0 (, γ ) ( f, γ ) S, () where ave[] measures the average energy in a specified region in the FR domain, S is the f, γ in the FR region specified by the points ( ) domain. A higher DR value indicates a better performance on energy concentration of the frequency-chirp rate. The signal, denoted as Example, in this simulation is expressed by the same expression as (), with the parameters of =, centroid frequencies f = 9.44 Hz, f = 8.56 Hz, f = 7.5 Hz, chirp rates γ = 7.06 Hz/s, γ = 8.6 Hz/s, γ = 8.6 Hz/s, A = for all, f s = 8 Hz and N = 5. Table shows the DR values obtained from Example by using different methods. These DR values are for signals without noise and therefore we consider them as ideal ones. It is seen that the DRs based on the LVD are significantly larger than the PFT and FrFT methods. It shows that the LVD obtains the best energy concentration in FR domain because of its capability of minimizing the interferences created by the window effects. This table also indicates that the DR decreases when the signal components increases. Fig. The DRs of the multi-component LFM signal in Example. Table DR omparison for Signals without Noise Method DR (=) (db) DR (=) (db) PFT FrFT LVD Fig. shows the performance of the DR in noise environments based on the Monte arlo simulation results for the signal specified in Example. In the figure, the DRs of the LVD are larger than those from the PFT and FrFT even in negative signal-to-noise ratio (SNR) range. It means that the detection performance based on the LVD should be better in low SNR environments since it has better capability of signal energy concentration than the other two methods. In the SNR range larger than 0 db, the DR values are approaching to those listed in Table. 4. The ILVD In this section, the experiments based on the ILVD are implemented, and the results are discussed and compared with its counterparts, WVD [] and LPTFT. The LPTFT [7, 8] is defined as * jθτ (, f ) X t, f = x t+ τ η τ e dτ, () ( ) ( ) ( ) L η τ is a window function in [ hh, ] M π ( τ τ Mτ ), θ (, τ f ) = f + f /! f / M!, and v- ector f = ( f, f,..., f M ) is a set of parameters which have to be estimated for the computation defined in (). The parameter f corresponds to where ( ) the IF of the signal and f i, for i =,,..., M, correspond to the ( i ) th derivatives of the IF. Since the input signal is an LFM signal, we set M = and the f corresponds to the chirp rate γ in (). In next simulation, we apply the real chirp rate on the LPTFT which is considered an ideal form of LPTFT, with a normalized rectangular window of length N /. 4.. TFR The simulation signal here is as the same form in () with =, centroid frequencies f = Hz, f = 5 Hz and f = 0 Hz, chirp rates γ =.5, γ = and γ = 7.5 Hz/s, the sampling frequency 5 ISBN:

5 Recent Researches in ircuits, Systems, ontrol and Signals frequency (Hz) frequency (Hz) WVD time (s) (a) WVD LPTFT f s = 8 Hz and N = 5, denoted as Example. Fig. 4 shows the TFRs of the three-component LFM signal based on WVD, LPTFT and ILVD. It is clear to see that the ILVD can provide a nice representation of time-frequency, while the WVD and LPTFT are all suffer from the cross-terms more or less. 4.. DR We also evaluate the energy concentration of the ILVD in terms of DR. Similar to the definition in equation (), we define the DR in TFR plane as following, DR (, ) ( ) ave TFR t f t, f S (, ) = 0log t f 0, (4) (, ) ( t, f ) ave TFR t f S where ave[ ] means the average energy, S denotes the signal domain in the TFR, i.e., the LFM signal lines here. Fig. 5 presents the DR values of Example based on WVD, LPTFT and ILVD. We can see in most SNR range the values of DR by ILVD are larger than by other two methods. Therefore the ILVD has a better performance in the issue of energy distribution concentration time (s) (b) LPTFT 0 ILVD 0 frequency (Hz) Fig. 5 The DRs of the multi-component LFM signal in Example based on WVD, LPTFT and ILVD time (s) (c) ILVD Fig. 4 The TFRs of the multi-component LFM signal in Example. 5 onclusion A frequency-chirp rate distribution called the LVD for LFM signals is introduced and some performance is analyzed in this paper. The LVD has better performance on FR resolution and signal energy concentration comparing to PFT and FrFT. Then a new time-frequency transform named ILVD based on LVD is introduced. Experimental results indicate that it can generate a clearer TFR and provide high- ISBN:

6 Recent Researches in ircuits, Systems, ontrol and Signals er value of DR than WVD and LPTFT even in negative SNR environments. References: [] L. ohen, Time-Frequency Analysis, Prentice Hall, NJ, 995. [] X. Lv, G. Bi,. Wan, and M. Xing, Lv's distribution: Principle, implementation, properties, and performance, IEEE Transactions on Signal Processing, vol. 59, no. 8, pp , Aug. 0. [] X. Xia, Discrete chirp-fourier transform and its application to chirp rate estimation, IEEE Transactions on Signal Processing, vol. 48, no., pp. --, Nov [4] Y. Wei and G. Bi, Efficient analysis of timevarying multicomponent signals with modified LPTFT, EURASIP Journal on Applied Signal Processing, vol. 005, no., pp. 6 68, Jan [5] F. Zhang, Y. hen, and G. Bi, Adaptive harmonic fractional Fourier transform, IEEE Signal Processing Letters, vol. 6, no., pp. 8--8, Nov [6] E. Sejdic, I. Djurovic, and L. Stanovic, Fractional Fourier transform as a signal processing tool: An overview of recent developments, Signal Processing, vol. 9, no. 6, pp , Jun. 0. [7] V. atovni, A new form of the Fourier transform for time-varying frequency estimateon, Signal Processing, vol. 47, no., pp , 995. [8] X. Li and G. Bi, Local polynomial Fourier transform: a review on recent developments and applications, Signal Processing, vol. 9, pp , 0. [9] S.M. Zabin and H.V. Poor, Efficient estimation of class a noise parameters via the EM algorithm, IEEE Transactions on Information Theory, vol. 7, no., pp , Jan 99. ISBN: