Classical Power Spectrum EstimationAbstract With the increasing need of spectrum, various computational methods and algorithms have been proposed in the literature. Keeping these views and facts of spectrum shaping capability by FRFT based windows we have proposed a closed form solution for Bartlett window in fractional domain. This may be useful for analysis of different upcoming generations of mobile communication in a better way which are based on OFDM technique. Moreover, it is useful for real-time processing of non-stationary signals. As per our best knowledge the closed form solution mentioned in this paper have not been reported in the literature till date.This paper focuses on classical period spectral estimation and moderu spectral estimation based on Burg algorithm. By comparing various algorithms in computational complexity and resolution, Burg algorithm was used to signal processing finally. Experimental and simulation results indicated that digital signal processing system would meet system requirements for measurement accuracy.Keywords periodogram spectral estimation ; Burg algorithmI. INTRODUCTIONWhen we expand the frequency response of any digital filter by means of Fourier series, we get impulse response of the digital filter in the form of coefficients of the Fourier series. But the resultant filter is unrealizable and also its impulse response in infinite in duration. If we directly truncate this series to a finite number of points we have to face with well known Gibbs phenomenon, so we modify the Fourier coefficients bymultiplying it with some finite weighing sequence called window functions, w(n). One desirable characteristics of the Fourier transform of most of the window functions comprises of a central or main lobe of small width containing most of its energy.. Also its side lobes should decay rapidly as the frequency tends to π.1. PERIODOGRAM SPECTRAL ESTIMATIONIF signal by A/u conversion obtain a group of sample data x(1), x(2) ... , through the power spectrum estimates, give the energy of analyzed signal with the frequency distribution and analyze the signal frequency components. Classical power spectrum estimation has two main methods, namely, direct method and auto-correlation method. They are kind of non-parametric methods whose features have nothing to do with any model parameters. Non-parametric spectrum estimation signal extends with N points for the cycle, so it is also known as periodogram. Direct method is regard the N-point observation data X N (n )of the random signal x(n) as an energy limited signal, having direct access to Fourier transform X N (e jw) of X N (n ), and then make the square o f amplitude divided by N as the real power spectrum estimation Of X N (n ).Spectral estimation methods include the following assumptions and steps.The stationary random signal X(n) is regarded as the state traversal , using a sample x(n) instead of X(n), and then use N observations xN(n) to estimate the power spectrum p(w) of x(n). Using the record of acontinuous signal x(t) to estimate P PER(w) , Also including the discretization(A/D), the necessary pretreatment (such as filtering).2. Auto-correlation methodAuto-correlation method (also called the indirect method), A first, X N(n) is estimated from auto-correlation function rem) , then according to Wiener-Khintchine theorem, to get power spectrum of X N(n) by Fourier transform Of r(m) as estimation of p(w) .Computation is not very big when M is small. When M is relatively large, especially near N-l, the estimated deviations of r(m) become larger, which make the estimated quality of power spectrum decline. Therefore, usually take M <<N-l in the auto-correlation method, that is, the maximum length 2N-l of auto-correlation function for truncated, its effect is equivalent to impose a window function, recorded as v( m) .It can be proved that the first window, the mean is equal the trueauto-correlation function r(m) multiplied by the triangular window. The triangular window is generated by data truncation with the width of 2N-1. Where v(m) is the second window of auto-correlation function with the width of 2M-I, M <<N-l. Since the width of v(m) is much smaller than w(m) , the main lobe width of the spectrum v(m) of w(m) will be much greater than spectrum W(w) . Thus, the effect on r(m) imposed by v(m) is equivalent to convolution for P per(w) and v(m) in the frequency domain. Whose role is smoothing periodogram. When M<<N-l, P BT (w) actually smooth periodogram. It can be achieved by directly multiplied by the data window in the x N(n), but this calculation more than the previous[3-4].3. Spectral Estimation of Burg AlgorithmAssumed that the research process x(n) is the output of a linear system H(Z) by stimulated an input sequence u(n), H(Z) is estimated by the known x(n) or r x(m) , and then estimates the power spectrum of x(n) from H(Z) .u(n) is a white noise sequence, the variance is 2, zero of A(Z) and B(Z) should be in the unit circle in order to ensure the stability of H(Z) and the smallest phase system.If known a1,a2.......a p,b1,b2......b q, the power spectrum can be obtained.The basic idea of Burg algorithm is to average power of forward and backward prediction error minimum. Burg definition of m-order forward and backward error is:The filter output can be obtained by substituting Levision recurrence formula. Defined average power of m-order forward and backward prediction error, if m p∂/m k∂=0, thatThe Burg recursive algorithm step is as follows. Firstly, initial value Po and reflection coefficient k m are calculated according to the above equation, then obtained filtering coefficient of forward prediction and prediction error power Pm . Finally, the filtering output is:Repeat the above step, until the prediction error power is no longer obviously reduced, when calculating the order of p,assumed that 111----<=m m m m m p p k p p α, where α is the controlling parameter. According to actual requirements, if the velocity error <0.3%, it is desirable α = 0.001, then the prediction error power P m is white noise variance σ2 . All the orders of AR parameters, and power spectrum was obtained [5-7] .4.Bartlett windowBartlett windowin time domain is given in Fig. 1 followed by MSLL plots and SLFOR plots in Fig. 2 and Fig. 3 respectively for a particular value of the tunable parameter a . In Fig. 4 and Fig. 5 , three dimensional plots of the Bartlett window taking amplitude and normalized frequency for different values of the tunable parameter a is drawn. Table I comprises of all the tabulated values of MSLL, HMLW, SLFOR, width of the main lobe at -3dB and -6dB down from the peak of the main lobe for different values ofa .The width of the main lobe limits the frequency resolution of the resultant windowed signal. As the width of the main lobe becomes narrow we are able to distinguish clearly between two adjacent placed frequency components. But on the other side narrowing of the main lobe results in spectral leakage i.e. energy of the window spreads into side lobes so a trade off is there and the window which is best suited for a particular application is chosen. We see as thevalue of a decreases from 0.9 to 0.1 the value of MSLL varies from -13 dB to -11.9 dB approximately and -3dB bandwidth and -6dB bandwidth decreases from 0.1441 to 0.02355 and 0.18825 and 0.0294 respectively. Side-lobe fall off-rate is also varied from -14dB/octave to -22dB/octave and then to -17.5 db/octave as a is decreased.Figure 1. (Time domain Bartlett window)Figure 2.(MSLL Plot for a =0.7)Figure 3.(SLFOR plot for a =0.7)Figure 4. (3- dim. Continuous values of FRFT Bartlett window)Figure 5. (3- dim. Continuous values of FRFT Bartlett window)II. SIMULATION ANALYSISSimulation results of several power spectrum algorithms are shown in Fig. 1-6. The following conclusions can be drawn through observing data processing results of various methods to different SNR.1) The frequency error is small using classical spectral estimation methods (FFT, periodogram) for different signal to noise ratio data estimation, but the resolution is low, not enough to smooth the power spectrum graphics.2) Modem Spectral Estimation of MUSIC method and the modified covariance method has higher resolution, smaller error in lower order, more smooth power spectrum, but has generally larger calculation.Figure I. Periodogram spectral power estimationFigure 2. Yule-walker power spectrum estimationFigure 3. Burg spectrum estimation(30-order)Figure 4. Burg spectrum estimation(50-order)Figure 5. Modified covariance power spectrum estimationFigure 6. MUSIC power spectrum estimation3)The processing results of Burg spectrum estimation (50 order) for high signal to noise ratio (10dB, 5dB) is good, smoother graphics, the maximum frequency of power spectrum can accurately identify. Peak deviation is small, the resolution can improve by increasing the order of AR model. Therefore, Burg algorithm is a more common method and has a better quality of spectral estimation to reach system requirements .III. CONCLUSIONSignal processing of speed radar generally uses the power spectrum estimation method. Classical method, whether direct or indirect methods, are available to Fast Fourier Transform (FFT) calculation. But the spectral resolution is low, the resolution is proportional to 27r / N . Compared with classical power spectrum estimation method, the parameter model power spectrum estimation has obvious advantages both the variance performance or in the resolution. Simulation result is shown that Burg algorithm has obvious advantages for spectral resolution andmodel order, which can meet system requirements for measurement accuracy.REFERENCES[1] Huaigao Zhou, Analysis of Multi-moving targets spectrum and Design of DSP-Based Radar. Hunan University School of Electrical and InfOimation Engineering.2002.[2] Guangshu HU,Digital signal processing -theOly. algorithm and implementation. Beijing, Tsinghua University Press,1997[3] S.M Kay. Theory and Application of Modem Spectral Estimation. Science Press, Bei jing, 1997,170-200[4] Kay S M.Malple S L Spectnun Analysis-a Modem Perspective 1981(04)[5] Zhenyang Wu.Application of Golay complementary sequences in the spectrU1ll analysis. Journal of Electronic Measurement and Instruments. 1996, (3)14-17[6] Feng Luo. Research on shOit sequence power spectnllll estimates based on the Burg algorithm. Journal of xidian university(natural science) ,2005. 32(5) : 724-728.[7] K. Singh, Performance of discrete fractional Fourier transform classes in signal processing applications, Ph.D. thesis, Thapar University, Patiala, 2006.[8] H.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, first ed., Wiley, New York, 2001.。
