Power Spectrum and Correlation (Vaseghi - Advanced Digital Signal Processing and Noise Reduction)

Advanced Digital Signal Processing and Noise Reduction, Second Edition.Saeed V. VaseghiCopyright © 2000 John Wiley & Sons LtdISBNs: 0-471-62692-9 (Hardback): 0-470-84162-1 (Electronic),Pjk ω0te9kω0t5HPOWER SPECTRUM AND CORRELATION9.19.29.39.49.59.69.7Power Spectrum and CorrelationFourier Series: Representation of Periodic SignalsFourier Transform: Representation of Aperiodic SignalsNon-Parametric Power Spectral EstimationModel-Based Power Spectral EstimationHigh Resolution Spectral Estimation Based on Subspace Eigen-AnalysisSummaryThe power spectrum reveals the existence, or the absence, of repetitivepatterns and correlation structures in a signal process.

Thesestructural patterns are important in a wide range of applications suchas data forecasting, signal coding, signal detection, radar, patternrecognition, and decision-making systems. The most common method ofspectral estimation is based on the fast Fourier transform (FFT). For manyapplications, FFT-based methods produce sufficiently good results.However, more advanced methods of spectral estimation can offer betterfrequency resolution, and less variance. This chapter begins with anintroduction to the Fourier series and transform and the basic principles ofspectral estimation. The classical methods for power spectrum estimationare based on periodograms.

Various methods of averaging periodograms,and their effects on the variance of spectral estimates, are considered. Wethen study the maximum entropy and the model-based spectral estimationmethods. We also consider several high-resolution spectral estimationmethods, based on eigen-analysis, for the estimation of sinusoids observedin additive white noise.Power Spectrum and Correlation2649.1 Power Spectrum and CorrelationThe power spectrum of a signal gives the distribution of the signal poweramong various frequencies.

The power spectrum is the Fourier transform ofthe correlation function, and reveals information on the correlation structureof the signal. The strength of the Fourier transform in signal analysis andpattern recognition is its ability to reveal spectral structures that may be usedto characterise a signal. This is illustrated in Figure 9.1 for the two extremecases of a sine wave and a purely random signal.

For a periodic signal, thepower is concentrated in extremely narrow bands of frequencies, indicatingthe existence of structure and the predictable character of the signal. In thecase of a pure sine wave as shown in Figure 9.1(a) the signal power isconcentrated in one frequency. For a purely random signal as shown inFigure 9.1(b) the signal power is spread equally in the frequency domain,indicating the lack of structure in the signal.In general, the more correlated or predictable a signal, the moreconcentrated its power spectrum, and conversely the more random orunpredictable a signal, the more spread its power spectrum. Therefore thepower spectrum of a signal can be used to deduce the existence of repetitivestructures or correlated patterns in the signal process. Such information iscrucial in detection, decision making and estimation problems, and insystems analysis.x(t)PXX(f)ft(a)PXX(f)x(t)t(b)Figure 9.1 The concentration/spread of power in frequency indicates thecorrelated or random character of a signal: (a) a predictable signal, (b) arandom signal.fFourier Series: Representation of Periodic Signals265,Psin(kω0t)cos(kω0t)jk ω0teKω0t5HtT0(a)(b)Figure 9.2 Fourier basis functions: (a) real and imaginary parts of a complexsinusoid, (b) vector representation of a complex exponential.9.2 Fourier Series: Representation of Periodic SignalsThe following three sinusoidal functions form the basis functions for theFourier analysis:x1 (t ) = cos ω 0 tx 2 (t ) = sinω 0 tx3 (t ) = cos ω 0 t + j sin ω 0 t = e jω0t(9.1)(9.2)(9.3)Figure 9.2(a) shows the cosine and the sine components of the complexexponential (cisoidal) signal of Equation (9.3), and Figure 9.2(b) shows avector representation of the complex exponential in a complex plane withreal (Re) and imaginary (Im) dimensions.

The Fourier basis functions areperiodic with an angular frequency of ω0 (rad/s) and a period ofT0=2π/ω0=1/F0, where F0 is the frequency (Hz). The following propertiesmake the sinusoids the ideal choice as the elementary building block basisfunctions for signal analysis and synthesis:(i) Orthogonality: two sinusoidal functions of different frequencieshave the following orthogonal property:Power Spectrum and Correlation266∞∞∞11∫−∞sin(ω1t ) sin(ω 2 t ) dt = 2 −∫∞cos(ω1 + ω 2 ) dt + 2 −∫∞cos(ω1 − ω 2 ) dt = 0(9.4)For harmonically related sinusoids, the integration can be takenover one period.

Similar equations can be derived for the product ofcosines, or sine and cosine, of different frequencies. Orthogonalityimplies that the sinusoidal basis functions are independent and canbe processed independently. For example, in a graphic equaliser,we can change the relative amplitudes of one set of frequencies,such as the bass, without affecting other frequencies, and in subband coding different frequency bands are coded independently andallocated different numbers of bits.(ii) Sinusoidal functions are infinitely differentiable.

This is important,as most signal analysis, synthesis and manipulation methodsrequire the signals to be differentiable.(iii) Sine and cosine signals of the same frequency have only a phasedifference of π/2 or equivalently a relative time delay of a quarterof one period i.e. T0/4.Associated with the complex exponential function e jω0t is a set ofharmonically related complex exponentials of the form[1, e ± jω0t ,e ± j 2ω0t ,e ± j 3ω0t , ](9.5)The set of exponential signals in Equation (9.5) are periodic with afundamental frequency ω0=2π/T0=2πF0, where T0 is the period and F0 is thefundamental frequency. These signals form the set of basis functions for theFourier analysis.

Any linear combination of these signals of the form∞∑ c k e jkω t0(9.6)k = −∞is also periodic with a period T0. Conversely any periodic signal x(t) can besynthesised from a linear combination of harmonically related exponentials.The Fourier series representation of a periodic signal is given by thefollowing synthesis and analysis equations:Fourier Transform: Representation of Aperiodic Signalsx(t ) =∞∑ ck e jkω t0k = − 1,0,1,(synthesis equation)267(9.7)k = −∞ck =1T0T0 / 2∫ x(t )e− jkω 0tdtk = − 1,0,1, (analysis equation)(9.8)−T0 / 2The complex-valued coefficient ck conveys the amplitude (a measure of thestrength) and the phase of the frequency content of the signal at kω0 (Hz).Note from Equation (9.8) that the coefficient ck may be interpreted as ameasure of the correlation of the signal x(t) and the complex exponentiale − jkω 0t .9.3 Fourier Transform: Representation of Aperiodic SignalsThe Fourier series representation of periodic signals consist of harmonicallyrelated spectral lines spaced at integer multiples of the fundamentalfrequency.

The Fourier representation of aperiodic signals can be developedby regarding an aperiodic signal as a special case of a periodic signal withan infinite period. If the period of a signal is infinite then the signal does notrepeat itself, and is aperiodic.Now consider the discrete spectra of a periodic signal with a period ofT0, as shown in Figure 9.3(a). As the period T0 is increased, the fundamentalfrequency F0=1/T0 decreases, and successive spectral lines become moreclosely spaced.

In the limit as the period tends to infinity (i.e. as the signalbecomes aperiodic), the discrete spectral lines merge and form a continuousspectrum. Therefore the Fourier equations for an aperiodic signal (known asthe Fourier transform) must reflect the fact that the frequency spectrum of anaperiodic signal is continuous. Hence, to obtain the Fourier transformrelation, the discrete-frequency variables and operations in the Fourier seriesEquations (9.7) and (9.8) should be replaced by their continuous-frequencycounterparts. That is, the discrete summation sign Σ should be replaced bythe continuous summation integral ∫ , the discrete harmonics of thefundamental frequency kF0 should be replaced by the continuous frequencyvariable f, and the discrete frequency spectrum ck should be replaced by acontinuous frequency spectrum say X ( f ) .Power Spectrum and Correlation268c(k)x(t)Ton(a)Toff1T0tkT0=Ton+ToffX(f)x(t)Toff = ∞(b)tfFigure 9.3 (a) A periodic pulse train and its line spectrum.

(b) A single pulse fromthe periodic train in (a) with an imagined “off” duration of infinity; its spectrum isthe envelope of the spectrum of the periodic signal in (a).The Fourier synthesis and analysis equations for aperiodic signals, the socalled Fourier transform pair, are given by∞x(t ) =∫ X ( f )ej 2πftdf(9.9)−∞∞X ( f ) = ∫ x(t )e − j 2πft dt(9.10)−∞Note from Equation (9.10), that X ( f ) may be interpreted as a measure of− j 2πftthe correlation of the signal x(t) and the complex sinusoid e.The condition for existence and computability of the Fourier transformintegral of a signal x(t) is that the signal must have finite energy:∞∫−∞x(t ) 2 dt < ∞(9.11)Fourier Transform: Representation of Aperiodic Signalsx(0)x(1)x(2)x(N–2)x(N – 1)269Discrete FourierTransform...N–1X(k) =∑2 π knx(m) e N–j...X(0)X(1)X(2)X(N – 2)X(N– 1)m=0Figure 9.4 Illustration of the DFT as a parallel-input, parallel-output processor.9.3.1 Discrete Fourier Transform (DFT)For a finite-duration, discrete-time signal x(m) of length N samples, thediscrete Fourier transform (DFT) is defined as N uniformly spaced spectralsamplesX (k ) =N −1∑ x(m) e − j (2π / N )mk ,k = 0, .

. ., N−1(9.12)m =0(see Figure9.4). The inverse discrete Fourier transform (IDFT) is given byx ( m) =1NN −1∑ X (k ) e j (2π / N )mk ,m= 0, . . ., N−1(9.13)k =0From Equation (9.13), the direct calculation of the Fourier transformrequires N(N−1) multiplications and a similar number of additions.Algorithms that reduce the computational complexity of the discrete Fouriertransform are known as fast Fourier transforms (FFT) methods. FFTmethods utilise the periodic and symmetric properties of e − j 2Œ1 to avoidredundant calculations.9.3.2 Time/Frequency Resolutions, The Uncertainty PrincipleSignals such as speech, music or image are composed of non-stationary (i.e.time-varying and/or space-varying) events.

For example, speech iscomposed of a string of short-duration sounds called phonemes, and anPower Spectrum and Correlation270image is composed of various objects. When using the DFT, it is desirableto have high enough time and space resolution in order to obtain the spectralcharacteristics of each individual elementary event or object in the inputsignal. However, there is a fundamental trade-off between the length, i.e. thetime or space resolution, of the input signal and the frequency resolution ofthe output spectrum. The DFT takes as the input a window of N uniformlyspaced time-domain samples [x(0), x(1), …, x(N−1)] of duration ∆T=N.Ts,and outputs N spectral samples [X(0), X(1), …, X(N−1)] spaced uniformlybetween zero Hz and the sampling frequency Fs=1/Ts Hz.