On estimating spectral moments in the presence of colored noise

Let{q^(1) (t)}, the signal, be a complex Gaussian process corrupted by additive Gaussian noise{q^(2) (t) }. Observations onp(t)q(t)andp(t) q^(2) (t)are assumed to be available wherep(t)is a smooth weighting function andq = q^(1) + q^(2). Using the Fourier transform of the samples ofp(t)q(t)andp(t) q^(2) (t), estimators are derived for estimating the mean frequency and spectral width of the unknown power spectrum of the unweighted signal process. The means and variances of these statistics are computed in general, and explicitly for nontrivial practical examples. Asymptotic formulas for the moment estimators as a function of the number of realizations, frequency resolution, signal-to-noise ratio and spectral width, and consistency of the estimators are some of the results that are discussed in detail.     

Robust prediction of band-limited signals from past samples (Corresp.)

For a complex-valued deterministic signal of finite energy band-limited to the normalized frequency band|w| leq piexplicit coefficients{a_{kn}}are found such that for anyTsatisfying0 < T leq 1/2,left| f(t)-sum^{2n}_{k=1}a_{kn}f(t - kT)right| leq E_{f}cdot beta^{n}whereE_{f}is the signal energy andbeta doteq 0.6863. Thus the estimate off(t)in terms of2npast samples taken at a rate equal to or in excess of twice the Nyquist rate converges uniformly at a geometric rate tof(t)on(- infty , infty). The suboptimal coefficients{a_{kn}}have the desirable property of being pure numbers independent of both the particular band-limited signal and of the selected sampling rate1/T. It is also shown that these same coefficients can be used to estimate the value ofx(t)of a wide-sense stationary random process in terms of past samples.     

Performance of Biphase Digital FM Transmission

The observation by several investigators that the spectrum of a digital FM signal is concentrated near the carrier frequency for small frequency deviation ratios (h) led Tjhung and Wittke [1] to investigate the probability of error to determine the optimum bandwidth-bit period (BT) product and optimumh. Their results using discriminator demodulation showed that for a probability of error of 10^-4the optimum values areh = 0.7andBT = 1.0. Further, their results showed that in some narrow-band cases digital FM is superior to PSK. The purpose of this correspondence item is to extend Tjhung and Wittke's results by applying their method to a biphase [2] baseband signal. Such a signal has the well-known advantage of facilitating synchronization since a zero-crossing occurs during each bit.     

The statistics of HF sea-echo Doppler spectra

Several important statistical properties of the HF sea echo and its Doppler power spectrum, which are useful in optimizing the design of radar oceanographic experiments, are established. First-and second-order theories show that the echo signal (e.g., the voltage) should be Gaussian; this is confirmed with experimental surface-wave data i) by comparison of the normalized standard deviation of the power spectrum at a given frequency with its predicted value of unity, and ii) by cumulative distributiun plots of measured spectral amplitudes on Rayleigh probability charts. The normalized standard deviation of the dominant absolute peak amplitudes of the power spectrum (which wander slightly in frequency) are shown from experimental data to besim 0.7for the first-order peaks andsim 0.5for the second-order peaks. The autocorrelation coefficient of the power spectra is derived from measured data and interpreted in terms of the spectral peak widths; from this information, the correlation time (or time between independent power spectrum samples) is shown to besim 25-50s for radar frequencies above 7 MHz. All of these statistical quantities are observed to be independent of sea state, scattering cell size, and relatively independent of radar operating frequency. These quantities are then used to establish the statistical error (and confidence interval) for radar remote sensing of sea state, and it is shown, for example, that 14 power spectral samples result in a sample average whose rms error about the true mean is 1.0 dB.     

FM-laser operation of the Nd:YAG laser

The FM-laser or frequency-sweeping mode of laser oscillation has been demonstrated in a Nd :YAG 1.06-μ laser with an intracavity LiNbO3phase modulator. The experimental results are in excellent agreement with the theoretical expressionDelta= (DeltaOmega/Deltanu) (delta/pi)where δ= peak single-pass phase retardation in the modulator,DeltaOmega= axial mode spacing,Deltanu=modulator detuning, and Δ=resulting FM index of the laser output. Modulation indices as large asDelta approx 230rad have been obtained, in which case the instantaneous laser frequency is sweeping over a full spectral range of2Delta cdot f_{m} approx 120GHz (≈ 4 cm^-1) at a repetition frequencyf_{m} approx 260MHz, with a time-bandwidth product per periodapprox 2Delta approx 460. The coherently mode-locked spectral bandwidth thus obtained in the FM-laser case is very much wider than can be achieved in the pulsed mode-locked case with the same Nd:YAG laser. Some possible ways of using this broad-band coherent FM spectrum are suggested.     

Alias-free randomly timed sampling of stochastic processes

The notion of alias-free sampling is generalized to apply to random processesx(t)sampled at random timest_n; sampling is said to be alias free relative to a family of spectra if any spectrum of the family can be recovered by a linear operation on the correlation sequence{r(n)}, wherer(n) = E[x(l_{m+n}) overline{x(t_m)}]. The actual sampling timest_nneed not be known to effect recovery of the spectrum ofx(t). Various alternative criteria for verifying alias-free sampling are developed. It is then shown that any spectrum whatsoever can be recovered if{t_n}is a Poisson point process on the positive (or negative) half-axis. A second example of alias-free sampling is provided for spectra on a finite interval by periodic sampling (fort leq t_oort geq t_o) in which samples are randomly independently skipped (expunged), such that the average sampling rate is an arbitrarily small fraction of the Nyquist rate. A third example shows that randomly jittered sampling at the Nyquist rate is alias free. Certain related open questions are discussed. These concern the practical problems involved in estimating a spectrum from imperfectly known{ r(n) }.     

On mean-square aliasing error in the cardinal series expansion of random processes (Corresp.)

An upper bound is derived for the mean-square error involved when a non-band-limited, wide-sense stationary random processx(t)(possessing an integrable power spectral density) is approximated by a cardinal series expansion of the formsum^{infty}_{-infty}x(n/2W)sinc2W(t-n/2W), a sampling expansion based on the choice of some nominal bandwidthW > 0. It is proved thatlim_{N rightarrow infty} E {|x(t) - x_{N}(t)|^{2}} leq frac{2}{pi}int_{| omega | > 2 pi W}S_{x}( omega) d omega,wherex_{N}(t) = sum_{-N}^{N}x(n/2W)sinc2W(t-n/2W), andS_{x}(omega)is the power spectral density forx(t). Further, the constant2/ piis shown to be the best possible one if a bound of this type (involving the power contained in the frequency region lying outside the arbitrarily chosen band) is to hold uniformly int. Possible reductions of the multiplicative constant as a function oftare also discussed, and a formula is given for the optimal value of this constant.     

Bounds on the accuracy attainable in the estimation of continuous random processes

The problem of estimating a message,a(t), which is a sample function from a continuous Gaussian random process is considered. The message to be estimated may be contained in the transmitted signal in a nonlinear manner. The signal is corrupted by additive noise before observation. The received waveform is available over some observation interval[T_{i}, T_{f}]. We want to estimatea(t)over the same interval. Instead of considering explicit estimation procedures, we find bounds on how well any procedure The principle results are as follows: 1) a lower bound on the mean-square estimation error. This bound is a generalization of bounds derived previously by Cramer, Rao, and Slepian for estimating finite sets of parameters. 2) The bound is evaluated for several practical examples. Possible extension and applications are discussed briefly.     

Stationary phase approximations of FM spectra

An approximation of the Fourier transform of an FM signal with modulation indexmis calculated by a stationary phase method. Whenm rightarrow inftythe error of amplitude and phase converges to zero,O(m^{-1/2}). If the spectrum obtained by the approximation is smoothed over a frequency interval equal to several bandwidths of the modulation, then the well-known adiabatic approximation of the FM spectrum is obtained. If the signal has hidden periodicities, the adiabatic approximation destroys the spectral fine structure, but the spectral lines can be calculated from the approximation formula when used without smoothing.     

On the detection of known binary signals in Gaussian noise of exponential covariance

The optimum test statistic for the detection of binary sure signals in stationary Gaussian noise takes a particularly simple form, that of a correlation integral, when the solution, denoted byq(t), of a given integral equation is well behaved(L_{2}). For the case of a rational noise spectrum, a solution of the integral equation can always be obtained if delta functions are admitted. However, it cannot be argued that the test statistic obtained by formally correlating the receiver input with aq(t)which is notL_{2}is optimum. In this paper, a rigorous derivation of the optimum test statistic for the case of exponentially correlated Gaussian noiseR(tau) = sigma^{2} e^{-alpha|tau|}is obtained. It is proved that for the correlation integral solution to yield the optimum test statistic whenq(t)is notL_{2}, it is sufficient that the binary signals have continuous third derivatives. Consideration is then given to the case where a, the bandwidth parameter of the exponentially correlated noise, is described statistically. The test statistic which is optimum in the Neyman-Pearson sense is formulated. Except for the fact that the receiver employsalpha_{infty}(which in general depends on the observed sample function) in place ofalpha, the operations of the optimum detector are unchanged by the uncertainty inalpha. It is then shown that almost all sample functions can be used to yield a perfect estimate ofalpha. Using this estimate ofalpha, a test statistic equivalent to the Neyman-Pearson statistic is obtained.     

A probabilistic analysis of time delay extraction by the cepstrum in stationary Gaussian noise

Often, the cepstrum has been used when estimating the time delay r between elements of a composite signal embedded in noise,n(t). Here a probabilistic analysis is conducted dealing with the effect of stationary Gaussian noise on the characteristics of such a nonlinear processor. The input noise,n(t)is reflected at the cepstrum's output as: 1) modulationM_{1}(omega)of sinusoidal information carrier with subsequent reduction in the height of echo peak attau; and 2) generation of background noiseshat{phi}(omega)andhat{psi}(omega)with tendency to obscure that echo peak. The resulting probabilities of these terms are derived. The expected mean and standard deviation of reduction in the peak level attaudue to noise are analytically described. Similar statistical measures on the background noise are also obtained. The results point out the dependence of the-statistical measures upon the pointwise variation of input signal to noise spectra. Thus the total signal power to total noise power(S/N)alone is an insufficient measure for setting the cepstrum performance, and relative bandwidths of signal and noise are also needed. The detection probabilityP_{D}of a peak corresponding totauis calculated for exponential spectra with equal bandwidths in signal and noise and with echo strengtha = 0.4; P_{D}is found to be close to one for anSINgreater than -4 dB; deterioration in detection is found to be rapid with thresholding near anS/N sim -6db whereP_{D}falls to 0.3.     

On codes with spectral nulls at rational submultiples of the symbol frequency

In digital data transmission (respectively, storage systems), line codes (respectively, recording codes) are used to tailor the spectrum of the encoded sequences to satisfy constraints imposed by the channel transfer characteristics or other system requirements. For instance, pilot tone insertion requires codes with zero mean and zero spectral density at tone frequencies. Embedded tracking/focus servo signals produce similar needs. Codes are studied with spectral nulls at frequenciesf=kf_{s}/n, wheref, is the symbol frequency andk, nare relatively prime integers withk leq n;in other words, nulls at rational submultiples of the symbol frequency. A necessary and sufficient condition is given for a null atfin the form of a finite discrete Fourier transform (DFT) running sum condition. A corollary of the result is the algebraic characterization of spectral nulls which can be simultaneously realized. Specializing to binary sequences, we describe canonical Mealy-type state diagrams (directed graphs with edges labeled by binary symbols) for each set of realizable spectral nulls. Using the canonical diagrams, we obtain a frequency domain characterization of the spectral null systems obtained by the technique of time domain interleaving.     

First and second passage times of Rayleigh processes (Corresp.)

The first and second passage times of a stationary Rayleigh processR(t,a)are discussed.R(t,a)represents the envelope of a stationary random process consisting of a sinusoidal signal of amplitude and frequencyf_{0}plus stationary Gaussian noise of unit variance having a narrow-band power spectral density which is symmetrical aboutf_{0}. Approximate integral equations are developed whose solutions yield approximate probability densities concerning the first and second passage times ofR(t,a). The resulting probability functions are presented in graphs for the case when the power spectral density of the noise is Gaussian. Related results concerning the approximate distribution function of the absolute minimum or absolute maximum ofR(t,a)in the closed interval[0,tau]are also presented. The exact probability densities are expressed in the form of an infinite series of multiple integrals.     

On the Schalkwijk-Kailath coding scheme with a peak energy constraint

In a recent series of papers, [2]-[4] Schalkwijk and Kailath have proposed a block coding scheme for transmission over the additive white Gaussian noise channel with one-sided spectral densityN_{0}using a noiseless delayless feedback link. The signals have bandwidthW (W leq infty)and average powerbar{P}. They show how to communicate at ratesR < C = W log (1 + bar{P}/N_{0}W), the channel capacity, with error probabilityP_{e} = exp {-e^{2(C-R)T+o(T)}}(whereTis the coding delay), a "double exponential" decay. In their scheme the signal energy (in aT-second transmission) is a random variable with only its expectation constrained to bebar{P}T. In this paper we consider the effect of imposing a peak energy constraint on the transmitter such that whenever the Schalkwijk-Kailath scheme requires energy exceeding abar{P}T(wherea > 1is a fixed parameter) transmission stops and an error is declared. We show that the error probability is degraded to a "single exponential" formP_{e} = e^{-E(a)T+o(T)}and find the exponentE(a). In the caseW = infty , E(a) = (a - 1)^{2}/4a C. For finiteW, E(a)is given by a more complicated expression.     

Recursive realization of finite impulse filters using finite field arithmetic

Recursive filter design techniques are described and developed for finite impulse filters using finite field arithmetic. The finite fields considered have the formGF(q^{2}), the Galois field ofq^{2}elements, and are analogous to the field of complex numbers whenqis a prime such that(-1)is not a quadratic residue. These filters can be designed to yield either a desired finite impulse or finite frequency response function. This filtering technique has other possible applications, including the encoding or decoding of information and signal design. Infinite signal trains can be decomposed naturally into orthogonal sequences which may be useful in the encoding and decoding process and may provide another approach to convolutional coding. Since the recursive filters developed here do not have the accumulation of round-off or truncation error that one might expect in recursive computations, such filters are noise-free transducers in the sense of Shannon.     

Optimum FIR Transmitter and Receiver Filters for Data Transmission Over Band-Limited Channels

The paper deals with the design of digital transmitter and receiver filters with finite impulse response (FIR) for data transmission over band-limited channels. The filters are matched and satisfy a zero intersymbol interference constraint when cascaded. For baseband transmission, the filters achieve optimum spectral concentration in the frequency range[-(1+beta)/2T, (1+beta)/2T]. Mathematically, the filter design leads to a generalized eigenvalue problem which is solved numerically by a projected gradient Procedure. For transmission over bandpass channels by combined amplitude and phase modulation, the design technique is modified so that filters with complex-valued impulse response and optimum spectral concentration in the range of positive bandpass frequencies[f_{c} - (1+beta)/2T, f_{c} + (1+beta)/2T]are obtained. In addition, the complex formulation allows the design of impulse responses with enhanced spectral attenuation in the corresponding range of negative frequencies in order to minimize imageband interference. Results are shown in terms of filter coefficients, signal spectra, and spectral concentrations obtained. For example, filters designed for a voiceband data modem operating at a symbol rate of 2400 baud achieve a spectral concentration of 98.5 percent with 24 coefficients andbeta = 0.1, and with only 0.001 percent of the total energy in the imageband region.     

Correlation of a signal with an amplitude-distorted form of itself

In comparing a signalf(t)with its amplitude-distorted formg(f(t)), whereg(cdot)is a monotonically increasing function of its argument, one is led to consider the correlation function begin{equation} R(s) triangleq int_{-infty}^{infty} dtg (f(t))f(t-s). end{equation} A rigorous proof is given of the inequalityR(s) leq R(O). Generalizations are presented for the cases of finite domains and of signals defined in two-dimensional space.     

A Phase Method Generation of Square-Law SSB Signals

An approximate single sideband signal for a square-law envelope detector (SL-SSB) has the formsqrt{1 + x(t)} cos {omega_{c}t + 0.5 arcsin[hat{x}(t)(1 + 0.5x(t))/(1 + x(t))]}, wherehat{x}(t)is the Hilbert transform of x(t). Signal SL-SSB is constructed in accordance with the phase method of SSB signal generation; however, before amplitude modulation a dc component is added to the modulating signal, and then the signal is squarerooted. In this manner a new SL-SSB modulation technique was obtained which is also suitable for C-SSB and AM stereo broadcasting. The SLSSB spectrum is given.     

Bounds for truncation error in sampling expansions of band-limited signals

Forf(t)a real-valued signal band-limited to- pi r leq omega leq pi r (0 < r < 1)and represented by its Fourier integral, upper bounds are established for the magnitude of the truncation error whenf(t)is approximated at a generic timetby an appropriate selection ofN_{1} + N_{2} + 1terms from its Shannon sampling series expansion, the latter expansion being associated with the full band[-pi, pi]and thus involving samples offtaken at the integer points. Results are presented for two cases: 1) the Fourier transformF(omega)is such that|F(omega)|^{2}is integrable on[-pi, pi r](finite energy case), and 2)|F(omega)|is integrable on[-pi r, pi r]. In case 1) it is shown that the truncation error magnitude is bounded above byg(r, t) cdot sqrt{E} cdot left( frac{1}{N_{1}} + frac{1}{N_{2}} right)whereEdenotes the signal energy andgis independent ofN_{1}, N_{2}and the particular band-limited signal being approximated. Correspondingly, in case 2) the error is bounded above byh(r, t) cdot M cdot left( frac{1}{N_{1}} + frac{1}{N_{2}} right)whereMis the maximum signal amplitude andhis independent ofN_{1}, N_{2}and the signal. These estimates possess the same asymptotic behavior as those exhibited earlier by Yao and Thomas [2], but are derived here using only real variable methods in conjunction with the signal representation. In case 1), the estimate obtained represents a sharpening of the Yao-Thomas bound for values ofrdose to unity.     

Design Criteria for Noncoherent Gaussian Channels with MFSK Signaling and Coding

This paper presents data and criteria to assess and guide the design of modems for coded noncoherent communication systems subject to practical system constraints of powerS, bandwidthW, noise spectral density N0, coherence time Tc, and number of orthogonal signalsM. Three basic receiver types are analyzed for the noncoherent multifrequency-shift keying (MFSK) additive white Gaussian noise channel: hard decision, unquantized (optimum), and quantized (soft decision). Channel capacity and computational cutoff rate Rcompare computed for each type and presented as functions of the predetection signal-to-noise ratioST/N_{0}and the number of orthogonal signalsM = 2TW. This relates the channel constraints of power, bandwidth, coherence time, and noise power to the optimum choice of signal durationT leq T_{c}and signal numberM.