The authors consider the problem of multichannel sampling for the case in which the input signal bandwidth is smaller than the maximum possible that allows exact recovery (from sub-Nyquist samples taken uniformly from each channel). They focus on this reduced bandwidth case for two channels. They show that, for this case, the conditions which the prefilters must satisfy are less restrictive in some parts of the frequency axis, compared to the full bandwidth case. For a given set of valid prefilters, there is a family of possible postfilters which allows exact recovery of the input, rather than one choice, as in the standard case. It is shown that, if the samples are corrupted by additive white random noise sequence values, the extra flexibility of the postfilters can be used to select an optimal choice that minimizes the reconstruction noise power. An example is provided in which a standard ill-posed reconstruction system can be replaced with one that leads to a well-posed reconstruction when the input is of reduced bandwidth.>

The generalized multichannel sampling theorem of A. Papoulis (1977) assumes a bandlimited signal simultaneously processed by a bank of m linear time-invariant (LTI) prefilters; the filter outputs are then sampled and the original signal recovered from these samples by means of m LTI postfilters. Consideration is given to the stability (well-posedness) of the signal reconstruction with respect to additive noise perturbations of the sample values. It is shown that the reconstruction is stable if and only if the postfilter transfer functions are square integrable, thus proving a converse to the ill-posedness result K.F. Cheung and R.J. Marks (1985) while giving an alternative proof of their theorem.>

A digital signal (x(n)) is used as a common input to a parallel bank of m digital linear shift-invariant (LSI) analysis filters. The m filter outputs are then both downsampled (decimated) by a factor of m and unsampled (interpolated) by the same factor m in each channel. The perfect reconstruction problem is that of finding digital synthesis filters in each channel such that an additive combination of these filter outputs results in restoration of the original sequence. It is shown that a solution of the perfect reconstruction problem in the digital domain can be obtained from the solution of a corresponding analog generalized sampling problem using a bandlimited input (which interpolates the discrete input at integer points) and analog filters that are simply related to the given digital filters of the maximally decimated filter bank.>

A deterministic two-dimensional bandlimited signal x(t/sub 1/, t/sub 2/) is used as a common input to m linear shift-invariant 2-D filters (channels). For m=m/sub 1/.m/sub 2/, the m channel outputs are 'downsampled' at rates in t/sub 1/ and t/sub 2/ that are reduced by factors of m/sub 1/ and m/sub 2/ respectively from the usual Nyquist rates required for input sample reconstruction. It is shown that if the transfer functions of the m-channels are 'independent', then the input x(t) is uniquely recoverable from the channel output samples taken at the indicated rates. Moreover, the recovery can be effected by the use of m linear shift-invariant post-filters.>

The classical problem of detecting a known signal in additive colored Gaussian noise on a finite interval is considered by using as "coordinates" (observables) the inner products formed between the received waveform and the elements of a simply constructed complete set of functions defined on the observation interval. Results of the Karhunen-Loève (K-L) approach are thereby obtained without the K-L expansion or any appeal to results from integral equation theory.

An alternate proof is given of a recent result due to A. Papoulis on predicting the current value of a wide-sense stationary band-limited random process in terms of its past samples. The possibility of such prediction is shown to be equivalent to the completeness of a certain set of complex exponentials on an interval, a property that also allows extension to the prediction of deterministic band-limited signals.

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 and\beta \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.

The first-order sampling of a bandpass signal when the upper cutoff frequency is a multiple of the bandwidth is reduced to an application of the low-pass sampling theorem. In the general case, a simple band-imbedding procedure restores the positioning constraint and yields the expansion.

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 form\sum^{\infty}_{-\infty}x(n/2W)sinc2W(t-n/2W), a sampling expansion based on the choice of some nominal bandwidthW > 0. It is proved that\lim_{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.

For a nondecreasing distortion characteristic\phi(\cdot)and a given signalx(\cdot), the "cross correlation" function defined byR_{\phi} (\tau) \triangleq \int_{-\infty}^{\infty} \phi[x(t)]x(t - \tau) dtis shown to satisfy the inequalityR_{\phi}(\tau) \leq R_{\phi}(0), for all\tau, generalizing an earlier result of Richardson that required\phi(\cdot)to be continuous and strictly increasing. The methods of the paper also show that, under weak conditions, \begin{equation} R_{\phi,\psi}(\tau) \triangleq \int_{-\infty}^{\infty} \phi[x(t)]\psi[x(t - \tau)] dt \leq R_{\phi,\psi}(0) \end{equation} when\psiis strictly increasing and\phiis nondecreasing. In the case of hounded signals (e.g., periodic functions), the appropriate cross correlation function is \begin{equation} \mathcal{R}_{\phi,\psi}(\tau} \triangleq \lim_{T \rightarrow \infty} (2T)^{-l} \int_{-T}^T \phi[x(t)]\psi[x(t - \tau)] dt. \end{equation} For this case it is shown that\mathcal{R}_{\phi,\psi} (\tau) \leq \mathcal{R}_{\phi,\psi}(0)for any nondecreasing (or nonincreasing) distortion functions\phiand\psi. The result is then applied to generalize an inequality on correlation functions for periodic signals due to Prosser. Noise signals are treated and inequalities of a similar nature are obtained for ensemble-average cross correlation functions under suitable hypotheses on the statistical properties of the noise. Inequalities of this type are the basis of a well-known method of estimating the unknown time delay of an observed signal. The extension to nondecreasing discontinuous distortion functions allows the use of hard limiting or quantization to facilitate the cross correlation calculation.

Forx(t)either a deterministic or stochastic signal band-limited to the normalized frequency interval\mid\omega\mid \leq \pi, explicit coefficients\{ a_{kn} \}are exhibited that have the property that \begin{equation} lim_{n \rightarrow \infty} \parallel x(t) - \sum_{1}^n a_{kn} x(t - kT) \parallel = 0 \end{equation} in an appropriate norm and for any constant intersample spacingTsatisfying0 < T < \fac{1}{2}; that is,x(t)may be approximated arbitrarily well by a linear combination of past samples taken at any constant rate that exceeds twice the associated Nyquist rate. Moreover, the approximation ofx(t)is uniform in the sense that the coefficients\{ a_{kn} \}do not depend on the detailed structure ofx(t)but are absolute constants for any choice ofT. The coefficients that are obtained provide a sharpening of a previous result by Wainstein and Zubakov where a rate in excess of three times the Nyquist rate was required.

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.

It is shown that the Shannon sampling expansion for finite-energy bandlimited signals is a special case of the Parseval relation for complex Fourier series.

The case ofnunity-variance random variablesx_{1}, x_{2},\cdots, x_{n}governed by the joint probability densityw(x_{1}, x_{2}, \cdots x_{n})is considered, where the density depends on the (normalized) cross-covariances\rho_{ij} = E [(x_{i}- \bar{x}_{i})(x_{j} - \bar{x}_{j})]. It is shown that the condition(^{\ast}) \frac{\delta}{\delta \rho_{ij}}\{E[f(x_{1}, x_{2}, \cdots , x_{n})\} = E|\frac{\delta^{2}}{\delta x_{i} \delta x_{j}}f(x_{1}, x_{2}, \cdots , x_{n})| \mbox{(i\neq j)}holds for an "arbitrary" functionf(x_{1}, x_{2}, \cdots , x_{n})ofnvariables if and only if the underlying densityw(x_{1}, x_{2}, \cdots , x_{n})is the usualn-dimensional Gaussian density for correlated random variables. This result establishes a generalized form of Price's theorem in which: 1) the relevant condition(^{\ast})subsumes Price's original condition; 2) the proof is accomplished without appeal to Laplace integral expansions; and 3) conditions referring to derivatives with respect to diagonal terms\rho_{ij}are avoided, so that the unity variance assumption can be retained.

An analysis is presented of an ideal two-channel cross-correlator in which each channel input consists of a deterministic signal combined additively with stationary Gaussian noise. It is assumed that all input quantities are bandlimited to some common passband,0 < \omega_{0}- \Omega/2 \leq |\omega | \leq \omega_{0} + \Omega/2, with angular bandwidth\Omega > 0. Moreover, the random noisesn_{1}(t)andn_{2}(t)are assumed jointly normal with crosscorrelation functionE[n_{1}(t)\cdot n_{2}(t+ \tau)]independent oft. After multiplication of the two composite inputs, the product process is passed through an ideal lowpass filter to produce the correlator output,G(t). The main result of the paper is an explicit determination of the characteristic function ofG(t)in closed form. This extends previous work by D. C. Cooper [1], who considered the same model with sinusoidal signals and a restricted form of dependency between the Gaussian noises in the two channels. The more general derivation given here makes use of canonical representations for the bandpass input quantities and exhibits the system output as a quadratic form in the (Gaussian) quadrature components. A recent result of Yu. S. Lezin [6] on the output probability density of an autocorrelator with bandpass inputs is shown to be a special case of the analysis.

Given two stationary random processesx_1 (t)andx_2 (t), the cross-correlation property of interest is the following: If one of the two processes is distorted by an instantaneous nonlinear device, then the cross correlation after the distortion is proportional to the cross-correlation function prior to the distortion. Using an expansion of the second-order joint probability distributionp(x_1, x_2)introduced by Barrett and Lampard, a necessary and sufficient condition for the above cross-correlation property is given in terms of requirements on the expansion coefficients. In certain cases, the constant of proportionality involved in the cross-correlation property is equal to the "equivalent gain" of the nonlinear device as defined by Booton. A necessary and sufficient condition for these two constants to be identical is formulated in terms of the expansion coefficients ofp(x_1, x_2). The class of distributions satisfying this condition is a subclass of the set of distributions for which the cross-correlation property is valid.