Instantaneous Power Spectrum Research Articles

Two-Dimensional Optical Processing Of One-Dimensional Acoustic Data

The concept of carrier-mean-frequency-selective convolution is introduced to solve the undersea problem of passive acoustic surveillance (PAS) and compared with the conventional notion of difference-frequency Doppler-corrected correlation. The former results in the cross-Wigner distribution function (WD), and the latter results in the cross-ambiguity function (AF). When the persistent time of a sound emitter is more important than the characteristic tone of the sound emitter, WD will be more useful than AF for PAS activity detection, and vice versa. Their mutual relationships with the instantaneous power spectrum (IPS) show the importance of the phase information that must be kept in any 2-D representation of a 1 -D signal. If a square-law detector is used, or an unsymmetric version of WD or AF is gener-ated, then one must produce the other 2-D representations directly, rather than transform one to the other.

Acoustooptic spectrum analysis of radar signals using an integrating photodetector array

This paper examines the instantaneous Fourier power spectrum for different types of radar signals such as cw, pulse-modulated cw, and linear FM signals using an acoustooptic spectrum analyzer. The effects on the time-integrated output intensity distribution, due to the truncation of the propagating acoustic signal by the finite aperture width of the Bragg cell and the amplitude weighting on the signal, are also analyzed. Some experimental results on pulse-modulated cw and linear FM signals are presented and then compared to theory.

Instantaneous power spectra

Conventional spectral analysis methods do not describe the distribution of signal power in both time and frequency simultaneously. Time dependent Fourier transforms are defined to extend the conventional theory to include instantaneous energy and power spectra for the past and future signal. These spectra are combined to define an instantaneous power spectrum (IPS) of the entire signal. The IPS is a noncausal mathematical property of the signal from which observable properties can be derived, e.g., the lagged window Fourier transform estimate of the time‐frequency variations of the signal. The IPS for a nonstationary process is shown to be given by the Fourier transform of Loeve’s generalized power spectrum of the process and to be a generalization of the Wiener–Kintchine theorem. It is also shown to represent the special cases of the stationary process, the locally stationary process and the deterministic process.

Statistical descriptors of time spread sonar targets

Sonar design and performance analysis requires a description of the distribution of echo energy in time parameterized by frequency and nominal incidence angle. The variability of echoes from sonar targets motivates a statistical description. The ensemble averaged instantaneous power spectrum of the target echo impulse response is the functional target descriptor which matches these requirements. The relationship of the instantaneous power spectrum to other statistical channel descriptors and conventional measures of target strength is discussed. Further interpretations of and measurement procedures for the instantaneous power spectra of the impulse response of random time spread transmission channels are provided. Higher order statistical descriptors and extension to frequency spread targets are commented on.

Impulse response approximation using an instantaneous power spectrum: W. D. Jennings, E. W. Purnell, and E. Holasek, Case Western Reserve University, School of Medicine, Department of Surgery, Division of Ophthalmology, Cleveland, OH 44106

Impulse response approximation using an instantaneous power spectrum: W. D. Jennings, E. W. Purnell, and E. Holasek, Case Western Reserve University, School of Medicine, Department of Surgery, Division of Ophthalmology, Cleveland, OH 44106

A nonstationary analysis of the electroencephalogram.

A statistical technique is described which allows description of the statistical characteristics of nonstationary electroencephalograms (EEG's). The EEG is investigated in terms of its nonstationary power spectrum. The instantaneous power spectra of certain transition processes in the EEG (the evolution and blocking of the alpha wave) are calculated and described on the time-frequency plane.

An Approach to Time-Varying Spectral Analysis

An approach to the time-varying spectral analysis for shock and vibration data is presented. The approach follows and extends Page’s concept of instantaneous power spectra. Various input-output relations for simple linear systems are derived. The error involved in the evaluation of the time-varying spectra of the response and conditions that reduce it are discussed in some detail. Numerical spectra and energy distribution function of four shock functions are provided and their significance is discussed.

Laterality Factors in the Perception of Sentences Varying in Semantic Constraint

The present experiment explored the differential rôle of the left and right ears in perceiving linguistic stimuli on the level of sentences. Sentences with identical syntactic structure but varying in semantic constraint [i.e., “The farmer plowed the field” (semantically well integrated) vs “The Indian hit the table” (semantically poorly integrated)] were presented to the left or right ear of 10 right-handed subjects. Listeners were required to listen to and then immediately repeat each sentence after presentation. A special speech-noise distortion circuit developed at the Communication Sciences Laboratory of The University of Michigan [M. A. H. O'Malley and G. E. Peterson, “An Experimental Method for Prosodic Analysis,” Phonetica 15, 1–13 (1966)] was used to produce a masking signal that provided a constant signal-to-noise ratio for the entire utterance. The circuit derives the speech masking signal from the utterance itself by destroying the segmental information although maintaining the instantaneous power spectrum and pitch information of the sentence. Masking output of the circuit was delivered to both ears of a listener while the speech signal was presented to only one ear at a time. This procedure provided an opportunity to examine the effect of speech distortion on the rôle of the two ears in perceiving sentences. Preliminary results indicate significant differences in immediate recall for ears—the right ear being more efficient than the left ear. Results of this pilot study indicate that: (1) differential ear preference is a function of cerebral dominance; and (2) there are differences between the two hemispheres in processing different types of linguistic information on the sentence level.

Signal energy distribution in time and frequency

The Fourier representation of signals and its relation to the signal structure in time and frequency, and more generally the inherent properties of phase-modulated signals, have received considerable attention in the past. These topics have led to such seemingly unrelated studies as the representation of a signal in a combined time-frequency plane, instantaneous power spectra, and the ambiguity function and its transform relations. It is shown in this paper that the studies can be unified by the introduction of the concept of the complex energy density function of a signal. The function is an extension and combination of the one-dimensional energy density functions in time and frequency, the energy density spectrum |\Psi(f)|^{2} , and energy density waveform |\psi (t)|^{2} . On the basis of the complex energy density function, the significance of complicated-appearing transform relations is readily understood. The new concept also conveys a good insight into the internal structure of phase-modulated signals.

The Formation of Simultaneous Radio Astronomical Images and Spectra by Electro-Optic Methods

This communication describes techniques, new to radio astronomy, which permit the analysis of electrical signals at radio-frequencies by optical methods.Two applications will be described: the first is to a spectrograph which gives the instantaneous power spectrum of a single broad-band electrical signal; the second is to the analysis of the signals from a number of aerials of an array in order to form a simultaneous image of the brightness distribution of the region of the sky under observation.

Detection of Known Signals in Nonstationary Noise

The basic design of a nonlinear, time-invariant filter is postulated for detecting signal pulses of known shape imbedded in nonstationary noise. The noise is a sample function of a Gaussian random process whose statistics are approximately constant during the length of a signal pulse. The parameters of the filter are optimized to maximize the output signal-to-noise ratio (SNR). The resulting nonlinear filter has the interesting property of approximating the performance of an adaptive filter in that it weights each frequency band of each input pulse by a factor that depends on the instantaneous noise power spectrum present at that time. The SNR at the output of the nonlinear filter is compared to that at the output of a matched filter. The relative performance of the nonlinear system is good when the signal pulses have large time-bandwidth products and the instantaneous noise power spectrum is colored in the signal pass band.

On the Concept of an Instantaneous Power Spectrum, and Its Relationship to the Autocorrelation Function

A definition sufficient to define a function of frequency ρ(t,f) which could be regarded as the instantaneous power spectrum was given by Page. This function is not unique, since it may have added to it a complementary function of frequency ρc(t,f) satisfying ∫ −∞∞ρc(T,f)df=0without changing the original signal in any way. It is shown that when a signal is observed by different observers, starting their observations at different times, they will not derive the same ``instantaneous power spectral function,'' and the differences between their results will be complementary functions as defined above. It is well known that the Fourier transform of the autocorrelation function is equal to the square of the magnitude of the spectral function. Analogously, a ``running autocorrelation function'' is defined and it is shown that the Fourier transform of its partial differential coefficient with respect to time is an instantaneous power spectral function.

Instantaneous Power Spectra

The intuitive concept of a changing spectrum is discussed. The instantaneous power spectrum is defined mathematically and used to make the intuitive concepts more precise. It depends upon the past history of a signal, but not upon the future. Integration of the instantaneous power spectrum over time yields the conventional energy spectrum. The instantaneous power spectrum of a random function may be averaged over the ensemble of functions, with a resulting stochastic average instantaneous power spectrum that is equal to the conventional time average power spectrum of a stochastic process.