Random Fading Research Articles

Probability of decoding error for random phase and Rayleigh fading channels

In this paper, the probability of error of two time-varying channels with memory, 1) the random phase channel and 2) the Rayleigh fading channel, is discussed. The input is assumed to be one of M equiprobable waveforms. An upper bound to the probability of error is derived by convetting the channel into a memoryless one by means of scrambling. A lower bound to the probability of error is derived by assuming that except for an additive noise, all the channel parameters are completely known at the receiver and, therefore, are not considered to be random variables any more. Following this assumption, it is shown that the channel is converted into a memoryless channel. For each one of the two channels, there is a region of SNR where the upper and lower bounds are close together and therefore yield a good estimate to the actual probability of error. Furthermore, using random coding and scrambling, this probability of error may actually be achieved (for a certain region of rates).

Ionospheric Scattering of Satellite Transmissions

RADIO waves from satellites may fade in both regular and random fashion when received with a linearly polarized aerial. Fading due to Faraday rotation or satellite rotation is an example of the former type. Random fades, up to several per second, are sometimes observed, and these have been attributed to scattering from the same kind of ionospheric irregularities as produce radio star scintillation. This fading is more rapid for satellites than for radio stars since it depends on satellite velocity rather than ionospheric drift velocity. Henceforth we shall be concerned only with the randomly fading component.

Relation of turbulence theory to ionospheric scatter propagation experiments

Summary— After a brief historical introduction the author considers the statistical behavior of ionospheric scatter signals. The random fading of the signals is taken to be suggestive of a scatter process, and predictions as to amplitude distribution (Rayleigh) and as to space or time correlations, made on this basis, are found to be in fair agreement with empirical results. It is noted that no attempts have as yet been made to predict the correlation coefficients. The scattering of electromagnetic waves by turbulent irregularities is discussed next. The Born approximation (single scattering) is employed. This together with the far-field approximation leads to an expression for the received power in terms of the spectrum of mean square deviations in electron density. Neglecting polarization effects and assuming isotropy and homogeneity of electron density deviations within the common volume, the received power is given by where PR and PT are the received and transmitted powers, respectively, ro is the classical radius of the electron, AR and GR are the effective area and gain of the receiving antenna, b is the vertical depth of the scattering layer, R2 is the distance from the scattering volume to the receiver, θ is the angle through which the energy is scattered, λ is the radio wavelength, and Γ (k) is the spectral representation as a function of wave-number magnitude only of the contributions to the mean square deviations in electron density. It is noted that this expression is the only linkage, so far, between turbulence theory and the results of radio experiments. Caution is advised in applying it because of the unsubstantiated assumptions employed in its derivation.

Study of “winds” in the F-region of the ionosphere during the unusual days in the IGY-calendar

The three-spaced-receiver fading records were taken at Banaras with vertical pulsed transmission at 3.8–4.2 Mc/s during the regular world days, world meteorological intervals following the IGY-calendar and also during the SWI's. The records gave information about the movements of the F-region of the ionosphere during these unusual days. The velocity and the direction of the “wind” were obtained by the method of similar fades and the cross-correlation method. The experimental arrangements and the methods of measurements have been described. Usually the velocity of the F-region “winds” was found to have high values during the world meteorological intervals. The sudden reversal in the direction of the F-region “wind” round the midnight hour was confirmed by the observations taken during the regular world days. The sudden reversal in the direction of the “wind” was found to be followed usually by extremely rapid random fading, indicating turbulence in the ionosphere. The fading records on the unusual days in the IGY-calendar have been classified and fully discussed with reference to the random and steady movements of the irregularities which exist for a short time in the F-region.

Statistical Analysis of Fading on Short-Wave Transmissions

In the present investigation on fading of radio signals in tropical countries, a statistical analysis is presented of the fading records taken on oblique incidence continuous wave modulated transmissions as well as on pulsed transmissions at vertical incidence. The amplitude distributions of some of the random fading curves have been worked out and are shown in this paper. It is observed that the probability distributions of the amplitude in such fading curves are the conventional Rayleigh, Gaussian and log normal type.Auto-correlograms of the fading curves have also been determined in a few cases, and are also presented.Fading records of the oblique incidence continuous wave transmissions have been compared with the ones taken simultaneously on vertical incidence pulsed transmission at the equivalent vertical incidence frequencies.

Investigation of horizontal drifts in the E region of the ionosphere in relation to random fading of radio waves

Regular determinations of E region wind velocities (V) and frequency of fading (N) were made from simultaneous fading records at three spaced receivers taken on two pulsed radio frequencies, 2.3 and 2.8 Mc/s respectively, between December 1954 and March 1955. A linear relation between V and N was obtained for each wavelength, and the experimental relation V = 1.86Nλ deduced. This result agrees with theory and the significance of the constant of proportionality is discussed.

Error rates in pulse position coding

An expression for the error rate in a system using a binary pulse position code is derived. In the system considered, the pulses amplitude modulate a carrier and the resultant signal is contaminated by additive Gaussian noise. At the receiver the pulses are recovered by an envelope detector. If synchronization errors and post-detection filtering are neglected, it is shown that the probability of a binary error is approximated well by 1/2 \exp (-a^2/2) , where a^2 is the peak input signal-to-noise power ratio. Finally, the error rate is derived for the case where the signal amplitude is subject to random fading. Some comparisons are made with error rates derived by Montgomery for other systems with and without carrier fading. It is found that when the signal is subject to fading the pulse position system is better than a comparable system using threshold detection.

An improved fading machine

A fading machine which formerly gave cyclic fading has been modified to give substantially random fading with the amplitude of the signal conforming to the Rayleigh distribution, as with long-distance h.f. radio signals. The fading is obtained by combining six components in random phase. The machine simulates propagation by up to three paths with a spread of up to 2 millisec in path-time delay and caters for dual diversity reception. It has proved to be a valuable tool in laboratory investigations of radiotelegraph equipment.

Ionospheric irregularities causing random fading of very low frequencies

The process by which random irregularities in the ionosphere cause random fading in a reflected radio wave at the ground may be divided into two parts: the scattering process within the layer which builds up an inhomogeneous field in free space immediately beneath it and the diffraction process by which this field propagates to the earth's surface, possibly changing in form as it travels. The extent of this change is evaluated by finding the correlation coefficients for three pairs of quantities measured experimentally at 75 and 150 kc/s frequency. These are 1. (a) the signal amplitude at two horizontally separated receivers 2. (b) the amplitudes of the first and second reflections 3. (c) the amplitude and phase of the first reflection. The diffractive changes are found to be considerable, but the wave emerging from the layer is probably modulated in phase rather than amplitude. A study of the variation of fading speed at 75 kc/s with height of reflection appears to confirm existing ideas concerning the presence of small-scale irregularities above a critical height in the lower E-region.

Some Studies on Random Fading Characteristics

The theory of random signals, as applied by various workers to the phenomenon of random fading of pulses singly reflected from the ionosphere, is considered critically. It is shown that, contrary to the prevalent idea, the velocity distribution of fading is independent of the power spectrum of the returned wave. This invalidates the ordinarily accepted explanation of certain observed departures from theory. An alternative explanation of these departures is given, which explanation is quantitative in character and can therefore be experimentally verified. The influence of a steady signal superposed on the randomly fading wave, on the velocity distribution of fading is investigated. It is found that unlike the effect on the amplitude distribution, the effect in this case is very small, resulting only in a very slight increase in the velocity of fading. An alternative method of determining the autocorrelation function of the fading pattern (from which the power spectrum of the returned wave can be derived by inverse Fourier transform) is given. This provides an alternative method of determining the velocity of drift of ionospheric irregularities. The method of deriving the theoretical power spectrum returned by drifting ionospheric irregularities has been improved upon, yielding a form of the power spectrum which appears to be more in accordance with observations.

The fading machine, and its use for the investigation of the effects of frequency-selective fading

The function of the fading machine is to simulate the frequency-selective fading which is characteristic of long-distance short-wave radio channels. Fading of this type is usually due to interference between waves which have traversed multiple paths between sender and receiver, and its effect is often to produce distortion of the modulation of radio waves. Many radio transmission systems have been devised with the object of minimizing the effects of selective fading; the fading machine enables different systems to be compared in the laboratory under precisely controlled fading conditions. The equipment described in the paper incorporates three transmission paths, the group time-delay differences between which may be varied in steps from 0 to 2 milliseconds. The phase differences between the paths may be varied manually (“static” fading) or continuously (“dynamic” fading), with fading rates ranging from 0.1 to 10 fades per second. Random noise, either fading or non-fading, may be included so as to synthesize a complete short-wave radio channel. The fading machine may also be used to simulate diversity reception obtained by spaced aerials. Examples are given of the use of the equipment to assess the merits of double-sideband, single-sideband and frequency-modulated transmission systems with telephony or telegraphy modulation, under conditions of severe selective fading and high noise level.

Notes on the Random Fading of 50-Megacycle Signals over Nonoptical Paths

To obtain data on the variation of field strength beyond the horizon, simultaneous recordings were made at three locations, one within the optical path of the transmitter, one 700 feet below the line of sight, and one 11,400 feet below the line of sight. All three locations were on the same linefrom the transmitter. Recordings extended over a two-week period, chosen at a time when atmospheric refraction was likely to be favorable. Analysis of the recorded data indicates several things of interest. The variation of field strength at each location was random and showed no correlation with any other location; the range of field-strength variation exceeded 49 decibels at the most remote location; maximum fields generally occurred at night; and previous data on the rate of attenuation beyond the horizon were confirmed.