SPECTRAL CHARACTERISTICS OF GASOUS MEDIA AND THEIR EFFECTS ON PROPAGATION OF ULTRA-WIDEBAND MILLIMETER WAVE RADIATION
Y. Pinhasi, A. Yahalom, O. Harpaz, G. Vilner
Dept. Of Electrical and Electronic Engineering –
The College of Judea and Sammaria
P.O. Box 3, Ariel 44837, ISRAEL
Abstract
The growing demand for broadband wireless communication links and the deficiency of wide frequency bands within the conventional spectrum, are leading one to seek for higher microwave and millimeter-wave spectrum at the Extremely High Frequencies (EHF) above 30GHz. One of the principal challenges in realizing modern wireless communication links at the EHF band is the effect emerging when the electromagnetic radiation propagates through the atmosphere. A general approach for analyzing wireless communication channels operating in the EHF band is presented. The theory is employed for analysis of communication channel operating at the EHF and utilizing pulse amplitude modulated signals. The atmospheric absorptive and dispersive effects on pulse propagation delay, width and distortion are studied.
1.Introduction
Beside to the fact that the EHF band (30-300GHz) is a wide, free of users frequency spectrum, it offers many advantages to wireless communication and RADAR systems:
· Broad bandwidths for high data rate information transfer
· High directivity and spatial resolution
· Low transmission power (due to high antenna gain)
· Low probability of interference/interception (LPI)
· Small antenna and equipment size
· No multipath fadings (although fading may cause by atmospheric conditions)
Among the practical advantages of using the EHF region for satellite communications systems is the ability to employ smaller transmitting and receiving antennas. This allows the use of a smaller satellite and a lighter launch vehicle.
One of the principal challenges in realizing modern wireless communication links at the EHF band are the effects emerging when the electromagnetic radiation propagates through the atmosphere. Figure 1 is a schematic illustration of a wireless communication line-of-sight (LOS) link, where the distance between the transmitter and receiver sites is d.
Fig. 1. Wireless line-of-sight communication link.
At frequencies below the EHF band, the ratio between the received power and the transmitted power in line-of-sight (LOS) radio links, operating at a frequency , is given by the well-known Friss transmission (‘link budget’) formula [1-2]:
(1)
where and are the transmitting and receiving antenna gains respectively, and is the speed of light. The above formula describes the ‘free-space loss’ due to diffraction of the transmitted radiation without considering atmospheric attenuation due to absorption and scattering and ignores multi-path effects along the path of propagation.
When a millimeter-wave radiation passes through the atmospheric medium, it suffers from a selective molecular absorption [3-8]. Several empirical and analytical models were suggested for estimating the millimeter and infrared wave transmission of the atmospheric medium. The transmission characteristics of the atmosphere at the extremely high frequencies (EHF), above 30GHz, as shown in Figure 3 was calculated with the millimeter propagation model (MPM), developed by Liebe [9-10]. Inspection of Figure 3.a reveals that the absorption peaks at 22GHz and 183GHz, where resonance absorption of water () occurs, as well as at 60GHz and 119GHz, absorption lines of the oxygen (). Between these frequencies, minimum attenuation is obtained at 35GHz (Ka-band), 94GHz (W-band), 130GHz and 220GHz, which are known as atmospheric transmission 'windows' [6].
The transmission characteristics are determined by weather conditions as temperature, pressure and humidity. The absorption is proportional to air density, and thus reduces with height. Attenuation due to fog, haze, clouds, rain and snow is one of the dominant causes of fading in wireless communication links operating in the EHF band [11-12]. Raindrops and dust scatter millimeter wave radiation, resulting in amplitude fluctuations and phase randomness in the received signal. This further degrades the availability and performance of the communication links. Sufficient fade margins are essential for a reliable system.
The inhomogeneous transmission in frequencies will cause absorptive and dispersive effects on the amplitude and phase of wide-band signals transmitted in the EHF band. The frequency response of the atmosphere plays a role as the data rate of a wireless digital radio channel is increased. The resulted amplitude and phase distortion leads to inter-symbol interference, and thus to an increase in the bit error rate (BER). These effects should be taken into account in the design of broadband communication systems, including careful use of adequate modulation, equalization and multiplexing techniques.
Several theoretical works have dealt with distortions appearing when a short pulse is propagating in absorptive and dipersive media including in gases and plasmas [13-19]. They studied the delay and pulse shape evolution along the path of propagation. Conditions for pulse compression and expansion were identified.
In this paper, we developed a general approach for analyzing wireless communication channels operating in the EHF band. The MPM model [9-10] is employed to calculate the characteristics of the atmosphere in the frequency domain. The theory was used to study the effects of the atmospheric medium on a radio link, shown in Figure 1. The data signal, represented by a complex envelope , modulates a carrier wave transmitted and propagates to the receiver sight through the atmosphere along a line-of-sight path. The demodulated in-phase and quadrature signals, retrieved at the receiver’s outputs, are examined to evaluate the expected performance of a high data rate digital wireless channel operating in the millimeter wavelengths.
2.Propagation of millimeter waves in gaseous media
The time dependent field represents an electromagnetic wave propagating in a medium. The Fourier transform of the field is:
. (2)
Propagation of electromagnetic waves in a medium can be viewed as transformation through a system (see Figure 2).
Fig. 2. Linear system representing propagation of electromagnetic waves
in a medium.
In the far field, transmission of a wave, radiated from a localized (point) isotropic source and propagating in a (homogeneous) medium is characterized in the frequency domain by the transfer function, derived in Appendix A:
. (3)
Here, is a frequency dependent propagation factor, where ε and µ are the permittvity and permeability of the medium, respectively.
The transfer function describes the frequency response of the medium. Its inverse Fourier transformation corresponds to the temporal impulse response . In a dielectric medium the permeability is equal to that of the vacuum and the permittivity is presented as . When the medium introduces losses and dispersion, the relative dielectric constant is a complex, frequency dependent function, which its real and imaginary parts satisfy the Kramers-Kronig relations [20]. The resulted index of refraction can be presented by:
, (4)
where is the complex refractivity of the molecules composing the air [11].
The propagation factor can be written in terms of the index of refraction:
. (5)
Substituting expressions (4) and (5) in equation (3), assuming horizontal propagation in an homogeneous medium, results in the transfer function:
, (6)
where is the attenuation factor; is the wavenumber of the propagating wave, shown in Figure 3.
The power transfer function along a propagation path given by:
(7)
is shown in Figure 4 for frequencies 35GHz and 94GHz, where minimum attenuation (‘transmission window’) is obtained, as well for 60GHz and 119GHz where maximum absorption of the oxygen molecules occurs.
(a)
(b)
Fig. 3. Millimeter wave (a) attenuation factor in [dB/Km] and (b) wavenumber increment in [rad/Km] for different values of relative humidity (RH).
Fig. 4. Power transfer at transmission windows 35GHz and 94GHz and at oxygen absorption lines 60GHz and 119GHz.
3. The propagation model
Assume that a carrier wave at is modulated by a wide-band signal :
(8)
as shown in Figure 1. Here is a complex envelope, representing base-band signal, where and are in-phase and quadrature information waveforms respectively. The Fourier transform of the transmitted field is:
(9)
where is the Fourier transform of the complex envelope . After propagating along a path with a distance d, the field is:
(10)
Since the transfer function is a Fourier transform of a real function, . Thus, the inverse Fourier transformation of Eq. (10), results in a received field given in the time domain by:
(11)
where the complex envelope of the signal obtained at the receiver sight is given by:
(12)
The formalism developed above is utilized in the following for analytical derivation and numerical calculations of the demodulated signal at the receiver sight. The flow chart in Figure 5 summarizes the procedure carried out for solving Eq. (12). Gaussian pulses were chosen as base-band digital signaling fed to the modulator in transmitter sight. The MPM model [10-12] is called for calculating the complex refractivity of the atmosphere, as required in the transfer function (6). Finally, the de-modulated complex signal and its related quadrature components and obtained at the receiver output are found, employing an algorithm of Fast Fourier Transformation (FFT) for solving equation (12).
4.Ultra-short pulse transmission
Assume that the transmitted waveform is a carrier modulated by a Gaussian envelope:
(13)
characterized by a standard deviation . Fourier transformation of the pulse results in a Gaussian line-shape in the frequency domain:
(14)
Fig. 5: The procedure of calculation of de-modulated quadrature signals received in a wireless communication channel.
shown in Figure 6. The corresponding standard deviation frequency bandwidth is . The full-width half-maximum (FWHM) is the –3dB bandwidth and is equal to .
Fig. 5. The normalized Gaussian lineshape.
In order to calculate the pulse shape after propagation along a horizontal path in the atmospheric medium, we substitute the Fourier transform (14) into expression (12). Analytical result can be found when the complex propagation factor (5) is approximated in the vicinity of the carrier frequency , by the second order Taylor expansion [14-16]:
(15)
where .
The second order approximation given in Eq.(15) can be used when the standard deviation of the Gaussian pulse satisfies
resulting in a complex envelope:
, (16)
where:
.
The expression obtained at (16) is valid when:
. (17)
This condition is always satisfied when . However, at frequencies where the attenuation curve is convex , as occurring in the vicinity of the absorption lines. In those cases, the analytical result is valid only when . Another interpretation of this result is that for a given initial pulse width , the distance should not exceed in order to keep the derivation (16) legitimate.
The magnitude (absolute value) of the complex envelope given in (16), has a Gaussian shape:
(18)
with temporal delay:
(19)
and standard deviation given by:
(20)
These results, which can be obtained from the solutions derived in [14,16], show that in the framework of the approximation (15), a Gaussian magnitude is preserved while propagating in the medium (although its typical width is varying) and thus can be retrieved as such by the receiver.
Since the pulse cannot propagate above the light speed, the time delay, given by (19) is always . This fact points out that the medium response should fulfill at any frequency. For a short distance (or a wide pulse), the time delay can be approximated by . This becomes the exact solution at attenuation peaks, where . When , the denominator of Eq. (19) may become arbitrary small, (however should be kept positive in order to satisfy the validity condition (17)). In that case the time delay is approximately (where must be ), resulting in and arbitrary long delay in the pulse arrival at distance d.
Examination of Eq. (20) reveals that when the pulse always expands along the path of propagation. However, for (e.g. in the vicinity of absorption frequencies), the pulse may become narrow while propagating in the atmospheric medium. Pulse compression occurs when . In both cases minimum pulse width is obtained when resulting .
For long propagation distances, the time delay approaches and the standard deviation .
Acknowledgements
The research was carried out in the framework of the Israeli Software Radio Consortium, supported by the MAGNET program of the State of Israel Ministry of Industry and Commerce. We also acknowledge the help and support of The Samuel Neaman Institute for Advanced Studies in Science and Technology.
Appendix A
In this appendix is we prove the validity of the transfer function given in equation (3). It is well known that electromagnetic fields obey the macroscopic Maxwell equations:
(A.1)
In the frequency domain the equations become:
(A.2)
in which . We assume the following relations between the displacement and electric fields:
(A.3)
For a non-magnetic material the Magnetic field and the density of magnetic induction are related by:
(A.4)
This results in the following set of equations:
(A.5)
which can be combined to a single equation for :
(A.6)
Defining the following quantities:
(A.7)
We obtain the equation in the form:
(A.8)
The corresponding equation for the fields green function is:
(A.9)
This equation can be written in terms of as:
(A.10)
Since the source of the Green function is a point source G must be spherically symmetric hence the equation takes the form:
. (A.11)
For we obtain:
. (A.12)
Which has the trivial solution:
. (A.13)
For the equation takes the form:
. (A.14)
With the well known solution:
(A.15)
Adjusting the two solutions yields:
(A.16)