In Chapters 2 and 3, we studied the prediction of the signal power received under free‐space propagation conditions and in the presence of the Earth, that is, obstacles, mountains, hills, buildings, and so on. Consequently, we predicted the received signal power due to reflection, refraction, diffraction and scattering of electromagnetic waves by the obstacles in the propagation path. However, the SNR, the principal figure‐of‐merit of a communication system, is determined not only by the received signal level but also the received noise and interference levels. Noise is defined as a random time‐varying electromagnetic phenomenon which may be superimposed on, or combined with, a wanted signal. Noise varies randomly with time and does not carry information. [1] There are many sources of noise in the RF band that are internal and external to the receiving system. Internal noise is mostly of thermal origin and is generated by the receiver itself. Antenna losses, feeder lines connecting the antenna to the receiver and the receiver itself are the major contributors of the internal noise. The external noise is collected by the antenna and may be due to: [1][2][3]

1. Emissions from atmospheric gases and hydrometeors (water vapour, oxygen, nitrogen, rain, snow);
2. Radiation from extra‐terrestrial sources such as galaxies, sun, moon and stars;
3. The Earth’s surface;
4. Atmospheric noise due to lightning discharges (atmospheric noise due to lightning);
5. Man‐made noise due to electrical machinery, power transmission lines, electronic equipment, engine ignition systems, and so on.

A receiver may also receive co‐channel transmissions whether intentionally or unintentionally radiated or spurious emissions by radars, radio/TV stations, multi‐user systems and jammers. Unlike noise, such signals carry information depending on their source and distance; hence interfere with the received signals.

A communication receiver should operate successfully in the presence of noise and interference since it can not distinguish between wanted and unwanted (noise and interference) signals impinging at the receive antenna in the frequency band of operation (Figure 4.1). The amount of noise and interference power coupled into a receiver depends not only on the antenna radiation pattern but also on the spatial and spectral distribution of the sources of noise and interference. The noise performance of a receiver is hence determined by the noise collected by its antenna and its internal noise.

In this chapter, we will be interested in the statistical characterization and the effects of internal and external noise in the SNR performance of a receiver.

4.1 Thermal Noise

Every object in the universe, with a temperature above 0 K, generates thermal noise which is caused by motion of electrons. The physical model for the thermal noise may be described, as shown in Figure 4.2, by considering a resistor R at temperature T above 0 K. The motions of the electrons in their orbits and occasional collisions between them give rise to a noise voltage v_n(t) which has a zero mean ; this implies that the polarity of the noise voltage has equal probabilities of being positive and negative. However, the mean‐square noise voltage (the noise variance), , generated at the terminals of a resistor R at temperature T (K) in a bandwidth Δf (Hz) is non‐zero. If a load resistor R_L is connected to the terminals of R, the power delivered to R_L is maximized when R_L = R and is given by Planck’s law: [4]

(4.1)

where k = 1.38 10⁻²³(J/K) denotes the Boltzmann’s constant and h = 6.6254 10⁻³⁴(J.s) is the Planck’s constant. At radio frequencies where hf << kT, one may use the binomial expansion (see (D.1)) to express the single‐sided power spectral density (PSD) of the thermal noise as

(4.2)

Figure 4.3 shows the variation of the noise PSD, given by (4.1), with frequency for various values of T. One may observe that the PSD of thermal noise is white (flat) in the RF frequencies of interest and may be approximated by (4.2) for f < 10¹² Hz at temperatures of practical interest. Hence, the double‐sided noise PSD, which applies for both positive and negative frequencies, is N₀/2.

Based on the above considerations, we define the random noise voltage w(t) across a 1Ω load resistor which is matched to the source resistor, hence (see Figure 4.2). The random noise voltage w(t) is described by a Gaussian probability density function (pdf) with zero mean and a variance equal to the double sided PSD N₀/2:

(4.3)

where R_w (τ) denotes the autocorrelation function of w(t). The white approximation for the PSD S_w(f) of the thermal noise defined by (4.2) implies that any two samples of w(t) are uncorrelated from each other unless they occur at exactly the same time; hence the autocorrelation function of w(t) is represented by a delta function in time:

(4.4)

4.2 Equivalent Noise Temperature

The physical model described above for the thermal noise generated by a resistance at temperature T may be generalized for modelling the thermal noise generated by any thermal noise source. Consider a narrowband noisy device with gain G and bandwidth B. The noise power kT_sB due to a resistance of temperature T_s at the input of the noisy device is amplified by the device of gain G so that the output noise power due to this resistance is given by kT_sBG (see Figure 4.4). On the other hand, the internal noise of the device also induces a noise power, say N_eo, at the output. Let T_e (K) denotes the temperature of an equivalent resistance which delivers the same average output noise power N_eo as the considered device in a given bandwidth B. Then, the total equivalent noise power at the device input is given by

(4.5)

The total equivalent noise power at the device output may be written as

(4.6)

where the device is now assumed to be noiseless. Note that T_e is not equal to the physical temperature of the noisy device but is related to it; it simply provides a measure of the noise power generated by the device. For example, let G = 20 dB, T_s = 290 K, B = 1 MHz and T_e = 580 K. The total equivalent input noise power N_i = k(T_s + T_e)B is equal to −139.2 dBW, and total output noise power N_out = N_i G is −119.2 dBW.

4.2.1 Equivalent Noise Temperature of Cascaded Subsystems

A communication receiver consists of cascaded devices, for example, antenna, RF amplifier, frequency convertor, demodulator, detector and audio amplifier, as shown in Figure 4.6. These devices may be active or passive but are all noisy with different noise levels. The internal noise of a receiver system of cascaded components may be determined in terms of the equivalent noise temperatures and gains of the individual components. Let G_i and T_ei, i = 1,2,…,n denote, respectively, power gains and equivalent noise temperatures of these components. Output noise power may be written as

(4.10)

If T_e denotes the equivalent noise temperature of the overall system, which has a power gain of , then the output noise power is given by

(4.11)

Equating (4.10) and (4.11) to each other, the equivalent noise temperature of the system is found to be

(4.12)

One may observe from (4.12) that the equivalent noise temperature T_e of the receiver is dominated by T_e1 and G₁, which usually correspond to the feeder line connecting the antenna to the receiver. In practice, low‐noise amplifiers (LNAs) are used between the receiving antenna and the feeder line in order to decrease the receiver noise, since low T_e1 and high G₁ decrease T_e significantly. One can also infer from (4.12) that the noise temperatures of the devices towards the end of the receiver do not have significant contribution to the receiver noise.

4.3 Noise Figure

Noise figure provides a measure of SNR degradation caused by a noisy device compared to a reference noise source at its input. Consider a noisy device with gain G as shown in Figure 4.7a. This device amplifies signal and noise powers S_i and N_i at its input so that the output signal and noise powers at the device output are given by GS_i and GN_i, respectively. In addition, the internal noise of the device also induces a noise power of N_eo at its output, which may be referred to its input as N_e = N_eo/G, as shown in Figure 4.7b.

The noise figure is defined as the ratio of the input SNR to the output SNR, for an input noise source at reference temperature T₀ = 290 K, that is, , which implies an input noise PSD of :

(4.13)

For a noiseless device, one has and . Since each device has T_e ≥ 0 K, noise figure is always higher than 0 dB; hence, higher the noise figure, noisier the device. Inserting the internal noise power of the considered device when referred to its input into (4.13), the noise figure and the equivalent noise temperature may be related to each other as follows:

(4.14)

Figure 4.8 shows the variation of the equivalent noise temperature T_e with noise figure F, usually expressed in dB. Note that T_e is less than 100 K for F < 1.5 dB; this is typical for a low‐noise amplifier which may require cooling. A typical LNA has a noise figure of ~3 dB, which implies an equivalent noise temperature of 290 K.

Choice of N_i/B = kT₀B, as a reference input noise PSD, is meaningful for systems operating at around T₀ = 290 (K), that is, for terrestrial applications. However, N_i/B = kT₀ is not a good reference for space applications, where the ambient temperature may be highly different from T₀ = 290 K. For a reference temperature T_A different from T₀, the corresponding noise figure is called as operating noise figure F_A:

(4.15)

It is clear from (4.15) that the value of noise figure depends on the assumption about the reference source temperature, but the value of the equivalent noise temperature T_e is independent of the reference source temperature.

Using (4.12) and (4.14), the equivalent noise figure F_e of cascaded subsystems may easily be written as

(4.16)

4.4 External Noise and Antenna Noise Temperature

Antenna noise temperature provides a measure of the noise power received by an antenna from external noise sources. It depends on the radiation pattern, the frequency of operation, the system bandwidth and spatial and spectral distribution of the noise sources surrounding an antenna. The external noise sources include the troposphere, the ionosphere, the Earth’s surface, the moon, the planets, the sun and the other stars and galaxies. Man‐made noise also contributes the antenna noise temperature.

The contribution of the atmospheric noise increases in proportion to the thickness of the atmosphere seen by the antenna; hence, it is minimum at zenith and increases with decreasing elevation angle. The clear sky has a background noise temperature of 3 K, which is attributed to the big bang. Contribution of planets, stars and galaxies depend on whether the antenna is looking towards them, the frequency of operation and the brightness of the source. In summary, the antenna noise temperature is determined by the received noise power from external noise sources, weighted by the antenna radiation pattern in frequency and space coordinates.

In telecommunication systems, transmitters are usually considered as point sources when they are located in the far‐field of the receiver. The Poynting vector intensity radiated by a point source decreases with the square of the distance as signal propagates away from the transmitter (see for example (2.13)). The power received by a receiver is proportional to the intercepted flux density in (W/m²) times the effective aperture of the receiver (see (2.96) and Figure 4.11). The noise sources may behave as point sources if their size is relatively small or their distance to the receiver is very high. However, in majority of cases, the noise sources are extended in space; hence they are analysed differently.

4.4.1 Point Noise Sources

if a noise source may be assumed as a point source, then it is meaningful to describe the received signals by the Poynting vector intensity (flux density) in W/m². Power flux density from a point source, which is assumed to be isotropic, is given by the Poynting vector intensity (see also (2.53))

(4.30)

where P_tn denotes the total radiated power by the noise source. When the source is not isotropic, the Poynting vector intensity changes with direction S_n(r, θ, ϕ). Then, the power flux through a solid angle dΩ (steradian) is given by (see Figure 4.11)

(4.31)

In spherical coordinates, the surface element dS is given by

(4.32)

The power received by an antenna located at distance r may be written as

(4.33)

where integration is over the antenna aperture A_A. Figure 4.11, (4.31) and (4.33) show that, in order to have a constant flux through a solid angle dΩ (steradian), the antenna size should be increased in proportion with r². If antenna size is kept constant, then the power received by the antenna (hence the power flux) will decrease in proportion with r². If the Poynting vector intensity may be assumed to be constant over the antenna aperture, then (4.33) simply reduces to (2.96), which states that the received power is equal to the product of the Poynting vector intensity and the effective receiving area of the antenna.

4.4.2 Extended Noise Sources and Brightness Temperature

In many cases, the sources contributing to the antenna noise can not be described as point sources since they are extended in space coordinates. Consider an extended noise source as shown in Figure 4.12 and a receiving antenna at distance r collects the noise power from all the extended source region.

The concept of blackbody radiation is suitable for characterizing radiation from extended noise sources. Although no perfect blackbody exists in nature, many objects (e.g., the sun, stars) behave approximately like blackbody over at least a range of frequencies. A blackbody is an idealized body which absorbs the electromagnetic energy at all frequencies impinging upon it. A blackbody in thermal equilibrium has the same rate of energy emission and absorption. According to the Planck’s Law, the brightness, that is, the power spectral density (PSD) radiated isotropically from a unit area of blackbody at temperature T (K) per solid angle, is given by [4]

(4.34)

where k = 1.38 10⁻²³(J/K) is the Boltzmann’s constant and h = 6.6254 10⁻³⁴(J.s) denotes the Planck’s constant. At RF frequencies, where hf << kT, one can use in (4.34) to get the Rayleigh‐Jeans approximation:

(4.35)

which is valid for all RF frequencies of interest, , (e.g., for for T = 300 K).

We now determine the power received from an extended noise source as shown in Figure 4.12 by a receiving antenna whose beam is confined to a solid angle dΩ. The antenna beam intercepts an extended noise source with brightness B(θ, ϕ), which implies that T in (4.34) and (4.35) is replaced by T_B(θ, ϕ), which is called the brightness temperature of the noise source in the direction {θ, ϕ}. If an antenna with effective receiving area A_e(θ, ϕ) is pointed in the direction (θ,ϕ), the power received per bandwidth Δf from an angular region defined by dΩ may be written as

(4.36)

where (2.98) is used to express the effective area of the receive antenna in terms of its gain. The factor ½ in front of the first expression in (4.36) accounts for the fact that the randomly polarized noise power is equally divided between two orthogonal polarizations and an antenna can extract noise power only from one of the two orthogonal polarizations.

In the special case where Rayleigh‐Jeans approximation is valid, that is, the noise PSD is flat in the frequency range of interest, then (4.36) for a lossless antenna simplifies to

(4.37)

Note that Δf is replaced by B, the bandwidth, since the PSD is flat. On the other hand, the gain G(θ, ϕ) is replaced by the directivity D(θ, ϕ) for the lossless antenna. It is clear from (4.37) that the brightness temperature of the extended noise source is weighted by the antenna radiation pattern to obtain the antenna noise temperature. In the special case where the brightness temperature does not vary with direction, , then the use of (2.70) leads to T_A = T_B for perfect polarization and impedance matching. Then, (4.36) reduces to (4.1), which gives the received power from a resistor at temperature T. As one may observe from (4.37), a highly directive receive antenna (with very narrow beamwidth), whose directivity D(θ, ϕ) approximated as a delta function in (θ, ϕ), may be used to map T_B(θ, ϕ). The brightness temperature of a noise source is not identical but is related to its physical temperature as given by

(4.38)

where ε(θ, ϕ) denotes the emissivity of the source, T_ph(θ, ϕ) is its physical temperature, ρ is the reflection coefficient and T_atm denotes the weighted average of the sky brightness temperature. The term ρ T_atm is more suited to describe the brightness temperature of the Earth’s surface and accounts for the reflection of the atmospheric noise. Up to about 100 GHz, but particularly below 10 GHz, the reflection coefficient ρ from the Earth’s surface is generally high and the emissivity is low. [1]

4.4.3 Antenna Noise Figure

The received noise power due to external sources is expressed in (4.37) in terms of the equivalent noise temperature of a lossless antenna. Similarly, the received noise power may also be measured by the equivalent noise figure F_A of a lossless antenna, which is defined by the ratio of the received noise power P_n to the reference noise power kT₀ B: [1]

(4.39)

where k = 1.38 × 10⁻²³ J/K is the Boltzmann’s constant and T₀ = 290 K is the reference ambient noise temperature. The noise figure F_A is related to the equivalent noise temperature T_A of a lossless antenna as [1]

(4.40)

In view of (4.37), the value of T_A is determined by the parameters of the noise source and the antenna radiation pattern. Using (4.39), the available noise power P_n from an equivalent lossless antenna may be written as

(4.41)

In some low‐frequencies applications, dipole antennas are used for signal reception. In these cases, the received signal and noise levels are determined by the received rms electric field intensity. Using given by (2.120) the received rms field intensity due to noise may be expressed as

(4.42)

where G_dBi denotes the gain of the receiving dipole antenna. It is evident from (4.41) and (4.42) that the antenna noise figure F_A, hence T_A, may be determined if the received noise power or the rms electric field intensity is known.

4.4.4 Effects of Lossy Propagation Medium on the Observed Brightness Temperature

The brightness temperature of external noise sources is observed through a propagation medium which attenuates signals and also generates noise; this affects the noise power received by the antenna. At frequencies higher than 10 GHz, rain attenuation and atmospheric absorption may cause significant degradation in the system performance due to increases both in system noise and in signal attenuation. For example, the propagation medium between the sky, of brightness temperature T_B(θ, ϕ), and the receive antenna attenuates signals and also changes the observed brightness temperature (see Figure 4.13). The observed brightness temperature may be expressed as

(4.43)

The medium temperature T_medium is usually related to the earth‐surface temperature T_surface as [8][9]

(4.44)

Here A(θ) denotes the signal attenuation at elevation angle θ due to absorption of electromagnetic waves by the gaseous constituents (oxygen, hydrogen, water vapour, and so on) of the atmosphere and/or attenuation due to precipitation, for example, rain, snow. If the loss in the direction of zenith is denoted by A_z, the loss at elevation angle θ is approximated by (3.55). Similarly, the attenuation due to rain is predicted by (3.74) or (3.77). Note that (4.43) accounts for the noise caused by a lossy propagation medium, for example, atmospheric absorption, rain, snow and water vapour. Since T_medium (~270 K) is ‘hotter’ than the brightness temperature of the sky (~3 K), a lossy propagation medium always increases the observed brightness temperature.

Figure 4.14 shows the variation of in (4.43) as a function of the the elevation angle θ. The loss is modeled as (dB) where A_z denotes the zenith attenuation of the lossy propagation medium. The physical temperature of the medium is given by (4.44) with T_surface = 290°K and , where the zenith brightness temperature is assumed to be . Note from (4.43) that if A(θ) = 1 (lossless medium), then . However, in a propagation medium with high losses, one has A(θ)>> 1 and . This implies that an antenna looking towards a region in the sky with high losses, for example, heavy rain intensity, suffers not only from heavy signal losses but also from the increased noise power received by the antenna.

4.4.5 Brightness Temperature of Some Extended Noise Sources

Any object in the universe radiates noise as long as its temperature is higher than 0°K. An antenna receives thermal radiation emanating from sources in its environment by weighting them with its radiation pattern. Since antennas are usually directed with an elevation angle, they see the sky with their main beam and some near‐in sidelobes. Thermal radiation from Earth’s surface contributes to the noise of a ground‐based antenna through its far‐out sidelobes. Compared with background ‘cold’ sky, with a brightness temperature of approximately 3 K, the ‘hot’ surface of the earth with 290 K brightness temperature provides the largest contribution to the antenna noise. Therefore, the suppression of antenna sidelobes is of great interest, especially in low‐noise receiver systems used in the area of telecommunications, radar, radioastronomy and remote sensing. Special attention should be paid for not directing the main lobe of the antenna to external noise sources like stars, sun and moon since this causes excessive increases in the antenna noise temperature.

Figure 4.17 shows the emissivity and the brightness temperature of a smooth water surface for vertical and horizontal polarizations and for two angles of incidence. The curves for fresh and sea water are indistinguishable for frequencies higher than 5 GHz. The emissivities (and hence the brightness temperatures) of land surfaces are higher than those of water surfaces due to smaller dielectric constant of land. Figure 4.18 shows the brightness temperature of a smooth ground surface for different moisture contents and for vertical, horizontal and circular polarizations. If the moisture content increases, the brightness temperature decreases. The brightness temperature increases as the surface becomes rougher. [1]

We now consider an Earth‐coverage antenna onboard a geostationary satellite that looks at the Earth. The radiation pattern of the Earth‐coverage antenna pattern is assumed to be

(4.52)

where θ denotes the angle off boresight and is the half‐power antenna beamwidth. Noting that the Earth, of 6370 km radius, is subtended by 17.43 degrees at the geostationary satellite, located at 35870 km from the surface of the Earth, the Earth fills the main beam of the Earth‐coverage satellite antenna between 3 dB points. As the satellite moves around its orbit, the receiving antenna sees the African (hot) land mass at 30⁰ East longitude and the (cold) Pacific at 180⁰ West to 150⁰ West longitude. Therefore, the antenna noise temperature provides a weighted average of the Earth’s brightness temperature. Brightness temperature increases with frequency largely due to atmospheric absorption. Curves shown in Figure 4.19 are based on the assumption that water vapour density is 2.5 g/m³ and 50% cloud coverage.

We now consider the case of a ground‐based antenna looking at the sky. Unless directed to a noise source like sun, moon, stars or a galaxy, the antenna sees the empty sky with brightness temperature of 3 K through the troposphere of T_medium = 274 K (see (4.44)). In view of (4.43), the cold troposphere masks the sky radiation and the brightness temperature observed by the receiver antenna is that of the troposphere. Figure 4.20 shows the brightness temperature of a ground‐based receiver excluding the sky noise contribution of 3 K or other extra‐terrestrial sources for frequencies between 1 and 60 GHz. The curves are calculated for several elevation angles and an average atmosphere (7.5 g/m³surface water vapour density, surface temperature of 288 K, and a scale height of 2 km for water vapour). Note that the variation of the brightness temperature is similar to that of the atmospheric absorption loss shown in Figure 3.23. Similarly, the variation of the brightness temperature with elevation angle follows the cosecant law for the atmospheric path attenuation in (3.55). This implies that the background sky noise temperature of 3 K is masked by the ‘hot’ troposphere and the noise temperature of an antenna on the Earth’s surface is determined by that of the atmospheric gases.

Figure 4.21 shows the brightness temperature of some extra‐terrestrial noise sources in the frequency range 0.1 to 100 GHz. For frequencies up to about 100 MHz, the median brightness temperature for galactic noise for a vertical antenna, neglecting ionospheric shielding, is given by. [1]

(4.53)

Below 2 GHz, one needs to be concerned with the Sun and the galaxy (the Milky Way), which appears as a broad belt of strong emission. Above 2 GHz, one needs to consider only the Sun and a few very strong non‐thermal sources such as Cassiopeia A, Cygnus A and X and the Crab nebula since the cosmic background contributes 3 K only and the Milky Way appears as a narrow zone of somewhat enhanced intensity. [1]

The noise due to active sun (with solar flares) is broadband and is much higher compared to those due to planets. Active sun refers to intense solar activity resulting in anomalously large solar flares for three‐year durations with a periodicity of 11 years. Figure 4.21 shows that the brightness temperature of the quiet sun is at 1 GHz, 10 GHz and 100 GHz, respectively. The brightness temperature of the Moon is ~210 K between 1−100 GHz when a 0.5 deg. beamwidth antenna points to the Moon. [10][11]

4.4.5.1 Frequency Dependence of the Antenna Noise Temperature

Figure 4.22 shows the maximum and minimum hourly median values of T_A in the frequency range 0.1Hz to10 kHz, based on measurements on all seasons and times of day on the entire Earth’s surface. In this frequency range, seasonal, diurnal or geographic variations are not significant. Larger variability in T_A in the 100−10000 Hz range is due to the variability of the cutoff frequency of the Earth‐ionosphere waveguide. Figure 4.23 covers the frequency range 10 kHz−100 MHz. Minimum and maximum values of the hourly median atmospheric noise correspond to values exceeded 99.5% and 0.5% of the hours, respectively. Atmospheric noise curves are based on measurements on all seasons and times of day on the entire Earth’s surface. In the frequency range up to 30 MHz (LF, MF and HF), the antenna noise and hence the receiver performance is dominated by the atmospheric noise. The noise level is so high that it is not feasible to invest for a low‐noise receiver.

In the VHF band (30−300 MHz), galactic noise is the dominant contributor. The antenna noise temperature may become 10 times higher when it looks towards the galactic center compared to the galactic pole. Average value of the galactic noise (over the entire sky) is given by the solid curves labelled galactic noise in Figures 4.23 and 4.24. Measurements indicate a ±2 dB variation about this curve, neglecting ionospheric shielding. The minimum galactic noise (narrow beam antenna towards galactic pole) is 3 dB below the solid galactic noise curve shown on Figure 4.24 and the maximum galactic noise for narrow beam antennas is shown via a dashed curve.

The UHF band (0.3−3 GHz) provides a low‐noise window for RF communications, since the antenna (external) noise is the lowest. In this band, absorption of the electromagnetic waves by the atmosphere and the water vapor are the major contributors to the antenna noise. Losses due to atmospheric absorption increase with the thickness of the atmosphere seen by the antenna main lobe and the elevation angle. In view of (4.43), this also implies increased antenna noise. Hence, a communication system with an antenna main beam directed close to the horizon has a higher antenna noise temperature compared to when it looks towards the zenith.

The antenna temperature, T_A, is the weighted average of the brightness temperature of the sky and ground by the antenna pattern (see (4.37)). For antennas whose patterns encompass a single source, the antenna temperature and brightness temperature are the same (curves C, D and E of Figure 4.24, for example). The majority of the results shown in Figures 4.22−4.24 are for omnidirectional antennas except unless otherwise noted. For very narrow‐beam HF antennas, however, studies have indicated that atmospheric noise due to lightning can vary as much as 10 dB (5 dB around the average T_A value shown) depending on antenna pointing direction, frequency and geographical location.

4.4.6 Man‐Made Noise

Median values of man‐made noise power for several environments is shown in Figure 4.25 which also includes a curve for galactic noise for comparison purposes. In all cases, the median value of T_A varies with frequency as

(4.54)

where c and d take the values given in Table 4.1. Note that (4.54) is valid in the range 0.3 to 250 MHz for all the considered environments except those of curves D and E in Figure 4.25. The man‐made noise considered here has a Gaussian distribution. Man‐made noise often has an impulsive component and this may have an important effect on the performance of radio systems and networks.

Environmental category	c	d
City (curve A)	7.68	2.77
Residential (curve B)	7.25	2.77
Rural (curve C)	6.72	2.77
Quiet rural (curve D)	5.36	2.86
Galactic noise (curve E)	5.20	2.30

4.5 System Noise Temperature

Besides the noise generated within the receiver system itself (including the noise due to antenna losses), an antenna collects the external noise by weighting with its radiation pattern. Thermal noise generated by a lossy antenna will also contribute to the antenna noise temperature. As shown in Figure 4.26, a lossy antenna is usually represented as an ideal antenna followed by a lossy device, with a loss factor , where η_L denotes the antenna loss efficiency (see (2.94)). The reference point for the overall noise level, which is usually measured by the noise temperature of a receiving system, is the terminals (front‐end) of a noise‐free receiving antenna. The system noise temperature T_S of a receiving terminal, shown in Figure 4.26, accounts for the receiver internal noise, the noise due to lossy feeder line, internal noise of a pre‐amplifier (optional), and the antenna noise (see (4.12)):

(4.55)

where {T_aL, T_ℓ, L_ℓ, F_ℓ} denote the equivalent noise temperature, the physical temperature, attenuation and the noise figure of the antenna losses. Similarly, {T_F, T_f, L_f, F_f} denote the equivalent noise temperature, the physical temperature, attenuation and the noise figure of the feeder line. They are related to each other by (4.27):

(4.56)

If , then (4.55) simplifies to

(4.57)

where from (4.40). The noise figure of the receiver system F_S, which is related to the system noise temperature T_S by (4.14), accounts for the effects of both both the internal and the external noise.

For the special case when there is no preamplifier, that is, , then (4.57) reduces to

(4.58)

where the system noise figure is given by . Furthermore, if the antenna noise temperature can be approximated as in the 1−10 GHz band (see Figure 4.24), then insertion of into (4.58) leads to

(4.59)

where the system noise figure is determined by the product of antenna losses, feeder cable losses and the receiver noise figure. This evidently shows the importance of keeping antenna and feeder line losses as low as possible.

Now consider a simplified baseline receiver system, as shown in Figure 4.27, by shifting the pre‐amplifier in Figure 4.26 to the receiver input and combining with the receiver and combining the antenna losses with the feeder‐line losses. Then, the considered receiver system may simply be described by a receiving antenna, of noise temperature T_A, connected to the receiver of noise figure F and gain G:

(4.60)

(see (4.16)) via a feeder line of physical temperature T_ph and loss L (see Example 4.4 and (4.58)):

(4.61)

The effective gain and the effective noise figure of the feeder line‐receiver combination are given by G/L and FL, respectively (see (4.58)). The effective noise temperature and the noise figure of this simplified baseline receiving system, shown in Figure 4.27, are given by

(4.62)

The term LT_R shows the equivalent noise temperature of the receiver referred to the input of the feeder line. The noise figure of the feeder‐receiver referred combination is given by LF for T_ph = T₀. If LF is high, then T_S is dominated by (LF−1)T₀ by masking the antenna noise temperature T_A. On the other hand, if LF is sufficiently low, then T_S approaches T_A. In this case, the noise of the line‐receiver combination becomes negligible.

The system noise temperature T_S may be written as when it is referred to the receiver input

(4.63)

This implies that the antenna noise temperature is seen at the receiver input just as the way the sky brightness temperature is seen at the receiver antenna through a lossy propagation medium (see (4.43)).

The SNR at the receiver output may be written as

(4.64)

where P_r denotes the received signal power and T_S is the noise temperature of the receiver system, both referred to the receiver front‐end. Note that (4.64) would be the same irrespective of whether it is referred to the front‐end, the receiver input or the receiver output:

(4.65)

since the signal and the noise would be amplified by the same factors at each point. On the other hand, the presence of the feeder lines and the receiver decrease the output SNR by a factor

(4.66)

When transmitter and receiver are in LOS of each other, the output SNR given by (4.65) may be written as

(4.67)

where the received power P_r is given by (2.117). Here, denotes the overall receiver efficiency accounting for polarization and impedance matching as well as implementation losses. A careful look at (4.67) shows that the SNR is determined by the EIRP, which is a transmitter parameter defined by (2.116), by the channel parameters, namely the bandwidth B and free‐space loss L_fs, depending on the range and the frequency, and the receiver parameters η_lossG_r/T_S, depending on the antenna size, frequency, receiver system noise and the receiver efficiency parameters. Here, G_r/T_S, usually expressed in dBi/K, is called the receiver figure of merit (FOM) and used as a quality measure for the receiver system (see Figure 4.16). Therefore, (4.67) clearly shows that the SNR at the receiver output requires a trade‐off between many system parameters related to the transmitter, the receiver and the channel.

4.6 Additive White Gaussian Noise Channel

The receiver system noise, which is described by T_S at the receiver front‐end (see Figure 4.27), accounts for the antenna noise, the feeder line noise and the receiver internal noise. Since these noise terms are uncorrelated with each other, their powers are additive. Hence, the receiver noise may be represented by an additive white Gaussian noise (AWGN), w(t), at the receiver input. In view of (4.3) and (4.4), the PSD of AWGN is white and it is described by a Gaussian pdf with zero mean and variance N₀/2. The AWGN is added to the signal at the receiver input. This channel, which is described by Figure 4.29, is called as AWGN channel under free space propagation conditions, when the signal at the receiver input is only an attenuated, delayed and phase shifted version of the transmitted signal. These parameters are generally incorporated into the AWGN term; the delay and phase shift due to channel do not affect the statistical characteristics of the Gaussian noise (see (4.3)) and signal attenuation may be accounted for by scaling the AWGN term so as to generate the desired SNR in the channel (see (3.5) and (3.6)). Therefore, the signal at the receiver input is qualified as ‘transmitted signal’ and ‘the received signal’ represents the sum of the ‘transmitted signal’ and the AWGN term, as shown in Figure 4.29. The received signal is hence described by a Gaussian pdf with its mean equal to the transmitted signal level and variance N₀/2, where is determined by the system noise temperature.

Note that the noise in some channels may be non‐Gaussian. The noise of non‐thermal nature also exists, for example, shot noise. However, the noise in the majority of communication channels of practical interest has thermal origin and can be modelled as AWGN.

References

1. [1] Recommendation ITU‐R P.372−10 (10/2009) Radio noise.
2. [2] Recommendation ITU‐R P.618−11 (09/2013) Propagation data and prediction methods required for the design of Earth‐space telecommunication systems.
3. [3] Recommendation ITU‐R P.530−15 (09/2013) Propagation data and prediction methods required for the design of terrestrial line‐of‐sight systems.
4. [4] J. D. Kraus, Radio Astronomy, McGraw Hill: New York, 1966.
5. [5] D. M. Pozar, Microwave and RF Design of Wireless Systems, Wiley: New York, 2001.
6. [6] A. B. Carlson, P.B. Crilly, and J. C. Rutledge, Communication systems, McGraw Hill: Boston, 2002.
7. [7] P. Z. Peebles, Jr., Probabiity, Random Variables, and Random Signal Processing (3rd ed.), Mc.Graw Hill: New York, 1993.
8. [8] R.L. Freeman, Telecommunication transmission Handbook (3rd ed.), John Wiley: New York, 1996.
9. [9] I. A. Glover and P. M. Grant, Digital Communications (2nd ed.), Pearson‐Prentice Hall: Harlow, 2004.
10. [10] C. Ho, A. Kantak, S. Slobin, and D. Morabito, Link analysis of a telecommunication system on earth, in geostationary orbit, and at the moon: Atmospheric attenuation and noise temperature effects, IPN Progress Report 42−168, Feb. 15, 2007.
11. [11] C. Ho, S. Slobin, A. Kantak, and S. Asmar, Solar brightness temperature and corresponding antenna noise temperature at microwave frequencies, IPN Progress Report 42−175, Nov. 15, 2008.
12. [12] A. C. Clark, Extra terrestrial relays‐can rocket stations give worldwide coverage, Wireless World, October 1945.
13. [13] J. G. Proakis and M. Salehi, Communication Systems Engineering (2nd ed.), Prentice Hall: New Jersey, 2002.

Problems

1. Consider the block‐diagram of a receiver as shown below.

   

   The received signal power at the input to the first amplifier is –90 dBm, and the received noise power spectral‐density is –170 dBm/Hz. The bandpass filter has a bandwidth of 10 MHz. Determine the SNR at the input to the demodulator.
2. An antenna with a gain of 40 dB is connected to a preamplifier (impedance‐matched) with a gain of 30 dB and a noise figure of 6 dB. The preamplifier output is connected to a receiver having a noise figure of 10 dB by means of a transmission line having a length of 20 m and an attenuation of 0.5dB/m. The antenna noise temperature is 50 K; the system bandwidth is 10 MHz and the wavelength of operation is 10 cm. Find the following parameters for the system:
   1. The system noise temperature T_s.
   2. What is the required incident power per unit area in order to give a signal‐to‐noise ratio of 30 dB at the receiver output.
   3. Find the corresponding signal‐to‐noise ratio at the receiver input.
   4. Solve the above problem again when the transmission line is inserted between the antenna and the preamplifier. Comment on the results obtained.
3. Assuming that the equivalent noise temperature and the gain of a broadband device are characterized by
   (4.84)
   where K is an arbitrary constant. Use (4.25) and the formulas from Appendix D to show that the mean equivalent noise temperature of this device is given by
   (4.85)
   The percentage bandwidth of a device is defined as the percent of the ratio of the bandwidth B = f₂−f₁ to the carrier frequency f₀. The bandwidth of a narrowband device is in the order of a several percent of the carrier frequency. Assuming that f₂/f₀ = f₀/f₁, plot the above formulas for the mean equivalent noise temperature as a function of 1 < f₂/f₀ < 1.3. Show that the narrowband assumption does not lead to significant errors in estimating the equivalent noise temperature of this device.
4. Using (D.56), show that the total brightness of a blackbody defined by (4.34) is proportional to the fourth‐power of it noise temperature:
   (4.86)
   where .
5. Transfer functions of devices vary with frequency. Consider that a source of PSD (W/Hz) is connected to the input of a baseband device with frequency‐dependent transfer function H(f).

   

   1. Express the PSD of the signal at the output of the device in terms of S(f) and H(f).
   2. Determine the total power available at the output of the device.
   3. Determine the total power available at the output of the device, P_o, if the source has an equivalent noise temperature of T_e.
   4. It is convenient to replace the actual frequency‐dependent gain characteristics by a filter with the noise bandwidth B_n determined by the condition of equal noise power output for the two cases: P₀ = kT_eB_nG, where G denotes the maximum gain of the device under consideration. Determine the noise bandwidth of a device if where f₀ = 1 MHz.
6. An antenna with noise temperature of T_A = 300K is connected to a receiver system which is formed by the cascade connection of a preamplifier (G₁, F₁ = 2, B > 10 MHz), mixer (G₂ = 1/2, F₂ = 6, B > 10 MHz) and amplifier (G₃ = 100, F₃ = 7, B = 10 MHz).
   1. What should be the gain G₁ of the preamplifier so as to keep the overall noise figure of the receiver less than or equal to 3.
   2. Find the system noise temperature when the preamplifier is fed by the antenna by using G₁ = 15 dB and the value of G₁ found in part (a).
   3. Find the noise power at the amplifier output for the two cases of part (b).
   4. Repeat part (b) if a feeder line with 2 dB loss connects the antenna to the preamplifier.
7. An antenna with 50K noise temperature feeds a cascade of two impedance‐matched attenuators. The first attenuator at a physical temperature of T₁ = 100 K and with a loss of L₁ = 2 is connected to the antenna. The second attenuator, at a physical temperature of T₂ = 290 K and loss L₂ = 1.4, drives the receiver of noise temperature T_R = 1000K, 3‐dB noise bandwidth of B = 1 MHz and G = 60 dB gain.
   1. Determine the effective noise temperatures of the two attenuators.
   2. Determine the effective noise temperature of the cascade.
   3. Determine the system noise temperature?
   4. Determine the noise figure and the noise temperature of this system having the antenna as its source.
   5. Determine the average noise power which is available at the system output.
8. An amplifier has B = 1 MHz bandwidth and standard (with a noise temperature of 290 K at its input) noise figure F = 4. The amplifier’s available output noise power is 0.1 μW when its input is connected to a radio receiving antenna with T_A = 100 K noise temperature. Determine the amplifier’s input effective noise temperature T_e, its operating noise figure F_op with an antenna of T_A = 100 K at its input, and its power gain.
9. Consider a system with 0.1 mW transmit power, 0dBi gain for transmit and receive antennas, operating at 500 MHz carrier frequency with 100 kHz bandwidth. The system operates in a suburban environment. Assume the receiver noise figure is 6 dB and the noise temperature of the receive antenna is 200K, independent of frequency.
   1. Determine the SNR received at a distance of 1km, assuming free space propagation?
   2. How does the SNR change when the carrier frequency is increased to 5 GHz?
   3. Explain why does the 5 GHz system show a significantly lower SNR? Is the receiver antenna noise temperature independent of frequency (see Figures 4.20 to 4.24)? For a given physical size, are the gains of transmit and receive antennas independent of frequency? In view of the above, what can you say about the frequency dependence of SNR?
10. A 15m diameter antenna, with 65 % aperture efficiency, is used to receive transmissions from a geostationary satellite operating at 10 GHz with an elevation angle of θ = 10⁰. When an antenna is pointed at zenith, the measured noise temperature at the feed output is 20K. Use Figure 4.20 for the brightness temperature of the sky. Ignore the noise temperature contribution from the antenna sidelobes at all angles.
    1. Assume that the atmospheric absorption loss can be modelled as where A_z denotes the atmospheric absorption loss at zenith. Determine A_z.
    2. Estimate the antenna noise temperature at an elevation angle of 10⁰
    3. Find the earth station G/T in clear sky in zenith direction and at 10⁰ elevation angle.
11. Consider a 12 GHz terrestrial link operating in free space propagation conditions. The signal power at the receiving antenna terminals is measured to be –42 dBm. The overall noise figure of the receiver is 6 dB and the noise bandwidth is 10 MHz. The clear sky antenna noise temperature at 12 GHz is given as 130 K.
    1. Estimate the clear sky SNR at the terminals of the receiving antenna assuming that the antenna has a loss efficiency of 97 % and is at a physical temperature of 300 K.
    2. Determine the effective SNR during a rainfall in the vicinity of the receiver, causing a signal attenuation of 3 dB. The physical temperature of the rain cloud is assumed to be 273 K.
12. The rain attenuation exceeded for 0.1% and 0.01% of the time is given by 1.1 dB and 7.5 dB, respectively, in a SATCOM link operating at 12 GHz. We assume that the brightness temperature of the clear sky is 3K and the physical temperature of the rain cloud is estimated by (4.44).
    1. Determine the corresponding increases in the receiver system noise temperature.
    2. Repeat (a) for 18 GHz.
13. The effect of rain on the signal attenuation and system noise temperature of a satellite ground terminal operating in 14/11 GHz band is given as follows:

    Time parameter exceeded (hours/year)	Rain attenuation (dB)	System noise temperature (K)
    _	0	250
    12	2	350
    6.5	4	420
    3.0	6	460
    1.8	8	480
    1.0	10	495
    0.5	12	510

    1. Determine the decrease in SNR for each time parameter with respect to the case of 0 dB rain attenuation.
    2. Determine the margin, which would ensure the SNR would be the same as in the case with no rain attenuation, for 99.99 % of the time.
14. The effective temperature of an antenna looking toward zenith is approximately T_A and is connected to a receiver by using a transmission line of loss L. The physical temperature of the transmission line (waveguide) is T_ph. The received signal power at the antenna terminals is S_i (W).
    1. Express the effective noise temperature, noise power and SNR at antenna and receiver terminals.
    2. Assuming that L = 0.1 dB/m, T_A = 5 K and T_ph = 300 K, determine the effective temperature at the receiver terminals when the length of the transmission line is 1 m and 30 m. Compare the values obtained with the receiver noise temperature of 100 K.
15. Consider a satellite at an altitude of 900 km transmitting at a carrier frequency of f = 1.5 GHz to a mobile user with a half‐wave dipole antenna. The satellite transmit power is P_t = 100 W, and the antenna gain is G_t = 30 dBi. Total system losses are assumed to be 4 dB.
    1. Find the received signal power at the mobile receiver antenna terminals
    2. Determine the system noise power if the receiver noise figure is 6 dB, antenna noise temperature 60 K and the receiver bandwidth is 1 MHz.
    3. Determine the SNR.
    4. Determine the decrease in received SNR in dB if there is 3 dB rain attenuation and the medium temperature is 290 K.
16. Antenna noise temperature of a mobile terminal is T_A = 290°K. The first amplifier stage of the radio provides a 20 dB gain and has a noise figure of 5. The second amplifier stage has 10 dB gain and a noise temperature of 3000°K. The contribution of other stages on the system noise is ignored.
    1. Determine the equivalent noise temperature of the first two stages of the radio and compare with system noise temperature.
    2. If the baseband processing of the radio requires an SNR of 10 dB in 10 kHz, what is the receiver sensitivity?
17. A satellite ground station sees the satellite through a cloud which induces a 1 dB loss to electromagnetic waves propagating through it. Assume that the brightness temperature due to background radiation is 30 K and the cloud has a physical temperature of 275 K.
    1. Determine the antenna noise temperature with and without the cloud. How much the antena noise temperature is increased due to the presence of the cloud?
    2. Determine the ratio of input SNRs with and without cloud. Compare the signal attenuation, the increase in the system noise temperature and the decrease in SNR due to the presence of the cloud.
    3. Repeat (b) assuming that the receiver has a noise temperature of 60 K. Compare the results obtained in (a) and (b).
18. A preamplifier with gain = 13 dB and noise figure = 1 dB is inserted between the antenna with T_A = 290 K and the receiver with 50 dB amplifier gain, 500 MHz bandwidth, 7 dB noise figure and 8 × 10⁻¹¹W input signal power.
    1. Determine the equivalent noise figure of the whole receiver. How much the noise figure is improved due to the use of the preamplifier?
    2. Determine the output SNR improvement due to the use of preamplifier.
    3. Repeat part (b) for T_A = 29 K and T_A = 2900 K. Determine the SNR improvement in dB.
    4. Discuss how the receiver noise figure with regard to the antenna noise temperature affects the SNR.
19. The effective noise temperature of an antenna looking toward empty sky is approximately 4 K. The antenna is connected to a receiver, which has a noise temperature of about 54 K, with a transmission line of physical temperature of 300 K. The transmission line has an attenuation coefficient 0.1 dB/m and its length is 10 m.
    1. Find the effective noise temperature at the receiver terminals and calculate the increase in the noise power in dB at the receiver input due to the use of the lossy cable.
    2. Repeat part (a) when there is a cloud in the boresight direction of the antenna which has a temperature of 250 K and causing a loss of 2 dB at the frequency of operation.
20. When a directional antenna, of gain 30 dBi, loss efficiency of η_L = 0.9 and physical temperature of 290°K, is pointing toward empty sky, the noise temperature falls to about 3°K at frequencies between 1 GHz and 10 GHz.
    1. If the antenna is connected directly to a low noise amplifier with a gain of 40 dB and a noise figure of 1.4, what is the system noise temperature?
    2. If a cable of 3 dB loss is inserted between the antenna and the receiver, determine the system noise temperature.
    3. Calculate the increase in noise power due to the use of the cable of 3 dB loss.
21. A microwave receiver used in satellite communications has an antenna with T_A = 20 K cascaded with an LNA with T_e = 6 K, G = 30 dB, an amplifier with F = 6 dB, G = 20 dB and a mixer + IF combination with F = 13 dB and G = 10 dB.
    1. Determine the equivalent noise figure and the noise temperature of the receiver (excluding the antenna). Compare the equivalent noise temperature of the receiver with that of the first amplifier. Calculate how many dB’s the noise power is increased compared with that of the first amplifier.
    2. Each stage of the receiver introduces some band limiting. However, it may be convenient to consider that all the stages of the system are of unlimited bandwidth and band limiting is done in a single IF filter at the output end of the IF amplifier. Under these circumstances, the noise input to this IF filter would have a white (uniform) power spectral density. Determine the noise power spectral density (W/Hz) present at the demodulator input.
    3. Explain whether the noise introduced by the demodulator or any succeeding decoder will further degrade the signal‐to‐noise ratio of the signal at the output of the receiver.
    4. Consider that the received signal has a baseband bandwidth of 4 MHz. If we want the signal to noise ratio at the demodulator input to be 20 dB, then find the signal power at the demodulator input? Determine the corresponding signal power at the receiver input (output of the receive antenna).
22. The noise figure of a cell phone receiver is specified as 16 dB. Since the internal noise is very high, the external antenna noise can be ignored. This receiver requires an SNR of 13 dB for reliable detection of a signal with 200 kHz bandwidth. Determine the receiver sensitivity.
23. A satellite is located at 10⁰ west longitude in the geosynchronous orbit. The transmitting ground station is located at 10⁰ east longitude and 30⁰ north latitude, and the receiving ground station is located at 33⁰ east longitude and 40⁰ north latitude. The uplink and downlink frequencies are 12GHz and 10GHz, respectively. All antennas are 3 m in diameter with 65% aperture efficiencies. Ignoring atmospheric absorption and noting that the transmission bandwidth in the downlink and the uplink are the same, the overall C/N ratio may be expressed in terms of the uplink and downlink carrier‐to‐noise ratios as
    
    1. Use (3.67) to determine the uplink and downlink ranges to the satellite.
    2. Find the uplink and downlink carrier‐to‐noise ratios so as to give an overall C/N ratio of 20 dB in a transmission bandwidth of 10 MHz. The G/T of satellite and ground‐station receive antennas are 4 dB and 7 dB, respectively.
    3. Using the C/N values for the uplink and downlink of part (a), determine the transmit powers for the ground station and the satellite.
24. In a SATCOM system, a ground receiver antenna has 64 dBi gain and 65% aperture efficiency. The relevant noise temperatures are given as follows:

    TS at 100 elevation angle:	234 K (clear‐sky conditions)
    Sky noise temperature at 100 elevation:	17 K (see Figure 4.20)
    Receiver noise temperature:	150 K

    During heavy rain, the slant path attenuation reaches 8 dB for 0.01 % of the year. Calculate the G/T_S for this percentage of the time and the corresponding reduction in the SNR.
25. Consider a mobile radio system operating at 900 MHz with 25 kHz bandwidth. Transmit and receive antenna gains are 8 dBi and 0 dBi, respectively and the antenna noise temperature is T_A = 300 K. Losses in cables, combiners, and so on at the transmitter are 3 dB. The voltage reflection coefficients at the transmit and receive side are given as 0.25. The received power in the channel is modelled by (3.92) with r₀ = 10 m and n = 4. The operating SNR at the receiver, with a 7 dB noise figure, is required to be 18 dB. Assuming a fading margin of 7 dB, determine the minimum transmit power for a coverage radius of 2 km.
26. A wireless microwave link, operating in the 10 GHz band with B = 10 MHz bandwidth, is used to relay TV signals at a range of 10 km. Transmitting and receiving antennas both have 15 dBi gains. For acceptable performance 30 dB SNR is required. Loss due to polarization mismatch is limited to 3 dB. Perfect impedance matching and free space propagation conditions are assumed.
    1. For T_A = 40 K and F_R = 5, determine the minimum required received power at the receiver antenna terminals.
    2. Determine the required transmit power level.
27. A transmitter and a receiver both on the surface of the earth establish a link between them by using the moon as a relay. Transmit and receive antenna gains are given by G_t and G_r, respectively. Transmitter and the receiver are located at distances d₁ and d₂ from the moon and are given by (m). The system operates with a transmit power P_t at f = 4 GHz and has a noise bandwidth of B Hz.

    The radar cross section (RCS) σ of the moon is reported to be 0.070 times the physical cross section. Since the diameter of the moon is (m), its physical cross section is (m²), and its RCS is approximately equal to (m²) in the frequency band of operation. Since the receive antenna ‘looks’ at the moon, the mean antenna noise temperature may be approximated by 210 K (see Figure 4.21). Differences in the lunar surface temperature may decrease its brightness temperature by a factor of two during the night and increases by the same factor during the day.

    1. Using (2.125), express the SNR at the output of the receiver with noise figure F and antenna noise temperature T_A in terms of the relevant system parameters.
    2. For F = 3, P_t = 1 MW, B = 30 kHz and transmit antennas with 65% aperture efficiencies and 3 m diameter antennas, determine the value of the SNR in dB.
    3. Calculate the variations with respect to the mean SNR due to night‐time and day‐time variations in the antenna noise temperature.
28. A police traffic‐control radar operating at 9.9 GHz has a transmit power of 0.1 W and uses dish antennas with 20 cm diameter and 65% aperture efficiency. The minimum SNR level required for reliable detection is 10 dB with a receiver system noise temperature of 2000 K. The system has a noise bandwidth of B = 1 kHz.
    1. Determine the maximum range at which cars with radar cross sections larger than 0.2 m² can be detected.
    2. A driver has a radar receiver which scans the radar frequency channels for detecting the presence of the police radar. At what range can the police radar be detected by this receiver with an antenna gain of 16 dBi and a signal sensitivity of –60 dBm.
29. Consider a LOS communication system operating at 9 GHz requires 10 dB SNR. The EIRP of the transmitter is 0 dBW. The receiving antenna diameter is 13 cm (with an aperture efficiency of 0.65), and the antenna noise temperature is 800 K. The distance between the transmitter and receiver is 12 km. The transmission bandwidth is B = 10 MHz.
    1. Determine the maximum allowable receiver noise figure
    2. Determine the value of the noise figure in part (a), when the transmission bandwidth is doubled.
    3. Determine the value of the noise figure in part (a), when the antenna diameter is doubled.
30. In order that a satellite keeps revolving in an orbit, the gravitational force should be balanced by the centripetal force
    
    where GM = 3.986 10¹⁴ (m³/s²), G is the gravitational constant, M is the Earth’s mass, m is the mass of the satellite, v is the tangential velocity of the satellite and r denotes the distance between the Earth’s center and the satellite. Inserting into the above equation, one can write the orbital period T, the time required for one complete revolution around earth in terms of h, the height above Earth, is as follows:
    
    where and R = 6371 km denotes the radius of the Earth.
    1. Plot the variation of T as a function of h, the height of the satellite above Earth’s surface and show that T increases with increasing height and becomes equal to 24 hours at a height of h = 35870 km. What is the period of a satellite at an orbital height of 900 km?
    2. Plot the variation of the tangential velocity v as a function of h and show that the velocity decreases with increasing orbital height.
31. A narrowband signal is to be transmitted a distance of 100 km over a wireline channel that has an attenuation of 1 dB/km. Assume that noise equivalent bandwidth of each repeater is B = 4 kHz and that N₀ = kT₀ = 4 10⁻²¹ W/Hz.
    1. Using (4.76), determine the transmitted power P_t required to achieve an SNR₀ = 30 dB at the output of a receiver with a 5 dB noise figure, if no relays are used.
    2. Repeat the calculation when a repeater is inserted every 10 km in the wireline channel, where the repeater has a gain of 10 dB and a 5 dB noise figure.
32. Consider a 802.11b WiFi uplink with the following parameters: f = 2.5 GHz, BS antenna gain of 5 dBi, and MS antenna gain of 0 dBi. Uplink transmit power is assumed to be 0.1 mW. The base station receiver has a noise figure of 7 dB, T_A = 290 K antenna noise temperature and a 4 MHz noise bandwidth. The indoor path loss is modelled by
    
    where X = 8 dB denotes the link margin in dB due to shadowing.
    1. For a distance of 20 m between the MS and the BS, calculate the SNR at the detector input of the BS receiver.
    2. How can the range be increased?
33. Specific attenuation due to rain can also be calculated by where R is the rain rate in mm/hr exceeded for p% of the time, and k and α depend on the rain drop size and frequency. Consider a terrestrial radio link system with a range of 20 km and operating at 12 GHz, for which k = 0.0178 and α = 1.209 (see Table 3.2). Consider a 30 mm/hr rain rate for 0.01% of the time.
    1. Determine the rain attenuation exceeded for 0.01% of the time.
    2. Ignoring the sky noise temperature of 3 K in (4.43) and (4.44), the increase in antenna noise temperature due to rain may be estimated as
       
       where A denotes the rain attenuation found in part (a). Determine the increase in received noise power, in dB, if the receiver system noise temperature is 200 K.
    3. Determine the degradation in signal to noise power ratio, SNR.
34. A satellite ground station is transmitting signals to a geostationary satellite located at a distance of 39000 km. For proper system operation, a power density of −120 dBW/m² is required at the satellite. Assume 2 dB loss for antenna pointing errors, polarization mismatch, and so on and 5 dB loss due to rain attenuation. Operating frequency is 14 GHz.
    1. Determine the required ground station EIRP and the transmitter power if the transmitting antenna has a gain of 40 dB.
    2. Determine the figure‐of‐merit of the satellite receiver antenna to achieve a C/N₀ = 95 dB.Hz. What is the SNR in a bandwidth of 40 MHz?
    3. If E_b/N₀ = 20 dB is required to achieve a certain probability of error, what is the maximum data rate which can be supported by this link?
35. The path loss model in 3GPP Release 9 (2010) is given as follows:
    
    where d is in meters and f denotes the frequency in GHz. An LTE macrocell BS has the following parameters: f = 2 GHz, P_t = 46 dBm, G_t = 15 dBi, the length of the grid range a = 500 meters. For LTE Femtocell BS one has f = 2 GHz, P_t = 21 dBm, G_t = 5 dBi, the length of the grid range a = 20 meters. Similarly, for 802.11b WiFi BS, f = 2.5 GHz, P_t = 20 dBm, G_t = 5 dBi, the length of the grid range a = 20 meters. Receive antenna gains for all users are assumed to be 2.3 dBi.

    Consider a WiFi receiver at a distance of d meters from the BS and receives at a data rate of R = 11 Mbps.

    1. Determine the bit energy E_b = P_r/R at the receiver located at a distance of d = 50 m from the WiFi BS.
    2. A WiFi link is considered to be good if the received E_b/N₀ ≥ 40 dB. Determine the requirement on the noise temperature of the WiFi receiver system. Comment if this is a tight requirement.