15.1.1 EIRP and Power Aperture

Commonly used performance measures for the transmitter system are the effective isotropic radiated power (EIRP) and power aperture. The EIRP is defined as the product of the power into an isotropic antenna and the antenna gain in a given direction relative to the isotropic antenna gain. Usually the EIRP is specified in the direction of the maximum antenna gain. The power aperture is defined as the product P_PAA_ta of the transmitter power and the transmit antenna area. The maximum gain along the antenna boresight axis corresponding to and the EIRP² is defined as

(15.8)

To express the communication range equation in terms of the EIRP, the transmitter loss (L_ts) is removed from the link RF losses resulting in the communication range equation

(15.9)

The communication range equation is expressed in terms of the power‐aperture product, P_PAA_ta, by substituting for G_t using (15.3), with the result

(15.10)

In this expression A_ta = A_a is the physical area of the transmit antenna. The power‐aperture product is used as a performance measure of the transmitter when the physical size of the antenna is an important consideration, for example, when the transmit power and physical size of the antenna tend to dominate the cost of the transmit system.

15.1.2 Signal‐to‐Noise Ratio

Expressing the range equation in terms of the signal‐to‐noise ratio, γ, is necessary when relating the system performance to the signal detection and false‐alarm probabilities or the acquisition and bit‐error probabilities. The noise floor at the input to the receiver is characterized in terms of the thermal noise power in the receiver noise bandwidth B_n and is given by

(15.11)

where  W/s‐°K is Boltzmann’s constant, T_o = 290°K is the standard temperature,³ B_n is the noise bandwidth in Hertz, and F_n is receiver noise figure. The receiver noise figure accounts for the additional noise contributed by the receiver relative to the intrinsic kTB_n thermal noise. The noise bandwidth, as used here, is related to the RF bandwidth at the input to the LNA with prefiltering frequency response H(f) and is defined as

(15.12)

where f_o is the frequency corresponding to the maximum of H(f).⁴ This definition of the noise bandwidth corresponds to an ideal unit amplitude rectangular noise filter having the same area as |H(f)|². Using these results and (15.6), the communication range equation is used to express the received signal‐to‐noise ratio as

(15.13)

where F_ns is the system noise figure that is somewhat higher than that of the receiver because of receiver antenna related noise sources as discussed in Section 15.2.3.

15.1.3 Maximum Range

The maximum range of a communication link is defined as the range beyond which the signal‐to‐noise ratio falls below the value required to maintain the specified system performance; typically this is viewed as the signal‐to‐noise ratio corresponding to the specified bit‐error probability; however, a more stressful requirement might be the signal‐to‐noise ratio required to support the waveform acquisition probability. In any event, using (15.5) to express the free‐space loss in terms of the range, (15.13) is then used to evaluate the maximum range to achieve the specified performance criteria in an additive white noise environment with the result

(15.14)

where γ_min is the minimum received signal‐to‐noise ratio corresponding to the system performance specifications.

15.2.1 Receiver Noise Figure

The receiver noise figure is developed in terms of the effective noise temperatures of the individual components of the receiver starting with those that contribute to the system gain. In these cases the input and output signal‐to‐noise ratios are defined by

(15.16)

and

(15.17)

where P_S is the signal power, T_o = 290°K is the standard temperature, B_n is the noise bandwidth, G_i is the gain of the device, and k is Boltzmann’s constant. The term G_ikT_oB_n is the output noise power resulting from the input noise power kT_oB_n and ΔP_ni is the output noise power contributed by the device.⁵ Using these results and (15.15), the noise figure of the device is expressed as

(15.18)

Upon defining the effective noise temperature as  °K the device contribution to the output noise is defined as so, in terms of the effective noise temperature, the device noise figure becomes

(15.19)

Charton points out that this characterization of the noise figure applies to a heterodyned receiver where the signal image is blocked or filtered and when the image is not blocked (15.19) is expressed as ; the factor of two results when the image power equals the desired signal power.

Because the noise figure of a device is usually specified by the manufacturer, the effective noise temperature is computed as

(15.20)

The interpretation of the effective noise temperature is depicted in Figure 15.1 in terms of noisy and noiseless devices both with gain G_i. In Figure 15.1a the output temperature is computed as , whereas, in the noiseless representation of Figure 15.1b, the output temperature is computed as . Therefore, using the equivalent noise temperature, the output temperature is obtained simply by multiplying the input temperature by the device gain. It will be seen that this also applies for lossy devices when the gains are replaced by the reciprocal of the losses, that is, for a lossy device with gain the gain is replaced by so the output temperature becomes .

Using these results, a cascade of successive amplifiers, such that i = 1, 2, 3, …, with different gains and noise figures, results in an overall receiver noise figure given by

(15.21)

or equivalently in terms of the effective noise temperatures

(15.22)

where T_rec is the effective receiver noise temperature given by

(15.23)

From these results it is apparent that the input device should be a high‐gain, LNA, in which case it essentially determines the receiver noise figure. For this reason, LNAs with high gain are typically used at the receiver input.

The noise figure of a lossy device, such as an attenuator, is evaluated in a similar way; however, there are two major differences to consider: the gain of the lossy device is less than unity so the loss is given by , the second consideration is that the physical temperature (T_pi) of the lossy device is typically used to characterize the effective noise temperature performance instead of the noise figure.⁶ The equivalent noisy and noiseless attenuators are shown in Figure 15.2 in terms of the physical and effective temperatures. For the noisy attenuator, the output temperature is given by and, for the equivalent noiseless attenuator, the output temperature is so, upon equating these results, the equivalent noise temperature for a lossy device is

(15.24)

Applying these results to a cascade of N attenuators i = 1, 2, …, N, as shown in Figure 15.2c, each with loss L_i and physical temperature T_pi, the effective noise temperature of the cascaded attenuators is

(15.25)

and the output temperature is computed as

(15.26)

Usually the cascade of receiver devices alternates between amplifiers and passive filters or attenuators as shown in Figure 15.3. In this context, evaluation of the overall system noise temperature and noise figure involves the appropriate application of the results in this section.

15.2.1.1 Example: Evaluation of Receiver Noise Figure and Temperature

As an example application of the results in Section 15.1.2, consider a receiver as shown in Figure 15.3 having a 20 dB gain LNA with a 2 dB noise figure. The LNA is located as close to the receive antenna output as possible, typically it is attached directly to the antenna waveguide, so that the loss prior to the LNA is associated directly with the antenna. The output of the LNA is coupled to the receivers first RF mixer, usually by a short length of coaxial cable. In the following example, the coaxial cable is assumed to have a physical temperature of T_p2 = 290°K and a loss of L₂ dB.

In examining this example, L₂ is considered to be the AGC control parameter. The signal from the coaxial cable is then input to the second amplifier with a gain of G₃ = 25 dB and a noise figure of F_n3 = 10 dB. This is in turn followed by an AGC gain control attenuator with L₄ = 0–70 dB of gain control and a physical temperature of T_p4 = 290°K. Finally, the output of the gain control is amplified by a G₅ = 40 dB gain amplifier with a noise figure of F_n5 = 10 dB. The receiver noise figure for this configuration is shown in Figure 15.4 for LNA gains of 20 and 40 dB as a function of the AGC attenuator setting (L₄) for various conditions of loss (L₂) between the LNA and the second amplifier.

In this example, the system noise figure for an LNA gain of 20 dB and low values of AGC attenuation is degraded from the 2 dB LNA noise figure by 0.25–0.95 dB for feed losses (L₂) between 0 and 6 dB. These low AGC attenuation levels are associated with very weak signals relative to the thermal noise floor (kT_oB_n) at the receiver input. As the input signal becomes stronger the AGC lowers the overall gain of the receiver by increasing the AGC attenuation to keep the output level constant. However, in spite of the increasing receiver noise figure, the resulting impact on the system performance is not significant because of the correspondingly high output signal‐to‐noise ratio associated with the higher input signal levels.

For example, consider the case with L₂ = 3 dB, a minimum detectable signal with the AGC attenuation L₄ = 0 dB, and a system requirement that the minimum output signal‐to‐noise is γ_o = 6 dB. Under these conditions the input signal‐to‐noise ratio is or 6 + 2.5 = 8.5 dB. When the input signal level increases by 20 dB the ideally adjusted AGC attenuation becomes 20 dB and the receiver noise figure is about 2.6 dB. Under these conditions the corresponding output signal‐to‐noise ratio is or 8.5 + 20 − 2.6 = 25.9 dB. As the input signal continues to increase another 20 dB the AGC attenuation becomes 40 dB and the receiver noise figure increases to 9 dB. In this case, the output signal‐to‐noise ratio is 39.5 dB. Therefore, in spite of the dramatic increase in the receiver noise figure from 2.5 to 9 dB, the system requirement that γ_o ≥ 6 dB is more than satisfied.

The curves in Figure 15.4 for the 40 dB LNA gain demonstrate how the sensitivity of the system noise figure is reduced by increasing the LNA gain. Furthermore, with the higher gain LNA the system noise figure is hardly degraded for AGC attenuations up to 20 dB even with a feed loss of 6 dB following the LNA. The higher LNA gain also provided for a wider dynamic range which may be desirable if the feed loss L₂ becomes excessive. From these examples it is clear that the loss between the LNA and the second amplifier plays a significant role in determining the receiver noise figure; however, the focus should be on impact of the receiver noise figure with the minimum detectable input signal, that is, when the AGC attenuation is 0 dB. For example, when the loss L₂ decreases to 0 dB the effective noise temperature becomes  = 0°K even though the device remains at the physical temperature of T_p2 °K. Therefore, the lossless feed does not influence the receiver noise performance. In this case, the receiver noise figure is influenced by the noise figure of the second amplifier, that is, F_n3, and, as the LNA gain is increased, this influence is diminished providing improved system performance with the minimum input signal power.

15.2.2 Antenna Temperature

With the evaluation of the receiver noise figure and temperatures in hand, the focus is turned to the antenna temperature T_a which is the remaining unknown parameter required to evaluate the system noise temperature and noise figure. The antenna temperature is characterized at the output of the antenna subsystem as indicated in Figure 15.5. The antenna is depicted as having a radome and a radiating dish with a feed horn as shown in Figure 15.5a. The feed horn is located at the focal point of the dish and, in the receive mode, is used to collect the incident electromagnetic wave energy. A short length of waveguide associated with the feed horn provides a flange for connecting the feed horn to the receiver subsystem. The radome protects the reflecting or radiating surface of the antenna from the elements of weather. Although the radome is optional, in Figure 15.5a the radome, reflecting dish, and antenna feed are considered to be an integral part of the antenna subsystem with their associated losses and physical temperatures. The overall antenna efficiency is made up of the aperture efficiency and the radiation efficiency. The radiation efficiency includes the spillover loss of a reflecting antenna, the reflector and feed ohmic losses and, if used, the radome loss [3]. These losses are included in the antenna gain and sidelobe measurements and are only used here to characterize the noise introduced by the antenna subsystem. Combining the antenna reflector loss with the antenna feed loss as the product of their individual losses, that is, is valid only if the reflector and feed are at the same temperature (see Problem 2). As indicated in Figure 15.5a, the external noise sources that are within the field of view of the antenna, including the antenna sidelobes, also contribute to antenna temperature. These noise sources include discrete sources (ds) such as stars and our Sun and Moon and background (bck) or distributed noise sources such as stellar gas, our galaxy, the Earth’s surface and thermal noise resulting from atmospheric absorption.

To begin the evaluation of the antenna noise temperature, the noise temperature of all external noise sources is characterized at the input to the antenna as . These noise sources are indicated in Figure 15.5b and are identified as follows: T_b is the noise temperature of distant noise sources entering through the main antenna beam, ρ_lsl is the fraction of the ground noise temperature seen by the lower sidelobes (lsl) of the antenna, ρ_usl is the fraction of the atmospheric noise temperature T_atm seen by the upper sidelobes (usl) of the antenna, and is the effective noise temperature of the antenna radome. The notations T_atm and T_gnd represent the respective physical temperatures of the atmosphere and ground. With the exception⁷ of , all of the temperatures at the antenna radome are dependent on the grazing (or elevation) angle and the antenna beamwidth. For example, as the antenna viewing angle is changed the temperature T_b will change as different noise sources come into view. Also, as the grazing angle decreases the factor ρ_lsl increases with a commensurate increase in the ground noise at the antenna radome; the factor ρ_usl will also fluctuate impacting the atmospheric noise at the radome. The factors ρ_lsl and ρ_usl are evaluated by integrating the appropriate noise temperatures over the sidelobes of the antenna beam in consideration of the attenuation with range. At the elevation angle θ_m, corresponding to the maximum ground noise temperature seen by the lower sidelobes, the ground temperature contribution factor to the antenna temperature is approximated as

(15.27)

where G_r(θ_m,φ) is the antenna gain corresponding to θ_m relative to the peak gain G_r(max). Example values are [2]: ρ_lsl = ρ_usl = 0.1 for grazing angles above 15° with ρ_lsl = 0.49 and ρ_usl = 0.1 for angles below 15°. The link analysis program discussed in Section 15.15 uses (15.27) with G_r(θ, ϕ) evaluated using the circular aperture antenna described in Section 15.3.2.

Referring once again to Figure 15.5b, the antenna temperature is characterized using the relationships developed in the preceding sections and is evaluated as

(15.28)

where T_ain is the temperature of the external noise sources. Evaluation of T_b is discussed in the following two sections and the culmination of the general case is expressed in (15.36) in Section 15.2.2.2. Section 15.2.3.1 provides an example involving the computation of the antenna temperature.

15.2.2.1 Contribution of a Single Sky‐Noise Source to the Antenna Temperature

The analysis of the antenna temperature is simplified by considering one external noise source. In the context of Figure 15.5, is defined as the temperature of either a distributed or a single discrete noise source, that is,  = T_bck or T_ds. In evaluating the antenna noise temperature due to a distant radiating source, the radiation intensity or brightness (b) of the source is characterized by the Rayleigh–Jeans approximation [4] of the radiated flux density per steradian, expressed as⁸

(15.29)

where k is Boltzmann’s constant. The approximation in (15.29) is valid if the ratio not to large which is the case for most applications. For example, the error is about 1% for f = 150 GHz and  = 300°K; if the error is unacceptable then Planck’s law must be used, for which the brightness is expressed as [5]

(15.30)

where h = 6.63e⁻³⁴ W‐s² is Planck’s constant, c = 3e⁸ m/s is the free‐space speed of light, the bandwidth B ≪ f, and k = Boltzmann’s constant, previously defined along with the remaining parameters. The factor of two in (15.29) and (15.30) results from the Omni polarized electrometric waves emitted from the radiating source; for a linearly polarized receiver antenna these results must be divided by two.

Considering a lossless path, corresponding to L_atm = 1 in Figure 15.5, with an effective aperture of A_e m² and one polarization, the total power density at the receiver antenna is evaluated as

(15.31)

Upon substituting (15.29) into (15.31), the temperature of the source at the antenna input becomes

(15.32)

where Ω_s is the solid angle of the sky‐noise source and Ω_a = λ²/A_e is the solid angle of the received antenna beam. Equation (15.32) applies for an antenna using single polarization; with dual polarization this result must be doubled. Also, as indicated, it applies when the solid angle of the source is less than that of the antenna, that is, Ω_s < Ω_a; when Ω_s ≥ Ω_a the result is

(15.33)

If the medium between the radiation source and the antenna exhibits a loss, for example, when L_atm > 1 with an effective temperature , then, using the previous results, the contribution of this external noise source at the antenna input is

(15.34)

15.2.2.2 Contribution of Multiple Sky‐Noise Sources to the Antenna Temperature

When all of the sky noise sources shown in Figure 15.5 are considered, the brightness temperature is evaluated by summing the background temperature with the temperatures of the discrete sources in consideration of overlapping regions as viewed by the antenna beam. To simplify this evaluation it is assumed that, except for the Sun, all of the discrete sources are completely within the antenna beam and do not overlap each other or the Sun. Evaluating spaced‐based cross‐links where the Sun may occupy a significant portion the beam area is a special case. Encounters involving the Moon simply involve replacing the Sun’s noise temperature with that of the Moon. With these considerations the brightness temperature of the noise sources with no atmospheric affects is computed as

(15.35)

and, by analogy with (15.34), when atmospheric affects are involved this result becomes

(15.36)

The noise power of the Sun varies approximately as the inverse square of the frequency. The piecewise linear dependence of a quiet Sun’s noise temperature on frequency is described by the approximate relationship

(15.37)

where f_g is the frequency in GHz. Equation (15.37) is based on the results of Kuiper [6] and Blake [5]. During sunspot activity, the noise temperature of the Sun may be 10²–10⁴ times higher than indicated by (15.37) lasting for several seconds followed by temperatures about 10 times higher lasting several hours.

The noise power from cosmic sources, such as our galaxy and interstellar gases, varies approximately as the inverse square of the frequency and the frequency dependence of the noise temperature varies approximately as [7] 1/f^2.5. For example, denoting the galactic noise temperature at 100 MHz as T₁₀₀, the galactic temperature at another frequency is expressed as [8]

(15.38)

Approximate maximum and minimum values of T₁₀₀ are given by Brown and Hazard [9] as 18,650 and 500°K with a geometric mean value of 3,050°K. The dependence of cosmic noise sources on frequency is discussed in more depth by Hogg and Mumford [10], Ko [11], Smerd [12], and Strum [13]. The inclusion of other discrete sources depends upon the knowledge of their noise characteristics and location relative to the receiver antenna beam pointing coordinates.

The antenna noise temperature is also a function of the loss through the atmosphere as characterized by the atmospheric noise temperature which is dependent on the elevation or grazing angle of the antenna, the temperature, T(r), and the absorption coefficient, α(r), at a range r from the antenna. The effective temperature of the atmosphere is evaluated using the relationship [14]

(15.39)

A simplified evaluation of (15.39) results by using the average temperature and Gardner [15] has reported that this average temperature is approximately 84% of the Earth’s surface temperature (T_gnd) so that .

15.2.3 System Noise Figure

The preceding analysis of the receiver and antenna subsystems, in terms of their noise temperatures and the receiver noise figure, was undertaken so that the system noise temperature and noise figure can be established to evaluate the performance of the entire receiver system. If the antenna is connected to the receiver input through the antenna to receiver feed subsystem, then the antenna temperature is transferred to the receiver input using the concepts developed in Section 15.2.1. In this case, the system noise figure at the receiver input is defined in terms of the corresponding system temperature so the sensitivity of the overall receiver performance is easily determined from knowledge of the antenna noise temperature and the antenna feed temperature and loss.

Consider the antenna and receiver system shown in Figure 15.6, where the antenna is connected to the receiver input amplifier or LNA using a coaxial cable or waveguide with a loss L_ar and physical temperature T_par °K. Using the antenna noise temperature (T_a) and the receiver noise figure (F_n), the receiver system temperature T_rs and the corresponding system noise figure F_ns at the input to the receiver are related as indicated in Figure 15.6. The relationship between T_rs and F_ns is similar to that expressed by (15.19), that is,

(15.40)

Therefore, upon evaluating the system noise temperature at the receiver input it is a simple matter to compute the system noise figure.

The evaluation of T_rs follows the procedures discussed in the preceding section with the result

(15.41)

where the effective noise temperature of the antenna‐to‐receiver feed is evaluated as

(15.42)

The receiver noise temperature is evaluated using (15.23) or, more directly, in terms of the receiver noise figure F_n using (15.22) as

(15.43)

Using (15.43) and (15.42) to compute T_rs, the system noise figure is evaluated using (15.40) and is plotted in Figure 15.7 as a function of the feed loss L_ar for several antenna temperatures. In this plot a receiver noise figure of 2 dB (1.58:1) is used and the physical temperature of the feed network is assumed to be T_parf = 290°K.

The asymptotic value of the system noise figure as L_ar approaches infinity is evaluated as  = 4.12 dB so the system noise figure is degraded by the physical temperature of the feed; this is of academic interest because the received signal power also experiences this loss and approaches zero as the feed loss increases. A more informative plot is the loss in the receiver noise figure as a function of the antenna noise temperature; this is shown in Figure 15.8 for various losses in the antenna‐to‐receiver feed with F_n = 2 dB and T_par = 290°K.

The receiver noise figure loss is characterized by substituting (15.42) into (15.41) and recognizing that the increase in the receiver temperature relative to T_rec is

(15.44)

The resulting loss in the receiver noise figure is defined as

(15.45)

Equation (15.45), expressed in decibels, is plotted in Figure 15.8 with (15.44) substituted for ΔT_rec.

A receiver noise figure loss of 2.12 dB is incurred for all feed losses when the antenna temperature is equal to the physical temperature of the feed, that is, T_a = T_par. Furthermore, as the feed loss increases, ΔT_rec approaches T_par independent of the antenna temperature so the noise figure loss corresponds to a constant loss given by

(15.46)

This result verifies the system noise figure in Figure 15.7, in that, as . Figures 15.7 and 15.8 are based on the unique condition that .

On occasions it is useful to refer the system noise temperature, as computed by (15.41), to the antenna input using the relationship

(15.47)

This section concludes by characterizing the gain temperature ratio (G/T) of the receive antenna under several conditions. The G/T ratio is a measure of the quality of the receive system much like the EIRP is a quality measure of the transmit system. The antenna G/T gain temperature figure of merit is typically expressed in units of dB/°K.

One application of the antenna system noise temperature referred to the antenna is to use the specification . This definition characterizes the antenna performance under operational conditions including various sky and ground noise sources and is used in the design and evaluation of the antenna subsystem. Another definition is when T_ain = 0. This definition provides a figure of merit for the antenna subsystem with no external noise sources as might be required for a performance acceptance test of the antenna.

15.2.3.1 Example: Evaluation of Antenna Temperature and Receiver G/T and C/N_o Ratios

Consider an Earth station antenna with a gain of G_a = 55 dB and a spot beam with beamwidth θ_B = 1° (0.0174 rad) viewing a satellite at a low elevation angle. The antenna does not use a radome; however, the reflector and feed losses are assumed to be L_arf = 0.5 dB with a physical temperature of T_parf = 290°K. The Sun, at a temperature of T_sun = 500,000°K, is totally within the satellite beamwidth and is viewed against cosmic background noise at a temperature of T_bck = 40°K. The ground temperature in the vicinity of the antenna is T_gnd = 290°K and Gardner’s approximation to the atmospheric temperature is used. The atmospheric loss is 2 dB, that is, L_atm = 1.585, and the lower and upper antenna sidelobe contribution factors to the ground and atmospheric noises are ρ_lsl = ρ_usl = 0.1. Furthermore, the antenna is connected to the receiver LNA through a feed with a physical temperature of T_ar = 290°K and a loss of 1 dB, that is, L_ar = 1.26; this loss results from a coaxial cable with an impedance mismatch. Applying the feed loss, gain, and AGC conditions used in the example in Section 15.2.1.1, with L₂ = 3 dB, LNA gain = 20 dB, and an AGC attenuation of 10 dB, the receiver noise figure is 2.5 dB.

Based on these conditions the antenna temperature (T_a), the antenna system noise temperature (T_as), and the receiver G/T ratio are evaluated. Some necessary constants are Sun’s radius is  km and the Sun’s distance from Earth is  km.

The first task is to determine the contribution of the sky noise temperature T_b at the antenna input. The solid angles of the Sun and antenna are

(15.48)

(15.49)

where is the diameter of the beam at a range of R_s; a spot beam has a circular pattern normal to the pointing direction. Because and L_atm = 1.585, the sky and atmospheric noise temperatures at the antenna input are computed using (15.36). In this computation, the effective atmospheric noise temperature, based on the ground noise, is approximated as

(15.50)

These results are used to compute the background noise temperature at the antenna input as

(15.51)

The ground and atmospheric noise temperature entering through the antenna sidelobes are and where T_atm = 660°K is the computed physical temperature of the atmosphere. From these results and (15.28) the total temperature at the antenna input is T_ain = 88,649°K.

Since there is no radome, only the effective antenna reflector and feed temperatures are involved in referring the antenna input temperature to the antenna output. Given the combined reflector and feed loss (L_arf) and physical temperature (T_parf), the effective antenna reflector and feed temperature is computed as

(15.52)

Using (15.52) with T_ain = 88,469°K and (15.28) with L_rad = 1.0 the antenna temperature is computed as

(15.53)

To determine the antenna system noise temperature the receiver noise temperature is required and, using (15.22) or (15.43) and the specified receiver noise figure, the result is

(15.54)

Using (15.53), (15.54), and (15.41), with a feed loss of 1 dB, temperature T_par = 290°K, L_ar = 1.26 and, from (15.42),  = 75.4°K, the receiver system temperature is evaluated as

(15.55)

The Sun has significantly degraded the system sensitivity as seen from the increase in the system noise figure F_ns = 23.4 dB. The system temperature referenced to the antenna input is computed using (15.47) with the result T_as = 79,400°K and, using this result, the antenna‐receiver system G/T_as ratio is found to be

(15.56)

This example demonstrates the procedures in the evaluation of the antenna G/T ratio as seen at the receiver input; it also demonstrates the horrific impact on the system performance when the antenna is pointed directly at the Sun. Problem 3 examines the impact of the Sun’s temperature when the antenna beam is partially illuminated by the Sun.

The receiver carrier power‐to‐noise density ratio, C/N_o, is an important receiver parameter, in that, the demodulator E_b/N_o performance is determined as E_b/N_o|_dB = C/N_o|_dB − 10log₁₀(R_b) where R_b is the user data rate. From the viewpoint of the link budget, E_b/N_o includes the theoretical value, corresponding to a specified P_be, and includes the demodulator losses. Based on (15.13) the C/N_o ratio for a point‐to‐point link is expressed in decibels, using previously defined parameters, as

(15.57)

where .

15.2.4 Remarks on the System Noise Figure

Often the minimum received power at the input to the LNA is specified along with a corresponding system performance requirement, for example, the bit‐error probability at a given E_b/N_o(req’d) signal‐to‐noise ratio. This defines the required receiver sensitivity and allows for the design and development of the receiver and demodulator to proceed without the antenna subsystem. Defining the minimum received power as

(15.58)

the minimum signal‐to‐noise ratio is evaluated using (15.13) and is expressed as

(15.59)

In this computation F_ns is the system noise figure so a specification of the system antenna temperature T_a must be provided. The temperature of the feed or cable required to connect the antenna to the LNA must also be considered in the computation of F_ns. With these caveats and the bandwidth expressed in terms of the bit rate B_n = R_b, and N_o = kT_oF_ns, (15.59) specifies the minimum (or theoretical) signal‐to‐noise ratio E_b/N_o(min) that is achievable based on the receiver sensitivity. Therefore, using these results, the receiver and demodulator design margin is

(15.60)

If the specified E_b/N_o corresponds to the theoretical value of P_be based on the selected waveform modulation and coding, then this margin must include all receiver and demodulator losses and hopefully some to spare. Examples of various receiver and demodulator losses are given in Sections 15.4–15.10 and the magnitude of the demodulator losses is identified in the simulations discussed in Section 15.14.

15.3.1 Rectangular Aperture Antenna Pattern

The antenna pattern for a linear one‐dimensional aperture of length L_a is characterized in terms of the far‐field electric intensity as

(15.65)

where, in general, W(ℓ) is the complex aperture weighting function and λ is the wavelength of the radiating frequency. The far field occurs when the range r ≫ λ and is formally characterized by the Fraunhofer region where the received electric field is represented by a plane wave; typically, the far field is considered to be r ≥ . Under this condition, the phase error at the 3 dB points of a uniformly illuminated aperture antenna corresponds to λ/16 wavelength. Upon defining the spatial frequency as ξ = sin(ϕ)/λ, (15.65) is recognized as the inverse Fourier transform so the weighting function can be determined by taking the Fourier transform of a specified electric field intensity with respect to the differential dξ = dsin(ϕ)/λ.

Evaluation of the electric field pattern for a uniformly illuminated aperture with constant phase, such that, W(ℓ) = 1/L_a, results in the sin(x)/x electric field intensity pattern given by

(15.66)

The magnitude‐squared |E(ϕ)|² of the electric field intensity results in the pattern of the radiated power that is used to characterize the antenna gain. The solid curve in Figure 15.9 is a plot of (15.66) in terms of |E(ϕ)|² in decibels and is the familiar sinc²(x) response function.⁹ The antenna gain is normalized for unit amplitude at ϕ = 0 so the peak gain is 0 dB. The abscissa is plotted as the normalized angle where L_λ = L_a/λ; however, the antenna beam angle is ϕ, evaluated as

(15.67)

where the approximation applies when L_λ ≫ 1, that is, for large apertures.

The cosine weighted aperture function is expressed as

(15.68)

where the factor results in the same aperture power as in the uniformly weighted case. Using (15.68) in the evaluation of (15.65) and defining L_λ = λ/L_a results in the electric field intensity pattern

(15.69)

Equation (15.69) is normalized by and plotted as the dashed curve in Figure 15.9. The response is identical to the previous applications of the cosine weighting function with a first spatial sidelobe level of −23 dB and null at L_λsin(ϕ) = 1.5.

The final antenna aperture to be considered is the triangular aperture with the weighting function given by

(15.70)

where, in this case, the results in the same aperture energy as the uniformly weighted aperture. Upon evaluation of (15.65) for the triangular aperture results in the electric field intensity given by

(15.71)

This result is plotted in Figure 15.9 as the dotted curve.

For the uniformly illuminated aperture antenna, the first spatial sidelobes are 13 dB below the main lobe; the distance between the first nulls, on either side of the main lobe, is λ/L_a radians; and the 3 dB beamwidth is 0.89λ/L_a radians (51°). These characteristics are summarized in Table 15.2 along with those of the cosine and triangular weighted apertures.

Aperture	Aperture Weight	Angle Between First Nulls (rad)	3 dB Beamwidth θB (rad)	First Sidelobe Level (dB)	Gain Loss (dB)
Uniform	1.0	2.0/Lλ	0.89/Lλ	−13.26	0.000
Cosine		3.0/Lλ	1.19/Lλ	−23.0	0.912
Triangular		4.0/Lλ	1.27/Lλ	−26.52	1.25

15.3.2 Circular Aperture Antenna Pattern

The circular aperture is evaluated by converting the aperture weighting function to polar coordinates with the result that the electric field intensity pattern is expressed as

(15.72)

Normally the aperture weighting is independent of the angle θ, that is, the illumination amplitude is constant around the aperture for 0 ≤ r ≤ D/2, and (15.72) simplifies to

(15.73)

where J_o(x) is the zero‐order Bessel function of the first kind. Evaluation of this result for a uniformly illuminated aperture of diameter D results in the expression [18]

(15.74)

where J₁(x) is the first‐order Bessel function of the first kind and . This result is normalized by πD²/4 to yield unit gain at E(ϕ = 0). The normalized radiated pattern for the uniformly weighted circular antenna is shown as the solid curve in Figure 15.10. This pattern is used in the link evaluation program to determine the ground noise dependence on the antenna elevation angle as discussed in Section 15.2.2. The 3‐dB beamwidth of this antenna is approximately 1.02/D_λ radians (58.4/D_λ degrees) and the first sidelobe is 17.6 dB below the peak gain. The aperture efficiency relative to the uniformly weighted linear aperture is 0.865 (−0.63 dB).

Silver [17] has developed expressions for the circular aperture electric field intensities using an aperture weighting function described by

(15.75)

where p ≥ 0 with the result

(15.76)

This result is normalized by πD²/4 so the value at E(ϕ = 0) is relative to the uniformly weighted aperture. Table 15.3 summarizes the approximate antenna characteristics for the circular aperture weighting functions considered.

Aperture	Aperture Weight	Angle Between First Nulls (rad)	3 dB Beamwidth θB (rad)	First Sidelobe Level (dB)	Gain (dB)
Uniform	1.0	2.42/Dλ	1.02/Dλ	−17.6	0.00
Silver, p = 1		3.28/Dλ	1.27/Dλ	−24.6	−1.25
Silver, p = 2		4.02/Dλ	1.48/Dλ	−30.6	−2.52

15.5.1 Antenna Polarization Loss and Isolation

In this section, the analysis of the antenna power loss and the isolation of an orthogonally polarized interfering signal are examined based on the work of Ippolito [31]. In this analysis, the polarization state of the received wave at the input to the receiver antenna, that is, after propagation over the range distance z_max between the transmitter and receiver, is specified together with the receive antenna polarization state. The interaction of the received wave and the antenna is based on their polarization states* and the ARs as characterized by the polarization mismatch factor 0 ≤ m_p ≤ 1. For arbitrary EP or LP states of the received wave and the receiver antenna, the polarization mismatch factor is given by

(15.95)

where r_x is the AR and x = (w,a) refers to the received wave or antenna, respectively. The sign of the AR is negative for right‐hand polarization and positive for left‐hand polarization. For LP, the sign is always positive. The rotation of the polarization axis is π/2 radians for VP and zero radians for HP and right- and left-hand polarizations. The AR is expressed in decibels as

(15.96)

The power at the output of the receive antenna is evaluated as

(15.97)

where PD is the received power density and A_e is the effective area of the receive antenna normal to the Poynting vector. The antenna polarization power loss, in decibels relative to an ideally matched antenna, is expressed as

(15.98)

The polarization mismatch factor for an elliptically polarized received wave with RHCP copolarization state and a vertically polarized receive antenna is shown in Figure 15.17 for wave ARs r_w = −1, −1.5, and −5 (0, 3.52, 40 dB). This example is not representative of an ideal situation, in that, a vertically polarized received wave^† is required to match the vertically polarized receive antenna. Referring to Figure 15.17, it is seen that a RHCP received wave r_w = −1 and |m_p(w,a)| = 0.5 results in a power loss of 3 dB and, as r_w becomes larger, the minimum loss approaches 0 dB and corresponds to the vertical y‐axis thus matching the vertically polarized receive antenna. An ideal circularly polarized antenna has an AR of r_a = 1 (0 dB) and a well‐designed antenna will have an AR of r_a = 1.19 (1.5 dB) and an ideal LP antenna has an AR of r_a = ∞ (∞ dB) with practical values ranging from r_a = 17.8 to 31.6 (25–30 dB).

The solid lines in Figure 15.18 show the maximum and minimum polarization losses as a function of the wave AR; the losses corresponding to the polarization mismatch factors in Figure 15.17 shown as the circled data points. These losses are orthogonal to each other with the minimum loss corresponding to the VP or the copolarized state and the maximum loss corresponding to the cross‐polarized state.

The antenna isolation is defined as the ratio of the power at the antenna copolarized output a_c and the power at the cross‐polarized output a_x. Given the total received signal power density the isolation is expressed as

(15.99)

Because any wave can be resolved into two orthogonal polarized states, denoted as w and w_o, (15.99) is rewritten as

(15.100)

where the second equality is obtained in a straightforward way with the constant xpd defined as the cross‐polarization discrimination, expressed as

(15.101)

Referring to (15.100) an ideal antenna occurs when all of the antenna co‐ and cross‐polarized outputs are equal to the respective co‐ and cross‐polarized inputs, such that, ; this also leads to the conditions with the result that the isolation for an ideal antenna is

(15.102)

The antenna mismatch factor, cross‐polarization discrimination, and isolation are examined in the following case study for an elliptical received wave with copolarization state LHCP and a receive antenna with a LHCP polarization state.

15.5.2 Case Study: Polarization Characteristics for a LHCP Antenna

This case study examines the antenna mismatch factor, the corresponding polarization loss, the cross‐polarization discrimination, and the isolation for a LHCP antenna with an elliptical received wave with a copolarization state that is matched to the receive antenna. The antenna is not ideal so the isolation asymptotically approaches the cross‐polarization discrimination function xfd for low values of xfd.

The antenna mismatch factor is evaluated using (15.95) for a specified r_a and parametric values of r_w with τ_a = 0 radians and variable angular displacements τ_w = θ*: 0 ≤ θ ≤ 2π radians. The results are plotted in Figures 15.19 and 15.20 corresponding to r_a = 1 and 1.5, respectively. With r_a = 1 the antenna mismatch factor appears as circles with radius ≤1. The case having unit radius corresponds the received wave being matched to the antenna with r_w = r_a = 1; the other cases result in a mismatched antenna and lower values of m_p corresponding to higher polarization losses. The results in Figure 15.20 represent a nonideal antenna with r_a = 1.5 and, although the mismatch factor for r_w = 1 is a circle it does not have unit magnitude and results in a polarization loss. The case with r_w = r_a = 1.5 is matched in the horizontal direction with increasing loss in the vertical direction. As the parameter r_w continues to increase the antenna mismatch factor decreases along both axes resulting in increasing polarization loss. The salient point is that the antenna mismatch factor results in zero loss when the received wave AR is equal to the antenna AR. The losses are shown in Figure 15.21 as a function of r_w, expressed in decibels, for r_a = 1 and 1.5. The circled data points represent the losses for the corresponding r_a and r_w conditions in the range of the abscissa.

Evaluation of the isolation of an ideal LHCP antenna involves evaluating the cross‐polarization discrimination xpd as expressed in (15.101) as a function of the wave axial r_w. The numerator of (15.101) is the copolarized antenna mismatch factor corresponding to wave AR r_a = 1 and the denominator corresponds to the cross‐polarized or orthogonal antenna mismatch factor with r_a = −1. In these evaluations, the antenna mismatch factors are independent of τ_a and τ_w and the resulting ideal antenna isolation is plotted in Figure 15.22 with the xpd(dB) = 10log₁₀(xpd) and the abscissa plotted in terms of the common logarithm as 20log₁₀(r_w). From this result it is seen that the isolation increases as the wave AR approaches 0 dB or r_w = 1. The value of xpd, plotted in Figure 15.22, is used to evaluate the isolation of a nonideal antenna using (15.100). In evaluating (15.100), the choice of the antenna copolarized and the cross‐polarized ARs r_c and r_x are based on the antenna design; the nonideal antenna isolation plotted in Figure 15.23 corresponds to r_c = 1.122 (1 dB) and various values of r_x in decibels. The selected values of the nonideal antenna ARs result in constant values of the antenna mismatch factors: m_p(w, a_c), m_p(w_o, a_x), m_p(w, a_x)_, and m_p(w_o, a_c). These factors influence the asymptotic convergence of the antenna isolation relative to that of the ideal antenna isolation. As r_x approaches 0 dB in Figure 15.23, the isolation is ultimately limited by the selection of r_c. The value of r_c = 1.122 results in a copolarization loss of 0.014 dB and a cross‐polarization loss of 48.8 dB.

15.6.1 Phase‐Noise Characterization

Oscillator phase noise [37, 39, 40] is characterized in terms of a log–log plot of the relative phase‐noise* PSD (S(f)) as a function of the frequency deviation f from the oscillator frequency f_o. The power density N_oϕ is defined relative to the oscillator carrier frequency power C and plotted with the ordinate specified as 10log(N_oϕ/C) dBc/Hz. The PSD is symmetrical about f_o and, normally, only the positive frequency portion is shown. A generic plot of a phase‐noise PSD specification S_s(f) is shown in Figure 15.24 indicating distinct line segments with frequency dependence k_m/f ^m corresponding to amplitude roll‐offs of 10 m in decibels per decade. The parameter k_m is a scaling factor that determines the dBc/Hz level for each segment. The lowest frequency contained in S_s(f) is typically 1–10 Hz because accurate measurements below 1 Hz are difficult to obtain and the phase noise in these ranges is generally removed in the demodulator by the phaselock loop (PLL) filter. The upper frequency of interest in the phase‐noise PSD is determined by the intermediate frequency (IF) filter bandwidth and ultimately by the symbol matched filter bandwidth.

Barnes [33] characterizes a low frequency segment for m = 4; however, this is seldom considered in applications involving a demodulator PLL.^† The phase noise corresponding to m = 3 is referred to as flicker noise and is related to abrupt phase changes associated with the oscillators inherent feedback circuit as it attempts to control the phase of the oscillators carrier frequency. The phase noise associated with m = 2 results from white and flicker noise sources within the oscillator. The phase noise associated with m = 1 is high frequency flicker noise and influences the m = 3 and m = 2 phase‐noise segments. The phase noise associated with m = 0 results from the receiver white noise density N_o and is the underlying noise term associated with the demodulators E_b/N_o ratio.

Typically, a coherent communication demodulator applies the received carrier‐modulated waveform to a PLL for phase and frequency tracking [41] that also tracks and removes the lower frequency phase‐noise terms that are within the loop bandwidth B_L. The PLL closed‐loop frequency response is characterized by the low‐pass function H(f) and the output phase noise results from the equivalent high‐pass response given by*

(15.103)

In the following analysis, the closed‐loop response for a second‐order loop, with damping factor , is used resulting in

(15.104)

Referring to (10.38) the natural resonant frequency of the closed‐loop response is related to the loop bandwidth B_L as

(15.105)

The response |T(f)|² is shown as the dotted curve in Figure 15.24 with a band‐reject frequency of B_L ≪ R_s Hz. Therefore, for low symbol rate modems the phase noise possesses a more severe performance issue. Typical values of the loop bandwidth are B_L = R_s/10 and R_s/100 for binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK) modulations, respectively.

The variance of the phase noise is computed as

(15.106)

where the individual noise variances are computed as

(15.107)

The integral in (15.107) is evaluated for m = 3 and 2 using the integral formula given by Gradshteyn and Ryzhik [42] and using (15.105), with f_n evaluated in terms of the symbol rate and the time‐bandwidth product , the noise variances are evaluated as

(15.108)

(15.109)

The integrals involving m = 1 and 0 with an infinite integration limit result in infinite variances so the integration must be performed over a finite bandwidth B ≥ R_s/2 Hz where the equality corresponds to the Nyquist bandwidth. In this evaluation the integral in (15.107) is defined as^†

(15.110)

with the integration limit B = R_s corresponding to the bandwidth of an integrate‐and‐dump (I&D) symbol detection filter. Therefore, for m = 1 and 0, (15.110) is evaluated using Mathcad^‡ symbolic processing and the results are expressed as

(15.111)

and

(15.112)

Using (15.105), the dependence on the normalized bandwidth B/R_s for is obtained by substituting

(15.113)

into Equations (15.111) and (15.112). The results are plotted in Figure 15.25 as a function of B/R_s for and . The numerical solutions to (15.110) for B/R_s = 0.5, 1, 4, and 8 are listed in Table 15.5.

Using these results, the phase‐noise variances for m = 1 and 0 are expressed as

(15.114)

and

(15.115)

The unknown parameters in these relationships are the scale factors k_m; however, with knowledge of the breakpoints and the additive constant phase‐noise PSD, the piecewise linear PSD shown in Figure 15.24 can be constructed. In the following section, these results are applied to a system specification of the phase noise and the results are used to evaluate the impact on the bit‐error probability of the communication link. The resulting performance loss is then used in establishing an overall link budget which is essentially the purpose of this chapter.

15.6.2 Phase‐Noise Evaluation Using System Specifications

The phase noise is frequently specified as part of an overall system requirements specification as in Figure 15.26.* The specification discussed in this section is intended to represent the demodulator heterodyning to baseband and is 10 dB lower than that specified for the receive terminal which has multiple heterodyning stages. Because the received and demodulator subsystems are typically developed by different contractors, separate subsystem phase‐noise specifications are required to provide for independent subsystem testing. The demodulator phase noise is intended not to impact the more sensitive phase noise of the receiver oscillators and frequency conversion stages.

These specifications apply to the 1/f³ and 1/f frequency dependencies so the composite linear spectral density is expressed as

(15.116)

The constants k_m, required for the evaluation of (15.116), are determined from the specifications as follows. Considering the specification shown in Figure 15.26 and repeated in Figure 15.27 as the solid lines; each linear segment

(15.117)

is plotted in terms of the common logarithm

(15.118)

where and are used as the abscissa in the spectral density plots. Therefore, using these results with m = 3, k₃ is evaluated by solving for K₃ in (15.118) under the condition F = 1. The corresponding values of L₃ and K₃ are −32 dBc/Hz and −2 dBc‐Hz²; in general K_m has units of dBc‐Hz^m−1. In a similar manner K₁ is found to be −42 dBc and K₀ = −112 dBc/Hz corresponding to the constant white noise spectral density. The corresponding values of k_m are as follows: , , and . The composite phase‐noise PSD, shown in Figure 15.27 as the dashed curve, is a plot of (15.116) using the computed values of k_m. The dotted curves represent the computed line segments based on K_m plotted over the range of abscissa values 1 ≤ F ≤ 8 with portions of these line segments corresponding to those of the specification.

Referring to (15.116) and evaluating the phase‐noise variances in (15.108), (15.114), and (15.115) with the values of k_m computed earlier for the receiver specification, the total untracked phase‐noise variance is evaluated using (15.106) with . The resulting untracked phase noise is plotted in Figure 15.28 with and 0.01 for the indicated normalized bandwidth ratios B/R_s. Figure 15.29 shows the untracked phase‐noise variance when B = R_s corresponding to the bandwidth of the I&D symbol detection filter.

15.6.3 Case Study: BPSK and QPSK Performance with Phase Noise

The performance of BPSK and QPSK waveform modulations is examined using the phase‐noise characteristics developed in the preceding section for the receiver subsystem phase‐noise PSD. The modulated waveforms are generated using the rect(t/T) weighting functions and detected using an I&D matched filter. The upper integration limit in the evaluation of the phase‐noise variance is B = R_s, where R_s corresponds to the noise bandwidth of the demodulator matched filter. The lower integration limit is determined by the high‐pass PLL transfer function T(f) expressed in (15.104). The low‐pass band‐reject bandwidth of T(f) is dependent on the loop time‐bandwidth product where B_L is the closed‐loop noise bandwidth. For BPSK and QPSK modulation, typical values* of ρ_L are 0.1 and 0.01, respectively; these values are used in the following performance evaluations.

In addition to the phase‐noise variance resulting from the receiver heterodyning operations, the PLL introduces phase jitter that is characterized by the phase variance expressed as [44]

(15.119)

where γ_L is the signal‐to‐noise ratio in the PLL bandwidth and γ_b = E_b/N_o is the signal‐to‐noise ratio in the matched filter bandwidth of R_s hertz. From this discussion two factors are in play that influence the performance of the BPSK and QPSK modulations. The first is that, for a given symbol rate, the lower integration limit in determining the phase‐noise variance is decreased by ρ_L. For a given bit rate, this is exacerbated with QPSK modulation since R_s = R_b/2. These factors become less significant as the data rate is increased and the phase noise is eventually influenced solely by the oscillator white noise density corresponding to k₀. The second factor influencing the performance is that the PLL phase noise is decreased by ρ_L with the advantage going to QPSK. In consideration of these issues, the total phase‐noise power measured in the matched filter bandwidth is given by

(15.120)

Because the oscillator phase is influenced by several phase‐noise sources, as indicated by terms giving rise to the k_m/f^m spectral density response, the phase‐noise random variable φ is generally characterized by the Gaussian distribution

(15.121)

This approximation and the limits require that . The bit‐error probability performance conditioned on the phase error φ for BPSK and QPSK is expressed, respectively, as

(15.122)

and

(15.123)

Using (15.121) with either (15.122) or (15.123) the resulting bit‐error probability is expressed as

(15.124)

The bit‐error probability expressed in (15.124) is evaluated using a 96‐term Gauss‐quadrature integration and the results are depicted in Figure 15.30. The robustness of BPSK modulation is evidenced by a loss of less than 0.1 dB even for data rates as low as 150 bps. For the 1024 kbps data rate, the QPSK modulation has a maximum loss of about 0.1 dB; however, the sensitivity to the phase noise is evident at 150 bps with a maximum loss of about 0.3 dB for the range of signal‐to‐noise ratios considered.

Figure 15.31 depicts the performance sensitivity when the phase noise results from the cascade of three subsystems: a transmitter, satellite repeater, and receiver, each with a phase‐noise specification corresponding to receiver subsystem. In effect, the phase noise is increased by 4.77 dB from that used in Figure 15.30. In this case, the BPSK performance for the 1024 kbps data rate is essentially unchanged; however, the QPSK performance is degraded by about 0.2 dB at P_be = 10⁻⁵ for 1024 kbps (512 ksps) and by about 0.5 dB for 150 bps (75 sps).

This procedure can be applied to higher order MPSK modulations (M > 4) with anticipated increases in the performance loss. In this regard, Baker [45] has analyzed the performance of 256‐ary and 1024‐ary QAM with phase noise in terms of the standard deviation σ_φ of the phase and, for , the losses at are about 0.6 and 5.5 dB, respectively.

15.10.1 Antenna Shaping Loss

Antenna shaping loss is based on aperture weighting function and is included in antenna gain.

15.10.2 Antenna Scallop Loss

Antenna scallop loss is associated with antenna beam loss during spatial antenna acquisition. Site loss is the antenna gain loss at a site location removed from the center of the beam. For Earth coverage fixed satellite beams, the site loss is compensated through beam shaping to ensure equal returns from all locations.

15.10.3 Frequency Scallop Loss

Frequency scallop losses are associated with frequency acquisition using discrete frequency scanning or a Fourier transform.

15.10.4 Signal Processing Loss

Signal processing losses are associated with discrete amplitude and time sampled quantization losses. Discrete amplitude sampling losses are minimized by using the maximum number of bits to describe the signal amplitude at various points along the processing path and the discrete‐time sampling losses are minimized through filter design and sampling frequency selection. Other sources of signal processing loss are associated with the limitation of various algorithms that approximate theoretical models; for example, the computation of decision thresholds based on a limited sample size.

15.11.1 Characterizing the TWTA Transfer Function

The characteristics of three TWTAs are examined with varying degrees of nonlinear signal distortion as identified by Saleh [48, 49]. The characteristics of interests are the AM‐PM transfer function in which the AM carrier input signal results in a nonlinear PM output signal; and the AM‐AM transfer function in which the AM carrier input signal results in a nonlinear AM output signal. These nonlinear transfer functions result in varying degrees of harmonic distortion at the output of the TWTA that cause co‐channel and adjacent channel interference (ACI). For clarity the three TWTAs, characterized by Saleh, are denoted as: TWTA No. 0 that is the least severe with zero AM‐PM phase distortion; TWTA No. 1 that exhibits a moderate AM‐PM phase distortion; and TWTA No. 2 that has severe AM‐PM phase distortion. All three result in nearly identical nonlinear AM‐AM distortion.

The comparison of the three traveling wave tube amplifiers is based on Saleh’s two‐parameter in‐phase and quadrature (I/Q) (or real and imaginary) curve‐fit functions expressed as

(15.135)

where α_p, β_p, α_q, and β_q are the curve fitting coefficients given in Table 15.7 for the three representative traveling wave tube (TWT) characteristics. The parameter r is the composite input signal level and determines the operating point on the TWT transfer function. In terms of the amplitude and phase responses, the corresponding I/Q components are expressed as and .

Plots of the curve fit results for the quadrature functions expressed in (15.135) are converted to the amplitude and phase functions and shown in Figure 15.35 using the parameter values listed in Table 15.7.

15.11.2 Evaluation of C/I and OBO

The evaluation of C/I and OBO discussed in this section is applied to a FDMA satellite communication system and follows the work of Saleh [49] using the I/Q responses for TWTs 0, 1, and 2 discussed earlier. The input signal is considered to be the sum of n independent phase‐modulated signals expressed as

(15.136)

where ω_i represents carriers in different FDMA bandwidths (B) occupying a total bandwidth of W = nB Hz. The output of the TWTA when operating in a FDMA network [50–52] is composed of distinct intermodulation tones with angular frequency ; the restriction ensures that all of the intermodulation products (IMPs) fall in the first spectral zone of each frequency band. The order of the IMP is given by , that is, only odd order IMPs are present. The value m = 1 represents the desired carrier term at the output of the nonlinear amplifier and all of the other terms, m = 3, 5, … are representative of output distortion tones in the users bandwidth. In the following analysis, the input carriers are assumed to have equal amplitudes with V_i = V :∀i and n ≫ m.

Based on this brief introduction, the C/I ratio at the output of the TWTA is given by

(15.137)

where v = 0 corresponds to the C/I ratio at the center of the users band and is the ratio at either edge of the users band. The parameter N_m,n(v) is the number of dominant m‐th order IMPs falling at location v and is given by [53]

(15.138)*

Where for and for . The terms P_m,n and Q_m,n represent the I/Q components of the m‐th order IMP, for which, Saleh developed the expressions

(15.139)

(15.140)

where

(15.141)

and

(15.142)^†

(15.143)^†

(15.144)^†

(15.145)^†

with for and . The parameter U is equal to the square root of the total average input power and the function E₁(z) is the exponential integral [54]

(15.146)

Saleh [49] provides an appendix that outlines a method for the numerical computation of (15.139) and (15.140) that avoids computational roundoff errors for large values of m. The solution involves rewriting the respective summations in the brackets { } of (15.139) and (15.140) as follows:

(15.147)

and

(15.148)

with the respective limit on m ≌ 82/X + 4.4 and 96/Y − 2.6. The F(z) is expressed as

(15.149)

and can be evaluated numerically using Laguerre–Gauss quadrature integration.

The OBO is defined as

(15.150)

where P(r) and Q(r) are the nonlinear output values corresponding to the operating point r of the input composite signal. These values are computed as given in Section 15.11.1 using the appropriate α_p, β_p, α_q, and β_q values from Table 15.7.

These relationships are evaluated for the TWTs 0, 1, and 2 and the results are shown in Figure 15.36 which relate the OBO dependence on C/I for the indicated IBO operating points. As the IBO increases the TWTs operate more in the linear region and away from saturation resulting in a higher C/I ratio. Figure 15.37 characterizes the OBO level as a function of the IBO for the three TWTs under consideration.

15.12.1 SSPA with Soft Saturation Response

In this example, the SSPA transfer function is characterized as having AM‐AM that is characterized in terms of the peak output voltage (V_po) as a function of the peak input voltage (V_pi). Because the AM‐PM response is zero the SSPA is considered to be ideal. The dashed curve in Figure 15.38 shows the measured voltage transfer function and the solid curve is based on a seventh‐order polynomial that is curve fit to the V_po vs. V_pi response. The modified response results in a constant gain over the linear range of the SSPA and a monotonically decreasing gain as the input signal level increases through the nonlinear range. The actual gain of the device is not important insofar as the performance simulation is concerned so the gain of the soft saturation response is purposely low to avoid dealing with large signal levels in the simulation program. The curve‐fit transfer function is expressed as

(15.151)

with the corresponding gain evaluated as 20log₁₀(V_po/V_pi). Using (15.151), the third‐order IMPs are characterized by the cubic term as

(15.152)

With the input power expressed as , Figure 15.39 shows the input–output power characteristics including the third‐order intermodulation power. The term is the output power corresponding to the 1‐dB gain compression identified in Figure 15.38, is the primary output power corresponding to the carrier power of the input signal under test, IP₃ is the third‐order intercept output power, and IM₃ is the ratio of the test signal carrier power to the third‐order intercept power resulting from the nonlinearity. Using the linear approximations to these power transfer functions, the third‐order intermodulation power ratio is defined as ratio P_o/P_o3 and is expressed in decibels as

(15.153)

Equation (15.153) is plotted in Figure 15.40 as the dotted theoretical curve for various OBO levels based on the SSPA characteristics described in Figures 15.38 and 15.39. The simulated intermodulation distortion discussed in Sections 15.13 and 15.14 is evaluated using a slightly different SSPA model and the theoretical and simulated intermodulation distortion are also shown in Figure 15.40 as the solid curve and the circled data points, respectively. Therefore, Figure 15.39 and Equation (15.153) serve as a general description for the evaluation of intermodulation distortion.

15.12.2 TWTA with Gain Compensated Response

The TWTA often use gain and phases compensation circuits to extend the linear range of the device. The resulting gain response rises to a slight peak before decreasing monotonically. The goal is to extend the 1 dB gain compression point and thereby extend the linear range. The gain compensated curve‐fit response for a TWTA is expressed as

(15.154)

and the phase transfer function is approximated as

(15.155)

It is left as an exercise (see Problem 13) to plot these responses and determine the peak input and output voltages that correspond to the 1 dB gain compression point.

15.13.1 Single‐Carrier Simulations

With a single carrier per channel (SCPC) simulation the limiter may occur in the transmitter HPA of a single‐channel‐modulated waveform or with multiple data channels that are time division multiplexed (TDM) onto a single carrier. The SCPC waveform also applies to a satellite HPA; however, the satellite often combines many carriers to form a FDM downlink.

Consider the SCPC‐modulated waveform is expressed as

(15.157)

The HPA or limiter operating points are characterized by (, ) and the mapping from (and to) the simulation signal levels is determined as shown in Figure 15.42. The limiter operating points are determined using the maximum peak output voltage of the limiter V_po(max) and a specified OBO level such that

(15.158)

The corresponding input is determined using the inverse of the polynomial expression of the limiter transfer function as indicated in Figure 15.41.

With the modulation function m(t) normalized to result in a unit power level, the power of the simulated waveform is simply the power in the carrier signal with peak voltage V_si, that is,

(15.159)

The mapping to the limiter input is determined by the voltage gain, expressed as,

(15.160)

Similarly, the output of the limiter is mapped back to the simulation signal level using the voltage gain

(15.161)

It is convenient in the simulation to let the input and output signal levels be equal so that P_so = P_si. That is, when the limiter is not used, the HPA appears as an ideal unit gain amplifier. Neglecting the various propagation and channel distortion losses, the received signal‐to‐noise ratio (γ_b) is established at the limiter output and the received noise power is computed as

(15.162)

15.13.2 Multiple Carrier Simulations

Multiple carrier simulations will occur in the transmitter HPA when several modulated carriers are combined at the IF to form a FDM waveform; this is also typical of multichannel satellite repeater downlinks. In these cases, the simulation is generally focusing on the performance of a particular carrier or signal and the limiter operating point must be established for the signal of interest.

Consider the composite signal described as

(15.163)

where

(15.164)

Considering s(t;n_o) to be the desired signal with the modulation term m(t;n_o) normalized to result in unit power, the desired signal power is then the carrier power

(15.165)

The limiter operating point for the composite signal is established as described earlier with the input power given by

(15.166)

However, the operating point for the desired signal is so the input signal mapping to the limiter is determined by the voltage gain

(15.167)

where the voltage gain is

(15.168)

Equation (15.168) maps the limiter composite input signal to the desired signal as indicated in Figure 15.41. The limiter output is mapped back to the simulation signal levels using the voltage gain

(15.169)

where the desired output signal voltage (n_o) is determined directly from the limiter transfer function using the input voltage (n_o). In this case, the signal‐to‐noise ratio (γ_b) for the desired signal is established at the limiter output using the noise power

(15.170)

where

(15.171)

The previous steps, required to setup and initialize the limiter processing, are summarized as follows:

• Given V_po(max) determine the composite signal operating point at the limiter output using the specified OBO
• Determine the output voltage gain G_vo that maps to the simulation voltage V_so
• For single‐carrier operation, compute the noise power for the specified signal‐to‐noise ratio γ_b
• Determine the corresponding composite signal operating point at the limiter input
• Determine the input voltage gain G_vi that maps the simulation voltage V_si to the limiter operating point

For multiple carriers, the following additional steps are required:

• Determine the voltage gain G_vn between the signal of interest V_si(n_o) and the composite signal V_si
• Using G_vn and determine the limiter input operating point for the desired signal
• Determine the limiter output voltage corresponding to the input

The final step involves computing the noise power using either (15.162) or (15.170) for a desired signal‐to‐noise ratio γ_b.

15.14.1 Simulation of Third‐ and Fifth‐Order Intermodulation Distortions Terms

The two‐tone intermodulation noise terms, resulting from the SSPA nonlinearity as characterized by (15.151), are evaluated as a function of OBO using the SSPA computer modeling described in Section 15.13. The third‐ and fifth‐order intermodulation noise terms are evaluated by applying two CW signals to the SSPA and examining the spectrum of the SSPA output. The simulation uses a received rms carrier signal level of 1 V that is mapped through the nonlinearity and back to the simulation level of 1 V rms.

The simulation uses a rectangular windowed SRRC frequency response with the QPSK‐modulated waveform operating at a baseband data rate of 4.8 K symbols per second with the adjacent channel at 19.2 Ksps. The adjacent channel is located at a positive carrier frequency in accordance with (15.172) using α = 0.4. The received CW tones for the intermodulation evaluation are obtained by using mark‐hold data on the I/Q rails. The intermodulation tones are clearly discernable in the inherently harmonic‐filled spectrum.

The spectrums are generated using an equivalent of N_s = 128 samples per symbol and averaging 20 spectrums with a cosine windowed fast Fourier transform (FFT) size of N_fft = 16,384 samples; the corresponding frequency resolution is 0.0375 Hz. The results depicted in Figure 15.44 are obtained using a linear PA and show the spectral content using the mark‐hold SRRC‐QPSK waveform to generate the two tones. Figure 15.45 shows the resulting two‐tone intermodulation performance for various OBO conditions; these results are plotted as the circled data points in Figure 15.40.

15.14.2 Spectrum Degradation with SSPA OBO

The spectrum of the SRRC‐QPSK‐modulated waveform is shown in Figure 15.46 for various OBO levels; the linear channel spectrum corresponds to an essentially infinite OBO condition. The spectrum results apply to an arbitrary symbol rate as suggested by the normalized abscissa. The OBO is defined relative to the 1 dB gain compression point of the SSPA and the 0 dB backoff case results in the peak carrier voltage being compressed by 1 dB with the average carrier voltage being 3 dB below the peak level. Therefore, a −3 dB OBO results in the average carrier voltage being compressed by 1 dB; this causes considerable compressing or clipping of the peak voltage. The increase in the modulation one‐sided spectral bandwidth beyond the specified limit of 1.4R_s/2 (α = 0.4) is influenced by the SRRC modulator weighted window duration that spans 12 symbols as discussed in Section 4.4.4.1. The impact of the OBO level on the performance with the adjacent channels is examined in the following section; however, it appears as though the OBO should be greater than 6 dB to preserve the excellent Nyquist bandwidth properties of the SRRC‐modulated waveform.

15.14.3 Bit‐Error Performance with OBO and Adjacent Channels

In the final section of this case study, the bit‐error performance of a SRRC‐BPSK‐modulated waveform is examined with and without adjacent channels. In this case the desired channel is centered at baseband with a symbol rate of R_s = 32 ksps and the adjacent channels are centered above and below the desired channel in accordance with (15.172) using α = 0.4. Two adjacent channel symbol rate conditions are examined, one with symmetrical adjacent channels with and the other with asymmetrical adjacent channels with . The Monte Carlo simulations are based on using 100K symbols for each signal‐to‐noise ratio ≤6 dB, otherwise, 1 M symbols for each signal‐to‐noise ratio is used. On occasions, when the Monte Carlo results appear inconsistent with the expected performance curve, 10 M symbols are used to establish the data point.

The performance in Figure 15.47 corresponds to the desired channel bit‐error performance without the adjacent channels. The dotted performance curve represents ideal antipodal signaling and is shown as a reference point when using a linear PA; the performance with 12 dB OBO is nearly the same as the ideal curve. The performance with OBO = 0 dB is only degraded by 0.2 dB at P_be = 10⁻⁵. The severe limiting with OBO = −3 dB is very much like that of an ideal hard limiter, in that, the SRRC AM is essentially removed, resulting in a performance degradation of about 0.9 dB.

In the performance evaluation with adjacent channels, each channel is generated using a separate transmitter with identical SSPAs and the channels are combined to form a FDMA channel; the SSPAs operate with identical OBO conditions. With this understanding, Figure 15.48 shows the performance with symmetrical adjacent channels with the 12 dB OBO condition having a negligible effect on the desired channel performance. The degradations with 0 and −3 dB OBO are about 0.5 and 1.8 dB, respectively.

The performance with asymmetrical adjacent channels with is evaluated under the conditions of constant E_b/N_o and C/N_o. If the higher data rate channels are modulated on a carrier with the same power as the desired channel, then the channel spectrum will be as shown in Figure 15.43 with the adjacent channel spectral levels lower than the desired channel spectral level. This corresponds to the constant C/N_o case that results in 6 dB less E_b/N_o than is available for the desired channel. This results in degradation of the bit‐error performance with low levels of OBO, although, the impact on the desired channel performance is less severe. On the other hand, if the adjacent channel carriers are increased by , then the spectrums in Figure 15.43 all have the same level and correspond to the constant E_b/N_o case. The increase in the carrier power results in a greater loss in the desired channel performance with low levels of OBO. The performance results for these two cases are shown in Figure 15.49 as the dashed and solid curves corresponding to the constant E_b/N_o and C/N_o cases, respectively. The performance difference in the degradations is about 2 and 1 dB for OBO = −3 and 0 dB, respectively, and is negligible for an OBO of 12 dB. Furthermore, the absolute performance loss decreases with increasing OBO and is negligible for OBO = 12 dB. The best performance for all channels is obtained when operating in the linear range of the SSPA under the constant E_b/N_o condition. The performance degradation can also be reduced, for a given OBO, by increasing the channel separation by Δf_max as indicated in (15.173). Therefore, there is a trade‐off between power and bandwidth.