Calculation Library

A collection of link budget and other RF calculations.

References:

• [1] L. W. Couch, “Digital & Analog Communication Systems,” 8th Ed., 2013.

• [2] M. Lindgren, “A 1296 MHz Earth–Moon–Earth Communication System,” 2015.

• [3] T. Pratt and J. E. Allnutt, “Satellite Communications,” 3rd Ed., 2019.

• [4] D. M. Pozar, “Microwave Engineering,” 4th Ed., 2012.

linkbudget.calc.antenna_noise_temp(attn_db, T_medium=270, coupling_eff=1.0)

Compute the antenna noise temperature

Follow the theory in Section 4.5.2 of [3].

Parameters:

• attn_db – Total path attenuation (in dB) experienced through the atmosphere, including clear air and rain attenuation.

• T_medium – Medium temperature (in K) assumed for the rain.

• coupling_eff – Sky noise coupling coefficient determining the fraction of incident sky noise energy output by the antenna.

Note

In [3], a medium temperature of 270 K is considered when computing the sky noise temperature for rain. In contrast, a temperature of 290 K is considered when analyzing clear sky. The rationale is to be confirmed.

linkbudget.calc.capacity(snr_db, bw)

Compute the channel capacity in bps

Parameters:

• snr_db – signal-to-noise ratio in dB.

• bw – nominal bandwidth.

Returns:

Capacity in bits per second (bps).

linkbudget.calc.carrier_eirp(sat_eirp, obo, peb=None, tp_bw=None)

Compute the carrier EIRP using transponder/amplifier information

Converts the saturated EIRP obtained by an amplifier or transponder into the corresponding carrier EIRP, after output backoff and power allocation.

There are two main use cases for this function:

1. To compute the carrier EIRP from the transponder’s saturated EIRP and carrier output backoff (OBO).

2. To compute the carrier EIRP based on the transponder’s saturated EIRP, the transponder’s OBO, the carrier’s power-equivalent bandwidth (PEB), and the transponder’s bandwidth.

In scenario 1, we start with the transponder’s EIRP in dBW obtained when its amplifier operates in saturation. This metric is often given as the transponder’s downlink EIRP at beam peak (BP) or beam center (BC). Then, assuming the transponder is shared by multiple FDMA carriers, we subtract the carrier OBO to obtain the EIRP assigned to the carrier of interest. In this case, note the carrier OBO is often a much larger value than the transponder OBO. The carrier OBO is easily in excess of 10 dB, depending on how much of the transponder it occupies. In contrast, as discussed in [3], the transponder OBO is typically within the range from 1 to 7 dB in FDMA systems.

The OBO interpretation changes in scenario 2 when considering FDMA systems. In this case, the OBO is interpreted as the transponder OBO, not the carrier OBO. Hence, the first step in the computation is to obtain the transponder’s operating EIRP, which is given by the saturated EIRP minus the transponder OBO. The second step is to allocate a fraction of the available transponder EIRP to the carrier of interest. This fraction is determined by the ratio between the PEB and the transponder bandwidth.

To compute scenario 1, call this function with parameter obo as the carrier OBO, while leaving the PEB undefined. To compute scenario 2, call this function with obo as the transponder OBO, while informing the PEB and transponder bandwidth. In any case, however, the sat_eirp parameter represents the transponder’s saturated EIRP in the direction of interest.

Note that, for convenience, the saturated EIRP is in the direction of interest, not necessarily at BP or BC. For example, it could be at beam edge (BE) or any arbitrary coverage contour. Furthermore, not this function takes the saturated EIRP (i.e., including the on-axis antenna gain), instead of the amplifier’s saturated output power. This is intentional because it is often easier to find the EIRP information for a given satellite contour than to find the transponder and antenna gain separately.

Lastly, note that both use cases are equivalent in TDMA systems. Since there is a single carrier in TDMA, the transponder OBO is the same as the carrier OBO. Furthermore, the PEB is the full transponder bandwidth in TDMA, so the peb parameter does not need to be informed.

Parameters:

• sat_eirp – Saturated amplifier/transponder EIRP in dBW in the direction if interest.

• obo – Output backoff in dB.

• peb – Power equivalent bandwidth if the carrier shares the amplifier/transponder with other FDMA carriers.

• tp_bw – Transponder bandwidth used when the PEB is given.

Returns:

Carrier EIRP in dBW.

linkbudget.calc.carrier_to_asi_ratio(rx_dish, long_separation, asi_eirp_ratio)

Compute the carrier to adjacent satellite interference (ASI) ratio

Parameters:

• rx_dish (Antenna) – Rx antenna object.

• long_separation (float) – Longitudinal separation in degrees between the wanted satellite and the adjacent interferer(s).

• asi_eirp_ratio (float) – Ratio between the aggregate adjacent downlink EIRP and the wanted signal’s EIRP.

Returns:

Carrier-to-ASI ratio in dB.

Return type:

(float)

linkbudget.calc.cascaded_input_noise_temp(temps, gains)

Equivalent input noise temperature of cascaded linear devices

Computes the overall equivalent input noise temperature for cascaded linear devices according to Equation 8-37 from [1].

Parameters:

• temps – List with the individual input noise temperatures (in K) corresponding to each of the devices in order from input to output.

• gains – List with the gains (in dB) of the cascaded linear devices, excluding the last, in the same order as given for the input noise temperatures.

Note

The list of gains should not include the gain of the last device in the chain because this gain is irrelevant for the computation.

Returns:

The overall input noise temperature in K.

linkbudget.calc.cascaded_noise_figure(nfs, gains)

Compute the overall noise figure of cascaded linear devices

Based on Equation 8-34 from [1].

Parameters:

• nfs – List with the noise figures (in dB) corresponding to each of the devices in order from input to output.

• gains – List with the gains (also in dB) of the cascaded linear devices, excluding the last, in the same order as given for the noise figures.

Note

The list of gains should not include the gain of the last device in the chain because this gain is irrelevant for the computation.

Returns:

The overall noise figure in dB.

linkbudget.calc.cnir(cnr, ci)

Compute the carrier to noise plus interference ratio

Based on the reciprocal CNR formula in Eq. (4.42) from [3].

Parameters:

• cnr (float) – Carrier-to-noise ratio in dB.

• ci (float) – Carrier-to-interference ratio in dB.

Returns:

Carrier to noise plus interference ratio C/(N+I) in dB.

Return type:

(float)

linkbudget.calc.cnr(P_rx_dbw, N_dbw)

Compute the carrier-to-noise ratio (CNR) in dB

Parameters:

• P_rx_dbw – Received (carrier) power in dBW

• N_dbw – Receiver noise power in dBW

Returns:

CNR in dB.

linkbudget.calc.coax_loss_nf(length_ft, Tl=290)

Compute the loss and noise figure of a coaxial RG6 transmission line

Parameters:

• length_ft – Line length in feet.

• Tl – temperature of the line in Kelvin.

Returns:

Tuple with line loss (dB) and noise figure (dB).

linkbudget.calc.eirp(tx_power, tx_dish_gain)

Compute the effective isotropically radiated power (EIRP)

EIRP (dB) = Tx Power (dB) + Tx Antenna Gain (dB).

Parameters:

• tx_power – Transmit power feeding the antenna (dBW).

• tx_dish_gain – Transmit antenna gain (dB).

Returns:

The EIRP in dBW.

linkbudget.calc.g_over_t(rx_ant_gain_db, T_sys_db)

Compute the G/T ratio in dB/K

Compute the ratio between the Rx antenna gain and the receiver system noise temperature, known as G/T. This ratio determines the quality of the satellite receiving system. The CNR is proportional to it, and the G/T ratio represents the part of the CNR that can be improved based on the receiver hardware alone. The other terms of the CNR are EIRP, slant range, noise bandwidth, and downlink frequency, which are usually fixed for a specific location and service.

Parameters:

• rx_ant_gain_db – Receiver antenna gain in dB.

• T_sys_db – Receiver system noise temperature in dBK.

Returns:

G/T in decibels with units of dBK^−1 (or dB/K).

linkbudget.calc.noise_fig_to_noise_temp(nf)

Convert noise figure to the effective input-noise temperature

Note the noise figure is always referenced to a noise source at the standard noise temperature of T0 = 290 K. In contrast, the noise temperature is independent of the temperature of the noise source.

Parameters:

nf – Noise figure in dB.

Returns:

Noise temperature in K.

linkbudget.calc.noise_power(T_sys_db, bw)

Compute the receiver noise power in dBW

According to Equation 8-40 in [1], the noise power is given by N = k*Tsyst*bw, where k is the Boltzmann constant, Tsyst is the receiver system noise temperature (in absolute units) and bw is the IF equivalent bandwidth in Hz.

Parameters:

• T_sys_db – Receiver system noise temperature in dBK.

• bw – Nominal signal bandwidth (also known as noise bandwidth).

Returns:

Receiver noise power in dBW.

linkbudget.calc.noise_temp_to_noise_fig(Te)

Convert an effective input-noise temperature to a noise figure in dB

Parameters:

Te – Noise temperature in K.

Returns:

Noise figure in dB.

linkbudget.calc.passive_attn_in_noise_temp(attn_db, T)

Input noise temperature of a passive two-port attenuator

Compute the equivalent input noise temperature of a passive two-port lossy component, such as a transmission line or waveguide.

When considering the equivalent input noise temperature, the model is as follows:

Signal --> Sum --> Noiseless Passive Component --> Signal + Noise Out
            ^
            |
            |
      Noise Source (Resistor at Temperature Te)

The noise source is at the input of the passive two-port element, and the latter is assumed noiseless but lossy with gain G or attenuation L. The noise produced by the hypothetical noise source, when amplified by G (or attenuated by L), yields the same noise power on the component’s output as the noisy (but otherwise equivalent) component produces by itself.

The output noise power produced by the noisy passive two-port component is modeled as equal to \(G k T_e B\), namely the input noise power \(k T_e B\) amplified by gain G. Finally, \(T_e\) is the physical temperature of a hypothetical resistor used to represent the noise source. The resistor produces noise power \(k T_e B\) over bandwidth B, and \(T_e\) is called the equivalent input noise temperature, given by:

\[T_e = (L - 1) T,\]

where \(T\) is the physical temperature of the two-port component, and L is the component’s attenuation (or loss) in absolute units. See Equation (8-31a) in [1] or Equation (10.15) in [4].

Parameters:

• attn_db (float) – Attenuation in dB.

• T (float) – Physical temperature of the attenuator.

Returns:

The equivalent input noise temperature of the passive two-port attenuator.

Return type:

float

linkbudget.calc.passive_attn_noise_fig(attn_db, T)

Noise figure of a passive two-port attenuator

Compute the noise figure of a passive two-port lossy component, such as a transmission line or waveguide.

The equivalent input noise temperature of such a component is equal to:

\[T_e = (L - 1) T,\]

where \(T\) is the physical temperature of the component, and L represents its attenuation (or loss) in absolute units.

Hence, the noise factor referenced to the standard temperature T0=290 K is equal to:

\[F = 1 + \frac{T_e}{T0} = 1 + \frac{T}{T0}(L - 1).\]

See Equation 8.32a in [1] or Equation 10.16 in [4].

Note

The noise factor approaches unit if the component’s physical temperature \(T\) is close to 290 K. This is typically the case in environments inhabitable by humans. In this scenario, the noise figure (in dB) is approximately equal to the device’s loss in dB. For instance, a waveguide or transmission line with 2 dB loss would have a noise figure close to 2 dB.

Parameters:

• attn_db (float) – Attenuation in dB.

• T (float) – Physical temperature of the attenuator.

Returns:

The noise figure of the passive two-port attenuator.

Return type:

float

linkbudget.calc.passive_attn_out_noise_temp(attn_db, T)

Output noise temperature of a passive two-port attenuator

Compute the equivalent output noise temperature of a passive two-port lossy component, such as a transmission line or waveguide.

When considering the equivalent output noise temperature, the model is as follows:

Signal --> Noiseless Passive Component --> Sum --> Signal + Noise Out
                                            ^
                                            |
                                            |
                      Noise Source (Resistor at Temperature Teo)

The noise source is at the output of the passive two-port element, and the latter is assumed noiseless but lossy with gain G or attenuation L. The hypothetical noise source produces the same noise power as the noisy (but otherwise equivalent) two-port component produces by itself.

The output noise power produced by the noisy passive two-port component is modeled as equal to \(k T_{eo} B\), and \(T_{eo}\) is called the equivalent output noise temperature. Finally, \(T_{eo}\) is interpreted as the physical temperature of a hypothetical resistor representing the noise source, and is given by:

\[T_{eo} = (1 - G) T,\]

where \(T\) is the physical temperature of the two-port component, and G is the component’s gain in absolute units (less than unit because the component is lossy). See Equation (4.21) in [3].

Note

The equivalent output noise temperature \(T_{eo}\) is equal to the equivalent input noise temperature \(T_e\) multiplied by the gain G of the lossy two-port component or network.

Parameters:

• attn_db (float) – Attenuation in dB.

• T (float) – Physical temperature of the attenuator.

Returns:

The equivalent output noise temperature of the passive two-port attenuator.

Return type:

float

linkbudget.calc.path_loss(d, freq, radar=False, obj_gain=None, bistatic=False, d_rx=None)

Calculate the free-space path loss (or transmission loss)

This function supports radar mode, in which case it computes the transmission loss considering the path loss in the forward and reverse paths (to and from the radar object) as well as the scattering at the radar object based on its radar cross section.

Parameters:

• d – Distance in meters between transmitter and receiver (e.g., satellite to ground station) or between the transmitter and the radar object.

• freq – Carrier frequency in Hz.

• radar – Radar mode.

• obj_gain – Gain of the radar object in dB.

• bistatic – Bistatic radar mode.

• d_rx – Bistatic radar mode only: distance between the radar object and the Rx that is not collocated with the Tx.

Returns:

Path loss in dB.

linkbudget.calc.radar_obj_gain(freq, rcs)

Compute the gain of the radar object

Parameters:

• freq – Carrier frequency in Hz.

• rcs – Radar cross section (RCS).

Returns:

Gain of the radar object in dB.

Note

The RCS definition repeated in [2] is the following: “the RCS of a radar object is the hypothetical area intercepting that amount of power which, when scattered isotropically, produces a power density at the receiver equal to that from the actual object.”

linkbudget.calc.rx_flux_density(eirp_db, distance, atm_loss_db=0)

Compute the received flux density in in dBW/m^2

Note the Rx flux density refers the incident power density (power per area) arriving at the receiving antenna, i.e., at the antenna’s input. In contrast, the value computed by function “rx_power” refers to the power collected by the antenna, i.e., at the antenna’s output.

Parameters:

• eirp_db – EIRP in dBW.

• distance – Distance between the Tx and Rx in meters.

• atm_loss_db – Total atmospheric loss in dB.

Returns:

Received flux density in dBW/m^2.

Note

The flux density unit of \(\text{dBW}/m^2\) should be interpreted as \(10\log10(\frac{W}{m^2})\), and not \(\frac{10\log10(W)}{m^2}\). With a flux density F in \(\text{dBW}/m^2\) and a given area A in \(m^2\), the Rx power can be obtained by \(F + 10\log10(A)\). Equivalently, if the area is given as \(A_{db}\) in \(\text{dBm}^2\) (decibels greater than 1 square meter), the Rx power can be obtained \(F + A_{db}\).

linkbudget.calc.rx_power(eirp_db, path_loss_db, rx_ant_gain_db, atm_loss_db=0, mispointing_db=0, feed_loss_db=0)

Compute the received carrier power in dBW

Parameters:

• eirp_db – EIRP in dBW.

• path_loss_db – Free-space path loss in dB.

• rx_ant_gain_db – Receiver antenna gain in dB.

• atm_loss_db – Total atmospheric loss in dB.

• mispointing_db – Antenna mispointing loss in dB.

• feed_loss_db – Antenna-to-LNA feed loss in dB.

Returns:

Received power in dBW.

linkbudget.calc.rx_sys_noise_temp(Tar, Te, feed_loss_db=0, feed_temp=290)

Compute the receiver system noise temperature

The receiver system noise temperature is equal to the effective input noise temperature (Te) of the entire receiver (seen as a blackbox) added to the antenna noise temperature (Tar). The Te term represents the noise introduced by the cascaded linear components of the receiver (e.g., LNB, coax line, and radio interface). Meanwhile, The Tar component represents the cosmic noise and Earth blackbody radiation captured by the antenna. Hence, the simplified model is as follows:

Rx Antenna (Tar) -----> Sum ----> Noise-free Gain Stage ---> Detector
                         ^
                         |
                         |
                  Receiver Noise (Te)

Note the antenna is not treated as another cascaded device within the receiver. Instead, the antenna noise adds to the equivalent input noise of the cascaded devices, see Figure 8-24 in [1]. This is because the observation point is at the input to the receiver system. As explained in [2], around equation 4.39, this is a convenient choice in terms of where the effective input noise temperature is observed.

By definition, an equivalent (or effective) input noise temperature represents the thermodynamic temperature of a noisy resistor connected to the input of a noiseless two-port element, which gives the same output noise power as the noisy (but otherwise equivalent) two-port element with an ideal noiseless source at its input [2]. Hence, if we group the entire receiver into a single equivalent two-port element, the Te term represents the noise generated by the entire receiver, which has power k*Te*B. When combined to the noise collected by the antenna, the result becomes the total system noise temperature.

Since the observation point is at the Rx input, the carrier-to-noise ratio (CNR) becomes the carrier power captured by the antenna (i.e., at the Rx input) divided by k*Te*B. If the observation point were instead at the detector’s input, both the carrier and noise powers would be multiplied by the combined gain of the cascaded linear components. Hence, the CNR would remain the same (see Equation 4.15 in [3]), which confirms the convenience of using the receiver’s input as the reference point.

Besides, note the input to the receiver typically represents the input to the LNA, which is assumed to be directly connected to the antenna. However, in practice, there is a lossy passive two-port element (waveguide or transmission line) connecting the antenna to the LNA. When the loss in this segment is non-negligible, the model becomes:

Antenna (Tar) --> Waveguide/Line --> Sum --> Noiseless Rx --> Detector
                                      ^
                                      |
                                      |
                               Receiver Noise (Te)

In this case, the lossy waveguide or transmission line attenuates the antenna noise, but also generates noise of its own. This function takes both effects into account when a non-zero feed loss is specified.

Parameters:

• Tar (float) – Antenna noise temperature in K.

• Te (float) – Effective input-noise temperature in K.

• feed_loss_db (float) – Loss (in dB) over the passive two-port feed connecting the antenna to the LNA.

• feed_temp (float) – Physical temperature of the feed.

Returns:

Receiver system noise temperature in K.

linkbudget.calc.spectral_density(power_dbw, bw, label='')

Compute the power spectral density (PSD) of a given signal

Assume the signal has a constant power spectral density over the given bandwidth. Equivalently, assume the signal behaves as white noise, i.e., with uncorrelated zero-mean samples in time.

Parameters:

• power_dbw – Signal power in dBW.

• bw – Bandwidth in Hz.

• label – Optional label used for logging.

Returns:

PSD in dBW/Hz.