Solution for Analyzing link budget for a medical body area network

Constants

P_TX_DB = -30  # dB
G_TX_DB = 2
G_RX_DB = 3
L_MISC_DB = 6
FLOOR_WIDTH = 100  # m
FLOOR_HEIGHT = 30
F = 2400e6  # Hz, Worst case
# We want to analyze the worst case.
# Worst case frequency is 2400 MHz because attenuation at higher frequencies is higher.

# For the thermal noise
BANDWIDTH = 2e6  # Hz
TEMPERATURE = 290  # K

# ITU indoor model
N_OFFICE_2GHZ = 30  # Office area 1.8-2 GHz
P_F_SAME_FLOOR = 0  # Same floor

DATA_RATE = 125e3  # bits
PACKET_SIZE = 32 * 8  # bits

Furthest distance from the receiver based on the diagonal of the floor

from sympy import sqrt

(distance := sqrt(FLOOR_WIDTH**2 + FLOOR_HEIGHT**2) / 2)

\[\displaystyle 5 \sqrt{109}\]

Define link budget equations in dB:

Free-space model

from sympy import Eq, log, symbols

l_fs, d, f = symbols(r"l_\mathrm{fs} d, f")
(l_fs_eq := Eq(l_fs, 20 * log(d, 10) + 20 * log(f, 10) - 147.55))

\[\displaystyle l_\mathrm{fs} = \frac{20 \log{\left(d \right)}}{\log{\left(10 \right)}} + \frac{20 \log{\left(f \right)}}{\log{\left(10 \right)}} - 147.55\]

ITU indoor model

l_itu, n, p_f = symbols(r"l_\mathrm{itu-indoor} n p_\mathrm{f}")
(l_itu_eq := Eq(l_itu, 20 * log(f, 10) + n * log(d, 10) + p_f - 28))

\[\displaystyle l_\mathrm{itu-indoor} = \frac{n \log{\left(d \right)}}{\log{\left(10 \right)}} + p_\mathrm{f} + \frac{20 \log{\left(f \right)}}{\log{\left(10 \right)}} - 28\]

\(p_\mathrm{rx}\)

snr, p_tx, g_tx, l_p, g_rx, l_misc, p_rx, l_thermal = symbols(
    r"""
    \mathrm{snr}
    p_\mathrm{tx}
    g_\mathrm{tx}
    l_\mathrm{p} 
    g_\mathrm{rx}
    l_\mathrm{misc}
    p_\mathrm{rx}
    l_\mathrm{thermal}
    """
)
(p_rx_eq := Eq(p_rx, p_tx + g_tx - l_p + g_rx - l_misc))

\[\displaystyle p_\mathrm{rx} = g_\mathrm{rx} + g_\mathrm{tx} - l_\mathrm{misc} - l_\mathrm{p} + p_\mathrm{tx}\]

Thermal noise

from math import log10

from scipy.constants import Boltzmann

B = 10 * log10(BANDWIDTH)  # dBHz
t = 10 * log10(TEMPERATURE)  # dBK
k = 10 * log10(Boltzmann)  # dBW/K/Hz
(l_thermal_db := B + t + k)

-140.9648872375883

SNR

(snr_eq := Eq(snr, p_rx - l_thermal))

\[\displaystyle \mathrm{snr} = - l_\mathrm{thermal} + p_\mathrm{rx}\]

Substitution and numeric evaluation free-space model

Substitute and use \(l_\mathrm{fs}\) as \(l_\mathrm{p}\).

subs = {
    p_tx: P_TX_DB,
    g_tx: G_TX_DB,
    d: distance,
    f: F,
    g_rx: G_RX_DB,
    l_misc: L_MISC_DB,
    l_thermal: l_thermal_db,
    p_rx: p_rx_eq.rhs,
}
subs_fs = subs.copy()
subs_fs.update(
    {
        l_p: l_fs_eq.rhs,
    }
)

\(p_\mathrm{rx}\)

(p_rx_evald := p_rx_eq.rhs.evalf(subs=subs_fs))

\[\displaystyle -105.407889900359\]

\(p_\mathrm{rx}\):

SNR

(snr_evald := snr_eq.rhs.evalf(subs=subs_fs))

\[\displaystyle 35.5569973372296\]

l_fs_eq.rhs.evalf(subs=subs_fs)

\[\displaystyle 74.4078899003587\]

import pandas as pd


def display_table(subs):
    data = [
        ["p_tx", P_TX_DB],
        ["g_tx", G_TX_DB],
        ["l_p", -int(l_p.evalf(subs=subs))],
        ["g_rx", G_RX_DB],
        ["l_misc", -L_MISC_DB],
        ["p_rx", int(p_rx_evald)],
        ["l_thermal", int(-l_thermal_db)],
        ["snr", int(snr_eq.rhs.evalf(subs=subs))],
    ]
    # TODO convert them to Latex symbols
    # using `p_tx` and then wrapping with $ works in Jupyter
    # but not in HTML output

    df = pd.DataFrame(data)
    df.columns = ["Parameter", "Value"]

    return df


display_table(subs_fs)

	Parameter	Value
0	p_tx	-30
1	g_tx	2
2	l_p	-74
3	g_rx	3
4	l_misc	-6
5	p_rx	-105
6	l_thermal	140
7	snr	35

Substitution and numeric evaluation ITU model

subs_itu = subs.copy()
subs_itu.update(
    {
        l_p: l_itu_eq.rhs,
        n: N_OFFICE_2GHZ,
        p_f: P_F_SAME_FLOOR,
    }
)
display_table(subs_itu)

	Parameter	Value
0	p_tx	-30
1	g_tx	2
2	l_p	-211
3	g_rx	3
4	l_misc	-6
5	p_rx	-105
6	l_thermal	140
7	snr	-101

Data rate sufficient?

(transmitted_bits_per_second := PACKET_SIZE * 10)

2560

print("Yes" if DATA_RATE >= transmitted_bits_per_second else "No")

Yes

Battery consumption analysis

(duty_cycle := transmitted_bits_per_second/ DATA_RATE )

0.02048

p_tx_linear = 10 ** (P_TX_DB / 10)  # W
energy_for_one_week = p_tx_linear * duty_cycle * 60 * 60 * 24 * 7
print(f"Energy needed for one week: {energy_for_one_week}")

Energy needed for one week: 12.386303999999999

RECHARGABLE_AA_BATTERY_ENERGY = 1.2 * 2.5 * 60 * 60  # 1.2 V * 2500 mAh
print("2 rechargable AA batteries' energy: ", 2 * RECHARGABLE_AA_BATTERY_ENERGY)

2 rechargable AA batteries' energy:  21600.0

print("Two AA sufficient?")
print("Yes" if 2 * RECHARGABLE_AA_BATTERY_ENERGY > energy_for_one_week else "No")

Two AA sufficient?
Yes

Is the modulation scheme appropriate?

# For 10^-10 we need 14 dB @QPSK
EB_PER_N0_DB = 14  # dB
R_B = 125e3  # bit
eb_per_n0 = 10 ** (EB_PER_N0_DB / 10)


n_0 = Boltzmann * TEMPERATURE
e_b = eb_per_n0 * n_0
p_rx_ = e_b * R_B
p_rx_db_ = 10 * log(p_rx_, 10)
print(f"Minimum P_rx ~{int(p_rx_db_)} dB")
print()

Minimum P_rx ~-139 dB

print("Modulation scheme appropriate for free-space?")
print("Yes" if p_rx.evalf(subs=subs_fs) > p_rx_db_ else "No")

Modulation scheme appropriate for free-space?
Yes

print("Modulation scheme appropriate for ITU indoor?")
print("Yes" if p_rx.evalf(subs=subs_itu) > p_rx_db_ else "No")

Modulation scheme appropriate for ITU indoor?
No

Conclusions

The device does not have enough link margin.

• LOS transmission feasible, but obstructed (non-LOS) transmission not possible

• Every room should have a receiver

One week runtime is not possible.

• The batteries could be changed more often

• The measurement frequency could be reduced.