Wireless Communication Principles and System Formulas

!!!!GENERAL!!!! a mod b = a - b * floor(a/b) x dBm = 10^(x/10) mW 10log(10 (mW) ) = 10 dB
!!PHY 2!! VARIABLES: P_r = Receiving power (W/mW) P_T = Sending power (W/ mW) G_T = Sending antenna gain G_R = Receiving Antenna Gain H_T = Sending antenna height (m) H_R = Receiving Antenna height (m) d = transmission distance (m) λ = c/f = wavelength (m) f = frequency (Hz) c = Speed of light = 3 * 10^8 (m/s) d_break = break point distance (m) N_T = number of transmitting antennas N_R = number of receiving antennas v = velocity (m/s)
Free space Path Loss (Friis Equation): P_R = P_T * G_T * G_R * (λ / (4πd))^2 unit antenna gain is 1. (IF isotropic antenna (theory) P_R = P_T * (c / (4πdf))^2 )
P_r / P_t = Total Path loss (db) TWO RAY: Time between first (Los) and last copy of NLos = Delay Spread P_r = P_T * G_T * G_R * ( (H_T * H_R) / d^2 )^2 40log_10(d) - 20log_10(h_t * h_r) = pathloss (dB)

if d is smaller than d_break, can't two ray and use FRIIS. Path Loss are absolute numbers. Can have constructive and destructive fading (強弱信號區)
Antennas: SISO (Single input single output), SIMO, MISO, MIMO (Multi in and out) SMALL SCALE FADING (Multi path, reflection and different path): LARGE SCALE FADING (Shadowing, mostly for shadow of building, and meters);RSS drop in shadow.
When Multi-path antenna spaced >λ/2, multipath signal from antennas can be uncorrelated. Space diversity: reliability; Space multiplexing: data rate Total independent paths = N_T * N_R
MULTIPLEXING: Degrees of freedom = min(N_T, N_R)
OFDMA: Each user has own subset of subcarriers for a few time slots; OFDM uses TDMA (e.G. WIFI) OFDMA allows Time + Freq DMA -> 2D->scheduling Channel bandwidth / subcarrier spacing = Number of subcarriers
Frequency: Low frequencies need larger antenna and spacing. vf/c = Doppler shift = Velocity * Frequency / (speed of light)
!!WLAN BASICS!! 802.11 b/g/n 2.4 GHz 802.11 a/n/ac 5 GHz 802.11 p (car to car) 5.9GHz 802.11 ad/ay 60GHz 802.11 ah (IoT) 8mmMHz (automation only)
2.4 GHz Channel overlap, max 3 unoverlapping channel possible. (1, 6(default), 11 most used) Most wifi ~100MHz are split into 14 channels Frequency 2401 to 2495, around 22MHz SIFS (e.G. ACK) < PIFS < DIFS
DCF backoff algorithm: init CW = CW_min after each collision, CW = min(2*CW + 1, CW_max)
VIRTUAL CARRIER SENSE: RTS = duration of SIF + CTS + SIF + Frame + SIF + ACK CTS = duration of SIF + Frame + SIF + ACK Frame = duration of Frame + SIF + ACK ACK = duration 0 MAC FRAME FIELDS:

DSSS data rate (802.11b): data rate = coding rate * chip rate (22) * (1 / chips per symbol) * bits per symbol (log_2(M QAM))
OFDM data rate (802.11a/g/n): amount of data subcarriers = if 20MHz, if 802.11a/g = 48, 802.11n = 52, if 40MHz, if 802.11n = 108 data rate = (amount of data subcarriers) * (coding rate) * (coded bits per subcarrier (log_2(M QAM))) / (4 or 3.6 depending 800ns or 400ns guard interval) *10^-6 = bits per second
CSI calculation ae^j (A = amplitude, theater = phase, j=sqrt(-1)) or a+jb (Amplitude = sqrt(a^2 + b^2), phase = arctan(b/a))
802.11e has direct link.


!!CELLULAR!! D = min distance between center of cells that use same frequencies (co-channels) R = radius of a cell d = distance between centers of connected cells d=R*sqrt(3), d<2R N = Number of cells in Cluster (repetitious pattern) 1/N = Frequency Reuse Factor, each cell in cluster use unique frequencies For hexagon cells, N is: I^2 + J^2 + (I*J), I, J = 0,1,2,... D/R = sqrt(3N) = Reuse Ratio D/d=sqrt(N) (Hexagon) Locate Co-channel Cell rule: each cell has 6 sides (60 degree), walk I, turn 60 degrees, then walk J steps (self not included as first step) CHANNELS DISTRIBUTION: K = T/N, T (Total channels), N(cluster size), K(number eof cahnnels per cell) S = number of sectors in cell, K = Number of frequency / channel allocations per cell N * S * K = Frequency reuse pattern
!!BLUETOOTH!! (Beware Byte OR bit) f_c = (2402 + k) MHz; k = 0...78 = carrier frequency k = channel index (79 1-MHz wide channels) Basic rate: 1Mbps | π/4-DQPSK: 2Mbps | 8DPSK: 3Mbps FREQUENCY HOPPING: 625πs slot use 312.5 πs (3200Hz) clock (1slot 2 clock ticks) TDD for both sides, Master starts in even numberd slots only. Slave only odd. Packets = 1 or 3 or 5 slots long (ONLY) Max FH (frequency hopping rate = 1600Hz, min 320Hz) 1/T_hop (how many second(s) between hops) = Frequency hopping rate (Hz/s^-1) T_hop = (x slot packet) x * 625 microsecond Assume Coding Rate 1 Each slot can store 625 bits Paging time < 1 second active device = 3 bits, parked = 8
!!RADAR!! B = bandwidth (Hz) C = Light speed = 3 * 10^8 m/s Range = TOF * C / 2 Resolution (m) = C / (2 * B) PULSE: bandwidth = 1/W W(s)(pulse width)
FMCW: ToF = time of flight f_b = beat frequency (Hz) Sweeping Duration = T(s) ToF=2R/C, Range = R (m) chrip slope: S = B / T (Hz/s), f_b/ToF = S

Resolution same. T = chirp duration
Range Resolution: ΔR = C / (2*B) = distance between two objects df = beat sequence difference dt = 2(Δd/c) ToF difference of 2 objects (df/dt) = S = B/T = (df) / (2 * Δd * c))
if df > 1/T, can distinguish two objects. if df = 1/T, Δd = C/(2*B)
!!IOT!! B = Bandwidth (Hz) T = Chirp duration (s) Frequency sweeping speed (k) = B / T (Hz/s) "X-ary modulation" log_2(x) = y bits per (symbol / chirp) SF bits per symbol = log_2 (2^SF); SF = log_2(M) 2^SF = M (modulation order, M) 2^SF / B = Symbol / chirp duration (s) LORA: Data rate = SF * (B / 2^SF) * Coding Rate (unit: bps) Lora min bandwidth = 125KHz, max = 500KHz possible CR = 4/5, 4/6, 4/7, 4/8 possible SF = 7 to 12
Energy{J} = V{V}[I_time{s} * I_current{A} + I_receive_time{s} * I _receive_current{A}]