Signal and system theory Convolution of signals x(t) h(t) = y(t): Fourier Transform: Communication Theory Summary of Important Definitions and Results X(ω) = X(ω) = y(t) = X(ω) = j x(t) e jωt dt, 0 Properties of the Fourier Transform: If x(t) X(ω) and v(t) V (ω), then: x(λ)h(t λ) dλ x(t) cos(ωt) dt, 0 x(t) sin(ωt) dt, for all signals ax(t) + bv(t) ax(ω) + bv (ω) x(t t 0 ) X(ω)e jωt 0 x( t) X( ω) x(at) 1 a X(ω ), if a > 0 a for even signals for odd signals t n x(t) j n dn dω X(ω) n x(t) e jω 0t X(ω ω 0 ), if ω 0 is real x(t) cos(ω 0 t) 1 [X(ω + ω 0) + X(ω ω 0 )] x(t) sin(ω 0 t) j [X(ω + ω 0) X(ω ω 0 )] t d n dt n x(t) (jω)n X(ω) x(λ)dλ 1 X(ω) + πx(0)δ(ω) jω x(t) v(t) X(ω)V (ω) x(t)v(t) 1 X(ω) V (ω) π X(t) πx( ω)
Common Fourier Transform Pairs: δ(t) 1 u(t) πδ(ω) + 1 jω cos(ω 0 t) π[δ(ω + ω 0 ) + δ(ω ω 0 )] sin(ω 0 t) jπ[δ(ω + ω 0 ) δ(ω ω 0 )] Autocorrelation function for deterministic signals: R x (τ) = R x (τ) = 1 T 0 T0 x(t)x(t + τ) dt, for an energy signal x(t) T 0 x(t)x(t + τ) dt, for a power signal x(t) with period T 0 Sampling and Quantization x δ (t) = δ(t nt s ), is a periodic impulse train of period T s n= x δ (t) 1 T s n= δ(f n T s ) SNR q = 3L, is the quantization SNR with an L-level quantizer Digital modulation and demodulation Q(x) = 1 exp( t π ) dt x If a 1 and a are the two possible sample values at the output of a digital receiver and σ0 is the variance of the output noise, the bit error probability P B is: a1 a P B = Q σ 0 When a matched filter is used, the error probability depends on the energy E d of the transmission signal-difference and the power spectral density N 0 / of the Gaussian noise as: ) P B = Q ( Ed The raised cosine (RC) filter with a bandwidth W and transmission rate 1/T = W 0 has the following characteristic: 1 for f < W 0 W H(f) = cos π f +W W 0 4 W W 0 for W 0 W < f < W 0 for f > W N 0
If the roll-off factor of the raised cosine pulse is r, then, W = (1 + r) T M-ary modulation and demodulation Symbol error probability of M-ary phase shift keying (MPSK) with energy E s per symbol in the presence of Gaussian noise with power spectral density N 0 / Watts/Hz is ( Es P E (M) Q sin( π ) N 0 M ) Bit error probability is P B = P E / log M Symbol error probability of non-coherently detected M-ary frequency shift keying (MFSK) with energy E s per symbol in the presence of Gaussian noise with power spectral density N 0 / Watts/Hz is Bit error probability is P B = M/ M 1 P E Communication link analysis P E (M) M 1 ( exp E ) s N 0 The power P r received by the receiver with antenna gain G r at a distance d away from a transmitter transmitting an electromagnetic signal (of wavelength λ) a power P t with an antenna gain of G t is: P r = P tg t G r λ (4πd) The gain G of an antenna is related to its effective area A e and wavelength λ of transmission as: G = 4πA e λ The link margin M of a communication link is related to the effective isotropic radiated power (EIRP), receiver antenna gain G r, required signal to noise ratio (E b /N 0 ) reqd, data rate R, Boltzmann s constant k, free space loss L s, other incidental losses L 0 and temperature T o (in Kelvin) as EIRP G r M = (E b /N 0 ) reqd RkT o L s L 0 The noise figure F and noise temperature T o of an amplifier are related as T o = (F 1)90 The noise figure F of a lossy line with loss factor L is F = L
The composite noise figure F comp of a cascade of n amplifiers with gains G i and noise figures F i is: F comp = F 1 + F 1 + F 3 1 F n 1 + G 1 G 1 G G 1 G G n 1 The effective bit error probability P B of a regenerative satellite repeater link is related to the bit error probabilities P u and P d of the uplink and downlink, respectively, as P B P u + P d The received signal-to-noise ratio (P r /N 0 ) ij at the jth receiver from the i-th transmitter (with power P i ) through a non-regenerative satellite repeater with total isotropic power EIRP s can be computed using γ j the downlink loss, A i the uplink loss, N s / the Gaussian noise power spectral density at the satellite, N g / the Gaussian noise power spectral density at the receiver, P T the total power received at the satellite, W the bandwidth, and, β = 1/(P T + N s W ) as: Pr = EIRP sγ j βa i P i N ij EIRP s γ j βn s + N g Multiple access techniques The normalized throughput ρ is related to the normalized traffic G on a pure ALOHA channel as: ρ = Ge G and has a maximum at G = 05 when ρ = 018 The normalized throughput ρ is related to the normalized traffic G on a slotted ALOHA channel as: ρ = Ge G and has a maximum at G = 1 when ρ = 037 Spread spectrum techniques The bit error probability P B of non-coherent BFSK with a partial band jammer that jams a fraction ρ of the spread bandwidth W ss is related to the energy per bit E b and jammer power spectral density J 0 over the bandwidth as: P B ρ ( exp ρe ) b, J 0 and the worst case jamming occurs when ρ = E b /J 0, assuming that E b /J 0 > The bit error probability P B of BPSK with a pulse jammer that jams at a duty cycle of ρ is related to the energy per bit E b and jammer power spectral density J 0 over the spread bandwidth W ss as: ( ) ρeb P B ρq, J 0
and the worst case jamming occurs when ρ = 0709 E b /J 0, assuming that E b /J 0 > 0709 The maximum number M of users that can be accommodated by a spread spectrum system is related to the spreading gain G p, the voice activity factor G V, antenna gain G A, asynchronism factor γ, outer cell interference factor H 0, and the signal to interference ratio (E b /I 0 ) reqd required for acceptable performance as: M = γg pg A G V (E b /I 0 ) reqd H 0