overview of 11.2-2
have two chirp sigs, need to find optimum rxer
choose Tb correctly to prevent non-causal filters and min delay
stuff goes into thresh device

hM = Acos(alpha1*(T-t)^2 + theta1) - Acos(alpha0*(T-t)^2 + theta0)
thresh: gamma = (Es1-Es0)/2
Es1 = int<0,T>(AAcos^2(alpha1*tt + theta1)dt) = magic here with fresnel integrals = AAT/2 + AA/4 * sqrt(pi/alpha1) * C(2Tsqrt(alpha1/pi)cos(2theta1) - S(2Tsqrt(alpha1/pi))sin(2theta1))

we get THREE sheets of paper to write stuff on for the final, DO IT

SNR is that crazy equation from last time with multiple sqrts and Esi
if sigs are equal energy, Es0=Es1=Eb -> SNR = sqrt(Eb(1-p)/n)
Pe = Q(SNR)
if sigs are antipodal, p=-1, Pe=Q(sqrt(2Eb/N))
if sigs are equal energy and ortho, p=0, Pe=Q(sqrt(Eb/N))

convolution sucks
equivalent rxers
any structure that provides the same decision statistic as the matched filter will have the same performance, but it may be easier to implement or analyze
let R(t) be input to matched filter, if output of filter after samp at T is Z(t) = int<inf>(R(u)hM(T-u)du), but hM(T-u) = s1(u)-s0(u) = s(u)
so Z(t) = int<inf>(R(u)s(u)du)

for time-limited sigs, Z=int<0,T>(R(t)s(t)dt)

ex11-3
s1=ApT(t)
s0=-ApT(t)
Eb = int<0,T>(AAdt) = AAT
set c=1/2A (chosen arbitrarily)
s = c2ApT(t) = pT(t) = rect((t-T/2)/T)
antipodal -> gamma=0, Z>0 -> s1, else -> s0
Pe = Q(2Eb/n)

ex11-4
s1 = e^(-t)rect((t-T/2)/T)
s0 = -rect((t-T/2)/T)
s = (e^(-t) + 1)rect((t-T/2)/T)

optimum rxer is usually implemented with correlation rxer


BPSK
sigs of form:
si = sqrt(2)cos(wct + phi)*pT'(t)
pT' is usually a rect pulse, but could be other signals
opt rxer multiplies by ref sig with matched phase that is the same as s1, integrate and into thresh
thresh at 0, because antipodal
freq easy to fix, may also get out of phase, have to fix with PLL

OOK - on-off
sigs are:
s1 = sqrt(2)cos(wct + phi)pT'(t)
s0 = 0

ex11-5 ook
let pT' = pT = rect
Es1 = magic = T + sin(2wct)cos(2phi)/2wc + sin(2phi)cos(2wct)/2wc, t between T and 0
if wcT = npi, all sin/cos stuff is 0
if wc >> 1/T, all sin/cos stuff is nearly 0
we will just claim that it's just T
Es0 = 0
gamma = T/2
Eb = T/2
Pe=Q(sqrt(T/2n))


for FSK
s0 = sqrt(2)cos((wc-delw/2)t)pT'(t)
s1 = sqrt(2)cos((wc+delw/2)t)pT'(t)
if delf=n/2T, the sigs are ortho
to min BW req, delf = 1/2T, which results in min shift keying MSK
opt rxer uses two different matched sigs multiplied by input, compares integration of mult to see which freq was txed