mean of Z is ^si(tm)
variance ends up at sigma^2 = n/2 * int<inf>(h^2(t)dt)

variance doesn't depend on which sig was sent
rxer is trying to make decision on symbol sent
to min the max prob of error, the thresh should be gamma=(mu0+mu1)/2
if sigs antipodal, gamma=0
Pe1 = Q((mu1-gamma)/sigma)
Pe0 = Q((mu0-gamma)/sigma)
if using minimax thresh, Pe0=Pe1=Pe=Q((mu1-mu0)/2sigma)

optimum filter, matched rxer
opt filter will min prob of error for given sig set
to min the prob of error, we must min sigma/variance and max diff of means

we find out through the notes that Pe is min when h(t) = hM(t) = c*(s1(tm-t) - s0(tm-t))
this is because the filter is matched to the transmitted signals

performance of system is unaffected by sample time tm, defined in matched filter and time the output is sampled, tm doesn't matter as long as it doesn't vary
non-causal filters are not allowed
tm must be greater than or equal to T
let tm=2T, additional delay of T
want tm = T so minimal delay if all sigs are time limited to T

snr can be written as SNR=sqrt((Es0 + Es1 - 2p*sqrt(Es0*Es1))/2n)
if sigs antipodal, p = -1 (rho, not a literal p, i'm lazy), best performance -> Pe=Q(sqrt(2Eb/n))
if sigs are equal energy and ortho, p=0 -> Pe=Q(sqrt(Eb/n))

example
S1 = e^-t * Pt(t) (1 between 0 and T, 0 otherwise)
S0 = -Pt(t)

find hM(t) matched filter impulse response

hM(t) = s1(T-t) - s0(T-t) = Pt(t) * (1+e^-(T-t))
hM(t) = Pt(t) * (1+e^(t-T))
find gamma thresh gamma = (mu0+mu1)/2
for a matched filter, mu0 = conv(s0,hM) at t=T = int<0,T>(s0(tau)hM(T-tau)dtau) = <s0,s1> - Es0
mu0 = conv(s1,hM) at t=T = Es1 - <s0,s1>
left with gamma = (Es1-Es0)/2
ONLY WORKS FOR MATCHED FILTERS
Es1 = int<0,T>(s1^2(t)dt) = (1-e^-2T)/2
Es0 = int<0,T>(s0^2(t)dt) = T
gamma = (1-e^-2T - 2T)/4
for T=1e-3, gamma = -5e-7

<s0,s1> = int<0,T>(e^-t *(-1) * dt) = e^(-T)-1 = -0.001
p = <s0,s1>/sqrt(Es1*Es0) = -1