correlation - measure of similarity between two signals
corr coeff rho = cos(theta) = <g,x>/(mag(g)mag(x)), -1<=p<=1
cross correlation function of two complex sigs is measure of similarity of two sigs as a f() of time shift tau
psizg(tau) = int<inf>(z(t+tau)g*(t)dt) = int<inf>(z(t)g*(t-tau)dt)
autocorrelation of a sig is correlation of a sig with itself
psiz(tau) = int<inf>(g(t+tau)g*(t)) = int<inf>(g(t)g*(t-tau))
ex2-5: cross correlation of s1(t) and s2(t) where s1 = 1 on [0,T] and s2 = 1 on (0,T/2] and -1 on (T/2,T]
<s1,s2> = int<0,T>(s1*s2*dt) = 0
psis1s2(tau) = int<inf>(s1(t+tau)s2(t)dt) = int<inf>(s1(t)s2(t-tau)dt)

tau> T, no intersection
tau< -T, no intersection
T>=tau>=T/2, f() = T-tau
T/2>tau>=0, f() = tau
0>tau>=-T/2, f() = tau
-T/2>tau>=-T, f() = T+tau
at 0 and elsewhere, f() = 0

autocorrelation: s1 vs itself, same as above
0>tau>=-T, f() = T+tau
T>tau>=0, f() = T-tau

autocorrelation of s2
-T/2>tau>=-T, f() = -tau-T
0>tau>=-T/2, f() = 3tau+T
T/2>tau>=0, f() = -3tau+T
T>tau>=T/2, f() = tau-T

fourier series
rep of waveform in frequency domain
trig form:
s = a0+sum<1,inf>(an*cos(2npi*f0*t) + bn*sin(2npi*f0*t))
a0 is DC term/avg value
a set and b set are freq components

compact trig fourier series:
s = C0+sum<1,inf>(Cn*cos(2npi*f0*t + thetan)
C0 = a0
Cn = sqrt(anan+bnbn)stuff

complex exponential fourier series
s = sum<inf>(Dne^(j2nf0*tpi))
Dn = 1/T0

a0 = 1/T0 * int<-T0/2,T0/2>(s(t)dt)
an = 2/T0 * int<-T0/2,T0/2>(s(t)cos(2npi*f0*t)dt)
bn = 2/T0 * int<-T0/2,T0/2>(s(t)sin(2npi*f0*t)dt)
if s is even, b's will all be 0
if s is odd, a's will be 0, including a0

ex7:stair step, 2,1,0, one unit at 2 and 1 repeats at 5 and -5
avg value = 3/5 = a0
an = 2/T0*int<T0>(s*cos(2npi*f0*t)dt)
an = 2/T0*int<0,1>(s*cos(2npi*f0*t)dt) + 2/T0*int<1,2>(s*cos(2npi*f0*t)dt)
an = 2/5(10/2npi * sin(2npi/5 * t) +...)
an = 2/npi * sin(2npi/5) + 1/npi(sin(4npi/5) - sin(2npi/5))
an = 1/npi(sin(2npi/5) + sin(4npi/5))
....
bn = 1/npi(2 - cos(2npi/5) - cos(4npi/5))