second class of the semester...
size of a signal
energy of a sig is int<inf>(|sig(t)|^2)
if amp doesn't go to 0 at inf, find power instead
power P = 1/T * int<-T/2,T/2>(Ccos^2(wt+theta)) = CC/2
Tw = 2pi

sum of sinusoids, ratio of frequencies must be rational to be periodic, rational ratio makes crossterm 0
2 cos with w1,w2,T1,T2, coeffs C1,C2
P = C1C1/2 + C2C2/2 + 0 (for cross term if ratio is rational)
P = (C1C1 + C2C2)/2

enormous range of power ratios expressable in terms of decibels w/o using massive or tiny numbers
decibel is dimensionless
dB = 10log(P1/P2)
P2 is the reference power
for gain, ApdB = 10log(Pout/Pin)
abs power gain is Pout/Pin
dBm is unit of measurement used to indicate ratio of a power level wrt a fixed ref level of 1mW
dBm = 10log(P/0.001W)

using dB allows adding instead of multiplying
ex2: conv the abs power ratio of 200 to a power gain in dB, 10log(200) = 10(2+0.3) = 23dB
ex3: 2 stage system w/ 2 amps and a filter, input P = 0.1mW, Ap1 = 100, Ap2 = 40, Ap3 = 0.25
	overall mult is 1000, 0.1mW*1K = 0.1W -> Pin = -10dBm, Pout = 20dbm
	different way: 100x -> 20dB, 40x -> 16dB, .25 -> -6dB, total is 30dB

signals versus vectors
a sig tha tis defined for a fininte number N of time instances can be written as a vector of dimension N
if g is defined over the time interval [a,b], then we can pick N points uniformly from this time interval
we can write signal vect <b>g</b> as an N-dimensional vector
allows us to use vec space ops for cont time sigs, for ex, the inner product of two vecs, projection, etc
if sig g is approx by another sig x as g ~~ cx then the optimum val of c that minimizes the energy of the error sig in this approx is given by
c = (int<t1,t2>(gx))/(int<t1,t2>(xx)) = 1/Ex * int<t1,t2>(gx)

int<t1,t2>(|x^2+y^2|) = int<t1,t2>(|y|^2) + int<>(|x|^2)

correlation of sigs
correlation is a measure of the similarity fo two sigs
for vecs, correlation coeff rho (p) is defined as p = cos(theta) = <g,x>/mag(g)mag(x) theta is angle between g and x
two vecs aligned in same dir, p=1, opposite dir, p=-1, orthogonal, p=0
for sigs the corr coeff is defined as p = 1/sqrt(Eg*Ex) * int<inf>(gx*)

ex4: s1(t) = Pt(t) = 1 on [0,T], 0 else
	s2(t) = sin(tpi/T) on [0,T], 0 else
	find the projection of s2 onto s1
	Es1 = int<0,T>(1^2) = T
	Es2 = int<0,T>(sin^2(tpi/T)) = T/2
	c = (2T/pi)/T = 2/pi
	p = (2T/pi)/sqrt(TT/2) = 2sqrt2/pi = 0.9