passed quiz 1 with flying colors
failed hw 2 with submarine colors

in PM, the angle, theta(t) is varied linearly with m(t)
the PM waveform is PM(t) = acos(wct + kp*m(t))
instantaneous angular freq is wi(t) = wc + kp*m'(t)

fm waveform: inst freq varies linearly with the modulating sig: wi(t) = wc + kf*m(t)
angle is theta = wct + kf*int<-inf,t>(m(alpha)dalpha)

ex5-1: sketech fm and pm waveforms for kf = pi*2e5, kp = 10pi, fc = 100mhz
m(t) = sawtooth with T=2e-4, A=1
generates a carrier with constantly varying freq for sawtooth, jumps in freq with derivative of sawtooth (square wave) has discontinuous freq shifts

ex5-2: square wave with < 50% duty cycle
causes output to switch between 2 freqs, fimin and fimax

problem with PM: instantaneous freq varies linearly with derivative of m, for a square wave, that means 0s and deltas
deltas manifest as an abrupt phase shift in digital signals
for digital messages, PSK and ASK are related

power of an angle modulated sig is A*A/2

carson's rule can be used to est the bw of a FM sig as Bfm = 2(delf + B) = 2(kfmp/2pi + B) where delf is the peak freq deviation given by delf = kf(mmax - mmin)/4pi

carson's rule can be written in terms fo the deviation ratio as Bfm = 2B(beta+1)

sampling theorem
lowpass sampling theorem
any sig g with no freq components above B hz can be reconstructed exactly from samples of g taken at a minimum rate of fs = 2B samples per second
to recover an undistorted version of the original sig, the sample values can be passed through an ideal lowpass filter of bw W hz where B<=W<=fs-B

ideal sampling
consider g with FT G(f) where mag(G) = 0 for abs(f) >= B
g is multiplied by ideal impulse sampling waveform: deltats(t) = sum<inf>(delta(t-nTs))
an & represents an overbar, in this case, a sampled signal
g&(t) = g(t)deltats(t) = sum<inf>(g(nTs)delta(t-nTs))


find G&(f) = F(g(t)deltats(t))
pulse train is periodic sig with T = Ts
get FS of pulse train
Dn = 1/Ts * int<Ts>(delta(t)e^(-j2ntpi*fs)dt) = 1/Ts
deltats(t) = sum<inf>(Dn*e^(j2ntpi*fs)) = Dn*sum<inf>(e^(j2ntpi*fs)) = 1/Ts * sum<inf>(e^(j2ntpi*fs))
G&(f) = F(g(t)deltats(t)) = 1/Ts * F(sum<inf>(g(t)*e^(j2ntpi*fs))) = 1/Ts * sum<inf>(F(g(t)*e^(j2ntpi*fs))) = 1/Ts * sum<inf>(G(f-nfs))