Dn = 1/T * int<T>(sig(t)*e^(-j2npi*f0*t1)dt)
fourier series of a pulse train
f0 = 1/T
Dn = 1/T int<T>(s(t)e^(-j2nt*f0*pi)dt) = -1/j2npi * (e^(jnf0pi*tau) - e^(-jnf0pi*tau)) = 1/npi * sin(npi*tau/T) = tau/T * sinc(npi*tau/T)
looking at matlab bs involving pulse trains and fourier decomposition
average normalized power: avg pwr taht s(t) provides to a resistive load multiplied by the resistance of the load
Ps = 1/T * int<T>(s^(2)(t)dt) = C0^2 + sum<1,inf>((Cn^2)/2)
units of avg normalized power are watt-ohms or volts^2
avg norm pwr can also be calculated form the complex exponential fourier series: Ps = sum<inf>(abs(Dn)^2)
shows how power is distributed through freq band
defined mathematically as Gs(f) = sum<inf>(abs(Cn)^2 * delta(f - nf0)), which is a succession of weighted delta functions
avg norm pqr of a real valued sig is Ps = int<inf>(Gs(f)df) = 2 int<0,inf>(Gs(f)df)

fourier transform
used to convert non-periodic energy sigs from the time domain to the freq domain
S(f) = int<inf>(s(t) * e^(-j2ftpi)dt)
s(f) = int<inf>(S(f) * e^(j2ftpi)df)
FT can be derived from the complex-exponential FS by letting the period of the sig go to infinity
s(t) can be replaced by an inf summation of sinusoids with freq differences between adjacent sinusoids infinitesimally small
soeffs of the sinusoids, S(f), are no longer scalar values (volts), but rather are densities
normalized energy: what is provided to the resistive load
Es = int<inf>(s^2(t)dt) = int<inf>(abs(S(f))^2 df)
norm energy spectral density of a sig describes the sig energy per unit bandwidth measured in joules/Hz and is defined as PSIs(f) = abs(S(f))^2 in volts^2 * sec/Hz
if real sig, abs(S) is even and Es = 2int<0,inf>(PSIs(f)df)

ex 3-1: find FT of a rect
F{Arect(t/tau)} = int<inf>(Arect(t/tau)e^(-j2ftpi)dt) = A/fpi * sin(fpi*tau) = A*tau*sinc(fpi*tau)
zero crossings are at n/tau
how to classify BW of a rect? use only the primary spectral lobe of the sinc which is -1/tau to 1/tau
time shift: s(t-t0) -> S(f)e^(-j2pi*f*t0)
freq shift
time scaling: s(at) -> 1/abs(a) * S(f/a)
freq scaling: S(af) -> ................
differentiation
integration
multiplication
duality
S(t)  <-> s(-f)

distortionless transmission and ideal filters
output has same shape as input, possibly diff amp and time delay
y(t) = kx(t-td)
k and td are constants
Y(f) = kX(f)e^(-j2fpi*td)
ideal filter: passes w/o distortion all freqs between f1 and f2 and has a zero mag response outside passband
for ideal lowpass filter, with cutoff freq of B Hz, mag response is a rect and the group delay td is constant for all passband freqs
H(f) = rect(f/2B)e^(-j2fpi*td)