distortionless xmission
output must have same shape as input
system response for such a system is H(f) = Ke^(-2fpi*t0)
group delay is defined as tg = -1/2pi * thetah'(f)
ideal low pass filter H(f) = rect(f/2B) * e^(-jf2pi*td)
	h(t) = 2Bsinc(2Bpi * (t - td))

sig distortion over a comm channel
linear distortion
	sig that is only nonzero in the time a<=t<=b has spectrum components with specific mag and phases that add up to the pulse g(t) over the interval a,b and add to zero outside that interval

if chan not ideal, components are multiplied by a factor which varies with freq
g will no longer add together to produce zero sig component outside the a,b interval,
pulse is spread in time, called dispersion
digital symbol xmitted over a dispersive channel spreads wider than its alloted time and interferes with adjacent sigs
this interference is called intersymbol interference (ISI)

ex3-2: linear distortion
channel freq response
H(f) = (1+kcos(2pifT))e^(-j2fpi*td) abs(f) < B
sig g is bw limited to B Hz
Y(f) = G(f)e^(-j2fpi*td) + k/2 * G(f)*(e^(-j2fTpi) + e^(-j2fTpi))*e^(-j2fpi*td)
Y(f) = G*e^(-j2fpi*td) + k/2 * (Ge^(-j2fpi * (td - T)) + Ge^(j2fpi * (td - T)))
g = g(t-td) + k/2 * (g(t-td+T) + g(t-td-T))
smears signal out, phase components modified



nonlinear distortion
usually caused by high power amps
consider a memoryless chan where the input g and output y are related by some nonlinear f(), y = f(g)
this equation can be expanded in a maclaurin series as y = a0 + a1g +a2gg + a3ggg...
if the bw of g is Bhz, the bw of gg will be 2Bhz and the 

ex3-3: nonlinear distortion
y = x + .000158x^2
x = 2000sinc(2000tpi)
from convolution table: sinc^2(Btpi) -> tri(f/2B)
Y = rect(f/2000) + 0.158*2*tri(f/4000)

distortion by multipath effects
multipath distortion occurs when xmitted sig arrives to rx by 2+ paths with diff delays
sigs rx from various paths can add constructively or destructively and the mag of the sum of the sigs will vary with freq
a multipath chan causes frequency-selective fading

ex3-4: multipath
x -> delay td  ---------------------+-> y
  \> attenuation -> delay td + tau /

y = x(t-td) + alpha*x(t-(td+tau))
Y = X(f)e^(-jf2pi*td) + alpha*X(f)e^(-jf2pi*(td+tau))
H(f) = Y/X = e^(-jf2pi*td) * (1+alpha*e^(-jf2pi*tau))
H(f) = e^(-jf2pi*td) * (1+alpha*(cos(2fpi*tau)-jsin(2fpi*tau)))
|H(f)| = sqrt((1+alpha*cos(2fpi*tau))^2 + alpha^2*sin^2(2fpi*tau)) = sqrt(1+alpha*alpha+2alpha*cos(2fpi*tau))
thetah = -2fpi*td + arctan((-alpha*sin(2fpi*tau))/(1 + alpha*cos(2fpi*tau)))
if f=n/2tau, n odd -> cos(2fpi*tau) = -1 (destructive interference)
	if alpha == 1 -> null frequency
if f=n/2tau, n even -> cos(2fpi*tau) = 1 (constructive interference, gain enhanced)



sig energy and energy spectral density