discrete-time signal - signal for which the value is only given at discrete times, often obtained by sampling a continuous signal
DT signals are represented as sequences of numbers at uniformly spaced instants in time (0, T, 2T, 3T, etc.)
x(t) -> x(nT) -> x[n]

advantages of DT systems:
	filters have greater degree of precision and stability
	can be reproduced exactly, no component tolerance issues
	more easily adapted. simiple source code changes can realize new functionality
	time sharing (time division multiplexing)
	low frequency filters become much easier
	small packaging and low power
	rapidly declining cost for more powerful hardware

DT impulse and unit step functions similar to analog (t=0 is always 1)
a CT exponential can be expressed as e^at = ((e^a)^t) = a^t

plot the following sequences and comment on whether or not the exponential is growning or decaying (Fig1)
graph of growing exponentials vs decaying exponentials (Fig2)
DT sinusoids: cos(wt+O) -> cos(OMEGAt + O)
important properties: not all DT sinusoids are periodic for example cos(2n)
DT signal is periodic if x[n] = x[n+No]
No = 2M*pi/OMEGA
pick smallest integer M that makes No an integer, where M is the number of samples
A CT sinusoid has a unizue waveform for every freq from DC to infinity
not the case with DT sinusoid cos(OMEGAn) = cos((OMEGA+2piM)n)
DT sinusoids only have unique waveforms for OMEGA within a 2pi range
energy of a DT signal is given by Ex = sum<-inf,inf>(abs(x[n])^2 
energy signal if finite
power signal if infinite

Px = lim<N->inf>(1/(2N+1)*sum<-N,N>(abs(x[n])^2))

det type of sig (energy, pwr, neither) 

as with CT sigs, there are several useful operations for DT sigs:
time shifting: y[n] = x[n-1]
time reversal: y[n] = x[-n]
time scaling: y[n] = x[an] (a>1 -> compression/downsampling, lose have the samples and squeeze in) (a<1 -> expansion, not upsampling, 0 goes in places where there was no data, no interpolation done)

design a constant alt autopilot sensor for an aircraft that determins the vert vel of the aircfaft (basic differentiator system)
input: aircfaft alt every T seconds
output: vertical velocity
v(nT) = dx(t)/dt|(t=nt) = lim<T,0>(x(nT)-x((n-1)T))/T
v[n] = lim<T,0>(x[n] - x[n-1])/T = (1/T)(x[n]-x[n-1])

6% interest per year, det discrete system rep for loan repayment
input: current monthly payment
output: mortgage balance at end of month
Fig3