test next week
given integral and FT tables
4 or 5 questions on test

practical sampling
instead of ideal impulse train, use short pulses
and on to PCM

Pulse Code modulation
in order to xmit an analog sig using digi comms, the sig must be sampled and quantized
quantizing rounds off the sampled sig value to one of the closest permissible numbers or quantized levels
let a message sig m lie in the range (-mp,mp). this range is partitioned into L subintervals, each of mag delv = 2mp/L
each sample amp is approx by the midpoint val of the subint in which the samp falls, thus each samp is approx to one of the L numbers
for PCM, the L levels are assigned a binary code of log2(L) bits

effects of quantizing
ideally, a sampled sig can be reconstructed precisely from just the sampled values, however, after quantizing the sig can't be recon exactly since info has been lost in rounding process
quantization error is diff between the sampled message sig and the quantized sig &m(t) (preceeding & means a hat: ^ like a following & means an overbar)
the signal q(t) is undesired and is known as quantization noise
to det the pwr (mean-square value) of q we can treat it as a random process where at each sample time Q=q(kTs) is the quantization error
assume the amp of the message sig is in the range +/-mp, at any sample time the message m(kTs) will be in one of the L intervals of width delv = 2mp/L
the error Q will be in the range +/- delv, and can be modeled as a uniform random var over this range
from random sig theory, the density of f() Q is fQ(q) = 1/2delv
and the quan noise power can be found as Nq = q&^2 = int<delv>(qqdv/delv)
the SNR of the system is So/No = (3LL(m&(t))^2)/mpmp

nonuniform quantization
uniform quantization levels are a problem for speech sigs since the sig power can vary widely and thus the quality of the sig
dynamic range of the sig is divided into nonuniform quan levels
any sample within a particular quan region is rounded off to the voltage val corresponding to the center of the region
sigs with lower mags have smaller quan levels than sigs with larger mags
useful in digitizing speech since humans perceive volume change on log rather than linearly
uniform quan is simpler to implement
nonuniform quan can be implemented directly or via compression:
	at transmitter, perform nonlin compression on the analog sig and then digitize the compressed sig using uniform quan, this warps sig prior to quantizing
	at the receiver convert the warped sig back into analog form and then decompress the sig using the inverse of the nonlin op 

this process is known as companding
yay for mulaw vs alaw
NA and japan use mulaw with mu=255, everywhere else uses alaw with A=87.6
using mulaw, output SNR is So/No = 3LL/(ln(1+mu)^2)

xmission bw for PCM
in bin pcm we assign n bits to each of the L quan levels: L=2^n
if the message sig is bandlimited to B, it requires a min of 2B samples per second which results in a total of 2nB bits/s
a max of 2B independent pieces of info per second can be xmitted, error free, over a noiseless chan of BW B
to transmit thsi pcm message will require a bw of Bt = nB Hz
the SNR after quan can be written as SNR = So/No = c2^(2n)
where c=3/ln(1+mu)^2 for ulaw and 3m&^2/mpmp for uniform quan

in dB, SNRdb = 10log10()

ex6-2: 3.4Khz voice signal
nyqist demands 6.8KHz sampling rate, actually use 8KHz (standard in telecom)
transmit using binary companded pcm w/ mu=100
if n=4, 16 levels = L
SNR = 3*16*16/ln(202) -> c = .141 -> alpha=10log10c
SNR = 15.49dB
Bt = 8K*4*1hz/2bits=32khz