#import "../template/lib.typ": *
#set page(paper: "a4")
#show: notes.with(
  title: [EE2T1],
  subtitle: [Telecommunication and Sensing],
  author: "Folkert Kevelam"
)

= Lecture 1 - Introduction

== Information

#definition[
  Information content:

  Information is related to probability: a less probable message contains
  more information

  $
    I_j = log_2(1/P_j) = - log_2(P_j) space "[bit]"
  $

  Information is additive:

  $
    I_(i j) &= log_2(1/(P_i P_J)) = -log_2(P_i)-log_2(P_j) \
    &= I_i + I_j
  $

  iff the messages are independent.
]

#definition[
  Source entropy, the average amount of information per message generated
  by a source:

  $
    H = sum_(j=1)^M P_j I_j = sum_(j=1)^M P_j log_2(1/P_j) space "[bit/symbol]"
  $

  In a binary system, the maximum source entropy will be when $P_1 = P_0 = 0.5$.
  The speed of a source:

  $
    R = H/T space "[bit/s]"
  $
]

#theorem[
  Shannon-Hartley theorem:

  $
    C = B dot log_2( 1 + S/N) space "[bit/s]"
  $

  - $C = "capacity [bit/s]"$
  - $B = "bandwidth [Hz]"$
  - $S/N = "ratio of signal power to the noise power"$
]

== Principles of range measurement

#definition[
  The transmitter "fires" a signal and the receiver measures the time delay
  $tau$ between the moments of transmission and reception of the echo.

  $
    2R = c dot tau arrow R = (c dot tau) / 2
  $

  with $c$ being the speed of light.
]

#definition[
  The ability of a radar to resolve two targets with a range difference
  $delta R$ is called *range resolution*.

  $
    delta R = (c dot tau_p) / 2 approx c/(2 B) space "[m]"
  $
]

== Modulation

#definition[
  Modulation: *manipulation of a signal waveform* to carry information, in
  order to transmit the signal at a specified frequency in the spectrum.

  $
    s(t) = R(t)cos(2 pi f_c t + phi(t))
  $

  with

  $
    R(t) = L{m(t)} space "linear modulation" \
    phi(t) = L{m(t)} space "angle modulation"
  $
]

== Practical signal waveforms

+ DC-value, mean value: $ w_(D C) = <w(t)> = lim_(T arrow infinity) 1/T integral_(-T/2)^(T/2) w(t) d t $
+ Instantaneous power: $ p(t) = v(t) dot i(t) $
+ Average power: $ P=<p(t)> = <v(t) dot i(t)> $
+ RMS-value: root-mean-square: $ w_(r m s) = sqrt(<w^2(t)>)  $
  For a resistive load: $ P = v_(r m s) i_(r m s) = (<v^2(t)>)/R = <i^2(t)>R $
+ Normalized power = power delivered to a $1 omega$ load.
  $ P = <w^2(t)> = lim_(T arrow infinity) 1/T integral_(-T/2)^(T/2) w^2(t) d t space "[W] = [J/s]" $
  $w(t)$ is a *power waveform* iff $0 < P < infinity$.
+ Normalized energy = energy dissipated in a $1 omega$ load.
  $ E = lim_(T arrow infinity) integral_(-T/2)^(T/2) w^2(t) d t space "[J]" $
  $w(t)$ is an *energy waveform* iff $0 < E < infinity$.

A signal waveform $w(t)$ cannot both be an *energy waveform* and a *power waveform*.
Practical waveforms are always energy waveforms.