#import "../template/lib.typ": *
#import "@preview/cetz:0.4.2"
#import "@preview/cetz-plot:0.1.3"
#set page(paper: "a4")
#show: notes.with(
  title: [WB2235],
  subtitle: [Signals and Systems],
  author: "Folkert Kevelam"
)

= Lecture 1 - Continuous and discrete-time signals

#definition[
  A signal is a variable changing in a relation to another variable.
]

== Continuous-time signals vs discrete-time signals

=== Signals in continuous-time

The independent variable $t$ exists in $RR$ and exists between $-infinity$
and $infinity$. *sampling* converts a continuous-time signal and converts it
into a discrete-time signal.

==== Properties

Total energy of $x(t)$ over interval $t_1 <= t <= t_2$

$
  integral_(t_1)^(t_2) |x(t)|^2 d t
$

Average power of $x(t)$ over interval $t_1 <= t <= t_2$

$ 
  1/(t_2 - t_1) integral_(t_1)^(t_2) |x(t)|^2 d t
$

==== Operations

- Time Shift: $x_2(t) = x(t - t_0)$
- Reverse: $x_2(t) = x(-t)$
- Scaling: $x_2(t) = x(alpha t)$ \
  $|alpha| > 1 arrow$ linearly compressed signals \
  $|alpha| < 1 arrow$ linearly stretched signals
- Time transformation: $x_2(t) = x(alpha t + beta)$ \
  First shift then scale

==== Periodic signals

#definition[
  A signal $x(t)$ is periodic with period $T$ if:

  $
    forall t: x(t + T) = x(t)
  $

  with $T in RR, T > 0$
]

==== Even and Odd Signals

#definition[
  A signal $x(t)$ is even if

  $
    forall t: x(-t) = x(t)
  $
]

#definition[
  A signal $x(t)$ is odd if

  $
    forall t: x(-t) = -x(t)
  $
]

=== Signals in discrete-time

The independent variable $n$ lives in $ZZ$ and exists between some finite
points. The book uses the syntax $x[n]$ to denote a discrete-signal in time.
A discrete-time signal only has a value at the sampling points. *interpolation*
converts a discrete-time signal to a continuous-time signal.

==== Inherently digital signals

when the independent variable is inherently discrete.

==== Properties

Total energy of $x[n]$ over interval $n_1 <= n <= n_2$ is

$
  sum_(n=n_1)^(n_2) |x[n]|^2
$

Average power of $x[n]$ over interval $n_1 <= n <= n_2$ is

$
  1/(n_2 - n_1 + 1) sum_(n=n_1)^(n_2) |x[n]|^2
$

==== Operations

- Time shift: $x_2[n] = x[n-n_0]$
- Reverse: $x_2[n] = x[-n]$
- Scaling: $x_2[n] = x[alpha n]$ \
  $|alpha| > 1 arrow$ linearly compressed signals \
  $|alpha| < 1 arrow$ linearly strecthed signals \
  when $alpha$ is larger than 1, lose information due to decimation. \
  when $alpha$ is smaller than 1, we need to interpolate to add more information.
- Time transformation: $x_2[n] = x[alpha n + beta]$

==== Periodic signals

#definition[
  A signal $x(n)$ is periodic with period $N$ if:

  $
    forall n: x[n + N] = x[t]
  $

  with $N in NN, N > 0$
]

==== Even and Odd Signals

#definition[
  A signal $x[n]$ is even if

  $
    forall n: x[-n] = x[n]
  $
]

#definition[
  A signal $x[n]$ is odd if

  $
    forall n: x[-t] = -x[n]
  $
]

=== Elementary Signals

==== Sinusoidal Signals

CT: $x(t) = A cos(omega t + phi)$
Same as: $x(t) = A sin(omega t + phi + pi/2)$

with:
- $A$ amplitude
- $omega$ angular frequency [radians/time]
- $phi$ phase [radians]

DT: $x[n] = A cos(omega n + phi)$
Same as: $x[n] = A sin(omega n + phi + pi/2)$

with:
- $A$ amplitude
- $omega$ angular frequency [radians/time]
- $phi$ phase [radians]

==== Exponential Signals

CT: $x(t) = C e^(alpha t)$

with:
- $C$ complex number
- $alpha$ complex number

in DT we change to base from $e^a$ to $alpha$.

Differences between CT and DT:

CT:

+ the larger the magnitude of $omega_0$, the higher the rate of oscillation.
+ All signals with a different $omega_0$ can be distinguished from eachother.

DT:

+ The rate of oscillation grows when the magnitude grows to an odd multiple
  of $pi$, and drops when the magnitude grows towards an even multiple of
  $pi$.

==== Unit impulse and unit step signals

DT:

Unit Impulse:

$ delta [n] = cases( 
  1 "if" n = 0,
  0 "if" n eq.not 0) $

Step:

$ u [n] = cases(
  1 "if" n >= 0,
  0 "if" n < 0) $

Relation:

$ delta [n] = u[n] - u[n-1] $

The DT unit impulse is the *first difference of the DT unit step*.\
The DT unit step is the *running sum of the DT unit impulse*.

$u[n] = sum_(k=0)^(infinity) delta [n-k]$

= Lecture 2: Continuous and Discrete Systems