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+#import "../template/lib.typ": *
+#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
+
+==== Unit impulse and unit step signals