172 lines
3.3 KiB
Plaintext
172 lines
3.3 KiB
Plaintext
#import "../template/lib.typ": *
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#set page(paper: "a4")
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#show: notes.with(
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title: [WB2235],
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subtitle: [Signals and Systems],
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author: "Folkert Kevelam"
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)
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= Lecture 1 - Continuous and discrete-time signals
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#definition[
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A signal is a variable changing in a relation to another variable.
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]
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== Continuous-time signals vs discrete-time signals
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=== Signals in continuous-time
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The independent variable $t$ exists in $RR$ and exists between $-infinity$
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and $infinity$. *sampling* converts a continuous-time signal and converts it
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into a discrete-time signal.
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==== Properties
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Total energy of $x(t)$ over interval $t_1 <= t <= t_2$
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$
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integral_(t_1)^(t_2) |x(t)|^2 d t
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$
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Average power of $x(t)$ over interval $t_1 <= t <= t_2$
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$
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1/(t_2 - t_1) integral_(t_1)^(t_2) |x(t)|^2 d t
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$
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==== Operations
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- Time Shift: $x_2(t) = x(t - t_0)$
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- Reverse: $x_2(t) = x(-t)$
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- Scaling: $x_2(t) = x(alpha t)$ \
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$|alpha| > 1 arrow$ linearly compressed signals \
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$|alpha| < 1 arrow$ linearly stretched signals
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- Time transformation: $x_2(t) = x(alpha t + beta)$ \
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First shift then scale
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==== Periodic signals
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#definition[
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A signal $x(t)$ is periodic with period $T$ if:
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$
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forall t: x(t + T) = x(t)
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$
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with $T in RR, T > 0$
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]
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==== Even and Odd Signals
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#definition[
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A signal $x(t)$ is even if
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$
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forall t: x(-t) = x(t)
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$
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]
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#definition[
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A signal $x(t)$ is odd if
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$
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forall t: x(-t) = -x(t)
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$
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]
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=== Signals in discrete-time
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The independent variable $n$ lives in $ZZ$ and exists between some finite
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points. The book uses the syntax $x[n]$ to denote a discrete-signal in time.
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A discrete-time signal only has a value at the sampling points. *interpolation*
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converts a discrete-time signal to a continuous-time signal.
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==== Inherently digital signals
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when the independent variable is inherently discrete.
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==== Properties
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Total energy of $x[n]$ over interval $n_1 <= n <= n_2$ is
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$
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sum_(n=n_1)^(n_2) |x[n]|^2
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$
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Average power of $x[n]$ over interval $n_1 <= n <= n_2$ is
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$
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1/(n_2 - n_1 + 1) sum_(n=n_1)^(n_2) |x[n]|^2
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$
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==== Operations
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- Time shift: $x_2[n] = x[n-n_0]$
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- Reverse: $x_2[n] = x[-n]$
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- Scaling: $x_2[n] = x[alpha n]$ \
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$|alpha| > 1 arrow$ linearly compressed signals \
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$|alpha| < 1 arrow$ linearly strecthed signals \
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when $alpha$ is larger than 1, lose information due to decimation. \
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when $alpha$ is smaller than 1, we need to interpolate to add more information.
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- Time transformation: $x_2[n] = x[alpha n + beta]$
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==== Periodic signals
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#definition[
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A signal $x(n)$ is periodic with period $N$ if:
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$
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forall n: x[n + N] = x[t]
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$
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with $N in NN, N > 0$
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]
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==== Even and Odd Signals
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#definition[
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A signal $x[n]$ is even if
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$
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forall n: x[-n] = x[n]
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$
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]
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#definition[
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A signal $x[n]$ is odd if
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$
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forall n: x[-t] = -x[n]
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$
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]
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=== Elementary Signals
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==== Sinusoidal Signals
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CT: $x(t) = A cos(omega t + phi)$
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Same as: $x(t) = A sin(omega t + phi + pi/2)$
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with:
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- $A$ amplitude
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- $omega$ angular frequency [radians/time]
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- $phi$ phase [radians]
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DT: $x[n] = A cos(omega n + phi)$
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Same as: $x[n] = A sin(omega n + phi + pi/2)$
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with:
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- $A$ amplitude
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- $omega$ angular frequency [radians/time]
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- $phi$ phase [radians]
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==== Exponential Signals
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CT: $x(t) = C e^(alpha t)$
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with:
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- $C$ complex number
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- $alpha$ complex number
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==== Unit impulse and unit step signals
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