semester06-signal_processing/assignment01
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finished exercise

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quirinecker
2026-04-26 22:39:48 +02:00
parent 930f7ae4a4
commit ec87b5838d
3 changed files with 9 additions and 9 deletions
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Assignment1_211.tex
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@@ -154,7 +154,7 @@ c_4 &= c_1 + c_2
\begin{align*}
c_5 &= c_1 \cdot c_2
= \left(\sqrt{34}\, e^{-j\,0.54}\right)\left(\frac{\sqrt{2}}{2} e^{-j \frac{3 \pi}{4}}\right)
\approx \frac{\sqrt{70}}{2} e^{-j\,2.8961} \\
\approx \sqrt{17} e^{-j\,2.8961} \\
\end{align*}
\begin{align*}
@@ -276,11 +276,11 @@ y_1(t) \ne y_2(t)
\[
y_1(t) &= \left(x(t - \alpha)\right)^2 \\
y_1(t) &= \left(x(t - \Delta)\right)^2 \\
\]
\[
y_2(t) &= y_1(t - \Delta) = \left(x(t - \alpha)\right)^2 \\
y_2(t) &= y_1(t - \Delta) = \left(x(t - \Delta)\right)^2 \\
\]
\[
@@ -290,18 +290,18 @@ y_1(t) \ne y_2(t)
\[
\implies \text{Time invariant}
\]
\item $y(t) = w(t) sin(\Omega_0 t)$
\item $y(t) = x(t) sin(\Omega_0 t)$
\[
y(t) &= x(t) \sin(n \Omega_0 t) \\
y(t) &= x(t) \sin(\Omega_0 t) \\
\]
\[
y_1(t) &= \left( \alpha \, x(t) \sin(n_1 \omega t) \right) + \left( \beta \, x(t) \sin(n_0 t) \right) \\
y_1(t) &= \left( \alpha \, x(t) \sin(\Omega_0 t) \right) + \left( \beta \, x(t) \sin(\Omega_0 t) \right) \\
\]
\[
y_t(t) &= \alpha \left( x(t) \sin(n_1 \omega t) \right) + \beta \left( x(t) \sin(n_0 t) \right) \\
y_t(t) &= \alpha \left( x(t) \sin(\Omega_0 t) \right) + \beta \left( x(t) \sin(\Omega_0 t) \right) \\
\]
\[
@@ -316,11 +316,11 @@ y_1(t) \ne y_2(t)
\[
y_1(t) &\coloneqq x(t - \alpha) \sin(\omega_0 t - \alpha) \\
y_1(t) = &\coloneqq x(t - \Delta) \sin(\Omega_0 t - \Delta) \\
\]
\[
y_2(t) &= y_1(t - \alpha) = x(t - \alpha) \sin(\omega_0 t - \alpha) \\
y_2(t) &= y_1(t - \Delta) = x(t - \Delta) \sin(\Omega_0 t - \Delta) \\
\]
\[