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Exercise

Use the Euler relation to show that

$\begin{align} \frac{e^{ix}+ e^{-ix}}{2} & = \frac{\cos(x) + i \sin(x) + \cos{x} - i \sin{x}}{2} \\ & = \cos(x) \end{align} \,$

Exercise

$\overline{e^{ix}} = \overline{\cos(x) + i \sin{x}} = \cos(x) - i \sin{x} = e^{-ix} \,$

Exercise

$\begin{align} \sin(x) \cos(y) + \cos(x) \sin(y) & = \frac{e^{ix} - e^{-ix}}{2i} \frac{e^{iy} + e^{-iy}}{2} + \frac{e^{ix} + e^{ix}}{2} \frac{e^{iy} - e^{-iy}}{2i} \\ & = \frac{e^{ix + iy} + e^{ix - iy} - e^{iy - ix} - e^{-iy -ix} + e^{ix + iy} - e^{ix - iy} + e^{iy - ix} - e^{-iy -ix}}{4i} \\ & = \frac{2 e^{ix - iy} - 2 e^{-ix - iy}}{4i} \\ & = \frac{e^{ix + iy} - e^{-i (x+y)}}{2i} \\ & = \sin(x+y) \end{align} \,$

Exercise

$\begin{align} \sin^2(x) + \cos^2(x) & = \left( \frac{e^{ix} - e^{-ix}}{2i} \right)^2 + \left( \frac{e^{ix} + e^{-ix}}{2} \right)^2\\ & = \left( \frac{e^{2ix} - 2 e^{ix - ix} + e^{-2ix}}{4 i^2} \right) + \left( \frac{e^{2ix} + 2 e^{ix - ix} + e^{-2ix}}{4} \right)\\ & = \frac{ -e^{2ix} + 2 - e^{-2ix} + e^{2ix} + 2 + e^{-2ix}}{4} \\ & = 1 \end{align}$

Exercise

Zu zeigen: $\sum_{t=0}^n q^t = \frac{1-q^{n+1}}{1-q}\,$

$\begin{align} \sum_{t=0}^n q^t & = 1 + q + q^2 + \ldots + q^n \\ (1-q) \sum_{t=0}^n q^t & = (1-q) \cdot 1 + (1-q) \cdot q + (1-q) \cdot q^2 + \ldots + (1-q) \cdot q^n \\ (1-q) \sum_{t=0}^n q^t & = 1 + (- q + q) + ( -q^2 + q^2) + \ldots + (-q^n + q^n) - q^{n+1} \\ (1-q) \sum_{t=0}^n q^t & = 1 - q^{n+1} \\ \sum_{t=0}^n q^t & = \frac{1 - q^{n+1}}{1-q} \\ \end{align} \,$

Exercise

Find another symmetric filter with k=2 that allows a quadratic trend $yt=a+bt+ct^2\,$ to pass without distortion.

$Y_t = a + bt +ct^2 +\epsilon_t \Rightarrow \operatorname{E}(Y_t) = a + bt + ct^2 \,$

$Z_t = \alpha Y_{t-2} + \beta Y_{t-1} + \gamma Y_t + \beta Y_{t+1} + \gamma Y_{t+2} \,$

$\begin{align} \operatorname{E}(Z_t) = & \alpha [a + b (t-2) + c(t-2)^2] + \\ & \beta[a + b (t-1) + c(t-1)^2] + \\ & \gamma[a + b t + ct^2] + \\ & \beta[a + b (t+1) + c(t+1)^2] + \\ & \alpha [a + b (t+2) + c(t+2)^2] \\ = & a + bt +ct^2 \end{align}$