# A Small Kit of $h(e^{-i\omega})$'s

In order to provide some feel for the effects of the various commonly used filters, Figure 4 reports the amplitude and phase of $h(e^{-i\omega})$ for various $h(L)$ lag distributions.

We have already calculated that for $h(L)=1-L$,

$$
h(e^{-i \omega})=2e^{i(\pi /2-\omega /2)}\sin(\omega /2),
$$

as the graphs confirm.

For $h(L)=1+L$, it is straightforward to calculate

$$
h(e^{-i\omega})=1+e^{-i\omega}=e^{-i\omega/2}(e^{+i\omega/2}+e^{-i\omega/2})=2e^{-i\omega/2}\cos (\omega/2),
$$

which again agrees with our graphs.

Notice that for $h(L)=(1-t_1 L - t_2 L^2)^{-1}$, we have chosen $(t_1,t_2)$ in the regions of peaked spectra of our Figure 2. Notice that as required, $h(e^{-i\omega})$ is characterized by peaks. (See Figure 2.)

Figure 4 collects the amplitude $|h(e^{-i\omega})|$ and phase
$\arg h(e^{-i\omega})$ of a kit of commonly used filters, grouped by family.
All are generated by
[`code/fig4_filter_kit.py`](https://github.com/thomassargent30/sargent-time-series/blob/main/code/fig4_filter_kit.py),
which evaluates $h(e^{-i\omega}) = \mathrm{num}(e^{-i\omega})/\mathrm{den}(e^{-i\omega})$
for $\omega \in [0, \pi]$.

```{figure} ../figures/fig4a_ma_filters.png
:name: fig-4a
:align: center
:width: 100%

**Figure 4a.** Moving-average / differencing filters. Note that $1 - L$ (a first
difference) suppresses low frequencies and amplifies high ones, while $1 + L$ does
the reverse; raising a filter to a power sharpens its effect.
```

```{figure} ../figures/fig4b_ar_filters.png
:name: fig-4b
:align: center
:width: 100%

**Figure 4b.** Autoregressive (inverse) filters. The filters
$(1 - t_1 L - t_2 L^2)^{-1}$ use $(t_1, t_2)$ from the peaked-spectrum region of
Figure 2; as required, $h(e^{-i\omega})$ then displays a peak — e.g.
$(1 - L + 0.8L^2)^{-1}$ shows a sharp resonance at an interior frequency.
```

```{figure} ../figures/fig4c_seasonal_filters.png
:name: fig-4c
:align: center
:width: 100%

**Figure 4c.** Seasonal filters in $L^{12}$. Over $[0, \pi]$ the amplitude shows
six ripples (the seasonal harmonics); $(1 - 0.9L^{12})^{-1}$ produces sharp peaks
at the seasonal frequencies, the comb-like response characteristic of seasonal
autoregressions.
```