Assuming our unit of type is the same [[Sampling|Sampling Interval]] (1 sample / unit time), the default samples are [[Discrete]] *ie.* [[Signal]] outputs are given at each sample time. So if the [[Signal]] was sampled from some analogue ([[Continuous Function|Continuous]]) [[Domain]] [[Signal]], then we have $X(t),t\in \mathcal{C}$ and $X_{t}=X(t T_{s}),t \in \Z^{+}$. >[!example] >[[CD Quality]] : $T_{s}= \frac{1}{44100}\pu{ \sec }$. Let $f_{s}$ and $\omega_{s}$ be the [[Sampling|Sample Frequency]] and [[Sampling|Sample Angular Frequency]]. With our unit time being our sample interval, the sample rate is $f_{s}=1$ eg. 1 sample per unit time. [[Finite]] Impulse Response [[Signal Filter|Filters]], also known as "feed-forward filters", are used in real-time filters for high frequency sampling. Generally, these type filters take a few (relative to the sample rate) samples and combine them via a [[Linear Combination]]. Linear combinations are used as having a linear system means for any arbitrary [[Continuous Function|Continuous]] single $X(t)$ we can decompose it as a linear combination of [[Fourier Basis|fourier bases]]. >[!example] Simple Delay by 1 Sample > $\huge \underbrace{y_{t}}_{\text{Outputs}} = \underbrace{x_{t}}_{\text{Inputs}} $ > We can also represent the whole ([[Discrete]]) signal as $\vec x$. >[!example] Slightly more Interesting: > $\huge \begin{align} > y_{t} &= x_{t} + x_{-1} > \end{align} > $ > > Lets see what happens to an input signal $\vec x$ given as a [[Phasor]] of [[Frequency]] $\omega$. > $\huge \begin{align} > x_{t} &= e^{i\omega t} \\ > y_{t} &= e^{i\omega t}+e^{i\omega \left( t-1 \right) } \\ > &= e^{i\omega t} \underbrace{\left( 1+e^{-iw} \right)}_{\text{Invariant under } t} > \end{align} $ > > This form of polar times some cartesian number comes up, and $\left( 1+e^{-i \omega} \right)$ is notated as $H(\omega)$ ([[Frequency Response Function]]). Note that $1+e^{-i\omega}$ is particular to this case. > > The magnitude response is > $\huge \begin{align} > r_{\omega}&=\sqrt{ (1+\cos \omega)^{2}+\sin(\omega)^{2} } \\ > &= \sqrt{ 1+2\cos\omega + \cos(\omega)^{2} + \sin(\omega)^{2} } \\ > &= \sqrt{ 2 } \sqrt{ 1+\cos \omega } > \end{align} > $ > > ![[Pasted image 20260929145129.png|500]]