# 1.8 Discrete-time processing of ct signals

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The module will provide analysis and examples of how a continuous-time signal is converted to a digital signal and processed.

## Analysis

${Y}_{c}()={H}_{\mathrm{LP}}()Y(T)$
where we know that $Y()=X()G()$ and $G()$ is the frequency response of the DT LTI system. Also, remember that $\equiv T$ So,
${Y}_{c}()={H}_{\mathrm{LP}}()G(T)X(T)$
where ${Y}_{c}()$ and ${H}_{\mathrm{LP}}()$ are CTFTs and $G(T)$ and $X(T)$ are DTFTs.
$X()=\frac{2\pi }{T}\sum$ X c 2 k T OR $X(T)=\frac{2\pi }{T}\sum$ X c k s
Therefore our final output signal, ${Y}_{c}()$ , will be:
${Y}_{c}()={H}_{\mathrm{LP}}()G(T)\frac{2\pi }{T}\sum$ X c k s
Now, if ${X}_{c}()$ is bandlimited to $\left[-\left(\frac{{}_{s}}{2}\right) , \frac{{}_{s}}{2}\right]$ and we use the usual lowpass reconstruction filter in the D/A, :

Then,

${Y}_{c}()=\begin{cases}G(T){X}_{c}() & \text{if \left|\right|< \frac{{}_{s}}{2}}\\ 0 & \text{otherwise}\end{cases}$

## Summary

For bandlimited signals sampled at or above the Nyquist rate, we can relate the input and output of the DSP systemby:

${Y}_{c}()={G}_{\mathrm{eff}}(){X}_{c}()$
where ${G}_{\mathrm{eff}}()=\begin{cases}G(T) & \text{if \left|\right|< \frac{{}_{s}}{2}}\\ 0 & \text{otherwise}\end{cases}$

## Note

${G}_{\mathrm{eff}}()$ is LTI if and only if the following two conditions are satisfied:

• $G()$ is LTI (in DT).
• ${X}_{c}(T)$ is bandlimited and sampling rate equal to or greater than Nyquist. For example, if we had a simplepulse described by ${X}_{c}(t)=u(t-{T}_{0})-u(t-{T}_{1})$ where ${T}_{1}> {T}_{0}$ . If the sampling period $T> {T}_{1}-{T}_{0}$ , then some samples might "miss" the pulse while othersmight not be "missed." This is what we term time-varying behavior .

If $\frac{2\pi }{T}> 2B$ and ${}_{1}< BT$ , determine and sketch ${Y}_{c}()$ using .

## Application: 60hz noise removal

Unfortunately, in real-world situations electrodes also pick up ambient 60 Hz signals from lights, computers, etc. . In fact, usually this "60 Hz noise" is much greater in amplitude than the EKG signal shown in . shows the EKG signal; it is barely noticeable as it has become overwhelmed by noise.

## Sampling period/rate

First we must note that $\left|Y()\right|$ is bandlimited to60 Hz. Therefore, the minimum rate should be 120 Hz. In order toget the best results we should set ${f}_{s}=\mathrm{240}\mathrm{Hz}$ . ${}_{s}=2\pi 240\frac{\mathrm{rad}}{s}$

## Digital filter

Therefore, we want to design a digital filter that will remove the 60Hz component and preserve the rest.

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