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,
:
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
.
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.
Dsp solution
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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