4.1 Discrete-time processing of ct signals

Dt processing of ct signals

Dsp system

Analysis

Then,

Summary

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

Note

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

• $G(\omega )$ 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 ${\omega }_1< BT$ , determine and sketch $Y_c(\Omega )$ 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.

Dsp solution

Sampling period/rate

First we must note that $\left|Y(\Omega )\right|$ is bandlimited to±60 Hz.Therefore, the minimum rate should be 120 Hz. In order to get the best results we should set $$f_s=\mathrm{240}\mathrm{Hz}$$ . $${\Omega }_s=2\pi 240\frac{\mathrm{rad}}{s}$$

Digital filter

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