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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.

Dt processing of ct signals

Dsp system

Analysis

Y c H LP Y T
where we know that Y X G and G is the frequency response of the DT LTI system. Also, remember that T So,
Y c H LP G T X T
where Y c and H LP are CTFTs and G T and X T are DTFTs.
X 2 T k X c 2 k T OR X T 2 T k X c k s
Therefore our final output signal, Y c , will be:
Y c H LP G T 2 T k X c k s
Now, if X c is bandlimited to s 2 s 2 and we use the usual lowpass reconstruction filter in the D/A, :

Then,

Y c G T X c s 2 0

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 eff X c
where G eff G T s 2 0

Note

G 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 2 T 2 B and 1 B T , determine and sketch Y c using .

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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.

Our EKG signal, y t , is overwhelmed by noise.

Dsp solution

Sampling period/rate

First we must note that Y is bandlimited to60 Hz. Therefore, the minimum rate should be 120 Hz. In order toget the best results we should set f s 240 Hz . s 2 240 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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Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
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