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Discrete-Time Processing of CT Signals

Module by: Robert Nowak

Summary: 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

Figure 1
DSP System
DSP System (sec9_fig1.png)

Analysis

Y c Ω= H LP ΩYΩT Y c Ω H LP Ω Y Ω T (1)
where we know that Yω=XωGω Y ω X ω G ω and Gω G ω is the frequency response of the DT LTI system. Also, remember that ω≡ΩT ω Ω T So,
Y c Ω= H LP ΩGΩTXΩT Y c Ω H LP Ω G Ω T X Ω T (2)
where Y c Ω Y c Ω and H LP Ω H LP Ω are CTFTs and GΩT G Ω T and XΩT X Ω T are DTFTs.

Recall:

Xω=2πT∑k=-∞∞ X c ω-2πkT X ω 2 T k X c ω 2 k T OR XΩT=2πT∑k=-∞∞ X c Ω-k Ω s X Ω T 2 T k X c Ω k Ω s
Therefore our final output signal, Y c Ω Y c Ω , will be:
Y c Ω= H LP ΩGΩT2πT∑k=-∞∞ X c Ω-k Ω s Y c Ω H LP Ω G Ω T 2 T k X c Ω k Ω s (3)
Now, if X c Ω X c Ω is bandlimited to - Ω s 2 Ω s 2 Ω s 2 Ω s 2 and we use the usual lowpass reconstruction filter in the D/A, Figure 2:

Figure 2
Figure 2 (sec9_fig2.png)

Then,

Y c Ω=GΩT X c Ωif|Ω|< Ω s 20otherwise Y c Ω G Ω T X c Ω Ω Ω s 2 0 (4)

Summary

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

Y c Ω= G eff Ω X c Ω Y c Ω G eff Ω X c Ω (5)
where G eff Ω=GΩTif|Ω|< Ω s 20otherwise G eff Ω G Ω T Ω Ω s 2 0

Figure 3
Figure 3 (sec9_fig3.png)

Note

G eff Ω G eff Ω is LTI if and only if the following two conditions are satisfied:

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

Example 1

Figure 4
Figure 4 (sec9_fig4.png)

If 2πT>2B 2 T 2 B and ω 1 <BT ω 1 B T , determine and sketch Y c Ω Y c Ω using Figure 4.

Application: 60Hz Noise Removal

Figure 5
Figure 5 (sec9_fig5.png)

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 Figure 5. Figure 6 shows the EKG signal; it is barely noticeable as it has become overwhelmed by noise.

Figure 6: Our EKG signal, yt y t , is overwhelmed by noise.
Figure 6 (sec9_fig6.png)

DSP Solution

Figure 7
Figure 7 (sec9_fig7.png)
Figure 8
Figure 8 (sec9_fig7b.png)

Sampling Period/Rate

First we must note that |YΩ| Y Ω 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 =240Hz f s 240 Hz . Ω s =2π240rads Ω s 2 240 rad s

Figure 9
Figure 9 (sec9_fig8.png)

Digital Filter

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

Figure 10
Figure 10 (sec9_figL.png)

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