Skip to content Skip to navigation

Connexions

You are here: Home » Content » Discrete-Time Processing of CT Signals

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

      What are tags? tag icon

      Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    • External bookmarks
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    		(0 ratings)

Recently Viewed

This feature requires Javascript to be enabled.

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.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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)

Comments, questions, feedback, criticisms?

Send feedback