DT Processing of CT Signals

DSP System

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

Then, Y c Ω G Ω T X c Ω Ω Ω s 2 0

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 where G eff Ω G Ω T Ω Ω s 2 0

Note G eff Ω is LTI if and only if the following two conditions are satisfied: G ω is LTI (in DT). 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 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 others might not be "missed." This is what we term time-varying behavior.

If 2 T 2 B and ω 1 B T , determine and sketch Y c Ω 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.

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

DSP Solution

Sampling Period/Rate First we must note that 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 240 Hz . Ω 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.