Discrete Time Processing of Continuous Time Signals 2.9 2002/08/08 2007/07/20 11:29:54.841 GMT-5 Justin Romberg jrom@ece.gatech.edu Prashant Singh prash@ece.rice.edu Richard G. Baraniuk richb@rice.edu Mariyah Poonawala mariyah@rice.edu continuous time discrete discrete time processing processing This module focus on the discrete time processing of continuous time signals.

DSP System

How is the CTFT of y(t) related to the CTFT of f(t) (Figure 1)? Let G ω = reconstruction filter freq. response Y ω G ω Yimp ω where Yimp ω is impulse sequence created from ys n . So, Y ω G ω Ys ω T G ω H ω T Fs ω T Y ω G ω H ω T 1 T r F ω F 2 r T Y ω 1 T G ω H ω T r F ω F 2 r T Now, lets assume that f(t) is bandlimited to T T Ωs 2 Ωs 2 and G ω is a perfect reconstruction filter. Then Y ω F ω H ω T ω T 0 Y ω has the same "bandlimit" as F ω . So, for bandlimited signals, and with a high enough sampling rate and a perfect reconstruction filter (Figure 2)

FT's of original (analog) signal f(t) and sampled version of f(t) respectively.

is equivalent to using an analog LTI filter (Figure 3)

Implementing a discrete time filter (H) in analog

where Ha ω H ω T ω T 0 So, by being careful we can implement LTI systems for bandlimited signals on our computer!!! Important note: Ha ω = filter induced by our system. Ha ω is LTI only if h, the DT system, is LTI F ω , the input, is bandlimited and the sample rate is high enough.