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Discrete Time Processing of Continuous Time Signals

Module by: Justin Romberg

Summary: This module focus on the discrete time processing of continuous time signals.

Figure 1: DSP System
Figure 1 (fig1.png)

How is the CTFT of y(t) related to the CTFT of f(t) (Figure 1)?

Let Gⅈω G ω = reconstruction filter freq. response Yⅈω=GⅈωYimpⅈω Y ω G ω Yimp ω where Yimpⅈω Yimp ω is impulse sequence created from ysn ys n . So, Yⅈω=GⅈωYsⅇⅈωT=GⅈωHⅇⅈωTFsⅇⅈωT Y ω G ω Ys ω T G ω H ω T Fs ω T Yⅈω=GⅈωHⅇⅈωT1T∑r=-∞∞FⅈωF2πrT Y ω G ω H ω T 1 T r F ω F 2 r T Yⅈω=1TGⅈωHⅇⅈωT∑r=-∞∞FⅈωF2πrT 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 -πTπT=-Ωs2Ωs2 T T Ωs 2 Ωs 2 and Gⅈω G ω is a perfect reconstruction filter. Then Yⅈω=FⅈωHⅇⅈωTif|ω|≤πT0otherwise Y ω F ω H ω T ω T 0

note:

Yⅈω Y ω has the same "bandlimit" as Fⅈω F ω .
So, for bandlimited signals, and with a high enough sampling rate and a perfect reconstruction filter (Figure 2)

Figure 2: FT's of original (analog) signal f(t) and sampled version of f(t) respectively.
Figure 2 (fig2.png)

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

Figure 3: Implementing a discrete time filter (H) in analog
Figure 3 (fig3.png)

where Haⅈω=HⅇⅈωTif|ω|≤πT0otherwise 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:

Haⅈω Ha ω = filter induced by our system.

Haⅈω Ha ω is LTI only if

  • hh, the DT system, is LTI
  • Fⅈω F ω , the input, is bandlimited and the sample rate is high enough.

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