Sampling 2.5 2002/08/08 2005/06/24 12:22:24.062 GMT-5 Justin Romberg jrom@rice.edu Prashant Singh prash@ece.rice.edu Richard G. Baraniuk richb@rice.edu Mariyah Poonawala mariyah@rice.edu continuous time CTFT discrete time DTFT This module deals with translating continuous time problem into discrete time

Introduction The digital computer can process discrete time signals using extremely flexible and powerful algorithms. However, most signals of interest are continuous time, which is how the almost always appear in nature. This module introduces the idea of translating continuous time problems into discrete time, and you can read on to learn more of the details and importance of sampling. Key Questions How do we turn a continuous time signal into a discrete time signal (sampling, A/D)? When can we reconstruct a CT signal exactly from its samples (reconstruction, D/A)? Manipulating the DT signal does what to the reconstructed signal?

Sampling Sampling (and reconstruction) are best understood in the frequency domain. We'll start by looking at some examples What CT signal f t has the CTFT shown below? f t 1 2 w F w w t

The CTFT of f t .

Hint: F w F 1 w F 2 w where the two parts of F w are:

f t 1 2 w F w w t What DT signal f s n has the DTFT shown below? f s n 1 2 w f s w w n

DTFT that is a periodic (with period 2 ) version of F w in .

Since F w 0 outside of -2 2 f t 1 2 w -2 2 F w w t Also, since we only use one interval to reconstruct f s n from its DTFT, we have f s n 1 2 w -2 2 f s w w n Since F w F s w on -2 2 f s n t n f t i.e. f s n is a sampled version of f t .

f t is the continuous-time signal above and f s n is the discrete-time, sampled version of f t

Generalization Of course, the results from the above examples can be generalized to any f t with F w 0 , w , where f t is bandlimited to .

F w is the CTFT of f t .

F s w is the DTFT of f s n .

F s w is a periodic (with period 2 ) version of F w . F s w is the DTFT of signal sampled at the integers. F w is the CTFT of signal. If f t is bandlimited to then the DTFT of the sampled version f s n f n is just a periodic (with period 2 ) version of F w .

Turning a Discrete Signal into a Continuous Signal Now, let's look at turning a DT signal into a continuous time signal. Let f s n be a DT signal with DTFT F s w

F s w is the DTFT of f s n .

Now, set f imp t n f s n δ t n The CT signal, f imp t , is non-zero only on the integers where there are impulses of height f s n .

What is the CTFT of f imp t ? f imp t n f s n δ t n F ∼ imp w t f imp t w t t n f s n δ t n w t n f s n t δ t n w t n f s n w n F s w So, the CTFT of f imp t is equal to the DTFT of f s n We used the sifting property to show t δ t n w t w n Now, given the samples f s n of a bandlimited to signal, our next step will be to see how we can reconstruct f t .

Block diagram showing the very basic steps used to reconstruct f t . Can we make our results equal f t exactly?