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Welcome to Week 6 of Digital Signal Processing.

Wow, time flies! We've already covered filters and saw a lot of cool applications. As you may have noticed, the more we try to do interesting stuff, the more often we have to bridge the gap between our discrete-time models and the "real world". On many occasions since the beginning of the course, we have informally mentioned the idea of sampling a real world signal or talked about ways to convert a discrete-time representation into, say, a sound that we could listen to.

This week, we will formally address the problem of the "interface" between reality and discrete time. The necessity of such an interface stems from the fact that the preferred mathematical abstraction for many natural phenomena is a continuous-time model -- the model of physics, calculus, electronics and so on.

In Module 6 we will introduce the continuous-time paradigm and we will study how to convert a discrete-time signal into a continuous-time signal (a process called interpolation) and vice versa (a process called sampling). The good news is that we will be able to prove very powerful mathematical results using Hilbert space theory -- all that initial work is going to be useful again! Even more importantly, the Sampling Theorem will show us that in most cases, if we process a sampled version of a signal and then interpolate the result, we are not going to lose any information; in practice, it's as if we never left the continuous-time domain at all, but with all the advantages of digital processing!

Finally, we will conclude with an extra section (6.X) that will show you another form of sampling (there are many!) In particular, we will look at how to measure rainfall in a weather station.

Indeed an intense module, but extraordinarily interesting both from the theoretical and from the applied point of view!

Day 11

Video lectures:

• 6.0 - Introduction
• 6.1 - The Continuous-Time Paradigm
• 6.2 - Interpolation
• 6.3 - The space of bandlimited signals

Lecture slides for Module 6: full version or trimmed version.

Signal of the day:

• Fukushima (IPython Notebook)

Numerical example:

• Interpolation and Sampling (Python), (access on GitHub)

Practice Homework:

• Practice Homework 11 (Solutions)

Homework (due December 7, 5:00pm CET):

• Homework 11
• Feedback 11

Programming assignments (due December 28, 5:00pm CET):

• Programming assignment 2

Day 12

Video lectures:

• 6.4 - Sampling and Aliasing - Introduction
• 6.5 - Sampling and Aliasing
• 6.6 - Discrete-time Processing and Continuous-time Signals
• 6.x - Another example of sampled acquisition

Lecture slides for Module 6: full version or trimmed version.

Practice Homework:

• Practice Homework 12 (Solutions)

Homework (due December 7, 5:00pm CET):

• Homework 12
• Feedback 12

Notes and external resources

For a more in-depth treatment of sampling and interpolation with a similar perspective, we recommend Chapter 5, ‘’Sampling and Interpolation’’, of the “Foundations of Signal Processing’’ book by Vetterli, Kovacevic and Goyal, downloadable here

For the original version of the Shannon sampling theorem, see here and look at the original paper (alternate link); the sampling theorem is Theorem 13 on p. 34. Regarding the paper, this is probably the most important paper written about communications, and it is both profound and readable, so truly a masterpiece!

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