{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

19_Sampling - Chapter 19 Sampling Chapter Outline 19.1...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
723 Chapter 19 Sampling Chapter Outline 19.1 FOURIER TRANSFORM OF A SAMPLED SIGNAL ............................ 725 19.1.1 Introduction ................................................................................................................................ 725 19.1.2 Ideal Sampling ......................................................................................................................... 725 19.1.3 Fourier Transform of a Continuous-Time Sampled Signal ................................. 726 19.2 RECONSTRUCTION OF SIGNALS FROM THEIR SAMPLES ............. 730 19.3 ALIASING AND THE NYQUIST SAMPLING THEOREM ................... 734 19.3.1 Introduction ................................................................................................................................ 734 19.3.2 A Sinusoid .................................................................................................................................. 734 19.3.3 General Case of Aliasing .................................................................................................... 737 19.3.4 Nyquist Sampling Theorem ............................................................................................... 739 19.4 ZERO-ORDER HOLD ......................................................................... 742 19.4.1 The Definition .......................................................................................................................... 742 19.4.2 Transfer Function of the ZOH ........................................................................................... 742 19.4.3 Frequency Response of a ZOH ........................................................................................ 744 19.4.4 Smoothing Filter ..................................................................................................................... 746 19.5 AN EXAMPLE .................................................................................... 746 19.5.1 Introduction ................................................................................................................................ 746 19.5.2 Fourier Transform of the Sampled Signal ................................................................... 747 19.5.3 Reconstruction of the Signal ............................................................................................. 749 19.5.4 Anti-Aliasing Filter ................................................................................................................ 750 19.5.5 MATLAB Experiments ........................................................................................................ 751 19.6 CHAPTER SUMMARY ........................................................................ 754 19.7 HOMEWORK FOR CHAPTER 19 ....................................................... 755 In Chapter 17 we introduced the concept of sampling. We also briefly described analog-to-digital converters and digital-to-analog converters, electronic devices which sample an electronic signal and reconstruct a signal from its samples. In this chapter we will explore this concept, sampling, in more depth. We will explore the relationship between a signal and its samples by developing the conditions under which it is possible to exactly reconstruct a signal from its samples. It is straightforward to extract the sampled signal from the original continuous-time signal. The exact relationship between the sampled version of the signal and the signal itself is more complicated, however. In this chapter we will give a complete discussion of the relationship between a signal and its samples. In this chapter we address the following question: “Can we recover a continuous- time signal from its samples?” The answer to this question is contained in the
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
724 Chapter 19 Sampling Nyquist sampling theorem. We will fully develop the concepts behind this theorem. This theorem will tell us what information is lost in the sampling process, and what information about the original signal can be extracted from the sample version of the signal. Hence, we can characterize those signals which can be exactly reconstructed from their samples. We can also characterize how the sampled signal is corrupted by those signal components which can’t be reconstructed (aliasing). This characterization explains how we can filter a signal before sampling so that the signal components of interest can be recovered from the (filtered) signal samples. These results are fundamental to the understanding of the computer processing of sampled signals. Nyquist’s sampling theorem suggests how we can exactly reconstruct a signal from its samples (when such reconstruction is possible). This reconstruction formula is generally not practical for the real-time reconstruction of a signal from its samples. Usually a DAC is used, which in signals and systems terminology is a zero-order hold (ZOH). In this chapter we develop a transfer function for a ZOH. This transfer function allows us to evaluate the distortion that is introduced by a ZOH when it is used to reconstruct a signals from its samples.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern