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Unformatted text preview: Chapter 11 Sampling and Reconstruction Digital hardware, including computers, take actions in discrete steps. So they can deal with discrete time signals, but they cannot directly handle the continuoustime signals that are prevalent in the physical world. This chapter is about the interface between these two worlds, one continuous, the other discrete. A discretetime signal is constructed by sampling a continuoustime signal, and a continuoustime signal is reconstructed by interpolating a discretetime signal. 11.1 Sampling A sampler for complexvalued signals is a system Sampler T : [ Reals → Complex ] → [ Integers → Complex ] , (11.1) where T is the sampling interval (it has units of seconds/sample). The system is depicted in figure 11.1 . The sampling frequency or sample rate is f s = 1 / T , in units of samples/second (or sometimes, Hertz), or ω s = 2 π / T , in units radians/second. If y = Sampler T ( x ) then y is defined by 2200 n ∈ Integers , y ( n ) = x ( nT ) . (11.2) 11.1.1 Sampling a sinusoid Let x : Reals → Reals be the sinusoidal signal 2200 t ∈ Reals , x ( t ) = cos ( 2 π ft ) , (11.3) y : Integers → Complex x : Reals → Complex Sampler T Figure 11.1: Sampler. 373 374 CHAPTER 11. SAMPLING AND RECONSTRUCTION Basics: Units Recall that frequency can be given with any of various units. The units of the f in ( 11.3 ) and ( 11.4 ) are Hertz, or cycles/second. In ( 11.3 ), it is sensible to give the frequency as ω = 2 π f , which has units of radians/second. The constant 2 π has units of radians/cycle, so the units work out. Moreover, the time argument t has units of seconds, so the argument to the cosine function, 2 π ft , has units of radians, as expected. In the discrete time case ( 11.4 ), it is sensible to give the frequency as 2 π fT , which has units of radians/sample. The sampling interval T has units of seconds/sample, so again the units work out. Moreover, the integer n has units of samples, so again the argument to the cosine function, 2 π fnT , has units of radians, as expected. In general, when discussing continuoustime signals and their sampled discrete time signals, it is important to be careful and consistent in the units used, or con siderable confusion can result. Many texts talk about normalized frequency when discussing discretetime signals, by which they simply mean frequency in units of radians/sample. This is normalized in the sense that it does not depend on the sampling interval. where f is the frequency of the sinewave in Hertz. Let y = Sampler T ( x ) . Then 2200 n ∈ Integers , y ( n ) = cos ( 2 π fnT ) . (11.4) Although this looks similar to the continuoustime sinusoid, there is a fundamental difference. Be cause the index n is discrete, it turns out that the frequency f is indistinguishable from frequency f + f s when looking at the discretetime signal. This phenomenon is called aliasing ....
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This note was uploaded on 09/29/2010 for the course EE 20N taught by Professor Ayazifar during the Spring '08 term at University of California, Berkeley.
 Spring '08
 Ayazifar

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