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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 continuous-time signals that are prevalent in the physical world. This chapter is about the interface between these two worlds, one continuous, the other discrete. A discrete-time signal is constructed by sampling a continuous-time signal, and a continuous-time signal is reconstructed by interpolating a discrete-time signal. 11.1 Sampling A sampler for complex-valued 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 continuous-time 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 discrete-time 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 continuous-time 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 discrete-time 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