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Unformatted text preview: Purdue University: ECE438  Digital Signal Processing with Applications 1 ECE438  Laboratory 4: Sampling and Reconstruction of ContinuousTime Signals October 6, 2010 1 Introduction It is often desired to analyze and process continuoustime signals using a computer. However, in order to process a continuoustime signal, it must first be digitized. This means that the continuoustime signal must be sampled and quantized, forming a digital signal that can be stored in a computer. Analog systems can be converted to their discretetime counterparts, and these digital systems then process discretetime signals to produce discretetime out puts. The digital output can then be converted back to an analog signal, or reconstructed , through a digitaltoanalog converter. Figure 1 illustrates an example, containing the three general components described above: a sampling system, a digital signal processor, and a reconstruction system. When designing such a system, it is essential to understand the effects of the sampling and reconstruction processes. Sampling and reconstruction may lead to different types of distortion, including lowpass filtering, aliasing, and quantization. The system designer must insure that these distortions are below acceptable levels, or are compensated through additional processing. Sampling System DSP Processor Reconstruction System x(t) y(n) z(n) s(t) Figure 1: Example of a typical digital signal processing system. Questions or comments concerning this laboratory should be directed to Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907; (765) 494 0340; [email protected] Purdue University: ECE438  Digital Signal Processing with Applications 2 1.1 Sampling Overview Sampling is simply the process of measuring the value of a continuoustime signal at certain instants of time. Typically, these measurements are uniformly separated by the sampling period, T s . If x ( t ) is the input signal, then the sampled signal, y ( n ), is as follows: y ( n ) = x ( t )  t = nT s . A critical question is the following: What sampling period, T s , is required to accurately represent the signal x ( t )? To answer this question, we need to look at the frequency domain representations of y ( n ) and x ( t ). Since y ( n ) is a discretetime signal, we represent its fre quency content with the discretetime Fourier transform (DTFT), Y ( e jω ). However, x ( t ) is a continuoustime signal, requiring the use of the continuoustime Fourier transform (CTFT), denoted as X ( f ). Fortunately, Y ( e jω ) can be written in terms of X ( f ): Y ( e jω ) = 1 T s ∞ summationdisplay k =∞ X ( f )  f = ω 2 πk 2 πTs = 1 T s ∞ summationdisplay k =∞ X parenleftBigg ω 2 πk 2 πT s parenrightBigg . (1) Consistent with the properties of the DTFT, Y ( e jω ) is periodic with a period 2 π . It is formed by rescaling the amplitude and frequency of X ( f ), and then repeating it in frequency...
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 Spring '08
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 Digital Signal Processing, Signal Processing, Purdue University, sampling rate

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