chap6slides - SAMPLING AND RECONSTRUCTION C Williams W...

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SAMPLING AND RECONSTRUCTION C. Williams & W. Alexander North Carolina State University, Raleigh, NC (USA) Fall 2011 C. Williams & W. Alexander (NCSU) SAMPLING AND RECONSTRUCTION Fall 2011 1 / 86
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Outline 1 Introduction 2 Ideal Reconstruction 3 Sampling of Bandpass Signals Uniform or First-Order Sampling Arbitrary Band Position In-Phase and Quad-Phase Components from Sampled Signals 4 Source Coding Delta Modulation 5 References C. Williams & W. Alexander (NCSU) SAMPLING AND RECONSTRUCTION Fall 2011 2 / 86
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Introduction Digital signal processing systems are used for many applications that involve continuous–time signals. C. Williams & W. Alexander (NCSU) SAMPLING AND RECONSTRUCTION Fall 2011 3 / 86
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Introduction Digital signal processing systems are used for many applications that involve continuous–time signals. These continuous–time signals are typically sampled and quantized to form discrete–time signals. C. Williams & W. Alexander (NCSU) SAMPLING AND RECONSTRUCTION Fall 2011 3 / 86
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Introduction Digital signal processing systems are used for many applications that involve continuous–time signals. These continuous–time signals are typically sampled and quantized to form discrete–time signals. This process is called analog to digital (A/D) conversion. C. Williams & W. Alexander (NCSU) SAMPLING AND RECONSTRUCTION Fall 2011 3 / 86
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Introduction A/D conversion is a very important process for many signal processing applications. Digital to analog (D/A) conversion is the process of forming a continuous–time signal from a sample sequence which is also very important for many applications. C. Williams & W. Alexander (NCSU) SAMPLING AND RECONSTRUCTION Fall 2011 4 / 86
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Introduction We often need to change the sample rate of sampled data for many digital signal processing applications. We either need to 1 reconstruct the signal of interest to obtain additional values, or 2 reduce the number of values used to represent the signal. C. Williams & W. Alexander (NCSU) SAMPLING AND RECONSTRUCTION Fall 2011 5 / 86
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Introduction When we reduce the sampling interval (increase the number of samples), we call it interpolation. When we increase the sampling interval (decrease the number of samples), we call it decimation. In either case, we need to be aware of the requirements to prevent aliasing as determined by the Sampling Theorem. We first discuss the ideal reconstruction of a sampled signal. C. Williams & W. Alexander (NCSU) SAMPLING AND RECONSTRUCTION Fall 2011 6 / 86
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Ideal Reconstruction Assume that we have sampled a continuous function x a ( t ) at F s = 1 T to obtain the sequence x ( n ) = x a ( nT ) . We can recover the original signal x ( t ) from the sampled sequence if aliasing does not occur. We will derive an equation for obtaining the the original continuous time signal x a ( t ) from x ( n ) . C. Williams & W. Alexander (NCSU) SAMPLING AND RECONSTRUCTION Fall 2011 7 / 86
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Ideal Reconstruction We know from the inverse Continuous-Time Fourier Transform that x a ( t ) is given as x a ( t ) = integraldisplay -∞ X a ( F ) e j 2 π Ft dF (1) Sampling x a ( t ) at t = nT yields x ( n ) x a ( nT ) = integraldisplay -∞ X a ( F ) e j 2 π Fn F s
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