10: Sampling and Reconstruction
- Page ID
- 22906
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 10.1: Signal Sampling
- This page covers the sampling process that converts continuous time signals into discrete signals through uniform value selection. It addresses the non-injective nature of this conversion, which leads to aliasing. Additionally, it discusses the frequency domain representation of sampled signals, emphasizing that bandlimited signals sampled at high rates can be accurately reconstructed, although the sampling process itself is usually not invertible.
- 10.2: Sampling Theorem
- This page discusses a theorem enabling the reconstruction of continuous signals from their discrete samples, facilitating digital signal processing. It highlights its practical applications, especially in psychoacoustics, where sampling rates align with human hearing, optimizing audio quality. The theorem establishes conditions for perfect reconstruction and underscores the relationship between continuous and discrete signals, contributing to various applications in signal processing.
- 10.3: Signal Reconstruction
- This page discusses sampling and reconstruction processes of signals, emphasizing the conversion between continuous and discrete signals. It explains the role of lowpass filters in reconstruction, introduces cardinal basis splines for smooth signal creation, and details how higher orders of splines yield closer resemblance to the sinc function.
- 10.4: Perfect Reconstruction
- This page discusses the perfect reconstruction of a bandlimited continuous time signal from samples taken above the Nyquist rate (twice the bandlimit). It highlights the use of the Whittaker-Shannon interpolation formula and an ideal lowpass filter based on the sinc function for reconstruction. Adhering to the sampling criteria allows for the original signal to be recreated using shifted and scaled sinc functions.
- 10.5: Aliasing Phenomena
- This page explains the Nyquist-Shannon sampling theorem and aliasing, which occurs when a signal is sampled below the Nyquist frequency, causing loss of information and the emergence of indistinguishable "aliases." This process is non-invertible, preventing perfect signal reconstruction via Whittaker-Shannon interpolation. To reduce aliasing in signal processing, anti-aliasing filters are essential.
- 10.6: Anti-Aliasing Filters
- This page addresses the importance of anti-aliasing filters in signal processing to avoid distortion when sampling bandlimited signals. It explains aliasing resulting from sampling below the Nyquist rate and describes the ideal lowpass filter as the best solution, despite practical challenges. Through examples, the text highlights how these filters are essential for accurate signal reconstruction by comparing errors with and without their use.
- 10.7: Discrete Time Processing of Continuous Time Signals
- This page covers the process of converting continuous time signals to discrete time signals using ADCs and DACs, emphasizing the significance of sampling above the Nyquist frequency to prevent aliasing and the role of anti-aliasing filters. It addresses quantization limitations, the requirement for causal filters, challenges in approximating ideal filters, and practical issues like quantization noise. The effectiveness of modern technologies in mitigating these challenges is also highlighted.