4: Time Domain Analysis of Discrete Time Systems
- Page ID
- 22864
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 4.1: Discrete Time Systems
- This page discusses discrete time systems, particularly focusing on linear and time-invariant (LTI) systems in digital signal processing. It explains linearity via additivity and homogeneity, and time invariance regarding consistent responses to time shifts. LTI systems enable easier computations and the use of frequency domain techniques.
- 4.2: Discrete Time Impulse Response
- This page explains that the output of a discrete-time linear time-invariant (LTI) system is determined by its impulse response and the input signal. The impulse response defines the system's reaction to a unit impulse and is used in convolution to analyze any input signal. The process involves breaking down the input into impulses, calculating individual responses, and combining them.
- 4.3: Discrete Time Convolution
- This page discusses convolution, a key concept in electrical engineering for analyzing linear time-invariant systems and their outputs based on impulse responses. It includes a graphical explanation of the convolution operation, which involves periodic extensions and point-wise products of functions. Circular convolution is also covered, allowing computations for periodic signals.
- 4.4: Properties of Discrete Time Convolution
- This page covers key properties of discrete time convolution in signal processing, including associativity, commutativity, and duration. It proves these properties mathematically, clarifying their significance in signal manipulation. The duration property states that the convolution of two signals equals the sum of their durations minus one.
- 4.5: Eigenfunctions of Discrete Time LTI Systems
- This page introduces linear time invariant (LTI) systems, explaining how eigenfunctions aid in calculating outputs from complex exponential inputs. It notes that the output is scaled by an eigenvalue derived from the impulse response, highlighting the usefulness of complex exponentials in analyzing discrete time signals through discrete time Fourier transforms and series for both aperiodic and periodic signals.
- 4.6: BIBO Stability of Discrete Time Systems
- This page discusses BIBO stability, which ensures that bounded inputs lead to bounded outputs in a system. It defines bounded signals as those that do not exceed a finite value. For discrete-time systems, this stability requires an absolutely summable impulse response. In the z-domain, BIBO stability is indicated by a region of convergence that includes the unit circle, which is essential for predictable system behavior with bounded signals.
- 4.7: Linear Constant Coefficient Difference Equations
- This page discusses difference equations, which model discrete-time systems and describe rate changes in quantities, such as fund growth with interest. It highlights linear constant coefficient difference equations, including their expressions and roles in applications like digital filters and the Fibonacci sequence. Understanding these equations is crucial in fields such as electrical engineering.
- 4.8: Solving Linear Constant Coefficient Difference Equations
- This page outlines the methods for solving linear constant coefficient difference equations, especially initial value problems. It highlights the need for general solutions, which include homogeneous and particular solutions, derived from the characteristic polynomial and unit impulse response, respectively. Examples such as the Fibonacci sequence and feedback systems elucidate these methods.