1: Discrete-Time Linear Systems
- Page ID
- 126572
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Course Description
This course introduces students to discrete-time signal analysis and linear systems. Topics include time-domain analysis of discrete-time linear time-invariant (LTI) systems; solutions of difference equations; system functions and digital filters; stability and causality; discrete-time Fourier series; discrete-time Fourier transforms and discrete Fourier transforms; z-transforms; sampling and the sampling theorem; discrete-time state equations; and communication systems. Students use analysis tools to design systems that meet functional specifications.
Course Learning Objectives
The objectives of the course are to learn to analyze existing discrete-time linear systems and to modify them or create new ones to satisfy specific requirements. By the end of this course, students should be able to:
- Prove whether systems are linear, time-invariant, stable, and causal. Explain the implications of these properties.
- Determine periodicity of discrete-time signals.
- Graph mathematical descriptions of discrete signals and write mathematical descriptions for graphed functions.
- Develop an accurate mathematical description of a discrete linear system and represent it in a block diagram. Determine the difference equation representing a discrete-time system from its block diagram representation.
- Explain the conditions required for discrete samples of a continuous-time signal to capture all of the information in the original signal. Sketch the resulting discrete-time frequency spectrum from the spectrum of the continuous-time signal.
- Translate among the different representations of a system, including difference equation, impulse response, frequency response, and transfer function.
- Convolve two discrete signals together either graphically or analytically.
- Compute the Discrete-Time Fourier Series (DTFS) for periodic signals and sketch their spectra. Explain conceptually how the DTFS relates to time-domain properties of the signal.
- Compute the Discrete-Time Fourier transform (DTFT) of basic signals such as the unit sample, sinusoids, and exponentials. Recognize and exploit DTFT properties to find the transform and inverse transform of more complicated signals.
- Apply the frequency response of a Linear-Time-Invariant (LTI) system to both the DTFT and DTFS to find the system output.
- Determine the transfer function of an LTI system using the z-transform, and solve for the system output for a given input, system description, and initial conditions.
- Determine system stability from the z-transform.
- Verify analytically obtained solutions using MATLAB and analyze system/signal behavior in time and frequency.