1.3: Convolution
- Page ID
- 126929
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives: Convolution
- Understand how the convolution equation is a consequence of linearity and time invariance.
- Practice computing convolutions using both the equation directly and the flip-and-shift (ticker tape) method.
- Begin to develop insight into how the optimal approach to computing a convolution depends on the signals being convolved.
YouTube Videos
LTI Systems and Convolution
Convolution Properties
Convolution Example - Flip and Shift
Convolution using Tickertape
Convolution with a Unit Step
Discrete-time Convolution Sum and Example
Summary
Convolution and LTI Systems
For discrete-time systems that are linear and time-invariant (LTI), the output \(y[n]\) can be expressed as the convolution of the input signal \(x[n]\) with the system’s impulse response \(h[n]\)
\[y[n] = x[n] * h[n]\]
where \(*\) denotes convolution.
Discrete-Time Convolution Equation
The discrete-time convolution sum is given by:
\[y[n] = \sum_{k=-\infty}^{\infty} x[k]\,h[n-k]\]
This equation follows directly from the properties of linearity and time invariance.
Flip-and-Shift (Ticker Tape) Method
An alternative way to compute convolution is the flip-and-shift method:
- Flip one of the signals (usually \(h[n]\)) to obtain \(h[-k]\).
- Shift the flipped signal by \(n\), giving \(h[n-k]\).
- Multiply point-by-point with \(x[k]\).
- Sum over all \(k\).
Figure 1.3.1: Example of the ticker-tape method of estimating the convolution between two discrete-time signals.
Choosing a Convolution Method
The most efficient way to compute a convolution depends on the characteristics of the signals involved:
- Finite-length signals often benefit from the flip-and-shift graphical approach.
- Signals with simple analytic expressions may be easier to compute directly using the convolution sum.
Properties
Commutative: The output (i.e., convolution) of an LTI system with input \(x[n]\) and unit impulse response \(h[n]\) is the same to the output of an LTI system with input \(h[n]\) and unit impulse response \(x[n]\)
\[x[n] * h[n] = h[n] * x[n]$$
Distributive: The output to \(x[n]\) convolved with two different unit impulse responses added together is the same to the output to \(x[n]\) convolved separately with each unit impulse response and then both convolutions added together after
\[x[n] * (h_1[n] + h_2[n]) = x[n] * h_1[n] + x[n] * h_2[n]$$
Figure 1.3.2: Flow diagram representing two parallel systems and their equivalent single systems
Associative: The output to \(x[n]\) convolved with two different unit impulse responses convolved together is the same to the output to \(x[n]\) convolved with either unit impulse response first, followed by that output being convolved with the other unit impulse response
\[x[n] * (h_1[n] * h_2[n]) = (x[n] * h_1[n]) * h_2[n] = (x[n] * h_2[n]) * h_1[n]$$
Figure 1.3.3: Flow diagram representing the associative property of convolution. Remember that the order of the LTI systems is irrelevant due to the commutative property.