Convolution
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Digital signal processing convolution, Study notes of Digital Signal Processing
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Typology: Study notes
2020/2021
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Convolution
Learning Objectives
- (^) Significance of zero state response
- (^) Role of convolution in finding zero state response
- (^) Understanding to compute the convolution by integration
- (^) Understand the graphical method of convolution
- (^) Earlier we have studied zero input response in 2.1 and 2.2 of text book.
- (^) Here we use impulse response (2.3) for calculating SECOND COMPONENT of system response i.e zero state response to be able to find TOTAL RESPONSE of system
- (^) Any Input signal X(t) can be represented as a Combination of pulses of very small width (may be modeled as impulses.
Modeling of signal by impulses
- (^) Any Input signal can be represented as a sum of pulses with width Δτ
- (^) recall that displacement of a pulse (time shifting ) is represented by
- (^) An other property of pulse is recalling from sampling concept impulse in ch 1
- (^) If Δτ͢ 0, this Pulse p(t) is considered as impulse
System response using impulse
response
This is also called convolutional Integral. In general for convolution between any two signals is represented as
Key Notes of Convolutional Integral
- (^) This is a kind of sweeping multiplication of two signals
- (^) Time in general sense / scale if represented by t whereas span of interaction on time scale is represented by τ
- (^) Integration is performed in τ whereas outcome y(t) is a function of time t.
Convolution table
- (^) A table has been developed to quickly lookup the outcomes of common functions. Observe table 2.
- (^) Practice example 2.5 and example 2.6 + E2.5 and E2.6 using help of table
Graphical meaning and procedure of
convolution
- (^) Graphical presentation of convolution is very interesting.
- (^) Lets try it on following links
- (^) https://mathworld.wolfram.com/Convolution.html
- (^) http://graphics.cs.brown.edu/research/exploratory/freeSoftware/ repository/edu/brown/cs/exploratories/applets/convolution/ convolution_guide.html
- (^) https://pages.jh.edu/~signals/convolve/index.html