1 / 17

Convolution

Convolution. A mathematical operator which computes the pointwise overlap between two functions. Discrete domain: Continuous domain: . Discrete domain. Basic steps Flip (reverse) one of the functions. Shift it along the time axis by one sample.

judith
Download Presentation

Convolution

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Convolution • A mathematical operator which computes the pointwise overlap between two functions. • Discrete domain: • Continuous domain:

  2. Discrete domain Basic steps • Flip (reverse) one of the functions. • Shift it along the time axis by one sample. • Pointwise multiply the corresponding values of the two digital functions. • Sum the products from step 3 to get one point of the digital convolution. • Repeat steps 1-4 to obtain the digital convolution at all times that the functions overlap.

  3. Continuous domain example

  4. Continuous domain example

  5. LTI (Linear Time-Invariant) Systems • Convolution can describe the effect of an LTI system on a signal • Assume we have an LTI system H, and its impulse response h[n] • Then if the input signal is x[n], the output signal is y[n] = x[n] * h[n] H x[n] y[n] = x[n]*h[n]

  6. Fourier Series • Most periodic functions can be expressed as a (infinite) linear combination of sines and cosines F(t) = a0 + a1cos (ωt) + b1sin(ωt) + a2cos (2ωt) + b2sin(2ωt) + … = F(t) is a periodic function with

  7. Most Functions? • F(t) is a periodic function with • must satisfy certain other conditions: • finite number of discontinuities within T • finite average within T • finite number of minima and maxima

  8. Calculate Coefficients

  9. Example • F(t) = square wave, with T=1.0s ( )

  10. Example

  11. Example

  12. Why Frequency Domain? • Allows efficient representation of a good approximation to the original function • Makes filtering easy • And a whole host of other reasons….

  13. But One Really Important One • Note that convolution in the time domain is equivalent to multiplication in the frequency domain (and vice versa)!

  14. Fourier Family

  15. Fourier Transform • Definitions: • Can be difficult to compute => Often rely upon table of transforms

  16. Delta function • Definition: • Often, the result of the Fourier Transform needs to be expressed in terms of the delta function

  17. Fourier Transform pairs •  •  • There is a duality in all transform pairs 

More Related