1 / 16

Lecture #11 EGR 261 – Signals and Systems

Lecture #11 EGR 261 – Signals and Systems. Read : Ch. 2, Sect. 1-8 in Linear Signals & Systems, 2 nd Ed. by Lathi Ch. 13, Sect. 6 in Electric Circuits, 9 th Ed . by Nilsson. Convolution In this class we will cover the following topics: Convolution – graphical approach

keola
Download Presentation

Lecture #11 EGR 261 – Signals and Systems

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. Lecture #11 EGR 261 – Signals and Systems Read: Ch. 2, Sect. 1-8 in Linear Signals & Systems, 2nd Ed. by Lathi Ch. 13, Sect. 6 in Electric Circuits, 9th Ed. by Nilsson • Convolution • In this class we will cover the following topics: • Convolution – graphical approach • Convolution – direct integration approach • Convolution – using Laplace transforms to bypass the convolution integral

  2. Lecture #11 EGR 261 – Signals and Systems Procedure for evaluating y(t) = x(t)*h(t) graphically (Note that x and h can be reversed in the steps below (commutative property), depending on which form of the convolution integral is used.) • Graph x() (with  on the horizontal-axis). • Invert h() to form h(-). Then shift h(-) along the  axis t seconds for form h(t - ). Note that as the time shift t varies, the waveform h(t - ) will “slide” across x() and in some cases the waveforms will overlap. • Determine the different ranges of t which result in unique overlapping portions of x() and h(t - ). For each range determine the area under the product of x() and h(to - ). This area is y(t) for the range. • 4. Compile the results of y(t) for each range and graph y(t). Which signal to invert and shift? Since x(t)*h(t) = h(t)*x(t), we can shift and delay either signal (commutative property). Hint: Shift and delay the simplest waveform

  3. y(t) = x(t)*h(t) 2 t 1 2 3 4 0 Lecture #11 EGR 261 – Signals and Systems Length of the response y(t) = x(t)*h(t) In general if the length of the input x(t) is T1 and the length of the impulse response h(t) is T2, then the length of y(t) = x(t)*h(t) will be T1 + T2. h(t) x(t) y(t) = x(t)*h(t) t t t T1 T2 T1 +T2 Example Recall the example worked in the last presentation where we determined y(t) = x(t)*h(t) as shown below. h(t) x(t) 2 1 t t 1 1 3 0 0 Length = 1 Length = 2 Length = 1+2 = 3

  4. Lecture #11 EGR 261 – Signals and Systems Example 1: Find y(t) = x(t)*h(t) using the graphical method. What should be the length of y(t)? h(t) x(t) 6 4 3 t t 2 6 5 4 1 0 0

  5. Lecture #11 EGR 261 – Signals and Systems Example 1: (continued)

  6. h(t) x(t) 2 2 t t 2 1 2 0 0 Lecture #11 EGR 261 – Signals and Systems Example 2: Find y(t) = x(t)*h(t) using the graphical method. Which waveform is easiest to shift and delay? What should be the length of y(t)?

  7. Lecture #11 EGR 261 – Signals and Systems Example 2: (continued)

  8. x(t) h(t) 1 t t -3 0 0 Lecture #11 EGR 261 – Signals and Systems Example 3 (problem 2.4-18d in Lathi text): Find y(t) = x(t)*h(t) using the graphical method. Which waveform is easiest to shift and delay?

  9. Lecture #11 EGR 261 – Signals and Systems Example 3: (continued)

  10. + 1  + x(t) = u(t+3)-u(t) y(t) 1 F _ _ Lecture #11 EGR 261 – Signals and Systems • Example 4: • Sketch x(t). • Find H(s) for the circuit shown below. Also find h(t) from H(s). • Find y(t) using Laplace transform techniques. • Look familiar? This is the same problem used in Example 3. Show that y(t) found using convolution is exactly the same as y(t) found using Laplace transforms.

  11. Lecture #11 EGR 261 – Signals and Systems Example 4: (continued)

  12. x(t) = (t)+2(t-1)+3(t-2) h(t) 2 t t 0 0 3 1 2 Lecture #11 EGR 261 – Signals and Systems Example 5: Find y(t) = x(t)*h(t) using the graphical method.

  13. Lecture #11 EGR 261 – Signals and Systems Example 5: (continued)

  14. u(t - ) = 1 only for  < t so change upper limit to t u() = 1 only for  > 0 so change lower limit to 0 A graphical approach would show that the functions overlap for 0 < t <  or else note that x(t) and y(t) both are non-zero for t > 0. so u(t)*u(t) = tu(t) Lecture #11 EGR 261 – Signals and Systems Convolution by direct integration Convolution can be performed by direct evaluation of the convolution integral (without sketching graphs) for fairly straightforward functions. This becomes difficult for piecewise-continuous functions, such as were used in many of our graphical examples. Example 6: Evaluate y(t) = x(t)*h(t) for: x(t) = u(t), h(t) = u(t) by directly evaluating the convolution integral rather than through a graphical approach.

  15. Lecture #11 EGR 261 – Signals and Systems Example 7: Evaluate y(t) = x(t)*h(t) for: x(t) = tu(t), h(t) = u(t) by directly evaluating the convolution integral rather than through a graphical approach. Example 8: Evaluate y(t) = x(t)*h(t) for: x(t) = e-tu(t), h(t) = u(t) by directly evaluating the convolution integral rather than through a graphical approach.

  16. Lecture #11 EGR 261 Signals and Systems Table 2-1 from Signals & Systems, 2nd Edition, by Lathi

More Related