{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# lecture02 - Lecture Notes 2 Lecture 2 Goals Be able to...

This preview shows pages 1–13. Sign up to view the full content.

Lecture Notes 2 Lecture 2: Goals Be able to calculate the output of a linear system (convolution) when the input is a specified function of time (or frequency). Be able to work in both time domain and frequency domain. Be able to determine the noise variance out of a linear system. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 1 / 174

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Lecture Notes 2 Linear Systems Filtering, Convolution, Correlation and Noise In most receivers in a digital communication system the received signal is filtered before a decision is made as to the data bit that is transmitted. The purpose of filtering is to remove as much of the noise as possible without removing any of the signal. a45 x ( t ) h ( t ) y ( t ) a45 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 2 / 174
Lecture Notes 2 Linear Systems Convolution Mathematically, filtering is the convolution of the input signal with the impulse response of the filter. That is, if the input to the filter is the signal x ( t ) and the impulse response of the filter is h ( t ) the output of the filter y ( t ) is given by Filter, Convolution y ( t ) = integraldisplay −∞ x ( t α ) h ( α ) d α = integraldisplay −∞ h ( t α ) x ( α ) d α The above mathematical operation on x ( t ) and h ( t ) is called the convolution of h with x . EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 3 / 174

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Lecture Notes 2 Linear Systems Graphical Convolution The convolution operation can be understood graphically. To do this we compute the output at different times. Combining the output at different times gives the total output. Consider the convolution of a rectangular pulse with a triangular impulse response filter. T t h ( t ) t T x ( t ) EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 4 / 174
Lecture Notes 2 Linear Systems Output at t = t 1 Consider the output of the convolution at time t = t 1 . y ( t 1 ) = integraldisplay h ( t 1 α ) x ( α ) d α First the function h ( α ) is flipped right to left to yield h ( α ) . T α h ( α ) α T T h ( α ) EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 5 / 174

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Lecture Notes 2 Linear Systems Output at t = t 1 Second, the function h ( α ) is shifted to the right by t 1 seconds T α h ( α ) t 1 T t 1 α h ( t 1 α ) EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 6 / 174
Lecture Notes 2 Linear Systems Output at t = t 1 Third, the flipped, shifted function h is correlated (multiplied and integrated) with the input x to give the output y ( t 1 ) at time t 1 . t 1 α h ( t 1 α ) x ( α ) y ( t ) t 1 t EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 7 / 174

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Lecture Notes 2 Linear Systems Output at t = t 0 t 0 α h ( t 0 α ) x ( α ) y ( t ) t 0 t EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 8 / 174
Lecture Notes 2 Linear Systems Output at t = t 1 t 1 α h ( t 1 α ) x ( α ) y ( t ) t 1 t EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 9 / 174

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Lecture Notes 2 Linear Systems Output at t = t 2 t 2 α h ( t 2 α ) x ( α ) y ( t ) t 2 t EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 10 / 174
Lecture Notes 2 Linear Systems Output at t = t 3 t 3 α h ( t 3 α ) x ( α ) y ( t ) t 3 t EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 11 / 174

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Lecture Notes 2 Linear Systems Output at t = t 4 t 4 α h ( t 4 α ) x ( α ) y ( t ) t 4 t EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 12 / 174
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern