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

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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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