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Lecture 1 - What Does a DSP System Look Like Inside Example...

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What Does a DSP System Look Like Inside ? Example: Numerical Integration (integrate a continuous – time signal using numerical methods) ( ) g179 t d ) ( y [n] = y[n-1]+ H x [n-1] A "Difference Equation"!! y (t) = g87 g87 x 0 By letting y [n] = y(nH) and x[n]=x(nH) What Does a DSP System Look Like Inside ? x [n] y[n] Example: Numerical Integration x [n-1] H x [n] Input Output H Delay 1 Sample g14 y [n-1] Delay 1 Sample Define: x [n] = Input at "current" sample index 'n' x [n-1] = Input at "previous" sample index (n-1) Define: y [n] = Output at "current" sample index 'n'
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How could we implement this Digital Filter in software?? // Repeat for the all samples for (n=0; n<number of samples; n++) // Handle the “initial conditions” for the 1 st output sample x_n_minus_1 = 0; y_n_minus_1 = 0; // Initialize variables, and other software “stuff ” H=numerical_integration_increment; Using C/ C++/MATLAB, etc. for (n 0; n number_of_samples; n ) { // Compute the new “current” output signal using diff. equ. y(n) = y_n_minus_1+H* x_n_minus_1; // Get a new “current” input sample and remember it as the “previous” input for next cycle x_n_minus_1 =x(n); // Remember the current output as the “previous” output for next cycle y_n_minus_1 = y(n); } TA D AAAA!!! EE 328 Discrete-Time Signals & Systems Discrete-Time Signals (Part 1) Sampled versions Of Continuous-Time Signals Sampled versions DISCRETE-TIME SIGNALS Where Do They Come From? t s x [n] x c (t) x [n] = x c ( t ) | t = nt s t s = "sampling interval" n = integer index Sequence of numeric values Defined only at particular (discrete) instants of time that are integer multiples of the sampling interval Discrete Time n Continuous Time Varying Analog Voltage Defined for any value of “t” x c (1.00219234) = 3.42598 t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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Notation / Conventions x c ( t ) Parentheses indicate continuous-time x [ n ] Square Brackets indicate discrete-time x [ n ] = { 6, 3, 7, 5, 2, 1, … } “Numeric Sequence Notation ”: Braces enclose sequence list g160 indicates index origin (n=0) Indicates sequence goes on to g102 (If not shown, assume all unspecified samples=0 n = { -2,-1, 0, 1, 2, 3, Notation / Conventions Once sampled, we loose our direct reference to time for each element in the sequence x[n] becomes “just a list of numbers” x[n] becomes just a list of numbers – Need to specify t s sampling interval to
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