ECE 485 Lesson 1

ECE 485 Lesson 1 - ECE 485 Advanced Engineering Mathematics...

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Unformatted text preview: ECE 485 Advanced Engineering Mathematics Lesson 1 Matrix Representation of Digital Filters Inner Product Matrix Multiplication Digital Filter Unit impulse response Unit sample response Digital Filter h(n) (n) h(n) 1 for n = 0 ( n) = 0 for n 0 Unit impulse sequence or unit sample 2 FIR Digital Filters hh ( n ) for 0 n 0 0 x ( n ) for 0 n Nh -1 N- 1 x h ( n) = x ( n) = elsewhere elsewhere 3 Response of Digital Filter to x(n) y ( n) = N x -1 k =0 h( n - k) x( k) n = 0,1, 2,K , N y - 1 N y = N x + Nh -1 4 Special Case Nh=3 and Nx=2 N y = N x + Nh -1 = 2 + 3 -1 = 4 y ( 1) = h ( 1) x ( 0 ) + h ( 0 ) x ( 1) y ( 3) = h ( 2 ) x ( 1) y ( 0) = h ( 0) x ( 0) y ( 2 ) = h ( 2 ) x ( 0 ) + h ( 1) x ( 1) 5 Inner or Dot Product y ( 1) = h ( 1) x ( 0 ) + h ( 0 ) x ( 1) h1 = ( 1) h h1 x = ( 1) h x ( 0 ) h ( 0) x = x ( 1) x ( 0 ) h ( 0) ) x ( 1) h ( 1) x ( 0+ h ( 0 ) x ( 1) y ( 1) = h1 x 6 The Output Samples h0 = ( 0 ) h h1 = ( 1) h 0 y ( 0 ) = h0 x h ( 0 ) y ( 1) = h1 x h2 = ( 2 ) h ( 1) y ( 2 ) = h2 x h h3 = h ( 2 ) y ( 3) = h3 x 0 7 The Output Vector y h h ( 0 ) 0 x 0 y h h ( 1) 1 x 1 y= = = x ( 2 ) 2 x 2 y h h y h h ( 3) 3 x 3 8 Matrix Form of Input-Output Relationship Toeplitz Matrix 0 y ( 0 ) h( 0 ) y (1) h(1) h( 0 ) x( 0 ) = y ( 2 ) h( 2 ) h(1) x(1) h( 2 ) y ( 3) 0 9 The Matrix Product A = BC The number of columns of B must equal to the number of rows of C. The number of rows of A is equal to the number of rows of B. The number of columns of A is equal to the number of columns of C. The element in the i-th row and j-th column of A is equal to the inner product of the i-th row of B and the j-th column of C. 10 Numerical Example h ( 0 ) = 3, h ( 1) = 2, h ( 2 ) = 1 x ( 0 ) = 1, x ( 1) = 1 ( 0) y 3 y 2 ( 1) = ( 2) y 1 y 0 ( 3) 0 3 3 1 5 = 1 2 3 1 1 11 Another Example 4 1 - 3 2 - 6 2 1 2 BC = 1 5 1 - 4 5 - 1 b 1 BC = b 2 [ c1 c 2 c 3 c 4 ] b 3 Column vector Row vector 12 Inner Product b = [ b1 b2 bn ] c1 c 2 c= c n b c bc =1 n Inner product of a row vector with a column vector 13 Matrix Multiplication b 1 BC = b 2 [ c1 b 3 b1 c 4 c2 c3 b 2 c 2 b 2 c 3 b 2 c4 b 3 c 2 b 3 c 3 b 3 c4 4 1 - 2 22 5 - 14 - 3 2 - 6 2 1 2 = 20 4 - 1 - 14 BC = 1 5 1 - 4 - 31 5 4 14 5 - 1 b 1 c1 c 4 ] = b 2 c1 b 3 c1 b1 c 2 b1 c 3 14 ...
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