linear_systems

# 24 matrix formulation of convolution from above we

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2.4 Matrix formulation of convolution From above, we have w [ m ]=( h v )[ m ] = N 1 k = 0 h [ m k ] v [ k ] . We want to write this operation as vector w = vector h vector v = L vector v . To make it easier to see, consider an input sequence of length n = 4 and an impulse response of length m = 3: vector v = v 0 v 1 v 2 v 3 vector h = h 0 h 1 h 2 .

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EE 505 B, Autumn, 2011 Linear Systems 6 We know that from the previous example that the output is of length 6, so w 0 w 1 w 2 w 3 w 4 w 5 = bracketleftBigg 6 × 4 matrix bracketrightBigg v 0 v 1 v 2 v 3 . The answer is w 0 w 1 w 2 w 3 w 4 w 5 = h 0 0 0 0 h 1 h 0 0 0 h 2 h 1 h 0 0 0 h 2 h 1 h 0 0 0 h 2 h 1 0 0 0 h 2 v 0 v 1 v 2 v 3 . To interpret this result, note that each column is a shifted impulse response, and for matrix multiplication, each column is scaled by the input vector vector v elements ( v k ). Compare this description with the description of convolution for discrete systems above. (They are the same.) The matrix above is a Toeplitz matrix, and has the general form vector w = h 0 0 · · · 0 0 h 1 h 0 · · · 0 0 h 2 h 1 · · · . . . . . . . . . h 2 · · · h 0 h M 2 . . . . . . h 1 h 0 h M 1 h M 2 · · · . . . h 1 0 h M 1 · · · h M 3 . . . 0 0 · · · h M 2 h M 3 . . . . . . . . . h M 1 h M 2 0 0 0 0 h M 1 bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright L is a M × N matrix v 0 v 1 v 2 . . . v N 1 . A note on Toeplitz matrices (you’ll see the relevance in a moment). For finite n , we don’t know of a closed from expression for the eigenvalues and eigenvectors of the Toeplitz matrix. However, for doubly infinite Toeplitz matrices, the eigenvectors are complex exponentials.
EE 505 B, Autumn, 2011 Linear Systems 7 Doubly infinite matrices are difficult to write, so we will concentrate on the cyclic convo- lution case. Actually, we care a lot about the cyclic case because it is cyclic (or circular) convolution which is related to the discrete Fourier transform (DFT), and the DFT is efficiently implemented by the fast Fourier transform (FFT). 2.5 Cyclic/circular convolution For N = 4, w 0 w 1 w 2 w 3 = h 0 h 3 h 2 h 1 h 1 h 0 h 3 h 2 h 2 h 1 h 0 h 3 h 3 h 2 h 1 h 0 v 0 v 1 v 2 v 3 . Cyclic convolution is used a lot in digital signal processing (e.g., EE 518). Regular (non-cyclic) convolution can be implemented using circular convolution (and therefore using the FFT) by zero padding the input and impulse response, if the impulse response and input are finite in length (e.g., for finite impulse response (FIR) systems, not infinite impulse response (IIR) systems).

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