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Represent a filter operation by a table (matrix) {xij}, where the rows are indexed by delay i and the columns by shift j, i.e., the row i is the coefficient ci for the term x(n-i), and the column 0 in row i is the msb of ci and column W-1 in row i is the lsb of ci , where W is the word length. • The row and column indexing starts at 0. • The...

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Signed Binary Digit (SBD) Addition/Subtraction • Y = Y+ - Y-, is a signed digit number, where Y+ and Y- are from the digit set {0, 1, , a}. • A signed digit number is thus subtraction of 2 unsigned conventional numbers. • Signed addition is given by: S = X + Y = X + Y+ - Y-, Þ S1 = X + Y+, S =...

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Steps for deriving a bit-serial IIR filter architecture: Ø A bit-level pipelined bit-serial implementation of the FIR section needs to be derived. Ø The input signal x(n) is added to the output of the bitserial FIR section w(n). Ø The resulting signal y(n) is connected to the signal y(n-1). Ø The number of delay elements in the edge marked ?D ...

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Many useful realizations contains roundoff nodes that are not connected to unit-delay branches. Thus these nodes (variables) do not appear in a state variable description and the scaling and roundoff noise computation methods can not be applied directly. • The SRP ( slow-down and retiming/pipelining) transformation technique can be used as a p...

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Scattered look-ahead pipelining: The denominator of the transfer function in (10.9) is transformed in a way that it contains the N terms . Equivalently, the state y(n) is computed in terms of N past scattered states y(n-M),y(n-2M), , and y(n-NM) • In scattered look-ahead, for each poles in the original filter, we introduce (M-1) canceling pole...

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Note: DCT is an orthogonal transform, i.e., the transformation matrix for IDCT is a scaled version of the transpose of that for the DCT and vice versa. Therefore, the DCT architecture can be obtained by “transposing” the IDCT, i.e., reversing the direction of the arrows in the flow graph of IDCT, and the IDCT can be obtained by “transposing” t...

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Example-7 (cont’d) – In this example, the Winograd convolution algorithm requires 5 multiplications and 11 additions compared with 6 multiplications and 2 additions for direct implementation • Notes: – The number of multiplications in Winograd algorithm is highly dependent on the degree of each . Therefore, the degree of m(p) should be as sm...

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Relating Systolic Designs Using Transformations : ØFIR systolic architectures obtained using the same projection vector and processor vector, but different scheduling vectors, can be derived from each other by using transformations like edge reversal, associativity, slow-down, retiming and pipelining. • Example 1 : R1 can be obtained from B2...

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Determine the minimum number of registers using lifetime analysis. • Input each variable at the time step corresponding to the beginning of its lifetime. If multiple variables are input in a given cycle, these are allocated to multiple registers with preference given to the variable with the longest lifetime. • Each variable is allocated in a...

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The original DFG cannot have sample period equal to the iteration bound because the iteration bound is not an integer. ØIf a critical loop bound is of the form tl/wl where tl and wl are mutually co-prime, then wl-unfolding should be used. ØIn the example tl = 60 and wl = 45, then tl/wl should be written as 4/3 and 3-unfolding should be used...

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