The frequency synthesis and mixing operation of Equation (1) accepts an input sequence with x-y coordinates [X I (n), Y I (n)] and an input frequency control word f, producing the output sequence [X O (n), Y O (n)].(1)The common implementation of (1) employs a DDFS to generate sin(2πnf) and cos(2πnf), feeding a complex multiplier (mixer) (Figure 7.5.1). Most DDFS architectures use lookup tables with pre-computed stored sine and cosine values. The exponential dependence between table size and phase word-length is a severe drawback for such implementations. This IC, based on the Figure 7.5.2 architecture, treats Equation (1) from the perspective of rotating a point (X I , Y I ) in the x-y plane around the origin by an angle θ = 2πnf, to reach (X O , Y O ). The overall rotation uses two stages, coarse and fine, along with accurate small-angle approximations for sine and cosine. These approximations eliminate the need for a large lookup table [2]. In Figure 7.5.2, the phase accumulator and W-bit normalized rotation angle θ = are standard DDFS components, as is the partitioning of θ's three MSBs to reduce the rotation angle to φ in the interval [0, ]. This is accomplished by conditionally interchanging the input coordinates (X I , Y I ), and re-creating the correct output point (X O , Y O ) via conditional interchange and negation operations. With an N-fractional-bit output precision, the partitioning of φ = φ M + φ L , where φ M denotes the +1 MSBs, allows Equation (1) to be implemented in two stages, the first rotating (X I , Y I ) by φ M to reach a point (X T , Y T ) and the second rotating (X T , Y T ) by φ L to obtain (X O , Y O ). Since φ L <2 -, the approximations sin φ L ≈ φ L and cos φ L ≈1-φ L eliminate the need to store sine and cosine values for φ L . For this IC, N = 11 and W = 15; hence φ has 13 bits. φ M requires +1=5b and the remaining 8 bits constitute φ L . The coarse lookup table size is 2 +1. Eliminating the lookup for φ L reduces overall table size from 2 12 + 1 = 4097 locations to 2 4 + 1 = 17 locations, a reduction factor of =241. The output error due to the approximations does not affect the 13b output data [2]. With W = 15 the worst-case spur is 90.3dBc [3].Sinθ 1 is defined by rounding sin φ M up to +1 fractional bits and cosθ 1 by truncating cosθ 1 to fractional bits. Using sinθ 1 and cosθ 1 in the coarse stage significantly reduces the number of partial products in multipliers #1 -#4. Compensation for these modifications is by adjusting the rotation angle and correcting the magnitude in the fine stage. The new coarse rotation angle is θ m =arctan . The fine stage rotation therefore becomes θ 1 = φ−θ m = φ L + φ m -θ m , which is obtained by storing φ m -θ m in the coarse lookup table and adding it to φ L . The fine stage magnitude adjustment introduces in Figure 7.5.2. The inputs of multipliers #5 -#9 are appropriately truncated to reduce the partial products while maintaining the desired output error bound. Figure 7.5.2 reflects these reductions.The following important points should be not...