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This paper describes the architecture and the IC implementation of a direct digital frequency synthesizer (DDFS) that is based on an angle rotation algorithm (similar to CORDIC). It is shown that the architecture can be implemented as a multiplierless, feedforward, and easily pipelineable datapath. A prototype IC has been designed, fabricated in 1.0-m CMOS, and tested. The IC produces 16-b sine and cosine outputs with a spurious-free dynamic range of more than 100 dBc. A 36-b frequency control word gives a tuning resolution of 0.0015 Hz at a 100-MHz sampling rate.

A quadrature direct digital frequency synthesizer @DFS) produces 16b sine and cosine outputs with a spurious-free dynamic range greater than lOOdBc and a tuningresolution of 0.0015 Hz at a sample rate of 1OOMHz. The prototype IC contains 58,000 transistors in a core area of 12mm2 in 1.0pm CMOS and dissipates 1.4W at 1OOMHz.Instead of table lookup, an angle-rotation algorithm implemented as a multiplierless feedforward datapath is used that allows easy pipelining and Limits the accumulation of roundoff errors 111. The modular architecture permits outputs of arbitrary precision by cascading enough angle rotation stages in the datapath.The computation of sine and cos6 can be viewed as the computation of the X-axis and the Y-axis coordinates of a phasor, initially coincident with the X-axis, rotated by an angle 8 on the unit circle. Cordic iteratively computes angle rotations and has been used for frequency synthesis [21. Cordic approximates a rotation by a sequence of N subrotations through angles Oi 80 that e = a1 + a, 8, + ... + % ON. If Bi = tan-' 2.' and CL, ~[ -l , 11, the datapath is multiplierless. The a,s are determined iteratively and require knowledge of the sign of the difference between two angles at each iteration. The throughput of the cordic datapath canbe improved by using redundant arithmetic (e.g., carry-save) eliminating the carry-propagation delay. However, detection of the sign in redundant arithmetic is non-trivial. This approach retains the simplicity of a cordic datapath while eliminating the anglecomputation hardware that would normally be required. The sine and cosine of any angle in [O, 2x1 can be obtained &om those of some different angle in [O,d4] by simple negation and/or interchange operations. An arbitrary angle 0 ( < 1 radian) can be represented as e = b, + b,B, + ... + b, e, where the bi are the angle binary representation bits and 0, = 2-, radians. Notice that the et are different from those of Cordic. The b, = 1 corresponds to a rotation by 0, = 2-, radians and b, = 0 corresponds to no rotation. b,e(O,l) are recodedin signed digits r&-l,l] where r, = 2bk.1 -1. An initial fixed rotation by el = (0.5 -0.5N+i) radians is necessary to ensure equality between the two representations. After recoding, a bit b, = 1 corresponds to a positive rotation by ek = 2P-l) radians and b, = 0 corresponds to a negative rotation by 8, = 2e-l)radians. The recoding (which need not be explicitly performed) is necessary to ensure a constant scale factor K in the datapath; this is most efficiently accounted for in the algorithm implementation by starting each angle rotation sequence a t the point %= K, Yo= 0.For an N-bit output resolution, the algorithm consists of an Nstage cascade with each stage implementing a positive or negative e, subrotation (Figure 1). The subrotation stages implementation can employ the approximation tan 8,58,=2-k for sufficiently small 0, (i.e., for sficiently large k), reducing the corresponding multiplication to a simple k-bit shift operation. Given a B-bit internal word length...

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