Question 10.1: Analyze the spurious tone locations for an accumulator with ...
Analyze the spurious tone locations for an accumulator with length N equal to 8, and FCW equal to (a) 96 and (b) 64.
(a) When FCW = 96, the spur period is given by
T_{spur}= \frac{2^{N} }{GCD(FCW, 2^{N} )}=\frac{256}{GCD(96, 256) } =\frac{256}{32}=8Assume that the accumulator starts with a value of zero. With input of 96, the accumulator values at each time step are given as follows: 0, 96, 192, 32, 128, 224, 64, 160, 0, 96, …,which shows that the accumulator value indeed repeats every eight clock cycles. The desired output frequency is given by
f_{o}= \frac{FCW . f_{clk}}{2^{N} } = \frac{96}{256} \cdot f_{clk}= 0.375 \cdot f_{clk}This corresponds to the overflow of the accumulator every 256/96 clock cycles. In addition, a secondary periodicity repeats every eight clock cycles, that is, at multiples of
f_{spur}= \frac{GCD(FCW, 2^{N} ) \cdot f_{clk} }{2^{N}}= \frac{f_{clk}}{8}= 0.125 \cdot f_{clk}The spurious tones are hence at the multiple of ±1/3 of the fundamental frequency f_{o}. In the example, there are 96/32 − 1 = 2 spurs between the harmonics of the output fundamental, and there are a total of 128/32 = 4 spurs in the Nyquist band from 0 to f_{clk}/2. The four spurs are located at f_{clk}/8, f_{clk}/4, 3 f_{clk}/8, and f_{clk}/2.
(b) When FCW = 64, without losing generality, assume the accumulator initial value is zero. The accumulator values at each time step are given as follows: 0, 64, 128, 192, 0, 64, …,which means that the accumulator repeats the same value after every overflow. Thus, there is no secondary periodicity except the desired output period of 4T_{clk} . Therefore; no spurs will be seen in the output spectrum.