Question E7H11:
What are the major spectral impurity components of direct digital synthesizers?  

Choices:  
A: Broadband noise  
B: Digital conversion noise  
C: Spurious signals at discrete frequencies  
D: Harmonics of the local oscillator    

Correct Answer is  C: Spurious signals at discrete frequencies  

To understand DDS spectral purity:

Why C is correct (Spurious signals at discrete frequencies)
- A DDS uses a phase accumulator, a phase-to-amplitude table, and a DAC. Finite word length and phase truncation create periodic quantization errors that are coherent with the tuning word and clock. These appear as discrete spurs in the spectrum.
- DAC nonlinearity and clock feedthrough also produce discrete spectral lines.
- Sampling images from the DAC occur at specific frequencies, also discrete.

Key relations:
- Output frequency: $$f_{out} = \frac{K}{2^M} \, f_{clk}$$
  where $K$ is the tuning word, $M$ is the accumulator width, and $f_{clk}$ is the reference clock.
- DAC image (reconstruction) frequencies: $$f = n\,f_{clk} \pm f_{out}, \quad n = 1,2,3,\dots$$
  These are discrete spurs requiring a low-pass reconstruction filter.
- Dithering trades spurs for a slightly higher noise floor by randomizing truncation errors.

Why the other choices are incorrect
- A: Broadband noise. DDS does have broadband components (e.g., quantization noise, jitter-induced phase noise), but the major, characteristic impurity is discrete spurious lines, not primarily broadband noise.
- B: Digital conversion noise. This vague term mainly refers to quantization noise of the DAC, which raises the noise floor but is not the dominant “signature” impurity compared to discrete spurs.
- D: Harmonics of the local oscillator. That wording fits mixers/PLLs. A DDS’s prominent issues are not simple LO harmonics; while clock-related images at multiples of $f_{clk}$ exist, the key impurity is discrete spurs tied to truncation and DAC imperfections.

Useful rules of thumb
- Ideal DAC quantization SNR (broadband floor): $$\mathrm{SNR} \approx 6.02\,B + 1.76 \text{ dB}$$ for a $B$-bit DAC.
- Jitter-limited SNR: $$\mathrm{SNR}_{jitter} \approx -20 \log\!\big(2\pi f_{out}\,\sigma_t\big) \text{ dB}$$ where $\sigma_t$ is clock jitter RMS.
- Low-pass reconstruction filter cutoff typically set just above $f_{out}$ and well below $f_{clk}/2$ to suppress $n\,f_{clk} \pm f_{out}$ images. Dithering can reduce prominent spurs by spreading them into the noise floor.