Question E7H10:
What information is contained in the lookup table of a direct digital synthesizer (DDS)?  

Choices:  
A: The phase relationship between a reference oscillator and the output waveform  
B: Amplitude values that represent the desired waveform  
C: The phase relationship between a voltage-controlled oscillator and the output waveform  
D: Frequently used receiver and transmitter frequencies    

Correct Answer is  B: Amplitude values that represent the desired waveform  

To synthesize a waveform, a DDS uses a phase accumulator feeding a phase-to-amplitude converter (a lookup table) and a DAC. The lookup table stores amplitude samples of the desired waveform (usually a sine), indexed by phase.

Why B is correct:
- The DDS lookup table contains amplitude values versus phase for the target waveform. The phase accumulator outputs a phase word; the table maps this phase to an amplitude that the DAC outputs. This produces the waveform sample-by-sample.

Why the other choices are incorrect:
- A: The phase relationship to a reference oscillator is not stored in the table. The reference clock sets the update rate; the phase relationship is determined by the phase accumulator and tuning word, not by lookup contents.
- C: A DDS does not use a VCO; that’s typical of PLL systems. Therefore, no VCO-to-output phase relationship is stored.
- D: Channel memories (frequently used frequencies) are separate from the DDS waveform lookup table. The DDS table is for waveform amplitudes, not frequency presets.

Useful formulas and concepts:
- Phase accumulator update:
  $$\phi_{n+1} = \phi_n + \Delta \phi \ (\text{mod } 2^N)$$
  where $N$ is the phase accumulator width (bits).
- Output frequency:
  $$f_{\text{out}} = \frac{\Delta \phi}{2^N}\, f_{\text{clk}}$$
  where $f_{\text{clk}}$ is the DDS system clock.
- Phase-to-amplitude mapping (lookup):
  $$A[n] = \sin\!\left(2\pi \frac{\phi_n}{2^N}\right)$$
  In practice, the LUT stores sampled sine values; often only a quarter-wave is stored and symmetry is used to reconstruct the rest to save memory.