# Direct-Sampling DACs in Theory and Application

Abstract: High-speed digital-to-analog converters (DACs) with high resolution in the 12- to 14-bit range enable new types of transmitter designs that employ a direct modulation scheme. In such designs, the modulated transmission signal is generated directly on the base frequency. This application note explains how direct-sampling RF DACs enable new communications systems such as cable infrastructure and explains how to synchronize multiple DACs that have multiplexed inputs (MUXDACs).

*Elektronik Informationen*magazine.

**Figure 1**shows a typical block diagram for an analog multicarrier QAM transmit path. To generate several QAM transmission channels, it is necessary to bring together several individual transmit chains with an adder. In addition, each RF modulator is supplied via its own synthesizer. Since each functional block has its own component tolerances, temperature drift, conversion losses etc., these must all be taken into account during system configuration by performing a tolerance calculation.

*Figure 1. Block diagram of an analog multicarrier QAM transmitter.*

**Figure 2**shows the block diagram for a digital multicarrier QAM transmit path. Generating 32 QAM channels, for example, does not require combining multiple transmit paths—a single transmit path can handle this. In this case, the transmission path consists of an RF DAC with a downstream filter and a VGA. This also eliminates the need for an RF modulator or synthesizer in each transmit branch.

*Figure 2. Block diagram for a digital multicarrier QAM transmitter with an RF DAC.*

**Figure 3**shows the output signal for a DAC. Here, this is a sequence of rectangle pulses, the amplitude of which matches the corresponding digital value. Since the duration of these rectangle pulses is finite, T

_{s}> 0, this results in an output spectrum. In the frequency domain, this output spectrum is described with the sinx/x function. The sinx/x function, which is also known as a sinc function, has zeros at the frequency f

_{s}= 1/T

_{s}.

*Figure 3. DAC output signal as a sequence of rectangle pulses. T*

_{s}= sample time._{0}has a spectral line at f

_{0}. If the sinus signal is now generated with a DAC, besides the spectral line at f

_{0}, additional frequency conversion products arise at higher frequencies (see

**Figure 4**).

*Figure 4. The sinus signal, described in the frequency range, and the output spectrum of a signal generated with a DAC.*

|K × f_{s} ± f_{0}| K = 1, 2, 3, … |
(Eq. 1) |

[(N - 1) × f_{s}/2, N × f_{s}/2] K = 1, 2, 3, … |
(Eq. 2) |

_{0}; instead, it also has further spectral components at higher frequencies. This means that the DAC’s output signal must be filtered. Besides these spurious emissions, additional frequency-conversion products arise. These are caused, for example, by the DAC’s nonlinear output characteristics.

**Figure 5**).

*Figure 5. The MAX5879 RF DAC’s impulse response for a) NRZ mode; b) RZ mode; c) RF mode; and d) RFZ mode.*

_{s}. The sinc function has its zeros at multiples of the update clock rate f

_{CLK}= 1/T

_{s}. When this impulse response is used, the following function arises for the DAC’s frequency response:

A_{NRZ} = A_{0}[sin(πf_{OUT}T_{s})/(πf_{OUT}T_{s})] |
(Eq. 3) |

_{OUT}is the DAC output frequency, T

_{s}= 1/f

_{CLK}is the DAC update clock rate, and A0 is the amplitude factor.

A_{RZ} = A_{0}/2[sin(πf_{OUT}T_{s}/2)/(πf_{OUT}T_{s}/2)] |
(Eq. 4) |

A_{RF} = A_{0}[sin(πf_{OUT}T_{s}/2)/(πf_{OUT}T_{s}/2) × sin(πf_{OUT}T_{s}/2)] |
(Eq. 5) |

A_{RFZ} = A_{0}/2[sin(πf_{OUT}T_{s}/4)/(πf_{OUT}T_{s}/4) × sin(πf_{OUT}T_{s}/4)] |
(Eq. 6) |

**Figure 6**shows the MAX5879’s frequency response with the four possible operational modes. The figure’s x axis shows the output frequency normalized to the input data sample rate. The range from 0 to 0.5 delineates the first Nyquist zone. The NRZ mode delivers the largest output signal in the first Nyquist zone. The RZ mode, comparatively, features the flattest frequency response in the first and third Nyquist zones. The RF mode is characterized by a maximum output power in the second and third Nyquist zones. Furthermore, the frequency response in the second Nyquist zone is marked by a flatter rise than the other two operational modes exhibit. Of all the operational modes, the RFZ mode has the flattest impulse response across all of the Nyquist zones.

*Figure 6. The MAX5879 RF DAC’s normalized frequency response for the four possible operational modes.*

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## Simplifying DAC Control with a Digital Upconverter (DUC)

## Clock Synchronization in Applications that Use Multiple DACs

- The relative phase angle of the rising clock edge must be detected.
- The relative phase angle between the individual DACs must be shifted until the DACs are synchronous with each other in the right phase relationship.

## Conclusion

#### References

- Kuckreja Ajay, Ostrem Geir, “High-Speed DACs ease transmitter designs,” Microwave & RF, August 2010
- Application note 3901, “Synchronizing Multiple High-Speed Multiplexed DACs for Transmit Applications.”
- Stanley Chen, Jan Rabea, “Direct Waveform Synthesis,” BWRC Summer Retreat 2008.