Flatten DAC frequency response
In a generic example, a DAC samples a digital baseband signal (Figure 1). The DAC's frequency response is not flat; it attenuates the analog output at higher frequencies. At 80% of f_{NYQUIST}, for instance (f_{NYQUIST}=f_{S}/2), the frequency response attenuates by 2.42 dB. That amount of loss is unacceptable for some broadband applications requiring a flat frequency response. Fortunately, however, several techniques can cope with the nonflat frequency response of a DAC. These techniques include increasing the DAC's update rate using interpolation techniques, preequalization filtering, and postequalization filtering, all of which reduce or eliminate the effects of the sinc rolloff.
Frequency response
To understand the nonflat frequency response of a DAC, consider the DAC input as a train of impulses in the time domain and a corresponding spectrum in the frequency domain (Figure 2). An actual DAC output is a "zeroorder hold" that holds the voltage constant for an update period of 1/f_{S}. In the frequency domain, this zeroorder hold introduces sin(x)/x, or aperture, distortion (Reference 1). The amplitude of the outputsignal spectrum multiplies by sin(x)/x (the sinc envelope), where x=πf/f_{S}, and
(1) 
describes the resulting frequency response (Figure 3). Thus, aperture distortion acts as a lowpass filter that attenuates image frequencies but also attenuates the desired inband signals.
The sin(x)/x (sinc) function is wellknown in digitalsignal processing. For DACs, the input is an impulse, and the output is a constantvoltage pulse with an update period of 1/f_{S} (the impulse response), whose amplitude changes abruptly in response to the next impulse at the input. You obtain the DAC's frequency response by taking the Fourier transform of the impulse response (a voltage pulse, Reference 2).
The desired signal frequency in the first Nyquist zone reflects as a mirror image into the second Nyquist zone between f_{S}/2 and f_{S}, but the sinc function attenuates its amplitude. Image signals also appear in higher Nyquist zones. In general, a lowpass or bandpass filter, often called a reconstruction filter, must remove or attenuate these image frequencies. Such filters are analogous to the antialiasing filter that an ADC often requires.
As the DAC output frequency approaches its update frequency, f_{S}, the frequency response approaches zero or null. The DAC's output attenuation therefore depends on its update rate. The 0.1dBfrequency flatness is about 0.17f_{NYQUIST}, where f_{NYQUIST}=f_{S}/2. As the output frequency approaches f_{S}/2, so does the first image frequency. As a result, the maximum usable DAC output frequency for systems in which filtering removes the image frequency is about 80% of f_{NYQUIST}.
The first image frequency is f_{IMAGE}=f_{S}–f_{OUT}. At f_{OUT}=0.8f_{NYQUIST}, f_{IMAGE}=1.2f_{NYQUIST}, leaving only 0.4f_{NYQUIST} between frequency tones for the filter to remove the image. Output frequencies higher than 80% of f_{NYQUIST} make it difficult for a filter to remove the images, but the reduction in usable frequency output allows for realizable reconstructionfilter designs.
Speed the update rate or interpolate?
At 80% of f_{NYQUIST}, the output amplitude attenuates by 2.42 dB. For broadband applications requiring a flat frequency response, that amount of attenuation is unacceptable. Because the DAC's output attenuation depends on its update rate, you can minimize the effect of sinc rolloff and push the 0.1dB flatness to a higher frequency simply by increasing the converter's update rate and keeping the inputsignal bandwidth unchanged.
Increasing the DAC's update rate not only reduces the effect of the nonflat frequency response, but also lowers the quantization noise floor and loosens requirements for the reconstruction filter. Drawbacks include a higher cost for the DAC, higher power consumption, and the need for faster data processing. The benefits of higher update rates are so important, however, that manufacturers are introducing interpolation techniques. Interpolating DACs offer all the benefits of higher update rates and keep the input data rate at a lower frequency.
Interpolation DACs include one or more digital filters that insert a sample after each data sample. In the time domain, the interpolator stuffs an extra data sample for every data sample entered, with a value interpolated between each pair of consecutive datasample values. The total number of data samples increases by a factor of two, so the DAC must update twice as fast.
One modern DAC, for example, incorporates three interpolation stages to achieve an 8× interpolation; the DAC's update rate is eight times the data rate (Reference 3). In the frequency domain, the sincfrequency response also moves out by a factor of eight, as does the effective image frequency, which loosens requirements for the reconstruction filter.
Preequalize?
Increasing the update rate reduces but does not eliminate the effect of sincfrequency rolloff. If you are already using the fastest DAC available, you must choose other techniques to make additional improvements. It is possible, for example, to design a digital filter whose frequency response is the inverse of the sinc function, that is, 1/sinc(x). In theory, such a preequalization filter exactly cancels the effect of the sincfrequency response, producing a perfectly flat overall frequency response. A preequalization filter filters the digital input data to equalize the baseband signal before it sends the data to the DAC. Removing all image frequencies at the DAC output allows original signal reconstruction without attenuation (Figure 4).
Any digital filter whose frequency response is the inverse of the sinc function will equalize the DAC's inherent sincfrequency response. Because the sincfrequency response is arbitrary, however, a FIR (finiteimpulseresponse) digital filter is preferable. Frequencysampling techniques are useful in designing the FIR filter. Assuming the signal is in the first Nyquist zone, you sample the frequency response, H(f), from dc to 0.5f_{S} (Figure 5). Then, using the inverseFourier transform, you transform the frequency sample points, H(k), to impulse responses in the time domain. The impulse response coefficients are:
(2) 
and
(3) 
where H(k) and k=0, 1, ... N–1 represent the ideal or targeted frequency response. The quantities h(n) and n=0, 1, ... N–1 are the impulse responses of H(k) in the time domain, and α=(N–1)/2. For a linearphase FIR filter with positive symmetry and even N, you can simplify h(n) using Equation 3. For odd N, the upper limit in the summation is (N–1)/2 (Reference 1).
Increasing the number of frequency sample points (N) of H(k) produces a frequency response closer to the targeted response. A filter with too few sample points reduces the effectiveness of the equalizer by producing a larger deviation from the target frequency response. On the other hand, a filter with too many sample points requires more digitalprocessing power. A good technique uses large N for computing h(n), truncates h(n) to a small number of points, and then applies a window to smooth h(n) and produce an accurate frequency response.
A sample filter uses N=800 to compute h(n) (Figure 6). You then truncate h(n) to only 100 points and apply a Blackman window to h(n). The frequency response for the combined FIR filter and DAC sinc response exhibits 0.1dB flatness nearly up to the Nyquist frequency (to approximately 96% of f_{NYQUIST}, where f_{NYQUIST}=f_{S}/2). In contrast, the uncompensated DAC response maintains 0.1dB flatness only to 17% of f_{NYQUIST}. Because the filter gain is greater than unity, you must take care that the filter's output amplitude does not exceed the DAC's maximum allowed input level.
After obtaining the impulseresponse coefficients, you can implement the FIR filter using a standard digitalprocessing technique. That is, h(n) filters the input signal data x(n):
(4) 
Dynamic performance for the compensated DAC is lower than that of the uncompensated DAC, because higher gain at the higher input frequencies requires that you intentionally lower the signal level to avoid clipping the input. Assuming the input is a single tone between dc and f_{MAX} (less than f_{S}/2), the attenuation depends on f_{MAX}:
(5) 
where V_{IC} is the input voltage for the compensated DAC, and V_{REF} is the reference voltage. If, for example, the maximum anticipated input frequency is f_{MAX}=0.8f_{NYQUIST}, you must attenuate the DAC input by V_{IC}=–2.4 dB below V_{REF}.
The resulting output amplitude is flat over frequency, representing perfect compensation, and equals the input amplitude of V_{OC}=V_{IC}=–2.4 dB below V_{REF}. You obtain output noise by integrating the noise power density from near dc to the reconstruction filter's cutoff frequency. DAC manufacturers also often specify SNR by integrating the noise out to f_{NYQUIST} without the use of a reconstruction filter:
(6) 
where N_{C} is the total noise power or voltage of the compensated DAC, and n_{Q}(f) is the DAC's output noise density, which is usually limited by quantization noise and thermal noise. The maximum SNR for the compensated DAC is constant and independent of frequency, but it depends on the maximum anticipated output frequency:
(7) 
where V_{OC} is the output amplitude. For the uncompensated DAC, the sinc envelope attenuates the output signal:
(8) 
Noise power for the uncompensated DAC is same as for the compensated DAC. Thus, the maximum uncompensatedDAC SNR is
(9) 
You can determine the degradation of the compensatedDAC SNR by dividing the SNRs:
(10) 
Degradation of the compensated DAC SNR, unlike that of the uncompensated DAC, is frequencydependent. Degradation is worse at frequencies lower than f_{MAX}.
Postequalize?
Another method of equalizing the DAC's sincfrequency response over the outputfrequency band of interest is to add an analog filter whose frequency response is approximately equal to the inversesinc function. Many such analogequalization filters exist for equalizing transmission lines and amplifiers, and you can adapt those equalization techniques for reducing the effect of a DAC's unwanted sinc response. The postequalization filter inserts after the DAC's reconstruction filter.
This application uses a simple active equalizer (Figure 7). For a given bandwidth, you choose R_{1}, R_{2}, and C_{1} so that the analog equalizer's frequency response cancels the DAC's sincfrequency response. Spicesimulation software can help optimize the frequency flatness for a given application. The frequency response for a typical analog equalizer shows that 0.1dB flatness extends to more than 50% of f_{NYQUIST}. Without the postequalization filter, 0.1dB flatness extends only to 17% of f_{NYQUIST}. Note that the maximum circuit gain is 1+R_{1}/R_{2}.

A postequalization filter affects the DAC's SNR because it amplifies the noise at higher frequencies. Assuming that quantization noise limits the noise in an uncompensated DAC, the sinx/x envelope attenuates both the output signal and the noise. With a postequalization filter, however, the outputsignal amplitude and noise density are constant over frequency, assuming perfect compensation. You obtain the output noise for the compensated and uncompensated DACs by integrating the noise power from near dc to f_{NYQUIST}:
(1115) 
and
(16) 
where H(f) is the frequency response for the postequalization filter, n_{Q}(f) is the noise power density, n_{QO} is the unattenuated quantizationnoise density near dc, and N_{C} and N_{U} are the total noise power of the compensated and uncompensated DACs, respectively. Maximum SNR normalizes to the reference voltage, V_{REF}. Remember that f_{NYQUIST} equals f_{S}/2. The SNRs are then:
(1718) 
Again, dividing the two SNRs gives the compensated SNR in terms of the uncompensated SNR. The maximum SNR degrades at lower frequencies but improves at higher frequencies:
(19) 
So far, you assume that the DAC's reconstruction filter is an ideal lowpass filter: Its frequency response is flat to f_{NYQUIST}, and then it drops abruptly to zero. In practice, a reconstruction filter also adds rolloff near its cutoff frequency. Accordingly, the preequalization and postequalization techniques can serve an additional purpose of equalizing any rolloff in the reconstruction filter.
Wrapping up
The effect of a DAC's inherent sincfrequency response attenuates output signals, especially at higher frequencies, and the resulting nonflat frequency response reduces the maximum useful bandwidth in broadband applications. Higher update rates flatten the frequency response but increase the DAC's cost and complexity.
The preequalization technique, which employs a digital filter to process the data before sending it to the DAC, offers 0.1dB frequency flatness to 96% of f_{NYQUIST} (f_{NYQUIST}=f_{S}/2) but requires additional digital processing. For comparison, an uncompensated DAC offers 0.1dB flatness only to 17% of f_{NYQUIST}. Another technique adds a postequalization analog filter to equalize the DAC's output and achieves 0.1dB flatness to 50% of f_{NYQUIST} but requires additional hardware. Both compensation techniques offer a lower SNR at low output frequencies.
References 

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