Design Guide
August 2003
 

Signal Reconstruction

Electronic Filter Design Guide

Digital to Analog Conversion (D/A)
As with input signals to A/D converters, waveforms created by D/A converters also exhibit errors. For each input digital data point, the D/A holds the corresponding value until the next sample period. Therefore, the output waveform exists as a sequence of steps. This output, a kind of "sample-and-hold" - is known as a "first-order hold".

Any step-function approximation of a smooth analog wave such as D/A output consists of a set of primary-frequency sinusoidals and their harmonics. To accurately recover the analog signal requires removing these harmonics, usually with a filter following the D/A. Such a filter features a very flat amplitude response in the pass-band and a rapid roll-off above fc. The stop-band floor must be deep enough to attenuate high-frequency component errors to below an LSB of the target system's A/D or D/A converter.

Roll-off need not be as sharp as an anti-alias prefilter, which must push the target system's useful bandwidth as close as possible to the Nyquist frequency. Even if the original signal bandwidth is 100% of Nyquist (an unrealizable goal without serious alias errors), the lowest undesirable frequency in the D/A output is the second harmonic. For reasons of convenience, many designers specify the same filter for both anti-alias and reconstruction. From an attenuation standpoint, however, this approach represents overkill. In addition, because the step-function D/A output includes fast rise and fall times, a softer roll-off, more linear phase filter (Bessel) would work better at this end of the process because it produces less ringing and overshoot than an elliptic or similar sharp-roll-off transfer function does.

According to Fourier-transform mathematics, a waveform reconstructed using a first-order hold exhibits an amplitude error (E) that varies as a function of frequency f and the sampling frequency fs, and whose magnitude is given by Figure 23.

E =
(Sin X)
X
, where X =
f
fs

Figure 23


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