So many filter types. So many choices. Can anyone help?
The purpose of the reconstruction filter is to suppress the Nyquist images present in the DDS output spectrum. Nyquist images appear at frequencies above 50% of Fs (Fs is the DDS system clock frequency or sample rate). As such, the filter must significantly suppress signals above 50% of Fs. Because all filters have a transition band, however, the maximum practical filter passband is at most 40% of Fs (passband and transition band are explained below).
Most DDS applications use a reconstruction filter of the L-C variety (inductors and capacitors), for which there are various design tools available to provide component values for a specific application. There are four general filter types based on the shape of their magnitude response: Butterworth, Chebyshev (Type I & II) and Cauer (or elliptical). The four responses appear in the accompanying PowerPoint slide. All four responses exhibit a passband, transition band and stopband. The passband is the frequency range over which the filter passes signals with minimal loss. The conventional frequency boundary of the passband is the 3dB attenuation point, but lesser values are sometimes used. The stopband is the frequency range over which the filter attenuates signals by no less than some specified amount (the stopband attenuation parameter). The transition band comprises frequencies between the boundaries of the passband and stopband. For a given filter type, the width of the transition band is inversely proportional to the filter order -- that is, higher order yields narrower transition band. Higher order, however, means more components, which implies increased cost and difficultly in mass producing. Higher order can also result in increased insertion loss due to the resistance of the inductor windings and greater difficulty in controlling the filter impedance.
The four response types compare as follows (assuming all four filters are of the same order):
Butterworth -- monotonically decreasing response with the flattest passband, but the slowest rolloff rate (implying the widest transition band).
Chebyshev I -- response ripples in the passband, a monotonically decreasing stopband, but a narrower transition band than the Butterworth response.
Chebyshev II -- monotonically decreasing passband, response ripples in the stopband with the same transition band as the Chebyshev I response.
Cauer -- response ripples in both the passband and stopband, but a narrower transition band than any of the other response types.
The vast majority of applications are best served by the Cauer response.
Applications that cannot tolerate any passband ripples, however, should rule out the Cauer and Chebyshev I responses because they both have response ripples in their passband. In this case, choose either the Butterworth or Chebyshev II, depending on whether the application can tolerate response ripples in the stopband. The tradeoff, of course, is that the Butterworth and Chebyshev II require a higher order filter than the Cauer for a given transition band width.
Some applications can tolerate a small amount of passband ripple. In such cases, one can still use a Cauer or Chebyshev I filter because passband ripple is a design parameter for these filter types. That is, most filter design tools allow the user to specify the maximum tolerable passband ripple.
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