Analog filtering
Theory
An 8x oversampled DA-conversion process will generate side bands at ± 8 fs, ± 16 fs etc., as shown in the following figure.
Figure 1: Spectrum of oversampled signal
By using this signal directly as output signal for amplification, intermodulation distortion may result. Therefore side bands must be filtered away, but in such a way that the filter doesn't detoriate the signal inside the audio band (from 10Hz to 20kHz).
The analogue filter is chosen to be situated directly after the dac, because this prevents any higher frequency signals to enter succeeding stages of the DAC (I-V conversion, driver stage). Filtering at an early stage also prevents resistive loads at the output of the dac caused by higher frequencies, which might affect its linearity.
We've decided to use a passive low-pass filter instead of an active filter containing opamps, as we had some negative sonic experiences with circuits containing opamps (e.g. as proposed by the data sheets of Burr Brown).
Designing a filter
The filter is dimensioned with the following contraints and goals in mind. When the frequence response of the filter is rather smooth, we assume that the human ear cannot or hardly detect attenuations smaller than 0.3dB in the audio band. The audibility of phase distortion is disputed very often [Dill94]. When using stationary signals, the audibility will be minimal, but when using short signals (e.g. impulses) phase distortion can be audible. This is because our ear doesn't respond to frequency information only, but acts like a time-frequency analyser. Linear phase distortion is not audible, as it can be considered as a constant delay for all frequencies. For constraints w.r.t. phase distortions we refer to [Blau78] where it is claimed that in very strict listening conditions and after intensive training we can detect group delays (derivate of phase to frequency) in the order of 0.4 ms. [Kohl94] reports that we cannot hear 0.2ms group delays or smaller when using headphones. Much attention must be paid to phase coherence between the left and right channel. Group delays of 10 μ s between left and right channel are said to be audible when using headphones (depending on the frequency, objects seem to move left or right in the stereo). In listening rooms which are not acoustical muted, group delays caused by loudspeakers and their reflections are definitely greater. In [Lips82] the audibility of phase distortion is expressed in terms of a filter parameter Q. Audible effects are reported for Q ≥ 1.
The goal of the analog filter is to attenuate the side bands as good as possible. We have no reference towards literature in which a minimal amount of attenuation needed to avoid inter-modulation caused by side band images is reported. The DAC design mentioned in [Stok94a] uses no filter at all, and good results are reported there. Simular experiments reported by [Ogie94] using a TDA1514 dac were rather dissapointing. Nevertheless, we expect that an attenuation of 60 dB of the first side band will be sufficiently large.
One kind of passive filters with a very flat response in the pass band and a very steep attenuation outside the pass band are the so called Butterworth filters. Frequence domain descriptions of Butterworth filters can be synthesized analysed with the Mathematica filter package by [Slan89]. In this package there is no functionality to construct these filters in terms of electronic components.
Based on the theory discussed in [Blin76] a new filter package which describes filters in terms of electrical components has been constructed [Heij94]. The idea behind the theory is as follows. An ideal low-pass filter has a frequence response of 1 below the cut-off frequency fc, and a response of 0 above fc (called g(x)). Such a low-pass filter cannot be realized in practice, and therefore must be approximated by a filter with a response given by f(x). The difference between f(x) and g(x) is called the error signal, and is given by: e(x) = f(x) - g(x). The error of a nth order filter can be minimized by first Taylor expanding e(x) around f = 0Hz, and then setting the first n-1 terms equal to zero. This will result in a Butterworth response. The nth order Taylor expansion of e(x) around 0 is given by: 
, which can be calculated using the computer program Mathematica.
Inside the package of [Heij94], first a frequence response H(s) of an nth order filter must be derived from an electrical circuit in terms of its electrical components. From this frequence response |H(jw)|2 can be calculated, and the expression 1 - |H(jw)|2 can be Taylor expanded, yielding the right expression for e(x). Setting the first n-1 coefficients equal to zero results in n-1 equalites in terms of the filter components. A 0.15dB allowable attenuation at 20kHz results in the nth equality. Solving this system of equalities will yield the filter component values for a Butterworth low-pass filter. See [Blin76] and [Heij94] for further details.
To find a feasible filter, we start to dimension a first order filter, proceed with a second and third order filter, and end with a fourth order filter. A template for the filters can be found in Figure 2. We've chosen this template for several reasons. First, starting with a capacitor to ground implies that for high frequencies the output impedance of the dac is practical zero (calculations showed that the output impedance of an LCL-filter was too high to guarantee linear behavior, also see IV-conversion for more details about the maximal impedance possible at the output pins of the dac). Secondly, using only L's and C's implies less components for a particular filter order than implementing it using RC and/or RL networks.
Figure 2: Analogue filter template
The response function Vdest / Iout of the filters are:
Fourth order: we'll save you this one (i.e. doesn't fit on one line) ...
Initially some filters have been dimensioned, built and auditioned using a different value for Rs ( = 670 Ω ) and Rd (39 Ω ). Despite the fact that this introduced some troubles for the Butterworth approximations in Mathematica, the filter is non-symmetrical. The values that have been obtained in this case can be found in the following table.
As can be observed from the table, the first and second order filters do not have enough attenuation at the first side band. Minimizing the second order error term for a third order filter can be established by taking C1 = Rd / Rs. C2. This results in relative large values of L1 (3 mH), which are rather difficult to obtain for this application (see also the next section). When C1 = 3 . C2, a local minimum for the second error term results. A fourth order filter is created by taking the second order filter of Table 1 as the 2nd stage of the filter, and calculating L1 for the first stage for different values of C1. A value of 180 μ F gives the best results. Remark that these filters are not exact Butterworth filters, but are very near approximations.
A symmetrical filter can be obtained by connecting a resistor between the output of the dac and ground. Using a 25x amplification, and assuming a 2VRMS = 2.8Vtt leads to a voltage of 0.112V that should result from I-V conversion. Given that the output current of the dac is 2mAtt, we need a resistance of56 Ω , which should be the resulting resistance of Rs and Rd . This means that Rd should be 112 Ω , and Rs should be 134 Ω (670 Ω // 134 Ω = 112 Ω ). One of the advantages of this circuit topology is that the filter will be less sensitive to the 10% max. changes in the value of the internal resistance of the dac. [Adam95] stresses the fact that the DAC should `see' a constant output resistance.
A third order Butterworth filter using these values of the resistors leads to a value of C1 = C2 = 39 nF, and L1 = 1mH.
Some typical figures of the filter can be found in Figure 3 to Figure 6.
Figure 6: Impedance of filter seen at the output of the dac
The attenuation of the filter at 330kHz is about 59dB.
W.r.t. the frequence range in which the filter components are applied, some observations can be made. First of all the capacitors and inductors used in the filter will not behave like ideal capacitors and inductors. The inductor for instance will in practice have a DC-resistance value. The inductor we use in our filter has a DC-resistance of 14 Ω , which results in the following frequence response.
Figure 7: Frequency response with DC-resistance of inductor taken into account
Comparing Figure 3 to Figure 7 shows that the DC-resistance of the inductor has an impact on the frequence response of the filter (appr. 0.1dB more attenuation at 20kHz).
Another point of concern is the properties of capacitors and inductors at high frequencies. The filter will be subject to high frequency signals. The dac has a settling time of 200ns for its full 2mA range, which means it can produce signals at 5 MHz at full scale inside its side bands. The typical behavior of capacitors as function of the frequency is shown in the Figure 8. The effect of a capacitor (1 / ω C ) starting to behave like an inductor ( ω L ) depends on its value, and starts at higher frequencies when the capacitor value is lower. For an inductor we observe a simular problem.
Observations
Audible comparison of the non-symmetrical filters to the symmetrical filter were in obvious favour of the symmetrical filter.
W.r.t. the inductors we tried out several types (HF, air core, magnetic core, hand-wounded). We can say that some brands seem to be much worse than others, resulting in a differences in frequence response (sibilance) and phase response (the timing is different). The inductors we use now are from a local supplier, and we don't know the brand. These are ordinary HF-types, but the ones which have a red background clearly sound worse that those with a white background. These differences among inductors are enormous (they have more impact than different filter topologies), and much care should be taken in choosing the right one!
Guy Adams from Audio Note has the same observation, and told us that he spent a great deal of time in finding the right inductors for the development of the Audio Note DACs. Also great care should be taken in the fact that the inductors pick up electro-magnetic fields from other circuitry. We had bad experiences with air-core inductors, as they caused a great amount of hum. If one likes to use these type of inductors, they should be shielded.
We did try out several capacitors and choosed a set of capacitors based on listening sessions. The types tried out were from Jensen paper-in-oil, Chateaux Roux, Intertechnik and Audyn Cap. We did found out that the Jensen paper-in-oil capacitors did gave us the most satisfying results.