12SOUND AND VIBRATION/DECEMBER 2002A key step in any digital processing ofreal world analog signals is convertingthe analog signals into digital form. Wesample continuous data and create a dis-crete signal. Unfortunately, sampling canintroduce aliasing, a nonlinear processwhich shifts frequencies. Aliasing is aninevitable result of both sampling andsample rate conversion. It can be ad-dressed with a properly designed anti-aliasing filter (AAF). An analog AAFmust be applied before the initial sam-pling process. If any sample rate conver-sion, such as decimation, is performed, adigital AAF must also be applied. AnAAF is one of the limiting factors in sys-tem performance. An improperly de-signed AAF, or improper application ofone, can introduce distortion artifactsthat may interfere with certain types ofanalysis. Modern AAF architectureseliminate these artifacts.The Nyquist sampling theorem definesthe minimum sampling frequency tocompletely represent a continuous signalwith a discrete one. If the sampling fre-quency is at least twice the highest fre-quency in the continuous baseband sig-nal, the samples can be used to exactlyreconstruct the continuous signal. A sinewave can be described by at least twosamples per cycle (consider drawing twodots on a picture of a single cycle, thentry and draw a single cycle of a differentfrequency that passes through the sametwo dots). Sampling at slightly less thantwo samples per cycle, however, is indis-tinguishable from sampling a sine waveclose to but below the original frequency.This is aliasing – the transformation ofhigh frequency information into false lowfrequencies that were not present in theoriginal signal. The Nyquist frequency,also called the folding frequency, is equalto half the sampling frequency fs and isthe demarcation between frequencies thatare correctly sampled and those that willcause aliases. Aliases will be ‘folded’from the Nyquist frequency back into theuseful frequency range. Thus a tone 1 kHzabove the Nyquist frequency will foldback to 1 kHz below, while a tone 1 kHzbelow the sampling frequency will ap-pear at 1 kHz as shown in Figure 1. Fre-quencies above the sampling frequencyare also folded back.Aliasing is irreversible. There is noway to examine the samples and deter-mine which content to ignore because itcame from aliased high frequencies.Aliasing can only be prevented by attenu-ating high frequency content before thesampling process as shown in Figure 2.To prevent aliasing completely, we mustapply a perfect filter that passes all en-ergy from DC to the highest frequency ofinterest and rejects all energy at theNyquist frequency and above. Unfortu-nately, perfect filters are not physicallyrealizable in analog or digital form.Physically realizable filters must havevariation in the passband, a smooth tran-sition from the passband to the stopband,and finite attenuation in the stopband.Therefore, we must design a filter withunity gain and low variation in the pass-band and with the lowest tolerable at-tenuation in the stopband.Note that finite attenuation means thatyou cannot eliminate aliasing, only re-duce it. Suppose you sample a signal thatcontains a 1 V tone at 1 kHz and a 1 Vtone at 39.9 kHz. You wish to analyze thedata to 20 kHz, so the sampling frequencyfs is 40 kHz. If the AAF gain is –80 dB inthe stopband (above 20 kHz) then thesampled signal will appear as a 1 V, 1 kHztone and a 0.1 mV, 100 Hz tone (the 39.9kHz tone aliases to 100 Hz and is attenu-ated 80 dB). The amplitude of the alias isdependant on the original amplitude ofthe out-of-band components and theamount of attenuation in the AAF. Theeffect is harder to analyze in the morerealistic case of broadband energy thatmust be rejected. All of the broadbandenergy will fold back into the analysisband. In general, the AAF attenuationmust be chosen considering the desirednoise floor and the frequency content ofthe energy that needs to be rejected.The next consideration is the width ofthe transition band. Consider designinga system with a useful frequency band-width of 20 kHz and 80 dB of alias pro-tection. If the sampling frequency is 40kHz, the AAF gain must change from 0 dBat 20 kHz to –80 dB at just over 20 kHz.We must increase the sampling frequencyto make the filter realizable. Consider asampling frequency fs = 51.2 kHz. TheNyquist frequency is 25.6 kHz, whichmeans that frequencies above 31.2 kHzwill fold back into the band of interest.Therefore, the AAF gain must go from 0dB at 20 kHz down to –80 dB at 31.2 kHz(see Figure 3). The region between thehighest useful frequency and the Nyquistfrequency is known as the guard band.Frequencies in this range will be attenu-ated and may suffer from aliasing, and areusually discarded in the presentation ofspectral results.So far we have considered the rejectionband attenuation and guardband width asperformance limits in AAF design. Threemore error sources are passband varia-tion, dispersion, and channel to channelmatch. These error sources are in thepassband, and are thus very important indetermining the overall performance ofthe system. Passband variation createsabsolute accuracy errors, while disper-sion, or nonconstant group delay, spreadsout signals over time. Channel to channelmatch is important when making crosschannel measurements. Low variationand low dispersion are both desirable,but are hard to achieve with high orderanalog filters. Channel to channel matchis compromised by analog componentvariations. In the above example, sam-pling at 51.2 kHz and keeping 20 kHz ofinformation required an 8th order ellip-tic filter. This filter has high variation anddispersion across the passband and is dif-ficult to fabricate.If we could increase the sampling rate,we could make the filter less aggressive,thus reducing the variation and disper-sion and making it easier to manufacture.We could even sample at a very high rate,then perform digital filtering (Figure 2).It is much easier to make low variationfilters digitally, and dispersion can beessentially eliminated. Digital filters aretrivially duplicated across channels. Thistechnique is used in a class of analog todigital converters called delta-sigmaADCs. For a 51.2 kHz sample rate, theseconverters often sample at 3.2768 MHz,making the Nyquist frequency 1.6384MHz. A 3rd order Butterworth filter pro-vides sufficient anti-alias protection inthis situation. A Butterworth filter ismaximally flat in the passband, has maxi-mally low dispersion, and is easier to