When you watch a movie, you probably have seen an example of aliasing. That’s what makes the spokes of a wagon wheel or the rotor blades of a helicopter appear to be rotating slowly backwards. In effect, the wheels are going faster than the frame rate of the movie film.
In data acquisition systems (DAS), aliasing occurs whenever an input signal has frequency components at or higher than half the sampling rate. These high-frequency components can result from the inherent noise of the system and from noise or interference not related to the DAS, including 50-Hz or 60-Hz pickup, broadcasting stations and mechanical vibrations. High-frequency components also are inherent in any sharp transitions of the measured signal.
If the signal is not correctly band-limited to eliminate these frequencies, they show up as aliases or spurious lower-frequency errors that cannot be distinguished from valid sampled data. The alias signals are actually at a higher frequency, but are converted by the sampling process to a false frequency below half the sampling rate.
For example, with a sampling rate of 1,000 Hz, a signal at 800 Hz will be aliased to 200 Hz, the false lower frequency. Consequently, aliasing is a phenomenon that occurs when a high-frequency component effectively takes on the identity of a lower frequency.
One solution to the aliasing problem is to sample the signal at a very high rate and then filter out the high frequencies with digital techniques. But such oversampling of data increases system costs by requiring faster A/D conversion for digital processing, more memory and higher bandwidth buses. It also leads to higher analysis costs by creating more data to process and interpret.
Signal Filtering
A more practical alternative limits the bandwidth of the signal to below one-half the sample rate with an anti-alias filter, which can be implemented on each input channel in front of the A/D converter. Anti-alias filtering must be done before the signal is sampled or multiplexed, since there is no way to retrieve the original signal once it has been digitized and aliased signals have been created.
When using an anti-alias filter to avoid aliasing, two processes must occur:
o As dictated by the Nyquist theorem, the input signal must be sampled at a rate of at least twice the highest frequency component of interest within the input signal.
o Any frequency components above half the sampling rate (also called the Nyquist frequency) must be eliminated by an anti-alias filter before sampling.
An anti-alias filter is a low-pass filter. This means that under ideal conditions it would exactly pass unchanged all slower signal components with frequencies from DC to the filter cutoff frequency. Faster components above that point would be totally eliminated, reducing the signal disturbance.
But real filters do not cut off sharply at an exact point. Instead, they gradually eliminate frequency components and exhibit a falloff or rolloff slope (Figure 1). These attenuation slopes typically range from 45 dB/octave to 120 dB/octave and bottom out at some finite value of stopband rejection, usually 75 dB to 100 dB.
A simple illustration of these processes can be seen in the case of the 800 Hz frequency aliasing to 200 Hz. Suppose that the 800 Hz is an unwanted interfering signal caused by a mechanical vibration. To prevent its alias from causing significant data errors at 200 Hz, the 800-Hz frequency must be removed by an anti-alias filter.
If the cutoff point is set near 450 Hz, a filter with a steep rolloff slope will eliminate the 800-Hz frequency, making the false 200-Hz frequency disappear. The input frequencies of interest below the filter cutoff (450 Hz) will still pass through the system unchanged.
Filter Selection
Among the types of filters more commonly used for anti-alias purposes are the elliptic or Cauer, Bessel and Butterworth. No filter is perfect, and different types of filters are imperfect in different ways.
The optimum type of filter for an application depends on which kinds of imperfections are most easily tolerated. Examples of imperfections include phase nonlinearity, gain error, passband ripple and droop, and wideband noise.
An extremely sharp cutoff-frequency rolloff makes the Cauer elliptic filter suitable for most anti-alias applications. Cauer filters also have good passband flatness and low wideband noise. But their nonuniform group delay can cause some overshoot or ringing in time-domain plots if the input signal has sharp transitions.
Although it has a slow rolloff, the Bessel filter has a uniform group delay with no ringing or overshoot. It is suited for time-domain anti-alias applications requiring minimum distortion of rapid slope changes.
A Butterworth filter is useful when maximum passband flatness is critical. It exhibits a moderate amount of overshoot in the phase response and, like the Bessel, has a gentler cutoff slope as compared to the elliptic.
Summary
The aliasing phenomenon becomes a problem in A/D conversion systems when an input signal contains frequency components above half the A/D sampling rate. These higher frequencies can fold over into the lower frequency spectrum and appear as erroneous signals that cannot be distinguished from valid sampled data. The best approach to eliminating false lower frequencies is to use an anti-alias filter, which inhibits aliasing by limiting the input signal bandwidth to below half the sampling rate.
About the Author
Rosita Zargar is an Applications Engineer at Alligator Technologies. She graduated from California State University with a B.S.E.E. degree. Alligator Technologies, 2900 Bristol St., Suite E-101, Costa Mesa, CA 92626, (714) 850-9984.
Instrumentation
Sidebar
Filter Terminology and Definitions
Aliasing
–A phenomenon that occurs during digital sampling (including A/D conversion) when a high-frequency signal or noise is converted by the sampling process to a false lower frequency below half the sampling rate.Attenuation Slope
–For a low-pass filter, the slope of the gain vs frequency curve in the region of rapidly increasing attenuation just above the cutoff frequency. Many filters have a gain in this region proportional to a negative power of frequency (that is, linear on a log-log graph). The slope is commonly expressed in decibels per octave, which indicates that the number of decibels increase for each doubling of frequency.
Most anti-alias filters have attenuation slopes ranging from 45 to 120 dB/octave. At 120 dB/octave, a disturbing input at a frequency of 1.7 times cutoff is attenuated by over 80 dB. That is, a 1-V disturbance is reduced to 100 µV, eliminating the disturbing signal for most purposes.
DC Offset
–The shift in the DC level of the signal. DC offset can be as much as 150 mV to 200 mV in some filters or as little as a fraction of a millivolt in other filters.Droop
–A monotonic decrease in passband gain with frequency. If present, it typically is included in the gain or ripple specification. See Gain Error and Passband Ripple.Gain Error
–The difference between the specified and actual passband gains. For typical filters, the gain error can be as high as a few percent. It can refer to frequencies far below the cutoff only or to the whole passband.
If a ripple is stated or if the term “DC gain error” is used, the gain error probably refers to low frequencies. If not, it likely refers to most of the passband or the entire passband and includes the ripple.
Group Delay
–The time delay through the filter for a finite-length burst (pulse) of a sine wave. A group delay which is not constant as the sine-wave frequency varies can cause overshoot or ringing in the passband.
At any sine-wave frequency, the filter group delay is the derivative of the filter phase shift with respect to frequency. As a result, a perfectly uniform group delay is equivalent to a perfectly linear phase response.
Harmonic Distortion
–The amount of distortion introduced by slight nonlinearities in the amplifiers of a filter circuit. This will vary drastically with signal amplitude, and to some extent, with frequency. For example, for peak-to-peak amplitudes of 80% of the full voltage range, typical distortion levels may be around 0.005% to 0.05% at mid frequencies. For levels closer to full scale, the distortion increases; for lower levels, it usually decreases.Low-Pass Filter
–A filter that ideally passes unchanged all slower frequency components from DC to the filter cutoff and totally eliminates faster components above that point. In real filters, the transition has a finite slope instead of being discontinuous, and the attenuation at high frequencies is not infinite.Noise and Interference
–Unwanted electrical signals that can come from the data acquisition system itself or from such external sources as 50-Hz or 60-Hz pickup, broadcasting stations and mechanical vibrations.Nyquist Frequency
–The frequency at one-half the sampling rate.Nyquist Theorem
–A mathematical theorem stating that a band-limited input signal can be recovered without distortion if it is sampled at a rate of at least twice the highest frequency component of interest within the signal.Passband Ripple
–The variation in gain over the passband, when the gain-vs-frequency response appears rippled instead of flat. Typical filters can have 1% or 2% ripple over most of the passband. If passband frequencies near the cutoff are included, the ripple can be significantly greater.Phase Linearity
–The linearity of a plot of phase shift vs frequency, the response of which should be a straight line. In practice, a filter designed for phase linearity can have passband phase linearity as low as a fraction of a degree, while other filters can have nonlinearities so great that the group delay changes by a factor as high as two or three over the passband.Stopband Attenuation
–For a low-pass filter, the attenuation at all frequencies above where the gain-vs-frequency response has finished its rapid falloff, typically 75 dB to 100 dB.Wideband Noise
–In a filter specification, the random noise generated by a filter over a band equal to or greater than the passband. This is often a large part of the total wideband noise at the filter output. Typical high-quality filters introduce 60 µV to 150 µVrms of wideband noise. Unlike resistor noise, changing the filter passband does not greatly change the total wideband rms value for switched-capacitor filter noise.
Copyright 1995 Nelson Publishing Inc.
September 1995
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