DIGITIZING SIGNALS AND ALIASING

Analog to Digital Conversion (A/D)

Most physical (real world) signals are analog. Operating on these signals efficiently often requires the filtering, sampling and digitizing of the analog data using A/D converters. The converted digital data may then be manipulated mathematically. Many data-acquisition systems must also construct a representation of the original signal from the digital data stream.

Unfortunately, sampling often sacrifices accuracy for the sake of convenience. The digital version of a signal may not resemble the original in some important respects. A graphic example is the movie scene that apparently shows wagon wheels or helicopter blades turning backwards. This erroneous image, known as an "alias", occurs because a "motion picture" camera actually samples continuous action into a series of stills, and the frame rate (commonly 24 or 30 frames per second) is not fast enough or is nearly an exact multiple of the object's rotation speed.

According to Nyquist's Theorem, accurately representing an analog signal with samples requires that the original signal's highest frequency component be less than the Nyquist frequency, which is at least half the sampling frequency. To correct the image in the movie example, the frame rate would have to exceed twice the rotation speed of the wheel (or its spokes) or of the helicopter blades. No practical data-acquisition system can sample fast enough to catch all of a real signal's components. Frequencies above Nyquist appear as false low-frequency aliases. As an example, Figure 1 shows the result of sampling a 900 Hz signal at 1 kHz.

Low-pass filters are by far the most common filter type, earning wide popularity in removing alias signals and for other aspects of data acquisition and signal conversion. For a low-pass filter, the pass-band extends from DC (0 Hz) to f_c and the stop-band lies above f_c .

In a high-pass filter, the pass-band lies above f_c , while the stop-band resides below that point.

Combining high-pass and low-pass technologies permits constructing band-pass and band-reject filters. Band-pass filters transmit only those signal components within a band around a center frequency f_o .An ideal band-pass filter would feature brick-wall transitions at f_L and f_H , rejecting all signal frequencies outside that range. Band-pass filter applications include situations that require extracting a specific tone, such as a test tone, from adjacent tones or broadband noise.

Band-reject (sometimes called band-stop or notch ) filters transmit all signals except those between f_H and f_L . These filters can remove a specific tone - such as a 50 or 60 Hz line frequency pickup - from other signals. A common application is medical instrumentation, where high-impedance sensors pick up line frequencies.

NON-IDEAL FILTERS (REAL WORLD)

Real-world signals contain both wanted and unwanted information. Therefore, some kind of filtering technique must separate the two before processing and analysis can begin. Real filters are far from ideal. They subject input signals to mathematical transfer functions with names like Butterworth, Bessel, constant delay and elliptic that only approximate ideal behavior. Instead of the sharply defined transition represented by ideal filters, real filters contain a transition region between the pass-band and the stop-band as shown in Figure 3.

Amplitude Response is defined as the ratio of the output amplitude to the input amplitude versus frequency and is usually plotted on a log/log scale as shown in Figure 5. Note how the steepness of the transition band slope (roll-off) increases as the number of poles increase.

All non-ideal filters introduce a time delay between the filter input and output terminals. This delay can be represented as a phase shift if a sine wave is passed through the filter. The extent of phase shift depends on the filter's transfer function. For most filter shapes, the amount of phase shift changes with the input signal frequency. The normal way of representing this change in phase is through the concept of Group Delay, the derivative of the phase shift through the filter with respect to frequency.

Group Delay (D) equation:

D =

d df

Group Delay

Group Delay is the phase slope on a linear phase vs. frequency plot. Figure 6 compares the group delay of some typical phase response curves.

Analog Filter Specifications

Low-Pass and High-Pass
In order to define the limits of the filter pass-band in real circuits, most filter specifications define the corner frequency (f_c), as the frequency where attenuation reaches -3 dB or for elliptic filters, the ripple frequency (f_r), the point where the response curve last passes through the specified pass-band ripple. Filter specifications may also include a shape factor (sf) requirement, which describes how fast signals roll-off during transition. The sf represents the ratio between the cutoff/ripple frequency and where the filter achieves a desired attenuation level, say (-80 dB).

Figure 7 is an elliptic filter that attenuates to -80 dB at 1.56 f_r, hence a shape factor of 1.56 to -80 dB. A high-pass filter with a 1.56 shape factor would achieve that same -80 dB of attenuation at f_r/1.56 or 0.64 f_r.

Also note that the elliptic transfer function attenuation floor is not infinite, but has notches and humps.

Filter Equations

Filter transfer functions relate filter output to input through polynomials in the Laplace-transform complex variable "S" as shown in Equation 1. Using the "S" domain may seem confusing, but allows both the amplitude and time response of a filter to be expressed in a simple format. A two pole, two zero, low-pass filter can be expressed as:

where: H_o = dc gain
Q = peaking factor at the corner frequency
_o = 2f_o = filter corner frequency
_n = filter notch frequency

Filters may include both Poles and Zeros. A Pole is any frequency that makes the denominator of the mathematical transfer function go to zero. A Zero is a frequency that makes the transfer-function numerator go to zero. Second-order transfer functions may contain two poles and up to two zeros. To achieve steeper roll-off, higher-order real filters usually include cascades of second-order and first-order filter stages.

To produce the phase and frequency response, "S" in the above equation is replaced by j . Consider the second-order function that produces the amplitude versus frequency curve in Figure 9.

Phase Filters
For some filter applications it is desirable to preserve a transient waveform while removing higher frequency noise components from the signal. If each of the frequency components of the input waveform (from the Fourier series or the Fourier transform) is phase shifted an amount linearly proportional to frequency, then they remain in the correct time relationship and sum together to create, at the output, the original waveform that was present at the input of the filter, with the higher frequencies components having been removed by the filter. When a filter has phase delay that varies linearly with frequency it is called a Linear Phase filter. A linear phase filter has a constant group delay, at least through the pass-band. Amplitude filters provide relatively constant group delay only from 0 Hz to about the mid pass-band frequency range peak near f_c.

As with amplitude filters, mathematicians have provided polynomial approximations of an ideal linear phase transfer function. The most common linear phase filter is based on Bessel (sometimes called Thompson) functions. Bessel filters provide very linear phase response and little delay distortion (constant group delay) in the pass-band. They show no overshoot in response to step input and roll-off monotonically in the stop-band. They also exhibit much slower attenuation in the transition-band than amplitude filters. Figure 13 presents amplitude and delay response curves for an 8-pole Bessel. Other types of phase filters include, constant-delay (a modified Bessel), equiripple phase, equiripple delay, and Gaussian transfer functions. They either have more pass-band amplitude roll-off for only a small improvement in phase linearity or only slightly less roll-off in the pass-band at the expense of degrading the phase linearity.

Amplitude equalization modifies the amplitude response of phase filters to produce a filter that is sometimes called a constant delay filter. Stop-band zeros on the j axis introduce attenuation notches in the stop-band, but contribute no phase or delay to the pass-band response. Figure 14 shows mirror-image right-half, left-half plane pass-band-zero pairs that modify amplitude response without additional phase or delay.

This technique can achieve a factor-of-two improvement in Bessel roll-off to a -80 dB floor, comparable to Butterworth-filter performance. For comparison, Figure 15 shows the amplitude response of an 8-pole Bessel, an 8-pole, 6-zero constant delay, and a 8-pole Butterworth response.

From a practical point-of-view, this technique allows filter and system designers an order-of-magnitude phase-linearity improvement over conventional amplitude transfer functions. Figure 16 illustrates the group delay of a 6-pole, 4-zero elliptic filter with and without a two-pole delay equalizer. The equalized plot is flatter over a larger portion of the pass-band at the expense of an increase in the amount of delay. The equalization process in this case increases total phase shift by as much as 360° at the cutoff frequency and by 720° at higher frequencies.

OUTPUT SIGNAL ERRORS

Besides inaccuracies of theoretical approximation, the most significant side effects of signal filtering are the following:

Settling time is not strictly an output signal error because it is mathematically related to the filter transfer function, but is usually deemed to be an undesirable filter side effect. All filters serve to delay the input signal by a certain minimum amount as well as increasing rise and fall time of any fast changing input signal. A general rule for settling time is that the more the filter approaches a "brick-wall" approximation, the longer it will take to settle. Therefore, an eight-pole filter will take longer to settle than a four-pole filter.

Step Response for amplitude type filters may exhibit substantial overshoot (ringing) when presented with a sudden change in voltage amplitude at the filter input. See Figure 17 for typical 8 pole transfer function step response curves.

Noise (noise created by both passive and semiconductor devices) is present at the output of any filter. In most cases, later filter stages remove stop-band noise from earlier stages, but they leave noise in the pass-band unaffected. High-Q filter stages amplify noise near their corner frequencies. In an active filter, for example, the noise spectrum in the stop-band is usually flat and low level, resulting largely from the output amplifier. At the low-frequency end of the pass-band, the noise spectrum is also flat, but with a magnitude two to four times the level of the stop-band noise. Near the corner frequency, noise levels peak at magnitudes that depend on the filter's transfer function. Elliptic filters, which feature high-Q last stages, produce noise peaks near the corner frequency of three to five times the level of low-frequency pass-band noise.

The importance of noise will depend on the system bandwidth and the level of signals passing through the filter. In digitizing systems, aliasing folds a frequency spectrum around each harmonic of the sampling frequency, and because these effects are additive, achieving the best data accuracy requires reducing broad-band-noise as much as possible.

Distortion, harmonics of the input signal's frequency components result from non-linearity in the filter circuit. These harmonics become inputs to the A/D converter, which digitizes them with the rest of the signal. As with broad-band-noise, each low-pass filter stage removes stop-band distortion components that the previous stage generates. Distortion levels vary with input-signal frequency, amplitude, transfer function, and corner frequency.

Total Harmonic Distortion (THD) is a specification often used as a single number representation of the distortion present in the output of an active circuit. It is the RMS sum of the individual harmonic distortions (i.e. 2^nd, 3^rd, --- etc.) that are created by the non-linearities of the active and passive components in the circuit when it is driven by a pure sinusoidal input at a given amplitude and frequency.

Harmonic distortion measurement requires a very low distortion sinusoidal input to the circuit, the removal of the fundamental frequency component from the output and the measurement of the amplitude of the remaining harmonics, which are typically 60 to 140 dB below that of the fundamental.

Spectrum analyzers and FFT instruments can measure individual harmonic components and can be used to calculate the THD. For active filters, the THD is usually specified in dBc (dB relative to the amplitude of the fundamental frequency component) and at a specific frequency and amplitude (ex. 10Vp-p @ 1.0kHz).

An RMS voltmeter can be used to measure the THD if, the fundamental frequency component can be removed by a notch filter to a level that is at least an order of magnitude below the largest harmonic component. However, that measurement will also include any noise that is within the bandwidth of the meter and is commonly referred to as the THD + NOISE or THD + N.

At lower frequencies, amplifiers have sufficient loop gain to reduce distortion to acceptable levels. For input frequencies near f_c, the filter removes second and higher order harmonics. Above the corner frequency, filter attenuation reduces the primary signal, which also reduces the distortion. However, if` the signal frequencies are well below the corner frequency and the signal has distortion, then that distortion will also reside in the filter's pass-band. Distortion components will affect the accuracy of the analog-to-digital signal conversion.

SELECTING THE RIGHT ANALOG FILTER

Choosing the correct filter shape for a particular application requires defining properties of the incoming signal that the filter must remove, as well as the properties that it must retain. In most situations, there is some overlap between these two areas, demanding a degree of compromise.

Time Domain Waveform Preservation
Filters for such applications feature linear phase response in the pass-band, and must not introduce ringing or overshoot. To preserve the signal waveform while removing undesired components, the filter must also pass many harmonics of the incoming signal's base frequency. "Noise" components that the filter removes must be at substantially higher frequencies than these necessary harmonics. Phase-derived filters, such as Bessel or constant-delay (equiripple-phase) and their amplitude-compensated derivatives, work best in these cases.

High Selectivity in the Frequency Domain
Situations where removal of undesired components is the overriding concern and some distortion in the time domain of the signal's shape is of less importance generally require sharper roll-off filters with Butterworth or elliptic transfer functions. Spectrum analysis, for example, involves only the amplitude of each frequency component of the input signal. Most voice and data transmission also requires integrity only of amplitudes, as do many forms of modal analysis, which determines resonant frequencies of structures and objects.

Compromise Filters
Although linear-phase filters preserve critical information, many applications also require rapid transition-band roll-off. A balance between these mutually exclusive requirements can often be achieved by phase-derived types and amplitude-compensated versions of phase filters. Applications for this approach include determining the direction of an object or signal source by analyzing the waveform from one or more receivers.