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Special Function Filters
By Rob Howald
Last month, we looked at one of a communication system’s most ubiquitous elements — the filter. This month, we wrap up our
two-part discussion of common filter applications with a look at more possible uses of these indispensable components.
Last month, we introduced filters. Unless your engineering degree is the honorary kind, such as a Hollywood star might earn for appearing in a big-budget science-fiction flick (I sure hope this doesn’t inspire another Star Trek movie), you’re probably familiar with filters in the context of engineering. This month, we’ll look at some special function
filters. Actually, one of the filters discussed last month falls into this category per my criteria. That would be the Nyquist filter, whose function is to shape a transmit and receive pulse to provide zero intersymbol interference (ISI).)
I’ll also include a brief discussion of digital filters. This topic was treated previously in “Building Blocks” as a more general discussion of digital signal processing (“
Now That You’ve Given Me All of
Those Bits
,”
Communication Systems Design,
Nov. 1998). Now, I know most of you copy and safely store each and every Building Blocks masterpiece. However, in the unlikely event the article is not readily available to you, a section of the earlier piece will be restated here. Also, I would highly recommend the
Communication Systems Design
Archives
for those who don’t already have it bookmarked. Archived issues are contained therein.
Loop Filters
The term “loop filter” normally refers to the important subsystem within a larger control system that helps determine the system’s key characteristics. The most interesting characteristics include the loop’s frequency response and/or the loop’s time domain response. In communications, these concepts often show up in two places: automatic gain control (AGC)
designs and phase-locked loops (PLLs).
Figure 1
shows a typical feedback control system. In PLLs, the controlled parameter through negative feedback is phase. PLLs have several uses. In communication receivers, they allow coherent communications by providing carrier and clock references. They are also used in transmitters to stabilize the carrier transmission, which assures that synchronization can be easily accomplished. Additionally, they are employed in
frequency-generation designs to manipulate the choice of frequency and phase noise performance.
In AGC subsystems, the controlled variable is output power. These circuits are important because they assure demodulation can be optimized within a reasonable, dynamic range of input level. There are compelling reasons to hit various elements of a receiver chain with a certain level range, such as a mixer with its intermodulation distortion (IMD) and leakage constraints, and an analog- to-digital converter
(ADC) with its degrading SNR with input back-off.
The control system uses the output of the system to provide information back at the input. The result is an adjustment that drives the difference between input and output to zero. Within the circuit, this feedback compares voltages that represent the output and input (reference) phases for PLLs, and the output level to reference level for AGC. The function of the loop filter is to process this error voltage in such a way that it keeps the system
stable (instability is always a threat in a feedback system) and provides the right response characteristics in time and frequency. The values of these parameters vary by application. As an example, the loop filter can assure that the error voltage is applied gradually to the variable gain element in the AGC system. This allows rapid input level variations to be ignored, such as undesired transients that do not need to be tracked. In addition, any amplitude modulation information on the signal that is too fast
will not be tracked out, since to do so would enable a loss of information.
On the PLL side, the loop filter again sets the dynamic characteristics of the phase at the output. Proper loop filter design determines, in this case, what portion of the reference and oscillator phases are present in the output signal, which essentially defines the stability of the output. Though the term can be confusing depending on what piece of literature you look at, the term “loop bandwidth” is a critical
parameter in the definition of the system response. The loop bandwidth is selected in conjunction with the application that sets the desired performance aspects. (Check out my PLL piece, “Give Thanks for Phase Locked Loops,” in the November 1997 issue, and the AGC piece, “Gaining Control,” in the October 1999 issue.)
Discriminator
While there is much more to
the circuitry, the fundamental concept at work in an FM discriminator (FM demodulation) is that of a system where the output amplitude is a function of frequency. For example, suppose some audio signal is used to FM modulate a carrier. In order to recover the audio undistorted (or, in the case of Barry Manilow, with lots of added distortion) we must design something that will respond to the frequency modulation of the carrier, which is a constant envelope. The information is encoded in the instantaneous
frequency of that carrier. A circuit with an amplitude that varies as a function of frequency is simply a filter. Think of it in terms of any generic filter type you are comfortable with, such as a resistor-capacitor (RC) lowpass filter. Now, assume some FM carrier is operated in the lowpass transition band of that filter, rather than in the middle of the passband. Then, the output voltage from the filter will vary as the frequency wobbles back and forth. This is a rudimentary (and not necessarily practical)
example, but it gets the point across.
Anti-alias
This is an oddly named filter that has become more common with digital infiltration. An anti-alias filter is just a fancy name for a lowpass filter. Calling it that, however, does not describe its functionality the way its name does. In the analog domain, an anti-alias filter removes out-of-band frequency content from the input of
an analog-to-digital converter (ADC). This is not particularly special, except that in this case the filter’s role is more critical than it would be for typical out-of-band interference rejection. This is because out-of-band signals at the input of an A/D converter can become in-band signals after conversion because of the folding over of the spectrum of signals and distortions in the discrete domain. Therefore, signals that would otherwise be of no consequence in the analog world can become crucial
when quantized. The job of the anti-alias filter, then, is to pummel these undesired components into submission before they get a chance to muck up the desired signal.
Choke
While “choke” has come to be synonymous with play-off ice hockey in Philadelphia, this term is actually common for describing DC filters. That is, a choke is often used to remove AC ripple on a DC
power line. Such ripple can come from the supply (which is fed with AC as an input), from the switching nature of some supplies, or from garbage picked up on a circuit board layout.
The variation of bias on an amplifier can cause modulation of the signal passing through, including AM-to-AM and AM-to-PM conversion. On an oscillator, PM sidebands can be created. Because we want a clean DC, chokes are very low frequency cut-off lowpass filters. This can be annoying because, to avoid DC loss in the lumped
elements, you sometimes get stuck with big old coils for inductors. In fact, choke often refers to a big inductor, since it performs such a function. Whereas the typical bypass capacitor on an IC serves a similar role (shunting ripple to ground), that function is typically aimed at higher frequencies. The really difficult, low frequency components that originate back in the supply usually require a more sophisticated filter design.
Notch
A notch filter is another name for a bandstop filter. Whereas a bandpass filter selects a desired band for passage, such as the tuner in your television set that chooses a channel, the bandstop filter rejects a chosen band. In many cases, the application of interest is for test purposes only. However, one practical application is older cable TV systems. Premium channels, for example, could be “trapped” out by placing notch filters at the
frequency of interest. If such a system existed in the nation’s capital, for example, and an occupant of the White House wanted an adult, I mean history channel, he or she would call the local cable company. The cable company would then remove the trap associated with the tap that drops signals into that dwelling, but something tells me they have it on a dish anyway.
Equalizers
An
equalizer, in the communication system sense of the word, is indeed a digital filter. Often adaptive, its job is to mitigate the frequency response distortions of a channel. Essentially, the equalizer provides spectral gain and attenuation corresponding with the channel irregularities and mitigates dispersion. This topic was addressed in a two-part “Building Blocks” series “The Great Equalizer,” in the July and
August 1997
issues of
Communication
Systems Design
.
Another type of equalizer is an all-pass phase equalizer. It is basically a filter used to mitigate group delay problems that often occur near the edges of the passband of highly selective filters with sharp transition regions (Chebyshev and elliptic filters are good practical examples). A phase equalizer is designed foremost for its phase response, with the goal of maintaining an all-pass amplitude response.
There are also amplitude equalizer filters. A common example is the
reconstruction filter after a digital-to-analog converter (DAC). The frequency response of the D/A operation has a (sin x/x) response. In order to be flat across the passband, an equalization (x/sin x) is added.
Filters for the bit-head
All filter types that can be designed analog can also, in principle, be done digitally. One key advantage of a digital over an analog filter is that
digital filters remove sensitivity to component tolerances, yielding robust, repeatable implementations. The two types of digital filters are infinite impulse response (IIR) and finite impulse response (FIR), also known as recursive and nonrecursive. A nonrecursive filter has an output based on present and past samples, while the recursive filter may reuse outputs. Thus, the nonrecursive filter has an FIR, and an input pulse driving the filter eventually decays to zero. This is not necessarily the case with the
IIR brand filter.
FIR
FIR filters have several important qualities. First, they are unconditionally stable. The FIR filter has an impulse response that is predictable and simply the length of the filter. There is great design flexibility in creating a desired amplitude and phase response. And, most importantly, the FIR filter can have a linear phase design within the passband. In
communication systems, this flat group delay in the passband means no frequency response dispersion, which can degrade the performance of the link by introducing ISI. While a sophisticated digital receiver can often handle mild distortions with equalization, the linear phase filter assures that none of the equalizer’s budget is wasted on contributions from digital filtering.
A drawback of the FIR design is the limited power of its transfer function. Because the IIR filter allows for feedback,
and an FIR does not, IIR math (no need to panic, there is no math on the horizon) has a discrete polynomial representation in both the numerator and denominator. Thus, for filters that can be either digital or analog, there are generally fewer taps for the same filter in an IIR relative to an FIR. The rational polynomial in the numerator and the denominator means that traditional analog filter design methods convert in a straightforward manner only to an IIR.
IIR
The IIR may look more like filter design to most analog designer types. However, this doesn’t make them better filters. Once a traditional function is defined that meets the desired specs (such as Butterworth or Chebyshev), there are two ways to pursue digital conversion. The two methods are called the impulse invariance method and the bilinear transform. They are essentially mathematical translations from the
continuous to the discrete domain required for digital.
One of the key differences is that the bilinear transform can be used for analog low, high, or band pass/stop conversions to digital versions of the same, something the impulse invariant method cannot accomplish. However, the bilinear transform also introduces nonlinearity, which warps the spectrum, and distortion can become an issue.
Other sins of IIR designs include numerical stability issues associated with nonlinear processes and the
feedback in IIR filters. IIR filters also suffer in the phase response category (where the FIRs were ideal). While every analog filter has, in principle, a digital equivalent, it is not necessarily practical or desirable to implement it as such. This discussion does not even address the frequency of operation issue which obviously constrains the digital approach.
So, we wrap up Filters 101 and 102. There are so many elements to the topic of filters and so much information available, it’s like asking
Alan Greenspan to explain how money works. I don’t want to give any unsolicited (or unprofitable) endorsements, particularly when my exposure does not cover every book or package available. However, if anyone is interested in sending an e-mail I can procrastinate over, I can recommend a few textbook and simulation favorites.
Rob Howald
is the director of systems engineering in the transmission network systems group at the Motorola Broadband Communications sector
(formerly General Instrument) in Horshan, PA. He has a BSEE and an MSEE from Villanova University, and received his PhD from Drexel University. He can be reached at
rhowald@gi.com
.
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