Building Blocks: (Now That You'Ve Given Me All Of These Bits . . .
There once was a time when the digital part of the receiver system was the logic threshold detect at the end of the chain that said TTL high or TTL low. Well, that gate has marched its way out of the corner and multiplied itself hundreds of thousands of times over in today’s systems where we now process
bits instead of waveforms.
By Rob Howald
I had a truly excellent textbook for undergraduate communication systems by one of the very best authors. It was Lahti’s
Modern Digital and Analog Communications Systems.
It taught me many important things and was the springboard to my launching into this specialty. It has only one shortcoming, one which I also learned from. I never use the word “modern” in a title. In my undergraduate days we split about 60/40 between study of
analog (60) and digital systems (40). That probably wasn’t even a fair shake then (1985 A.D.), but likely not uncommon given all of the legacy analog systems that still exist. I wasn’t bothered by the mix, since I thought FM circuits were cool (what a nerd) and the satellite comm-type stuff that I did during my study used FM, as well as frequency shift keying. I was young and stupid (as opposed to old and stupid), and figured, surely, satellite equals high tech. I found that, in fact, some of the
hardware is state-of-the-art. However, more of it, by testing and qualification necessity, is behind the technology curve. And satellite channels, from a communication theory standpoint, can be among the more straightforward. My graduate course work, which followed not long after, heavily favored digital communications, but digital signal processing (DSP) was still separately dealt with in all circumstances. This may be the right way to approach the two subjects in an academic setting, but they are
inseparable today in “modern,” real systems.
Earlier this year, we gave a kind of introduction to data conversion, in the form of back-to-back columns in June and July. (“Conversion is Key,”
Communication Systems Design,
June 1998, pp. 12-19, and “Converter Concepts and Specs,”
Communication Systems Design,
July 1998, pp. 14-19.) This series stirred up the most interest to date in my e-mail bag, and I got some nice comments. For the most part, they came from people who
were not blood relatives, and/or looking for a job. As part of that two-part feature, I had indicated that in a future piece I would chat about DSP itself. To understand the state of DSP technology, and of implemented hardware, such as digital receivers, it is important to understand what goes on after the A/D conversion, and, if applicable, before the D/A conversion.
DSP finds a home in modern systems
One of the nice things about studying DSP in the context of communication systems is that it
doesn’t generally require you to introduce yourself to a whole new array of functional building blocks. Indeed, the pieces of the puzzle required for a sophisticated digital communications system have always contained all of the same basic elements. Consider such a generic digital communications system as shown in Figure 1. Now, to be fair, DSP has provided a kind of paradigm shift in how we look at the importance of each block in areas such as forward error correction (FEC) and equalization. This is not
because these were invented for DSP, but because once DSP became practical to implement at the data rates and clock speeds necessary, these functions became a whole lot more powerful. (For the true masochist, there is a tutorial column associated with FEC – “Springing Forward to Higher Performance” [
Communication Systems Design
, March 1997, pp.14-18], and a two-part piece associated with equalizers –“ Taming the Unruly Channel, Part 1” [
Communication Systems Design
,
July 1997, pp. 18-22] and “The Great Equalizer” [
Commun-ication Systems Design
, August 1997, pp. 14-18].) Adaptive equalization can be considered a function borne directly out of progress in DSP. High-density QAM modems become eminently more practical to implement using the discrete domain. Recall from the prior conversion discussions, discrete time or the discrete domain is what we call continuous values sampled in time. Consider a digital communications system such as Figure 1, and
the variety of other traditional functions, shown directly or not: bit mapping, encoding, randomization, symbol detection, synchronization (clock, carrier, block), filtering, and frequency translation.
We have discussed on multiple occasions the advantages of using digital techniques over analog.ones. Based on the premise that nearly any function has improved overall characteristics if performed digitally, the question becomes: Which of the above functions can be performed in the discrete domain? It is
now apparent why the progress in converters towards higher and higher speeds is very important. Virtually the only thing left out of the mix in necessary functions that cannot be done digitally is RF or microwave functionality. Even in that case, however, the “margin” of the RF monopoly is shrinking as conversions can now take place at hundreds of megabits-per-second. Thus, digitization has quietly crept into traditional RF locations where IF processing took place at 44 MHz. In fact, in part two
of the converter discussions, the bandpass sampling route to frequency conversion was addressed as one of the neat little niches made possible by today’s high-speed, high-performance A/D converters. In the remainder of this discussion, we will apply some focus to one of the most recognizable of functions, which can be embedded in digital communication ICs: filtering.
Modern filters — the digital way
At the risk of sounding like the biggest engineering geek ever, I must say that one of the
few items of my undergraduate education that I remember thinking was really interesting was digital filter design. I still remember passing up the weekly Thursday night beer ball bash to master filter design from scratch for an upcoming exam. Is there a greater sacrifice as an undergrad? I thought it was pretty intriguing how you could take a waveform, convert it to a bunch of ones and zeroes, send it through some shift registers, multiply and add operations, and presto — it was filtered. Indeed, the
whole analysis technique, the z-transform, is built around the variable
z = exp(jwT).
The mathematics of the discrete domain involves manipulation of equations associated with exponential powers of
z
, where the exponent for this transform represents the discrete sample delay or advance of
z
. For example,
1/z
would represent a one-sample delay. Performing these simple little processing steps on the discrete signal opens up grand possibilities in practical design. Now, I could take a
whole column to introduce the z-transform and its important properties, but you would be bored. Instead, I will try to impart a feel for the topic in absence of these important mathematical details. The z-transform does for discrete filter design what the Fourier transform does for analog. In fact, it probably does a little more because of its ability to easily represent stability, the time delay function, and the frequency response in a compact fashion.
First, note that all the filter types that can be
designed in analog — lowpass, bandpass, etc. — can also be done in digital. Analog component tolerances and sensitivities to such give way to robust, repeatable digital implementations, one of digital’s key advantages. There are no magical advances to designing filters with extreme characteristics. Typically, the trade is from lots of components to lots of digital processing operations (where multiplications are the least desirable). There are two distinct flavors of digital filters —
infinite impulse response (IIR) and finite impulse response (FIR), also referred to as recursive and non-recursive filters. The latter pair of names is more descriptive, as a nonrecursive filter has an output based on its present and past inputs, while the recursive filter also (recursively) may re-use outputs, current or past. Because of this, a nonrecursive filter has a FIR. That is, an input pulse exciting the filter produces an output that eventually terminates to zero. This is not necessarily the case
with the recursive, IIR brand. In Figure 2, a FIR filter is shown. Compare this to the topology in Figure 3 – an IIR filter.
FIR
There are many reasons to like the FIR version. Very importantly, it is unconditionally stable. Stable IIR filters can be designed, but when the design goes from utopia to finite precision reality, errors due to limitations in the numerical representation can send the IIR design off to the races. The FIR filter has an impulse response that is just the length of the
filter. There is great flexibility in both desired amplitude and phase response of the filter. Most importantly, the FIR filter is a linear phase design within the passband. Put another way, perhaps more familiar to some, it has flat group delay variation, or zero group delay variation (GDV), in the passband. From the communication systems perspective, the importance of this is that there is no residual clean-up required from the (assumed present) equalizer. If the design has an FIR filter at all, it is likely
to be part of a system that contains a generally adaptive equalizer. This design will eat up none of the equalizer “budget.” If there is no equalizer, then the GDV advantage cannot be overstated for high-bandwidth efficiency modulation. The exact linear phase response is the major strength of the FIR design, and it therefore follows that when linear phase is necessary, FIR filters are employed.
That is all swell. So, where’s the rub? There is a constrained nature to the topology in Figure 2.
An IIR filter can have both the tapped-delay line structure of Figure 2, and the output-to-input feedback of Figure 3, and it will still be IIR. The Figure 2 structure is it for FIR. In filter-speak, (for any fellow RF weenies remaining in modern engineering), the IIR design can contain poles and zeroes, and the FIR only poles. I can’t expound much more, without some z-transform jibberish. Think of it as a transfer function. If you want to create a specific response with a transfer function, the FIR
math tries to do it with a power series expression (numerator only), while the IIR math has a polynomial representation in both numerator and denominator. Mainly, what all of this means is two things. First, there are many more “taps” or delay elements with FIR compared to the corresponding IIR. Put another way, the FIR filter length and impulse response are much longer, requiring more calculations and more memory, not to mention a large time delay. The second item of significance is that the
well-developed, traditional methods of analog filter design cannot be drawn upon to convert to a digital FIR, only a digital IIR. This even includes the low-pass-to-whatever transformations commonly used to turn normalized lowpass functions into alternate types. Applying these transformations to a FIR low-pass filter leaves you with an IIR filter.
The design of a FIR filter does have a little bit of that “black magic” to it that may make RF filterers feel at home. Since the most straightforward
description of a FIR filter is via its impulse response, the cleanest way to design a FIR filter is to start with the desired frequency response, and then determine the impulse response. Unfortunately, this approach doesn’t work as easily as hoped, a common ailment of most direct design strategies. To have a finite impulse response, we need an infinite frequency response. To sidestep this predicament, even if we define via simulation an “infinite” frequency response, the time domain response is
truncated to something reasonable. The truncation typically includes a windowing function, which smoothes the on-to-off coefficients by multiplicative weighting, so they don’t show up in the actual response as ripple. There are about five common window functions, including rectangular (i.e., no weighting). Windowing is another topic completely unto itself aimed at providing the most efficient representation in terms of length, with the least distortion. It’s an out-of-scope subject here (I
don’t do windows except for the benefit of Bill Gates).
There is one other “fuzzy” design approach for FIR, called the Parks-McClellan method. Because you can count on being stuck with some ripples like a Chebyshev design, this approach provides a means to control that parameter in the context of meeting the other important filter parameters. The P-M algorithm methodically determines via computer manipulation which coefficient set will do the job.
IIR
For the designer weaned on Ls and Cs to
do his filter designs (or at least Cs and op amps), the IIR design hits a comfort zone. The filter designer can go about his filter design and simulation as always (my little-known tool recommendation: Aviles Technologies). It is not necessary to do so, as direct design is available. Once the traditional Butterworth, Chebyshev, or whatever function is defined that meets the desired specs, there are a couple of common avenues to pursue. The goal is to map an infinite frequency region to the finite discrete
frequency domain, which repeats as a function of sampling time. The two methods are called the impulse invariance method and the bilinear transform. The first approach simply converts from s-domain (Laplace, continuous time domain) to z-domain using, for example, known transform tables and relationships. It amounts to making the IIR impulse response equal to the sampled values of the analog filter’s impulse response. The impulse invariance method does create a theoretically stable response (low-pass
only), but keep in mind the note about non-idealities. If a filter other than low-pass is required, a digital domain translation, obtainable from a table or PC program, can be employed. All bets are off for stability in that case. This method works if aliasing is small, due in part to distortions around the folding frequency.
The bilinear transform uses the arctangent function to map from infinite to finite. It also distorts the spectrum to a degree. The bilinear transform is powerfully easy to use,
creating a digital design that meets specs developed in the analog world first. It has no alias problem, because the mapping function is defined only up to the sampling frequency. However, the nonlinearity warps the spectrum, so frequency response distortion can be an issue. One of the other advantages this method has, speaking completely in the IIR world, is that the 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 do.
The plusses and minuses of the IIR have been half-covered. Most importantly, it has a nice legacy of analog design to relate to. There are generally less taps for the same filter in an IIR relative to a FIR, so highly selective filters would lend themselves better to IIR. The implementation is slightly more complex. There is the stability issue associated with nonlinear numerical processes. These nonlinearities also can have undesirable effects because of the feedback in IIR filters. Finally,
IIR filters suffer in the phase response category. If the IIR filter is stable, including within the implementation reality, then it is a logical choice if linear phase is not a requirement.
Skills of the modern engineer
As you might expect, DSP design, FPGAs, and VLSI ASIC design have been fantastic for the progress and economics of communication systems, as well as for the economics of the engineering community. Skill in these areas tends to attract big, fat salaries, and bonus figures. Most of the
skills necessary weren’t even available to be taught through my first stint in graduate school in the late 80s. As an 80’s student, as far as today’s grad is concerned, I might as well be sharing the bench with Shockley. Your engineering age must be counted in something close to dog years in this business, relative to the pace of technology. Digital processing is one of those areas where a periodic update rate will assure that you can stay only a few footsteps behind the curve, or at worst, a
few paw prints.
Rob Howald is a staff engineer in the transmission network systems group at General Instrument in Hatboro, PA. He has a BSEE and an MSEE from Villanova University, and received his Ph.D from Drexel University. He can be
reached at rhowald@gi.com.
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