“Filters are generally used to limit the bandwidth of signals, right? So filtering and bandlimiting are synonymous, right?”—say any number of engineers, after a filtering presentation or lecture.
I love it when people try to catch me out. No, really. When you’re young, the gaps in your knowledge are empty; just voids waiting to be filled. When you’re old—though I prefer the epithet “experienced”—the gaps can still be there. But like the huge regions of non-coding DNA in chromosomes, these gaps are sometimes filled by junk sequences that look like knowledge, but are just nonsense. So when you’re older and experienced, you have to learn in a different way, which is to expose your (supposed) knowledge to brighter, younger minds to see if they can find the stuff you think you know, but don’t.
Okay, way off topic, except to say that for many years, I was so close to my work on filters that I never really stopped to think why I would get so riled when someone said, “Oh, I put a little RC in to filter out some noise” and I would think, “First-order RC? That’s NOT a filter!”
But if it’s not a filter, what is it? And if it isn’t filtering, what is it doing?
Here’s how I think of it. Filtering and bandlimiting are both processes that affect the frequency response of a circuit or a block in a system. “Frequency response” is common shorthand for how the gain of that circuit or system varies with frequency. It’s not a single value; rather, it’s a function of frequency, often plotted graphically to aid visualization. Bandlimiting, as a term, means doing something to limit, or constrain, the bandwidth of that block. And bandwidth is a single, scalar measure—it’s a simpler concept than frequency response as it’s just a number.
Filter vs. Bandlimiter
So for me, the distinction between a filter and a bandlimiter is this: A filter has an intentional effect on the frequency response of the block, whereas a bandlimiter has an intentional effect on the bandwidth of the block. I’ve written before that if you leave out (or misjudge) bandlimiting, performance will suffer; if you leave out filtering, function will suffer. Let me give a simple example.
Imagine a world where there’s only one radio station. Your receiver is a long, long way from its transmitter, and you only have a very approximate idea of the frequency upon which they transmit their programmes (that’s the British way of spelling this particular use of the word).
For this example, let’s assume that they are using AM. If you erect a broadband antenna and connect it to a suitable detector, chances are that you’ll be able to capture their signal. But you’ll also collect innumerable sources of noise thrown at you by the rest of the Universe. This noise will be spread over the entire frequency range over which your antenna and detector are active, and it just might drown out what you wanted to hear from the radio station.
Once you’ve got a somewhat better idea of what frequency the radio station broadcasts on, you can limit the bandwidth of your receive channel to a progressively narrower—yet not particularly well-defined—band around that frequency. The narrower you make the bandwidth, the more of the Universe’s interference you’ll be able to keep from messing up your reception. The quality of your reception is likely to be a smooth function of the reduction in bandwidth that you impose. The rate at which the attenuation of your bandlimiting circuit increases at frequencies away from your center frequency need not be great.
What’s important is the noise bandwidth of the circuit, which you can calculate by integrating the noise power that makes it through the bandlimiter, over frequencies from about zero to about infinity. The only real noise power contributions are those fairly close to the center frequency, and even with only modest components, you can quickly get that bandwidth down to suitable levels.
Now imagine that a second radio station (and a third, and so on…) starts up, pretty close in frequency to the original one. A new problem emerges—your antenna and receiver can pick up both stations at once, and with AM, you end up listening to the sum of their broadcasts. Adjusting the bandwidth of your system with simple components doesn’t help you here, because no matter how small you make the bandwidth, the resulting circuits don’t have the discrimination to separate the transmissions of the two stations.
The problem you now have is not one of performance—how much noise is there in the background. It’s a functionality issue: How do you receive the wanted transmission and not the unwanted one? In more modern communication systems, the reception could well be completely broken.
In this case, a bandlimiter isn’t good enough. You need… a filter.
Becoming More Receptive
The filter design task is a little more complicated. You have to specify both the range of frequencies that you really want (passband) and the range of frequencies that you don’t want that must be rejected (stopband). The whole arsenal of filter design is intended to help you with this problem.
As you make this filter progressively sharper, you may well find that, suddenly, the system starts working properly and you can hear what you want. Definitely not a smooth function.
Now, when you’ve built this filter, it also has a bandwidth, defined by the range of frequencies you decided to let through. You can perform the same integration over frequency as you did before to get this single scalar result. But in this particular example, you didn’t design the filter to have that bandwidth; it’s a side effect.
That’s a pretty long-winded way of expressing what I hope is actually a fairly simple point. If you just have to control the amount of noise passing through your system in a way that can be captured by a scalar figure of merit, then bandlimiting is what you need. That’s what all of the little RC circuits do in innumerable analog signal chains you’ll see in daily engineering life. They control the amount of high-frequency noise that gets into the input of an ADC, alias down, and make your system look noisier than you thought it should be. They dim down the high-frequency frizz on an output from your system that, even if it doesn’t cause a problem, makes it hard to see what’s happening at lower frequencies when you probe with a scope that’s faster than you really need (which is basically all scopes these days). These are the little tweaks that help us get our performance right. And I still consider that they’re not really filters.
When your system will simply break unless you can stop certain frequencies from passing through a block, well, that’s when you need a filter—not just a few non-critical Rs and Cs. And that’s where life gets interesting. For a Filter Wizard, anyway!