## Sensing/Conditioning

# Which filters are noisier - analog or digital? part 1

**The Filter Wizard takes a look at some fundamental noise mechanisms in filters, using SPICE to illustrate the performance limits you can expect. This article will concentrate on analog filters.**

Give me a lever long enough, Archimedes is said to have observed, ...and I shall move the world. He didnt have the benefit of our modern understanding of materials, which suggests that this boast might be hard to fulfill without a generous supply of Unobtanium. If you drew Archimedess experimental setup on paper, it might look just like a ball of rock being lifted on the plank of a see-saw (thats a teeter-totter, to US readers) but theres a little matter of scaling. Lets say you double the size of everything the radius of the rock, and the width and thickness of the plank. The strength of the plank increases by a factor of four (it goes as width*thickness, roughly) but the load on the plank goes up eight-fold. Keep scaling up, and eventually youll exceed the capabilities of any plank material.

Whats the connection with noisy filters? Well, it turns out that things here dont scale quite the way you might expect, either. In this two-part Filter Wizard, well look at some fundamental noise mechanisms in filters, using SPICE to illustrate the performance limits you can expect. Well concentrate on analog filters in this part, with the biggest bombshell being reserved for Part 2s look at digital filters. Yes, digital filters generate noise too! And sometimes in unexpectedly, unacceptably large amounts. Well see in Part 2 how SPICE noise simulation can be used on digital filters, permitting a direct apples-to-apples comparison of analog and digital filtering solutions.

By the way, if youve got a practical interest in low-noise filters, I recommend that you get Doug Selfs new book The Design of Active Crossovers. While it is focused on a specific audio application, the material has wide applicability, and it is a great new addition to the canon of real-world filter cookbooks. In this article Ill take a slightly more fundamental look at noise issues in analog filter design, to set the scene for an eventual show-down with digital filters. Which will win? How is it going to end? Youll just have to wait for Part 2!

Im going to concentrate on lowpass filters, whose noise bandwidths are essentially determined by their filter responses. Also true of bandpass filters, this isnt the case with highpass and bandstop filters, whose noise levels are generally determined by the bandwidths of the amplifiers used.

So, where does noise come from in an analog filter? Well, some of it comes from the op amp(s), for sure. Each active filter topology has its own particular noise gain, which causes the inherent input voltage noise of the op amp to be frequency-shaped in a way that is related to but not identical to the actual shape of the signal transfer function that the filter creates. This is a fundamental insight useful across all of circuit design that a circuit doesnt always process its own internal noise in the same way that it processes the inputs you apply to it.

Many op amps have not only an input voltage noise source but a noise current source as well. The consequence is that any finite source impedance attached to an input of the filters amplifier, and therefore passing this noise current, creates an additional noise voltage that also contributes to the overall filter noise. Each topology again has its own signature here, also dependent on the apparent impedances of the resistor-capacitor networks that are hung on the amplifier.

Last, but definitely not always least, is the fundamental noise contribution from the

passive components themselves resistors and capacitors, since were not considering any exotic active filters that include inductors. Now, there are two ways of looking at this noise that might seem to be diametrically opposed, but in fact spring from the same well of physical law.

One common perspective holds that Johnson noise in resistors is the only source of noise energy in a circuit, and that capacitors are noise-free and only affect circuit noise behavior through the interaction of their frequency-dependent impedances with the circuit resistances. Resistance to current flow is a bulk property of matter; anything made out of matter is at a non-zero absolute temperature, and hence the thermal energy in the structure of a resistor smacks free charges around and causes a varying noise potential across any two points in the conductor. This gives answers that match experiment, and is the way that SPICE calculates noise in a circuit. In a SPICE noise simulation, the contribution of the Johnson noise of each resistor in turn at the output is calculated using a linear AC analysis, and all the noises are squared and added. The square root of this sum is the overall rms total noise voltage.

Theres another approach, though, used frequently by IC designers. The need for it can be deduced from the thought experiment in which you allow the value of a resistor to tend towards infinity. Since the Johnson noise voltage density of a resistor is proportional to the square root of the resistance value, its clearly troublesome that the noise voltage of an infinite resistor calculates out as infinite. It may be OK for theoretical physicists to juggle with infinities as part of their day jobs, but in the everyday engineered world we dont see high voltages appear across the terminals of very high value resistors.

In this second approach, we ignore the resistors and concentrate on the capacitors. The usual way of introducing this idea is to consider a resistor R and a capacitor C in parallel, and contemplate the noise voltage of the combination. The total noise voltage is the product of two pieces: the noise voltage density, which is proportional to sqrt(R), and the square root of the measurement bandwidth, which is proportional to 1/sqrt(RC). So the total noise is proportional to 1/sqrt(C) and the value of the resistor doesnt even enter into it! As the resistor value changes, the spectral density of the noise changes, but the total integrated noise value is constant. This applies for arbitrarily high values of resistance, which of course result in arbitrarily low values of bandwidth, and noise voltages that could be moving very slowly indeed.

This is not to say that capacitors actually **generate** noise. The residual uncertainty of the voltage on a capacitor in parallel with an infinitely large resistor (the combination hence having zero bandwidth) is an indication of the Johnson noise that it was exposed to at some time in the past, when the voltage was defined by contact with a circuit in which the resistance was finite. The capacitor has sampled the Johnson noise of the resistor in the circuit that charged it, and now its holding it for you.

Whats interesting is that this behaviour generalizes to **any** RC network, including active ones. If we hold constant the capacitor values in an active filter, and scale the cutoff frequency with those resistors that determine it, the total noise contribution from the passive network stays constant.

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