## Sensing/Conditioning

# Which filters are noisier – analog or digital? part 2

**The Filter Wizard takes a look at some fundamental noise mechanisms in filters and poses the question: Can we make useful noise level predictions if our filters are implemented digitally?**

In Part 1 of this pair of articles, we ran SPICE noise simulations on a simple second order lowpass filter. We saw that there is something fundamental about the 'hold' that the filter's capacitor network has over the total output noise level. Scaling all the filter's resistors by a constant factor, to change the cutoff frequency of the filter without changing any of the capacitor values, leaves the total noise voltage **unchanged**. With practical amplifiers, the noise level is degraded from the ideal case, but it's still pretty straightforward to predict what they will be.

Can we make useful noise level predictions if our filters are implemented digitally? In modern electronic product design, one can often make a choice between analog and digital signal processing. With analog processing, the filtering and other signal manipulation is done before converting the signal to digital (if it's indeed ever converted). The digital model involves converting as early as possible in the signal chain, doing the processing in the digital domain, and then perhaps converting back to analog.

The device I spend most time solving people's problems with, Cypress's PSoC3, has op amps for constructing analog active filters, and also a fast digital filter engine that can implement a wide range of filters in the digital domain. To help make the choice, systems engineers need a reliable method for directly comparing the noise performance of analog and digital filtering approaches.

We're taught that going digital creates quantization noise, which is the per-sample error involved in fitting a value of arbitrarily high precision into a lower-resolution number system, usually an N-bit binary system with 2^N available states. For any real-world signal, this error is completely uncorrelated with the actual signal and therefore can be treated as random noise, whose value is uniformly distributed between -0.5 LSB and +0.5 LSB. Textbooks demonstrate both that this noise is 'white, i.e. has a frequency-independent spectral density, and also that the rms value is {LSB}/sqrt(12), when integrated from DC to Nyquist.

Quantization noise is different from analog noise in one particular way: its deterministic. Process an identical signal a second time in your system and youll get the same error again. In an analog system, the noise is different every time.

To level the playing field in our comparison between analog and digital filters, lets assume that input signals are converted to digital, either after the analog filter or before the digital filter, using enough bits of resolution that this input quantization noise can be neglected. Ill use 20 bits of input resolution in the examples here. Also that our signal processing is done to at least this resolution, if not higher, so that theres no internal reduction of resolution in the digital case that might affect the results. Ill use 24-bit internal arithmetic, which is the signal path width of PSoC3s filter block.

With all this, you might be thinking that Im setting up a false dilemma. Digital filters contain neither noisy op amps nor noisy passive components. So, if weve taken the input quantization noise and signal levels into account, arent our digital filters going to be essentially perfect and noise-free, running with 24-bit arithmetic on 20-bit data?

Its a fair question. If the only noise that a digital filter implementation suffered from was the basic quantization noise associated with the resolution of its processing path, wed have little to worry about. But and heres the 'kicker this **isnt** the case, at least, not for the filter topology that youre most likely to use because its the one given in all the textbooks and filter design packages. That topology is the Direct Form filter, the most common form of which is shown in Figure 1. Well soon see that we do indeed have a noise problem; first, we need to figure out how to actually analyze the circuit.

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