# When modelling noise,how do they decide whether additive or multiplicative?

How does one take a decision of whether noise is additive or multiplicative?

It will depend on what you are modelling in your system. Consider an active radar system. There are both additive noise components and multiplicative noise components. What distinguishes them in this case:

1. If I increase the transmit pulse power by a factor of $K$ and the SNR improves by a factor of $K$ then the noise is additive
2. If I increase the transmit pulse power by a factor of $K$ and the SNR does not change then the noise is multiplicative

To give some examples to clarify:

1. Thermal noise at the input of the ADC is generally additive - noise level does not generally change with power of the received signal, so SNR will change with transmitted/received power.
2. Clipping in the ADC (distortion) is non-linear - it is more multiplicative than additive. Increasing the receive power won't affect the SNR (assuming clipping still occurs).
3. Phase noise in the ADC (timing errors) is multiplicative
4. Speckle noise - Presence of a lot of random scatters in radar resolution cell is usually modeled as multiplicative - increasing the transmit power does not reduce the speckle effect.

Another common example is in active Sonar. In near range there is a lot of reverberation (volume reverberation), but as the transmitted pulse travels away from the sonar the reverberation drops and the system performance becomes limited by the ambient noise. In these two cases the sonar is operating in reverberation limited mode and noise limited modes respectively.

1. Reverberation - Multiplicative - Increasing transmit power offers no improvement in SNR.
2. Noise Limited - Additive noise - Increasing the transmit power improves the SNR.

This assumes a very simplistic model of underwater sound propagation. You will have other wave propagation effects mostly due to the sound velocity profile (Reverberation at surface/bottom, sound channels, convergence zones, etc).

So within a single system you can have both. In what mode of operation will determine which type of noise dominates. It should also be noted that the noise may not be entirely multiplicative or entirely additive. You can have noise which contains elements of both. In this situation you will get some SNR improvement but it may not be linearly related to the increase in transmit power.

• Nice answer! +1 – Peter K. Apr 21 '16 at 15:07

It really will depend on the application. One way to think about it is: what is the relationship between what you can measure and what you want to know?

If what you want to know is almost the same as what you can measure, but there are spurious peaks or troughs around "the truth", then additive noise is probably what you should select.

More generally, additive noise is likely if there is a linear (or affine) relationship between the value and the measurement.

If what you want to know is related to what you can measure by the energy in the measurements, then you may want to choose multiplicative noise.

More generally, multiplicative noise is likely if there is a (highly) nonlinear relationship between the value and the measurement.