The document discusses nonlinear signal processing and nonlinear filtering techniques. It begins by explaining that many common signal processing operations are nonlinear, such as rectifying, quantization, power estimation, and modulation. It then discusses several examples of nonlinear signal processing applications, including Bayesian filtering, particle filtering, Kalman filtering, median filtering, fuzzy logic, and artificial neural networks. The document focuses on explaining the Kalman filter, how it works, and the Kalman filtering algorithm. It then shifts to discussing nonlinear systems and introduces the extended Kalman filter for estimating states of nonlinear systems. Finally, it discusses using dynamic mode decomposition to initialize the extended Kalman filter for improved state estimation of nonlinear systems.

1. 1
DebrajBanerjee
debrajbanerjee2020@gmail.com

2. In reality many common signal processing operations are nonlinear :
• rectifying
• quantization
• power estimation
• modulation
• demodulation
• mixing signals (frequency translation)
• correlating
Nonlinear Signal Processing Nonlinear Processing of Signals
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Filtering a signal with fixed coefficients is linear, while using an adaptive
filter, having variable coefficients, can be regarding as a nonlinear
operation.

3. There is an almost infinite number of nonlinear signal processing applications; a few examples
are :
 Bayesians Filtering
 Particle Filtering
 Kalman Filtering
 Smooth Variable-structure Filtering
 Median Filter
 Fuzzy Logic
 Artificial Neural Networks (ANNs)
 Deep-Learning based Filtering
Kalman Filtering
Some of these are devices or algorithms that are quite easy to implement using DSP
techniques, but would be almost impossible to build in practice using classical analog
methods.
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4. What is a Kalman Filter ?
The Kalman filter is a set of mathematical
equations that provides an efficient
computational (recursive) means to estimate the
state of a dynamic system, in a way that
minimizes the mean of the squared error.
The filter is very powerful in the sense that it
supports estimations of past, present, and even
future states, and it can do so even when the
precise nature of the modeled system is
unknown.
■ Tracking objects (e.g., missiles, faces, heads, hands)
■ All forms of navigation (aerospace, land, and
marine)
■ Economics
■ Fuzzy logic
■ Neural network training
■ The detection of underground radioactivity
■ Many computer vision applications
■ Many more
The principal uses of Kalman filtering have been in
'Modern' control systems, in the tracking and
navigation of all sorts of vehicles, and predictive
design of estimation and control systems.
Applications :
Kalman Filtering
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5. The Kalman filter estimates the state of a dynamic system. This dynamic system can
be disturbed by some noise, mostly assumed as white noise. To improve the
estimated state, the Kalman filter uses measurements that are related to the state
but disturbed as well.
How does Kalman filter work ?
Example of a
moving car
Thus the Kalman filter consists of two steps:
 The prediction
 The correction
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6. The Kalman Filtering Algorithm[1.1]
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7. Quick Example : After running Kalman Filtering Algorithm :
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8. We have discussed state estimation for linear
systems. But what if we want to estimate the states of
a nonlinear system? As a matter of fact, almost all
real engineering processes are nonlinear. Some can
be approximated by linear systems but some
cannot.This was recognized early in the history of
Kalman filters and led to the development of the
"extended Kalman filter" which is simply an
extension of linear Kalman filter theory to nonlinear
systems.
A (discrete) Nonlinear system is simply a process that
can be described by the following two equations:
Shift to Nonlinear Systems :
Now for most of the nonlinear
systems the system noise &
measurement noise can be
modelled using zero-mean
gaussian random process
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9. The Extended Kalman Filtering Algorithm[1.2]
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Extensions

10. Quick Example :
After running Extended Kalman Filtering Algorithm :
Consider the following second order nonlinear system:
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11. Dynamic Mode Decomposition[2]
Dynamic Systems :
A dynamic system is a system or process in which
motion occurs, or includes active forces, as opposed to
static conditions with no motion.
Dynamic Systems are generally governed/described
mathematically through differential equations of the
following form :
Experiment : Collect Data :
discrete analogue
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12. Extended Kalman Filtering using DMD
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13. vectorize
column wise
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15. Quick Example : Initialisation using DMD Algorithm:
Ns = 6
DMD
covariance matrix
P0
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16. After running Extended Kalman Filtering Algorithm :
(with system obtained by DMD)
After running Extended Kalman Filtering Algorithm :
(on same nonlinear model)
Error :
Error :
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17. Advantages :
 Works better than Extended Kalman Filter in case of nonlinear systems.
 Estimation gets more accurate with time.
 Pre-predicted DMD models (from ideal system) will boost both accuracy & efficiency.
 Extended DMD model also takes care of the s-correlations between the different state-space
signals.
Disadvantages :
 Computationally more expensive than the Extended Kalman Filter.
 Sufficient initial estimation of state-space signals is needed.
 Estimation accuracy depends on the initial number of estimated samples.
Improvements :
 Implementation of Pi-DMD[3], C-DMD[4] etc. will result in more accurate modeling of the
nonlinear system.
 Implementation of Optimized-DMD[5] will lead to better noise cancellation effect.
 Estimating the DMD model after each iteration of Kalman Filtering can increase the accuracy
when the exact system model is unknown. 17

18. Bibliography :
[1] Setoodeh, P., Habibi, S., & Haykin, S. (2022, March 24). Nonlinear Filters: Theory and
Applications (1st ed.). Wiley.
[2] Chen, X. (2022, September 6). Dynamic Mode Decomposition for Multivariate Time Series
Forecasting. Medium. Retrieved September 18, 2022, from
https://towardsdatascience.com/dynamic-mode-decomposition-for-multivariate-time-series-
forecasting-415d30086b4b
[3] Baddoo, P. J. (2021, December 8). Physics-informed dynamic mode decomposition (piDMD).
arXiv.org. Retrieved September 18, 2022, from https://arxiv.org/abs/2112.04307
[4] Brunton, S. L., & Kutz, J. N. (2022). Data-Driven Science and Engineering: Machine Learning,
Dynamical Systems, and Control. Cambridge University Press.
[5] Travis Askham and J. Nathan Kutz. Variable projection methods for an optimized dynamic
mode decomposition. SIAM Journal on Applied Dynamical Systems, 17 (1):380–416, 2018.
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19. THANK YOU
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