10
votes
Accepted
Why is not Fourier Transform Good for Non-linear Processes
Because complex exponentials $e^{\jmath \omega t}$, which are results of Fourier transform, are the eigenfunctions for linear, time invariant (LTI) systems. See eigenfunction of LTI. Also see this ...
- 116
7
votes
Accepted
Volterra Kernel Convolution Method
Yes the MATLAB code is correct. Be careful though, the bandwidth of the signal squared is twice that of the signal itself, which may lead to aliasing if the sampling frequency is too low compared to ...
- 13.6k
7
votes
Accepted
Is there a way to compute the spectrum effect of a non-linear function?
is there any way to calculate the rate at which the harmonics decrease in power for a given function
Yes, but it's complicated and typically not worth the bother. If the non-linearity is static, i.e. ...
- 47k
6
votes
Why is aliasing inherently non-linear?
I think you mean "images", not "aliases". They become aliases if there is foldover from resampling.
It's because you are not adding two signals, $x(t)$ and $\operatorname{III}(t)$, you are ...
- 21.1k
6
votes
Is there a way to compute the spectrum effect of a non-linear function?
Stealing$^\dagger$ from this answer:
For non-linear functions that admit a series expansion (e.g.
Taylor/Maclaurin), you can get a decent intuition for how fast the
harmonics decay. The Maclaurin ...
5
votes
How to determine if the system is linear or nonlinear
As Matt L. says you'll need to check for homogeneity and, possibly, additivity.
Homogeneity
That test says that if:
$$
y[n] = f(x[n])
$$
then
$$
A \cdot y[n] = f(A \cdot x[n])
$$
for all scalar $A$.
...
- 25.9k
5
votes
Accepted
Power spectral density of $\left(x(t)\right)^2$?
Since the question has been raised as to whether the hint that I had given to the OP in a comment on the original question was appropriate for a newcomer to signal processing, here goes.
Stripped of ...
- 20.7k
5
votes
Accepted
What subclass of nonlinear systems can be represented by Volterra series?
The system must be time invariant and smooth in the functional derivative sense. That doesn't guarantee that the Volterra series converges (like with Taylor series, there are pathological counter ...
- 4,593
5
votes
As of 2019, which discrete nonlinear, time-invariant systems with memory are considered "easy" to model and identify?
It is very strange phenomena that one object is completely dropped out of attention of researchers. It is Urysohn operator. First of all Urysohn is equivalent to multiple parallel Hammersteins and ...
- 301
5
votes
Modeling Analog Filters for Software Implementation
Modeling real analog components with their non-linear behavior can be a challenge to do digitally. I suggest starting with wave digital filters. Quoting from the linked source:
A Wave Digital Filter ...
- 1,183
4
votes
Why is aliasing inherently non-linear?
Hints:
Can an LTI system generate components in some frequency $\omega_0$ if the input signal $x(n)$ was such that $X(e^{j\omega_0})=0$?
Does aliasing do such thing?
The answers to these questions ...
- 5,040
4
votes
Accepted
Create distortion from basic linear (and non-linear if neccessary) DSP elements
Using those four basic elements will allow you to implement linear systems, which can change the magnitude and phase of the input signal, but which will not add the harmonics that are expected from a ...
- 91.1k
4
votes
Is it possible to use a Volterra series to generate subharmonics?
Many years later I was asked to write some of what I've learned on this. The short answer is that it depends on what the term "subharmonics" means - things are very different if we're ...
- 507
4
votes
Accepted
Dynamic convolution vs Volterra series
A flavor of dynamic convolution (is that a trademark by the way?) has a different impulse response $g_i$ associated with each range of instantaneous input. A number of ranges can be defined by fuzzy ...
- 13.6k
4
votes
As of 2019, which discrete nonlinear, time-invariant systems with memory are considered "easy" to model and identify?
The easiest is Urysohn adaptive filter:
http://www.ezcodesample.com/UAF/UAF.html
It can build nonlinear model by few lines of code. The theoretical details can be found here http://www.ezcodesample....
- 301
4
votes
Accepted
How to find the output mean and autocorrelation of a non-linear system
The OP's updated working is incorrect. Following up what Hilmar suggested gives
\begin{align}
Y(t) &= a\left(X(t)\right)^2\\
&= a\left(S(t) + N(t)\right)^2\\
&= a\left(S(t)\right)^2 + 2aS(...
- 20.7k
3
votes
Volterra Kernel Convolution Method
I think actually your code is not correctly implementing the equation you gave. What you are actually imeplementing in matlab is this:
$$ y(n) = \sum_{m_1=0}^{M-1} h_1(m_1)x(n-m_1) + \sum_{m_1=0}^{M-...
- 423
3
votes
Power spectral density of $\left(x(t)\right)^2$?
For this problem you can't use the formula involving $|H(f)|^2$ because it only applies to linear time-invariant (LTI) systems, and a squarer is obviously a non-linear system.
The only way to solve ...
- 91.1k
3
votes
Accepted
Why harmonic components appear only after a certain level when a signal is clipped?
Is this a well-known phenomenon?
Yes, of course. You will see harmonics as soon as your clip point is lower than the maximum amplitude in the time domain. The latter is a function of the relative ...
- 47k
3
votes
Why only odd harmonics after non-linear amplification?
An ideal amplifier would have a transfer characteristic of $f(x)=Ax$: the input signal comes out amplified and otherwise undistorted. A real amplifier will deviate from this and go into saturation. We ...
- 2,463
3
votes
SRF-PLL discretization problem
I faced the same problem in the past. Perhaps there is a way without adding a delay but I haven't found it.
You need to realize that your 3 first solutions (delay after vq, delay at the delta_freq ...
- 3,787
3
votes
Accepted
How to linearize this state space model and write it in discrete form?
You have
$$
f\left(\mathbf x, u\right) = \begin{bmatrix}\frac{-1}{T}\tau+\frac{K}{T} u \\ \frac{\tau}{mr} \\ 0 \end{bmatrix} \tag a
$$
From which you (eventually) derive
$$
\mathbf {A}_d=\begin{...
- 12.9k
3
votes
Accepted
How to avoid harmonic distortions in a DAC?
Frequency Planning in Radio Design
The OP has clarified in comments that his question is focused on what would I believe would be commonly referred to as frequency planning in the process of radio ...
- 53.8k
2
votes
Accepted
Nonlinear time-invariant frequency doubler
As already suggested by Robert and Olli, a system that maps $x(t)=k\cos(2\pi f_0t)$ to $y(t)=k\cos(4\pi f_0 t)$ can be formalized as
$$y(t)=|x(t)|_{max}\left(2\left(\frac{x(t)}{|x(t)|_{max}}\right)^2-...
- 91.1k
2
votes
Accepted
The frequency spectrum of a static non-linearity driven by colored noise
If your nonlinearity can be expressed as a polynomial (i.e., in terms of addition and multiplication), you can make use of:
The linearity of the Fourier transform, i.e., if $f$ and $g$ are (benign) ...
- 138
2
votes
Accepted
Discrete time non-linear time invariant system dynamics descriptions (state-space or input-output relationship)
Well, any input-output representation obviously admits a state-sapce form. for your equation in $y[k]$ you can easily construct one as follows. Create a "shift" system (an integrator chain) as
$$
\...
- 768
2
votes
Dynamic convolution vs Volterra series
Dynamic convolution is a model for systems that can be written as follows:
$$ S[x] = \sum_k (L_k \circ N_k)[x]$$
where $L_k$ are linear time-invariant systems and $N_k$ are non-linear memoryless ...
- 4,593
2
votes
Accepted
What is meant by "Cubic Difference Frequency"?
After skimming through the paper, I can see more clearly now. The measure $D_3$ quantifies the relative strength of the 3rd order intermodulation product. If two sinusoidal signals with frequencies $...
- 91.1k
2
votes
Determining invertibility of a system
In general there is no systematic way and you simply have to analyze the given system. In the case of the system in your question, it's easy to see that it can't be invertible, because the output is ...
- 91.1k
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