[Fourier transform and its application study notes] 25. Linear systems, transfer functions, eigenvalues

Matrix convolution, discrete finite-dimensional linear time invariant systems

This is the same description as in the previous lesson continuous infinite dimensional linear time invariant system: when and only if the linear operator is expressed in convolution, the system is a linear time-invariant system (LTI systems).

$\underline{w} = Av = \underline{h}* \underline{v}$

The above equation expresses the linear time-invariant system of discrete finite dimension, which can express the product of pulse response and input matrix, and also can express the convolution between matrices.

Let's use an example to deepen our understanding of linear time-invariant systems.

For example, a LTI system

$\underline{w} = Av = \underline{h}* \underline{v} \qquad, \underline{h}=\begin{pmatrix}1\\ 2\\ 3\\ 4\\\end{pmatrix}$

The LTI system matrix $a$ is obtained.

Based on the knowledge learned in the previous lesson, we know that the matrix $a$ is the impulse response of the LTI system, i.e.

$\begin{align*}
A
&=a[\underline{\delta}_0,\underline{\delta}_1,\underline{\delta} _2,\underline{\delta}_3] \ \
&=[a\underline{\delta}_0,a\underline{\delta}_1,a\underline{\delta}_2,a\ underline{\delta}_3]\\
&=[\underline{h}*\underline{\delta}_0,\underline{h}*\underline{\delta}_1,\ underline{h}*\underline{\delta}_2,\underline{h}*\underline{\delta}_3]\\
&=\left[
\begin{bmatrix}
1\\
2\\
3\\
4
\end{bmatrix}
\begin{bmatrix}
4\\
1\\
2\\
3
\end{bmatrix}
\ Begin{bmatrix}
3\\
4\\
1\\
2
\end{bmatrix}
\begin{bmatrix}
4\\
1\\
2\\
3
\end {Bmatrix}
\right] \qquad \delta\ shift\ property\\
&=\begin{bmatrix}
1 &4  &3  &2 \
2 &am p;1  &4  &3 \
3 &2  &1  &4 \
4 &3  &2  &1
\end {Bmatrix}
\end{align*}$

This type of matrix is called the cyclic matrix (circulant matrix), which has periodicity. The periodicity here means that each column of the matrix is shifted from the previous column.

Eigenvalue of linear time invariant system, feature function/eigenvector continuous infinite dimension space

In a continuous infinite-dimensional space, the LTI system has the following representation

$w (t) = Lv (t) = (H*V) (t) $

Its Fourier transform is

$\mathcal{f}w = \mathcal{f}h\mathcal{f}v$

Converted to the following symbol representation

$W (s) = H (s) V (s) $

where $h (s) $ is called the transfer function

The characteristic function of LTI system is complex exponential function

In the discussion of matrix multiplication, we introduced eigenvectors and eigenvalues, and now we are talking about continuous infinite dimensional spaces, where similar concepts are introduced: feature functions.

The definition of a feature function is: In an LTI system, if there is $lv (t) = \lambda V (t) $, then $v (x) $ is the characteristic function of the LTI system, and $\lambda$ is the corresponding eigenvalue.

So why is the characteristic function of the LTI system a complex exponential function?

The conclusions are derived from the following derivation:

Input complex exponential function $e^{2\pi I\nu t}$ for an LTI system, which behaves in the frequency domain as

$\begin{align*}
W (s)
&=h (s) V (s) \ \
&=h (s) \mathcal{f} (e^{2\pi i\nu t}) \ \
&=h (s) \delta (s-\nu) \ \
&=h (\NU) \delta (s-\nu) \qquad (\delta\ shift\ property)
\end{align*}$

It behaves in the time domain,

$w (t) = Le^{2\pi i\nu T} = H (\nu) e^{2\pi i\nu T} \qquad H (\nu) \ is\ constant$

By the above equation we know that in the LTI system

• Complex exponential $e^{2\pi i\nu t}$ is its characteristic function
• $H (\NU) = (\mathcal{f}h) (\NU) $ is the corresponding characteristic value

So does LTI have other feature functions?

Here we take $v (t) = cos (2\pi \nu t) $ as input to see if it is a feature function.

$\begin{align*}
Lv (t)
&=lcos (2\pi i\nu t) \ \
&=l\frac{1}{2}\left (e^{2\pi i\nu t}+e^{-2\pi i\nu t}\right) \ \
&=\frac{1}{2}\left (le^{2\pi i\nu t}+le^{-2\pi i\nu t}\right) \ \
&=\frac{1}{2}\left (H (\nu) e^{2\pi i\nu t}+h (-\nu) e^{-2\pi i\nu t}\right) \ \
&=\frac{1}{2}\left (H (\nu) e^{2\pi i\nu t}+\overline{h (\nu)}e^{-2\pi i\nu t}\right) \qquad H (t) \ is\ real-valued\, it ' S\ symmetric\ in\ frequency,h (-\nu) =\overline{h (\NU)}\\
&=\frac{1}{2}\left (H (\nu) e^{2\pi i\nu t}+\overline{h (\nu) e^{2\pi i\nu t}}\right) \ \
&=\frac{1}{2}\times 2re\left (H (\nu) e^{2\pi i\nu t}\right) \qquad (A+BI) + (A-BI) =2a,re\ means\ real\ part\\
&=re\left (| H (\nu) |e^{i\phi (\nu)}e^{2\pi i\nu t}\right) \qquad H (\nu) =| H (\NU) |e^{i\phi (\NU)}\\
&=| H (\NU) | Re\left (E^{i (2\pi \nu t+\phi (\nu))}\right) \ \
&=| H (\NU) | Re\left (cos (2\pi\nu t+\phi (\NU)) +isin (2\pi\nu T+\phi (\NU)) \right) \qquad eular\ fomular\\
&=| H (\NU) | Re\left (cos (2\pi\nu t+\phi (\NU)) \right)
\end{align*}$

As a result, in LTI, $cos (2\pi\nu t) $ cannot be converted to $\lambda cos (2\pi\nu t) $, so it is not a feature function.

In fact, only the complex exponential function is the feature function of LTI.

Discrete finite dimensional space

In the discrete finite dimension space, the LTI system has the following representation

$\UNDERLINE{W} = l\underline{v} = \underline{h}* \underline{v}$

Its discrete Fourier transform is

$\UNDERLINE{\MATHCAL{F}W} = (\underline{\mathcal{f}h}) (\underline{\mathcal{f}v}) $

Converted to the following symbol representation

$\UNDERLINE{W} = \underline{h}\underline{v}$

LTI's eigenvector is a complex exponential vector

The conclusions are derived from the following derivation

Input complex exponential vector $\underline{\omega}^{k}$ for discrete LTI systems, i.e.

$\UNDERLINE{V} = \underline{\omega}^{k} = \left (1,e^{2\pi i\frac{k}{n}},e^{2\pi I\frac{2k}{n}},..., e^{2\pi i\frac{(N-1) K}{n}}\right) $

Their performance in the frequency domain is

$\begin{align*}
\UNDERLINE{\MATHCAL{F}}\UNDERLINE{W}[M]
&=\underline{\mathcal{f}h}[m]\underline{\mathcal{f}\omega}^k[m]\\
&=\underline{\mathcal{f}h}[m]n\underline{\delta}[m-k]\\
&=\underline{\mathcal{f}h}[k]n\underline{\delta}[m-k]\\
&=\UNDERLINE{H}[K]N\UNDERLINE{\DELTA}[M-K]
\end{align*}$

They behave in the time domain (IDFT the results above)

$\UNDERLINE{W}[M] = \underline{h}[k]\underline{\omega}^{k}[m]$

That

$\underline{w} = L\underline{\omega}^k = \underline{h}[k]\underline{\omega}^{k}$

We know from the above results that in the LTI system

• The eigenvector is $\underline{\omega}^k$, which is the eigenvector base of the LTI $\underline{\omega}$, $k $ can be any integer
• The corresponding characteristic value is $\underline{h}[k]$

The following is an example of a discrete finite-dimension LTI system eigenvalue

The LTI system is provided as follows

$\UNDERLINE{W} = l\underline{v} = \underline{h}* \underline{v}\qquad \qquad \underline{h} = \left (1, 2, 3, 4\right) $

Eigenvalue is $\underline{h}[k] = \underline{\mathcal{f}h}[k]$

First we need to find out $\underline{h}$

$\underline{h} = \underline{\mathcal{f}h} = \displaystyle{\sum_{k=0}^{3}\underline{h}[k]\underline{\omega}^{-k}} = ( 10,-2+2i,-2,-2-2i) $

After this result, we can select the corresponding $\underline{h}[k]$ ($\underline{h}$ is the periodic loop) according to the $k$ value in the eigenvector $\underline{\omega}^k$.

[Fourier transform and its application study notes] 25. Linear systems, transfer functions, eigenvalues