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All eigenfunctions of an LTI system can be described in terms of complex exponentials, and complex exponentials form a complete basis of the signal space. However, if you have a system that is degenerate, meaning you have eigensubspaces of dimension >1, then the eigenvectors to the corresponding eigenvalue are all linear combination of vectors from the ...


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For discrete data both are the same - Finding set of orthogonal directions which maximizes the Variance (Energy) of data along them. Sometimes those are called the natural axis of the. Since we're dealing with variance it is only natural both are calculated from the covariance matrix of data. You may encounter places where KL might be even defined on the ...


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For any convolution matrix (even truncated), your diagonal entries would be $h_0$; hence, the characteristic polynomial, setting $b_0 = h_0-\lambda$, \begin{align}|H-\lambda I| = \begin{vmatrix} b_0 & 0 & 0\\h_1 & b_0 & 0\\h_2 & h_1 & b_0 \end{vmatrix} &= b_0\,\begin{vmatrix} b_0 & 0\\h_1 & b_0 \end{vmatrix} - 0\,\begin{...


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First, this question is probably better for MATH.SE, but I'll give it a shot. It's been a long long time since I did this stuff. If $N > M$: 1) $C^*$ has rank M. 2) $R_x^{-1}$ has rank M. (or it wouldn't exist as an inverse) 3) Therefore $C^*R_x^{-1}$ has rank M, since $R_x^{-1}$ is a full rank square matrix. 4) $C$ has rank M 5) $C^*R_x^{-1}C$ ...


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My first swing at the answer had some very incorrect claims. I do not have access to the article, so I am inferring some things from the portion posted in the question. NOTA BENE: My arguments assume that the eigenvectors of $\mathbf{R}$ are arranged so that the first $n$ belong to the signal subspace and that the last $m-n$ belong to the noise subspace. ...


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Given no formal system model in the question, I will outline in words what each does and the relation between them. Matched Filter: The MF maximizes SNR when the signal is in additive Gaussian noise. You can go back and look at the derivation of the MF, but it does not include any mention of interference. During the derivation, there is a step where we say ...


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Thanks for your answer Cedron! Taking your same assumption $N \leq M$, and by definition of PD, $x^* R_x^{-1} x > 0\quad \forall x \in \mathbb{R}^M \setminus \{0\}$ and since $C$ is full rank $\dim (\mathcal{R}(C)) = \min(M,N)= N$. By the rank-nulity theorem, $\dim (\mathcal{R}(C)) + \dim(\mathcal{N}(A))=N$, so we have that the null-space is trivial. This ...


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I've had issue with this -- please see the errata of the reference. The eigenfunctions are not scaled in the original version of the book. This errata should solve it.


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The eigenvalues belong to the same set of quartic roots of unity verifying $\lambda^4=1$, whatever the order of the DFT. For more details on their multiplicity, you can read: Eigenvectors and Functions of the Discrete Fourier Transform, 1982, Dickinson and Steiglitz (online).


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You can write $$ R=YY^H $$ where $Y$ is a matrix of size $N\times N_f$ and $N$ is the dimension of $y_k$. $Y$ contains all the measured $y_k$ as its columns. Then, the rank of $R$ is upper bounded by $N_f$. In particular, if $N_f<N$, $R$ will always be a singular matrix. So, if you have too few measurements, you will likely run into the problem you ...


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The piece I was missing was the distribution of the initial phase values $\varphi_1$ and $\varphi_2$. It is standard to assume that these are uniformly distributed [^]. This leads to: $$ \mathbf{R_x} = \mathbf{E[xx^H]} = \mathbf{S \Lambda S^H} $$ where $\mathbf{\Lambda} = \begin{bmatrix} A_1^2 & 0 \\ 0 & A_2^2 \end{bmatrix}$ and $\mathbf{S} = \begin{...


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It is an essential component in Principal Component Analysis(PCA) that allows to reduce the number of dimensions by projecting less dimensions of this sort of data, thus narrowing it down faster to finding a pattern. The projection operation characterizes an individual face by a weighted sum of the Eigen faces features and so to recognize a particular face ...


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I will try giving you some intuition. The SVD says each matrix can be decomposed into 3 operations - Rotation, Stretching (Scaling) and the another Rotation. What matters is which directions are scaled and how. Directions are vectors (Pointing some direction). The SVD has many uses in Linear Algebra. One its most known use is Low Rank Approximation of a ...


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For any arbitrary LTI sytem, the complex exponential is, to the best of my knowledge, the only known eigensignal. On the other hand, consider the ideal LPF. The $\operatorname{sinc}$ function: $$\operatorname{sinc}(t) \triangleq \frac{\sin(\pi t)}{\pi t}$$ can easily be seen to be an eigen signal. This points to the existence of LTI systems (such as the ...


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