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Geometric representation of a signal using basis functions

Feels like a bit of a trick question. Your answer is indeed correct. $\Psi_1$ ...$\Psi_3$ are an orthonormal basis but it's an incomplete basis. Any signal you can construct with this basis has 4 ...
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Geometric representation of a signal using basis functions

Here is a graphical explanation. Sorry, I have depicted $-x(t)$ in red, and the $\psi_k$ in black. In gray, the area of the product on sub-intervals. Positive when $-x(t)$ and $\psi_k$ have the same ...
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The inverse of an orthogonal matrix is its transpose

An orthogonal matrix has orthogal columns, i.e. the scalar product of two different columns is zero (the case $i\neq j$). For the case $i=j$ you have $a_i^Ta_i=\|a_i\|^2>0$. So, all you can say ...
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Relationship between input and output sequence in Hartley transformation

Wikipedia's entry for the discrete Hartley transform shows states that the $\mathsf{DHT}$ is, up to a scaling, its own inverse. If $x$ is a vector with $N$ entries and $y$ is its discrete Hartley ...
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Obtaining normalized matrix for the Haar Wavelet Transform

After a few quick calculations, it seems to me that the trouble comes from poor notations for the root in your reference. If you read, in the final normalized matrix, $\sqrt{8/64}$ and $\sqrt{2/4}$ ...
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What's the meaning of the continuity in spectrum analysis?

Let $v(x, k)$ be an orthonormal set of basis functions such that $$\int_{-\infty}^{+\infty} v(x,a) v^*(x,b) dx = \delta_{a,b}$$ where $\delta_{a,b}$ s the Kronecker delta. I believe your question ...

Partitioning the energy of a signal between its components

Say you decompose the signal $x(t)$ into two components: $$s(t) = x(t) + y(t).$$ As you say, there are myriad ways to do this. The energy of $s(t)$ is $$E_s = \int_{-\infty}^\infty s^2(t) dt.$$ If we ...
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What's a Normalized function?

Orthogonal means for two (real) vectors $u$ and $v$ that the scalar product vanishes: $$\lt \overline{u},v\gt = 0$$ Under this definition, any $0$ vector is trivially orthogonal to others. As the ...
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What's a Normalized function?

Similar to a a vector that is normal when it's magnitude is 1, which is the inner product with itself. In the same fashion, a function is normal over a defined range $[t_1,t_2]$ when the root of the ...
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Orthogonal basis of signal space and the projection of white noise

Leave time dimension aside for a moment and deal with just one symbol at a time. In a PAM modulation the generated functions are not ortho and not normal. Actually, in a 4-PAM modulation you have four ...
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The inverse of an orthogonal matrix is its transpose

Suppose you have a set $n$ vectors $o_k$ in $\mathbb{R}^{m,1}$ (length $m$, column-style), pairwise orthonormal, i.e. orthogonal with unit norm. Stack them side by side in a matrix A = \left[o_1,\...
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Relationship between input and output sequence in Hartley transformation

The Hartley transform is an involution: it is (up to a scale factor) its own inverse. The classical discrete Hartley transform of order $N$ is such that $H_N^{-1} = \frac{1}{N}H_N$. Be careful with ...
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What's the meaning of the continuity in spectrum analysis?

Great question, and a really tricky one to get your head around. I think your dual use of the word distance, and the comparison of vectors in an infinite function space to a vector in a finite-support ...
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