I will introduce some terminology and intuition that will be helpful when reading other references. It will be neither complete nor completely rigorous. The measures that we first encounter in real ...

I think that considering the DFT from a linear algebraic point of view has some value, so I will attempt to introduce the foundations. We will assume that our signal is a vector of $N$ complex ...

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 ...

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 ...

Let $x$ and $y$ be signals of $N$ samples each, numbered as $x(0),\ldots,x(N-1)$. Then their DFTs are $X$ and $Y$, which also have $N$ entries each: \begin{eqnarray} X(k) &=& \sum_{n=0}^{N-1}x(...

Let me state at the beginning that the details that make this rigorous do not bring any extra understanding of the statistical behavior of signals, so the desire to use the Dirac delta distribution is ...

In the absence of more details, I assume that the phase $\Theta$ is a random variable and $\Theta$ is uniformly distributed on the interval $[0,2\pi]$, so that its probability density function is $\...

The DFT of a real sequence is complex-valued. The array output by scipy.fftpack.rfft consists of the real part of the 0th entry, followed by the imaginary part of the 0th entry, followed by the real ...

Note that the original mathematical back-projection method assumes that the value of the Radon transform is known for all lines. This is an infinite amount of information, so it is pure mathematics at ...

As I mentioned in my comment, I think there is a typographical error in the equation. I think it is supposed to be \begin{equation} y[k] = \sum_{i=1}^{M}a_iy[k-i] + \sum_{j=1}^{N}b_jx[k-j]. \end{...

BOTTOM LINE UP FRONT: I think the exponential decay growth in $\left<|x(t)|^2\right>$ can be shown in the frequency domain only if the "boundary terms" are nonzero when we compute the ...

Given the definition of the correlation matrix $\mathbf{R}_{\mathbf{x}}$ here, I am assuming that $\mathsf{E}[\mathbf{x}] = \mathbf{0}$. I do this because the correlation matrix is usually defined as $...

Given speech samples, the LPC computation yields linear prediction coefficients $a_1,\ldots,a_p$. These describe a dependence model for a few samples. It is assumed that for short-enough collections ...

The root-raised-cosine pulse-shape is widely used because using the root-raised-cosine as both pulse shape and as matched filter yields the raised-cosine pulse-shape, and the root-raised-cosine and ...

Previous answers have already discussed some of the mathematical issues that arise after the Fourier transforms in the question, so I will try to impart some more easily earned intuition. Even if we ...

I am suspicious due to the claim that $\mathcal{R}_0$ is the inverse of $\mathcal{D}_0$. Decimation is not invertible. Once samples/entries are deleted, those values are forgotten. They cannot be ...