# Questions tagged [math]

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21k views

### What is the first derivative of Dirac delta function?

Could you please help me in a simple way, what is the first derivative of a Dirac delta function? I found this answer: The informal answer is a positive Delta function immediately followed by a ...
1k views

### Mathematical question that comes out of using bilinear transform

So this is related to the Cookbook and I tried solving it maybe two decades ago, gave up, and was reminded of the unsolved problem. But it's pretty damn straight forward, but I still got slogged down ...
7k views

### Looking for an arcsin algorithm

Does anyone have a simple algorithm for computing a reasonably accurate arcsine? By "simple" I mean some sort of polynomial that requires <= 5 multiplies per output sample. And by "reasonably ...
2k views

### Functional analysis for signal processing engineers

It seems that the most advanced pure mathematics course most EE engineers take is Fourier analysis, and after that it's basically 'applied' courses. There's probably a good reason for this, but I'm ...
2k views

### Replacing "e" in Euler's formula with another number

Does Euler's formula remain valid if we use any real number other than the constant $e$? For example replacing $e$ with 5 would make the formula look like this: $5^{it}$. I tried this idea in ...
4k views

### What is the math behind median filter's noise reduction property?

I am interested in understanding the mathematical reason for why does applying a median filter on an image (or signal) result in reduction of noise.
2k views

### Books/resources for implementing various mathematical functions in fixed point arithmetic for DSP purposes

I'm looking for books or resources that cover the following in detail: implementing mathematical functions (e.g., logarithm, exponential, sine, cosine, inverse) in fixed point arithmetic for DSP ...
6k views

### Phase correlation vs. normalized cross-correlation

I asked this over at Mathematics Stack Exchange, but since this sort of lies on the border of the questions normally asked over there and the questions you see over here I'll ask it here as well. (As ...
577 views

### Upper Bound for the DFT (FFT) Coefficients of a Bounded Signal

I have a discrete signal $x[n]$ of length say $1024$, such that : $$-1 \leq x[n] \leq 1, \qquad \forall n$$ and let $y[k]$ be its DFT. Is there an upper bound for $|y[k]|$ ? (i.e. for the DFT of a ...
84 views

### Why Does the Median Filter Minimize the Absolute Value Error $L_1$ Cost Function?

I can easily prove that the mean filter minimizes the square error $L_2$ cost function using simple calculus. However, how do you prove that the median filter is optimal with respect the absolute ...
125 views

### Why Cramér spectral representation and not DTFT for stochastic process

In a lot of time-series analysis references I find (written by mathematicians or statisticians rather than engineers), I find the following signal decomposition for a stochastic process, termed the &...
178 views

### Minimum Mean Square Estimator - Equivalent Expressions to Minimize

Given $M \in \mathbb{R}^{N \times N}$ which is a Positive Definite Matrix. Let $\hat{x}$ the MMSE of $x$ given $z$, namely $\hat{x} = \mathbb{E} \left[ x \mid z \right]$. Prove the ...
711 views

### Laplacian of Gaussian Approximation and Gaussian Blur as the Solution of Heat Equation

While I was reading SIFT paper(Lowe, 2004), I came across the method that he apply "heat diffusion equation" to Gaussian function to derive that $$\frac{∂G}{∂σ} = σ∇^2G$$ I searched Wikipedia and ...
507 views

### Derivation of fixed-point $\tt atan2$ with self-normalization

I'm trying to understand the maths behind this Fast fixed-point $\tt atan2$ calculation with self-normalization. In particular, equation $(2)$ for theta1 appears to ...
1k views

### Design ${L}_{2}$ Norm Optimal Infinite Impulse Response (IIR) Filters

It is widely known that matching a FIR filter of fixed length to a band model is an unconstrained QP-problem. The MATLAB function firls() implements a solution to ...
149 views

### Weighted Sum of Auto Correlation - Lower Bound

Given a vector $v$ with elements ${\left\{ {v}_{n} \right\}}_{n = - \infty}^{\infty}$ and denoting ${v}_{n}^{\left( k \right)} = {v}_{n - k}$, namely, a shifted vector by $k$ elements (Mind ...
15k views

### How to Calculate Gaussian Kernel for a Small Support Size?

How to calculate the values of Gaussian kernel? I think I understand the principle of it weighting the center pixel as the means, and those around it according to the $\sigma$ but what would each ...
904 views

### How moving part pixel intensity values of video frames becomes dominant compared to stationary part intensities in reconstructed frames?

Hello everyone i want to do dynamic texture video sementation using the Fourier transform in MATLAB. I am applying 3-D fft on dynamic texture video frames (using matlab function 'fftn') and ...
2k views

### State Estimation by steady state Kalman Filter

I am working with a discret Kalman filter on a System $x_{k+1}=A_k x_k+B_k u_k+\omega_k$ $y_k=C_k x_k+\upsilon_k$ $E[\omega_k\omega_k^T]=Q$ $E[\upsilon_k \upsilon_k^T]=R$ I have estimated the ...
998 views

### What is the meaning of this notation?

I have seen this notation in the book Schaums-Outline a couple of times now and I can't quite understand it. I'd very much appreciate if someone explained what it means. I am thinking that it means ...
284 views

456 views

### Mathematical relationship between highpass and lowpass filtering

Let $g, h_{HP}, h_{LP}: \mathbb{R} \rightarrow \mathbb{R}$ and $G, H_{HP}, H_{LP}$ denote their continuous Fourier transforms under the Fourier operator $\mathcal{F}$. Let $*$ denote the continuous ...
147 views

### Independence of Noise at Each DFT Output

My math may be a little rusty, so I would like confirmation or correction or my calculations here. Given white noise samples, $x_i$, which are IID and zero-mean, and variance $\sigma^2_x$. I want to ...
77 views

### Is it possible to compute the PSD of s(t)^2 given the PSD of s(t)?

Given a well-behaved signal $s(t)$ given by, $s(t) = \sum_{i}^{N} \cos( \omega_it + \phi_i)$ with a large $N$ and a well-known power spectral density (PSD), $\tilde{S_1}(\omega)$. Is it possible to ...
339 views

### SIFT: why it is invariant to affine tranlsation?

I am studying SIFT. I am a little confused in some of Lowe's algorithm. In order to have a descriptor invariant to translations like scaling, we must get rid of the scaling factor in LoG (Laplacian ...
704 views

### DFT of pure sinusoidal wave

I'm writing a program in which you can synthesize waves by adding to a sound's Fourier transform, and then inverse the transform to get the modified sound. In order to do this, I need to know what to ...
476 views

### Coordinate system in digital image processing

I am trying to understand image coordinate system where I read that it is reversed. So if $f(x,y)$ is actually $f(y,x)$ where $f(\cdot, \cdot)$ is the image. I am a bit confused and would appreciate ...
100 views

### Digital integrator

I have been implementing a control software where I need to calculate a magnetic flux based on the measurement of the phase voltages of a three phase grid (basically three sinewaves) according to the ...
101 views

231 views

### Absolute value based AM envelope detection viewed in the frequency domain

In my understanding when a passband AM signal is demodulated with an envelope detector taking the absolute value restores the original baseband version of the signal and the low-pass filter removes ...
125 views

### How to make the $\ell_2$ norm of all columns and rows of an $n \times n$ matrix equal to $\sqrt{n}$?

I have an $n \times n$ matrix and I would like its columns and rows to have $\ell_2$ norm equal to $\sqrt{n}$. Is this possible?
90 views

### Showing processes as an Equation

I have developed an algorithm that counts the number of times a particular block (within a 2D block) crosses zero. Here's an example: ...
6k views

### Convolution of two rectangular pulses intuition

From what I understand the convolution of the impulse response of a system with the input to that system gives the output. Now if the impulse response is a rectangular function and the input is also ...
55 views

### Rearranging exp terms

I have some problems rearranging exp terms to receive a cos or sin. I guess the solution is silly but I‘m not able to spot my mistake. I know the solution (since i looked it up with wolfram alpha) ...
1k views

### The Mathematical Relationship Between Derivative and Convolution

What I'm looking for is the mathematical proof of why we can calculate derivatives as a simple convolution of a mask, and how do we get that mask? I know it has something to do with how the derivative ...
I am confused on how to solve this problem, does anyone have any ideas? https://www.merl.com/publications/docs/TR2004-043.pdf For a Haar filter $H \in \mathbb{R}^{3 \times 5}$ defined as below: What ...
I have been studying the digital audio processing by using the book <Designing Audio Effect Plugins in C++>. For analog Sinusoid: Complex Sinusoid = $e^{jωt}$ Delayed Sinusoid = \$e^{jω(t−n)} = e^...