Questions tagged [math]
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167
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Showing analytically that sampling exactly 1 period of a sinusoid yields a spectrum with no lobes in the DFT
The DTFT of a discrete sinusoid $f[k] = \sin(\omega_0 k)$ is
$$F(\Omega)=i\pi(\delta[\Omega-\omega_0] + \delta[\Omega+\omega_0]), \: \: \: \Omega \in [-\pi, \pi)$$
The DTFT of a rect function $w[k] = ...
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Complex cross-correlation: conjugate first or second argument?
If I understand correctly the definition of cross-correlation, the complex conjugate is not applied to the same argument depending on whether we're working with deterministic signals or with ...
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Wavelets vs Windowed Fourier
Are there any good simple examples where wavelet scalograms provide more information than windowed fourier?
P.s It would be even cooler if there was an example of what cannot be obtained from a ...
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The order of filtering and convolution
Assume I have a source signal s(t) to pass through a system. I record the signal at system output place, which is denoted as d(t)...
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447
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Convolution of sinusoid and running sum filter of order 5
Problem
Compute analytically the convolution between the impulse response of a running sum filter of order 5 and a sinusoid $\sin(20\pi t)$ sampled with $f_s = 50 \: \text{Hz}$.
Attempt
The impulse ...
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Bit-wise MAP criterion for Bi-AWGN Channel
I am trying to derive an expression for the bit-wise MAP criterion for the binary AWGN channel.
I were able to derive the codeword ML criterion, but now using the LLRs obtained from the codeword ML ...
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Efficient computation of 2^x (antilog2) for fixed-point
I've created a log2 AGC for an FPGA project, similar to Dan Boschen's solution in this question. I implemented a hardware version of log2 based on Dan's linked page in the previous question - ...
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How to know if a continuous function can be represented by a finite sum of sinusoids?
I have a lack of mathematical knowledge, and notably in mathematical vocabulary, so maybe a similar question exists but with a different wording.
What I want to know, is actually how to know if a ...
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Why is there a tradeoff between simplicity of a window function and its ability to be adjustable and controllable?
While studying and applying different window functions in a project I am working on, I found that known windows (here is an example paper which cataloged numerous windows) suffer from 2 kinds of trade-...
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What is the effect of multiplication on the value of discontinuities?
I see that the values of a distribution at discontinuities are essentialy equal to the mean of the right-hand and left-hand limit values when dealing with distributions in relation to the Fourier ...
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How to handle undefined functions in a convolution?
When convolving with undefined functions, do we just treat them like variables? (eg. How do I handle $u \left( k \right)$ and $u \left( n - k \right)$ since they are just step functions with no ...
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How to handle imaginary numbers with total square error questions
I am pretty new to algrebra so if u can use simple terms and explain a bit more it would be helpfull. Ive been trying to use imaginary numbers to get the total square error of a graph of a non linear ...
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Improving Modeling of Thermal Noise Behavior at Antenna
Background
I recently asked this question over on Electrical Engineering Stack Exchange. On the advice of some commenters there, I've broken off those pieces which are appropriate for asking as ...
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Improving Modeling of Thermal Noise Propagation Through a Signal Chain
Background
I recently asked this question over on Electrical Engineering Stack Exchange. On the advice of some commenters there, I've broken off those pieces which are appropriate for asking as ...
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How can I show that an LTI system can be expressed as a difference equation?
I'm in the process of re-learning DSP (not a subject I've visited since University) and in quite a few resources I see this general form of a DT-LTI difference equation:
$$y[n] + a_1y[n-1] + a_2y[n-2] ...
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35
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Verifying Linearity and Shift Invariance Under Summation
I am having some trouble working through verifying linearity and shift invariance when the transformation is under a summation.
The given transformation is as follows:
$$
y(m,n)=\sum_{i=-1}^{i=1}\sum_{...
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Ambiguity function volume
I find myself confused how two properties of the ambiguity function relate to one another.
The properties in question are (presented here from wikipedia, but similar formulas can be found in books by ...
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Under what conditions does DFT(f(x)) = f(DFT(x)) hold?
I have encountered a text (a specification) in which $$\text{FFT}\left(f\left(x\right)\right) = f\left(\text{FFT}\left(x\right)\right)$$ is clearly presumed (for the functions they are applying).
Thus,...
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Time shifting a signal in python
I am currently modeling FMCW (linear FM) and SFCW (stepped FM) chirps in python. For my project, I need to simulate those signals as a transmit chirp and a received one, scattered from point targets ...
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Pole Quantization Patterns in 2nd Order IIR Resonators
Background
My typical approach to fixed point design for digital filters is to iteratively scale and increase quantization while comparing the fixed point simulation to the floating point design ...
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Fourier transform why can I convert one of the axes into an imaginary number?
Contextualizing
This question is inspired by the following video:
https://www.youtube.com/watch?v=-qgreAUpPwM&t=60s&ab_channel=3Blue1Brown
I own a sign, a drawing of a square with 200 points ...
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Why Hilbert Transform is terrible choice for amplitude demodulation of broadband signals?
A reference answer empirically demonstrates that Hilbert envelope does not work well for the (amplitude) demodulation of a broadband signal. I am looking for the math which explains why...
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Get IMU acceleration of a direction when tilted
I have an IMU that is placed on the body. My goal is to calculate the position data of the vertical upward movement. Since the sensor is never perfectly aligned and will always have a little tilt, I ...
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Confusion Understanding the mathematical expression of duality property of dft?
Duality Property for DFT
Above dsp.se question provides good understanding about dft duality property but i am having difficulty understanding its mathematical expression because on Google when i try ...
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What *is* pitch-shift?
As I understand it, pitch-shifting (without changing speed) and time-scaling (without changing pitch) are two sides of the same coin, because if we can do one we can get the other through resampling ...
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How to Get Started On Demodulating Signals Such As Amplitude Modulation and Pulse Width Modulation
I'm looking to understand how signal modulation/demodulation works.
My goal is to be able to demodulate a 315 MHz key fob and be able to tell what buttons are being pressed so I can have a Python ...
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460
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How to Find the Kernel of the Convolution for Linear Interpolation?
I'm trying to solve the following exercise:
Image A was doubled by linear interpolation. The magnification was performed in two stages. In the first stage, add about zero pixel to the image between ...
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I'm having problems simplifying this discrete-time fourier tranform
I have this problem, and I can't get to the solution.
$$X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} {(0.6)^{|n|}[u(n + 10) − u(n − 11)]}e^{-j\omega n}$$
The solution is
$$X(e^{j\omega}) = \frac{0.64 − ...
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How do I carry out SRT Radix-4 fixed-point division with more than a few bits?
I've managed to get a hand on how SRT division works, but I haven't been able to construct a working example. My current sticking point is that e.g. the Pentium floating-point division bug came about ...
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what are the application of fourier transform in finding frequency response? [duplicate]
i need the application of Fourier transform in finding frequency response in mathematical terms
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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 ...
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False positive rate with CRC
What is the technique to calculate the false positive rate for an N-bit CRC?
Example: Communication system sends M-bit packets with a N-bit CRC (M >> N) appended to each packet. What is the ...
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Problem with the output self implemented “ Iterative FFT DIT Radix2 ” algorithm matlab
I´m trying to implement my own FFT with the iterative Radix2 algorithm (I even thought about using the recursive one but it seems Simulink doesn´t allow me to). I tried all of the possible variations ...
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Parseval's Theorem for Time Domain Data with DC
I have read several times that the variance for time domain data with a zero mean is equal to the integral of the power spectral density divided by N.
From wikipedia, the discrete form of Parseval's ...
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Different mathematical signal models for different applications
I am looking for some interesting and physically meaningful applications of different signal models. I am currently working with complex analytic signal model given below, but I couldn't come up with ...
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Backpropagation in a Network [closed]
Hi does anyone know how to solve this. Im really struggling to figuring this one out.
I am unclear of how the backpropagation part would work in this network and how to calculate it for each weight. ...
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Calculating Haar features
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 ...
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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 ...
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Question about delayed sampled sinusoid math expression
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^...
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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 ...
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mathematical expression to detect modulated data within a vector
Assume I have such modulated data which is, for example, $x=0.7+0.7i$. That modulated data is encapsulated in a vector as below:
OR
where $c$ is any constant number, let’s say, for example: $c=0.7$ ...
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Effectively extracting “real” data from a noisy dataset
This question was originally posed to Math SE. It was suggested that DSP SE would be more suitable.
Background & Motivation:
I have three lists of timestamps (UNIX timestamps, plus a subsecond ...
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Is it important to learn to find de period of the sum fo two DT signals?
The fundamental period (N) of the sum of two DT signals could be the LCM of the fundamental periods. However, this is not always the case. I would like to know wether, during engineering design or ...
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When is Markov a Martingale
I have two questions and I am very confused about the concepts
Can a Markov process of order one also be a a Martingale?
Is any Markov process of order one also a Martingale?
For 1. I would say yes, ...
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The Math of Histogram Matching
Please help solve the problem in the picture. I am confused due to the piecewise nature of the density functions. If it was a simple curved PDF, I would simply integrate over full range and use the ...
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Problems computing the DFT of finite length sequence
I am having trouble finding the same answer as the solution manual for this sequence.
The problem asks to compute the DFT of
$$
x[n] = \begin{cases}
1 & \text{for even } n \in \{0\...
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Linear correlation of two vectors?
We had a lecture on digital watermark detection using linear correlation. Here it was explained that the linear correlation of two vectors is equal to their scalar product divided by the dimension. So,...
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Relation between the matrix trace and the amplitude of each element
Assume a diagonal matrix $\mathbf X$ whose size $N\times N$ and its diagonal elements are $0.5 + 0.5i$, and the vector $\mathbf p$ of size $N\times 1$ whose elements have similar amplitude.
I have ...
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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 ...
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SNR for the magnitude of a complex signal
The SNR for a complex valued signal is presented by Dilip Sarwate in: https://dsp.stackexchange.com/a/42359
Following the same steps I would like to estimate the SNR for the magnitude of a complex ...
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