Questions tagged [discrete-signals]
A discrete signal or discrete-time signal is a time series consisting of a sequence of quantities.
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Z-Transform of $ x(n+2)u(n) $
I don't know where I went wrong, this might be a dumb question. I am solving for the z-transform of $$ y(n) = x(n+2)u(n) $$.
To solve, I used the z-transform equation,
$$ Y(z) = \sum_{n=-\infty}^\...
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How to impose power constraint on an encoder's output while training a deep NN?
A deep encoder which takes in a 4 bit input gives out an 8-dimensional vector. If I want the power of this 8-D vector constrained to be 1, then I should multiply all the elements by A/B? Where A is ...
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Discrete Fourier Transform of the Gaussian
Cross-posted from here
I encountered the following question in a Digital Image Processing examination:
Find the 2D DFT of $\frac{1}{2 \pi \sigma^2} e^{-\frac{(x - x_0)^2 + (y - y_0)^2}{2 \sigma^2}}$ ...
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Finding zero input response with only one initial condition
I have to find the Zero input response of the following system if $y[-2]= 0.5$
$$x[n] - 3y[n-1] - 4y[n-2] = 0$$
I am not sure if this is the correct way since every other example I have done had two ...
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For unit step $g(t) = u(t)$, why does $G(z) = \frac{z}{1-z}$, whereas $G(s) = \frac{1}{s}$?
Edit:
The source of my confusion is over the existence of multiple s-z mappings.
After researching these mappings and where they came from, I couldn't find why $z=e^{sT}$ can be ignored and replaced ...
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Allan deviation correct with one-side noise but wrog with two-side noise. Why?
I'm developing a Simulink model for noise according to the description of the algorithm of Accelerometer in this page: IMU simulation model.
My system takes the RandomWalk (defined in [unit], being ...
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Finding Discrete Fourier Transform (DFT) for different DFT size
$N$ is an even integer, $x[n]$ is a finite length signal over the interval $n \in [0,N-1]$, and $X[k]$ is the $N$-point DFT of $x[n]$. Analytically find the DFT of sequence below in terms of $X[k]$. ...
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Implementing point-wise multiplication as a convolution
As the title says, I'm looking for a way to implement the point-wise multiplication of 2 signals somehow as a convolution.
Here's why:
In my system, I have an input signal $x[n]$ which has been ...
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What is the rationale behind the "same" mode of discrete convolution?
Having read up on discrete convolution and how it is implemented, I've started to think about which "mode" is applicable to which situation. For two signals ...
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Calculating the main lobe width of Dirichlet kernel
The Dirichlet kernel of order $N$ is defined as:
$$
D(\theta,N)=\frac{\sin(N\theta /2)}{\sin(\theta /2)}
$$
We know that the main lobe widths between zero crossings is: $4\pi/N$
I was wondering if ...
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Proper audio noise and artifact detection approach
I'm working on developing an audio device testing suite (for speakers and like). I'm looking for approaches to detecting two types of noise: static such as white or pink noise and artifacts such as ...
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Fourier Transform and Music Analysis
I am a senior in high-school and am currently trying to conduct an exploration on Fourier Analysis, specifically using the Discrete Fourier Transform to analyze a chord played on my piano. Basically, ...
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CFO Estimation in LoRa Chirp Signal (Preamble part)
I am trying to estimate CFO in LoRa chirp signal (preamble part). I have seen the discussion about CFO on this forum but it is mainly related to CFO estimation in OFDM.
I want to estimate the CFO in a ...
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Why is the DC component of discrete fourier transform not the same as the signal's arithmetic mean?
In this question we have a mathematical proof that the DC component of normalized discrete Fourier transform should be the same as the signal's arithmetic mean. However, in the following example I ...
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Why dft sine plot is so strange
Why does my dft plot of $\sin(x)$ look so strange? It goes to negative and then inverts closer to the edge.
...
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Practical Implications of DFT Eigenvector Formulations
Different sets of eigenvectors for the Discrete Fourier Transform (DFT) are well-established.
Are there any potential practical implications associated with deriving explicit formulations for these ...
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Determine if the sequence is time invariant or not
Determine if the following sequence is time invariant or not:
$$y[n]= n^2x[n-2]$$
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Why is numpy.angle returning values on real numbers?
This question is a follow up on previous. As I am analyzing the sampled data I need to know if it is complex or real. This is important for me since I want to transform my two-sided to a one-sided PSD....
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Approximating variable-length coding by "aggregating" fixed-length coding
Consider the problem where I want to quantize a univariate standard Gaussian with two bits, and the objective is to minimize MSE reconstruction loss. According to these slides, the Lloyd algorithm and ...
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Is this system time variant or time invariant
I'm studying DSP from a particular book.
There is an exercise that examines the properties of some given systems.
The system I'm referring to is:
...
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How can I estimate the 2 parameters of this signal?
I have a digital signal of fixed length (e.g. 100 samples). Somewhere within this signal is a contiguous region characterized by "low variance". The remainder of the signal is characterized ...
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Infinite Sum of Future Signal Values of an LTI System
Let $x[n]$ be the input and $y[n]$ be the output of an LTI system. That is,
$y[n]=\displaystyle\sum_{k=-\infty}^{\infty} h[n-k]\space x[k]$
where $h[\cdot]$ is the impulse response function. We are ...
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How to calculate The Signal-to-Noise Ratio (SNR) in dB (different units)?
Recently, I've ventured into a new domain that I intend to incorporate into my work. After thorough research and consultation, I've found this model to calculate the data transmission rate (Capacity).
...
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Change pitch of voice
I am programming a voice changer (in c++). I need only to change the pitch of the voice. Unfortunately, I absolutely lack theoretical knowledge and hence I could not find information on the Internet :(...
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Proof of time-invariance of a discrete-time system
I have two systems that are not time-invariant.
From inserting a shifted signal, I can tell that it is not time-invariant, but I couldn't prove rigorously why it isn't time-invariant. I got confused ...
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How to define Q matrix (covariance matrix for process noise) in EKF
I am implementing an Extended Kalman Filter (EKF). I have developed the model for my system and I know must find the Q matrix. I have searched all over the web and noone explains how to define this ...
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Discrete Time Filtered White Noise
Theoretically, the variance of a filtered white noise should be:
$$\sigma^2 = \int_{-\infty}^\infty \frac{N_0}{2}|H(f)|^2\,\mathrm df = \int_{-\infty}^\infty \text{PSD}_{\text{new}}(f)\,\mathrm df$$
...
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Difference between ADC dynamic range and voltage resolution?
Say I have a 10-bit SAR ADC with a $5 \ \text{V}$ reference. I calculate its resolution to be
$$\Delta V_\text{lsb} = \frac{5 \ \text{V}}{2^{10}} = 4.88 \ \text{mV} \approx 5 \ \text{mV}. $$
So for a $...
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Avoiding latency distortion at high denoise levels with DWT
I am denoising biological signals using the DWT, and for UI reasons would prefer the smoother waveform afforded by denoise level 5. However, higher denoise levels seem to distort the latency of ...
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Impulse Response for an Input-Output Pair
Given an input-output pair of a LTI system
\begin{gather*}
x[ n] \ =\ 2\delta [ n+2] -\delta [ n+1] +\delta [ n-1]\\
y[ n] \ =\ 4 \delta [ n+2] +\ 4\delta [ n+1] -\delta [ n-1]
\end{gather*}
My ...
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If the convolution of two signals is a unit impulse, what does this tell us?
I have two discrete-time LTI systems whose transfer functions satisfy $h_1[n] * h_2[n]= \delta[n]$. We also know that system 1 is causal and stable. Does $h_1[n] * h_2[n]= \delta[n]$ tell us anything ...
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What is the possible application of eigenvalues?
I am a PhD in mathematics. Recently, we made an attempt to compute the eigenvalues of non-normalized discrete sine and cosine transforms. Surprisingly, the issue regarding three particular types, DCT-...
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How to calculate the following mutual infomation between $X$ and $Y$ in matlab or python?
Recently I meet the following question: Let $X\in {\mu_1,\mu_2,\cdots,\mu_m}$, $Y \in {R_1,R_2,\cdots,R_n}$, where $\mu_i, i = 1,2,\cdots,m$ is means of $m$ gauss probablity density function, $Y$ is ...
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Is this proof valid?
Consider a discrete-time sequence, $y[n]$, defined as:
$$y[n] = \frac 12 x[n] + \frac 12 (-1)^n\: x[n]$$
where $x[n]$ is another discrete-time sequence.
The DTFT of $y[n]$ defined as $Y(e^{j\omega}) = ...
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Programming a delay in Matlab
Let $x(t)$ a signal of duration $T$, the sampling frequency is $f_s$.
I know the signal $x(t+\tau)$ orresponds to a signal delayed by a delay $\tau$. In Matlab, the signal $x(t+\tau)$ is given by: $[...
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Discrete time Fourier transform of an exponential decaying sigal [duplicate]
I have a fundamental question about the discrete-time Fourier transform. I used two methods but got two results.
Background knowledge
The discrete time signal is given by:
$$x[ n ] = x[ {n{T_s}} ] = x(...
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How to normalize FIR filter gain from the amplitude response in frequency domain?
I've got a set of measured data (amplitude and phase responses) of a filter. And I want to normalize its gain to be unit (equal to 1) for further processing. I know that we can convert to time-domain ...
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Input (Auto)Correlation Matrix for LMS Adaptive Algorithm
I read quite a few posts regarding the autocorrelation matrix and although some of them were of the LMS adaptive algorithm application, I didn't find an answer to my question.
How do I calculate R; ...
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Does convolution of delayed sequences produces a delayed output?
Say, I have $x[n]$ and $y[n]$ be periodic sequences of period $N$. Let us define $z[n] =x[n] \ast y[n]$ where $\ast$ denote the circular convolution operation.
What is $x[n+\delta] \ast y[n+\delta]?...
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Does filtering by wavelet decomposition and reconstruction introduce a phase shift in the filtered signal?
I am removing low frequency noise from a signal using Matlab using the DWT and then reconstruction at a specific level:
I am removing the approximation signal at level 10 which approximates the low ...
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How can I apply a phase shift to an LFM pulse?
I am modelling radar scenarios in python and am trying to make a model which retains phase information. To do so, I'm envisioning something which works like this:
Generate samples of an LFM waveform ...
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How to discretize a continous signal
I want to transform the continuous signal $$u(t)=u_0+\hat{u}\cdot \cos(2\pi \frac{t}{T_0})$$ to a discrete signal. The discrete signal should contain $N$ values and exactly $m$ periods (starting with $...
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Difference between H2 control and input shaping
Let us assume a system with external disturbance to reject. The system that is used for controlling has vibrations, that we would like to avoid.
We assume moreover that the dynamics are known (10% ...
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Effect of windowing on signals
I have an input signal of frequency 1000Hz. Sampling frequency=16kHz. I compute the FFT on 128 input samples and plot the magnitude spectrum. I am not doing the normalization by 128.
My two sided ...
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Replicating SDR transmitter and receiver (shifting of a quadrature signal)
I am trying to implement an SDR transmitter and receiver based on the description here.
...
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Unwindowed STFT of sine, closed form solution and insights (sliding FFT)
I seek to calculate, mathematically, the unwindowed Short-Time Fourier Transform of
$$
\cos(2\pi f t + \phi)
$$
i.e. any arbitrary real-valued sine: any frequency, duration, phase shift, and number of ...
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How should I go about calculating sines or cosines without the errors introduced solely by the transcendence of pi?
This may be irrelevant for many people but I'm asking this question because I don't need to do these things in real time, I don't care if modifying a 1-minute sound recording takes 5 minutes to ...
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Not getting the same step response from Laplace transform and it's respective difference equation
I am trying to simulate a plant on a microcontroller. The transfer function of the plant is
$$ G_{p} \left( s \right) = \frac{2}{\left( s + 3 \right) \left( s - 1 \right)} \tag{1} \label{1}$$
The step ...
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Find constants such that the Nyquist criterion is satisfied [closed]
I have been given the following spectrum, and I want to find a and b such the pulse satisfies the Nyquist criterion, given that the bandwidth of the pulse is 2 kHz and the sampling frequency is 3 kHz. ...
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How to calculate the derivative of a sequence (discrete function) with respect to another discrete function?
In Learning Recursive Filters for Low-Level Vision via a Hybrid Neural Network, the hidden state of the recurrent neural network (RNN), after some simplification, presented as
$$
h[k] = (1-p)\cdot x[k]...
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