Questions tagged [integration]

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4
votes
1answer
135 views

Numericaly computing an integral in MATLAB [closed]

I have the following integral $$\int_{-\infty}^{\infty} g(t-\tau) \, \mathrm d\tau$$ where $g(\cdot)$ is a known function, e.g., $g(t) = \sin(t)$, and I want to evaluate its value with MATLAB. How ...
4
votes
2answers
6k views

Removing drift from integration of accelerometer data

I am trying to get a positional data from the accelerometer data using the following steps: Re-zero the accelerometer value Removing mean from accelerometer value First integration of accelerometer ...
4
votes
2answers
1k views

Deriving the integration property of the Fourier Transform

I want to derive the property of the Fourier Transform that states that if $X(j\omega) = \mathcal{F} (x(t))$ then $$\mathcal{F} \left( \int_{-\infty}^{t} x(\tau) \mathrm{d} \tau \right) = \frac{1}{j\...
3
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5answers
2k views

Weird behaviour converting velocity to displacement

I'm trying to convert accelerometer data from accelerations to displacement, I assume that the initial velocity is zero and the initial position is also zero. Here's my code, ...
3
votes
2answers
124 views

Running Integral of sine and cosine functions

In typical signal processing course we were taught that the integral of signal $x(t)$ is given by $$y(t) = \int_{-\infty}^{t}x(\tau) d\tau$$ How can we use this definition to evaluate the integrals of ...
3
votes
1answer
258 views

Mutual information of $ \infty $-PAM

I'm trying to compute using matlab the mutual information for an $ \infty $-PAM input (the limit of a very dense PAM constellation) for a range of snr and I got stuck. I'm working with a real-valued ...
3
votes
1answer
102 views

Dimensional analysis of integrated white noise process

This question is somewhat related to this post. Let us consider we have a white noise current source $i_n(t)$, with a variance $\sigma_i^2$, and mean, $\mu_n=0$. Assume that this current is passed ...
2
votes
3answers
103 views

Derivative of delta function

I've come across the below. Can you please help me understand the proof for this? $$\int_{-\infty}^{\infty}x(t)\delta'(t-2.5)dt=-\frac{dx(t)}{dt}{\Big |}_{t=2.5}$$
2
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2answers
196 views

Why is $\int^\infty _{0^-}\delta(t-nT)e^{-st}dt = e^{-nsT}$?

I'm currently in the process of going over the $\mathcal Z$-transform and more specifically its derivation. I understand and I am able to follow it up until the final step whereby involving the ...
2
votes
2answers
2k views

Integration of square wave

trying to program an integrator. My input is a square wave and my expected output should be a triangle wave. However, whenever I pass it through my low pass filter algorithm (just a 2nd order ...
2
votes
3answers
607 views

integration property of fourier series

Please help me sort this issue out. The integration property in Fourier series is as follows: So, for proving the above property, i took this approach: This is where my doubt is. Some books and ...
2
votes
1answer
178 views

Orthogonal signal generator using integer arithmetic

I have a problem with implementing an orthogonal signal generator (OSG) algorithm on a microcontroller using integer arithmetic. I use this algorithm for a single-phase phase-locked loop (PLL) ...
2
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0answers
249 views

Designing a practical integrator

One way to describe a practical integrator ("leaky integrator") is $$ H(s) = \frac{g R}{1 + sRC} $$ I am trying to understand how to choose the values $g$, $R$ and $C$ because in practice, I will ...
1
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2answers
1k views

z-Transform Methods: Definition vs. Integration Rule

The definition of the z-transform is defined as $z = e^{sT}$ where "s" is complex frequency for continuous-time systems and "T" is the sample period. Why are rules such as the forward rectangular ...
1
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1answer
112 views

How does integration of a signal change the spectrum?

It's a very basic question, but I havn't found the question on google or dsp.stackexchange.com: How does integraion of a signal change the spectrum?
1
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1answer
111 views

Convolution Integral of Harmonic Signal (Cosine) with the Sinc Function

I was asked to show that this convolution integral results in the answers also given in the image. Not quite sure how to approach this integral, everything seems to be coupled together. Does anyone ...
1
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1answer
225 views

How to convolve an arbitrary signal with a causal decaying exponential?

I need to simplify the following convolution $$x(t)\star [e^{-2t} u(t)]$$ where $u(t)$ is the unit step function. I'm very confused with this. Using the definition of convolution of continuous-time ...
1
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1answer
171 views

Integral of an upsampled signal, without actually resampling it

Given a signal X which is sampled at a certain frequency. The value we currently compute is given as the integral of the upsampled signal. Thus: Y = X but 100 times upsampled, by means of sinc ...
1
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2answers
97 views

Help with finishing this integral, to obtain the power spectral density of a pure cosine wave

I am trying to evaluate the power spectral density $S_{xx}(f)$ of a cosine signal $x(t) = A\cos(2\pi f_0t)$, by starting from its definition for deterministic power signals $$S_{xx}(f) = \lim_{T\...
1
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2answers
1k views

Why is the Fourier transform valid only for absolutely integrable signals?

Why is the Fourier transform valid only for absolutely integrable signals? For example, why can't we do the Fourier transform of exponential order functions?
1
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1answer
62 views

How to model a generic low frequency signal?

I'm trying to apply Fourier analysis to a specific problem I have. I have essentially an integral like the following $$ \int_{\Omega} f(t) g(t) dt $$ And I'm trying to assume that $g$ is a narrow ...
1
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1answer
741 views

Differential Equation and Transfer Function of Multiple Summer Integrator Block Diagram

I've been working on this more than 5 hour, I know how to solve feedback, summer, but was not able to solve this one, I've tried to simplify it by choose A(i) to be the first output of first summer ...
1
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2answers
193 views

What does the $n$ do in $1/(j \omega)^n$

I'm currently working through a research paper on beam forming. In this paper a magnitude compensation is introduced to compensate for frequency dependency. Due to other calculations low frequencies ...
1
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1answer
142 views

Graph of $[u(\tau)-u(\tau-4)]\cdot[u(t-\tau)-u(t-\tau-4)]$

I'm trying to visualize what would the following signal be like $$[u(\tau)-u(\tau-4)]\cdot[u(t-\tau)-u(t-\tau-4)]$$ Consider that $\tau$ is the independent variable here, $t$ is the shift variable. ...
1
vote
1answer
127 views

Evaluating the inverse $\mathcal Z$-transform on the unit circle

I am trying to understand the math. The inverse $\mathcal Z$-transform is given by: $$x[n] = \displaystyle\frac{1}{j2\pi} \int_cX(z)z^{n-1}dz$$ where $\displaystyle \int_c$ is a contour integral. ...
1
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1answer
2k views

Inverse z transform - contour integration

Here is my task: Find inverse z transform of $X(z)=\frac{1}{2-3z}$, if $|z|>\frac{2}{3}$ I need to find it using definition formula, $x(n)=\frac{1}{2\pi j}\oint_{C}^{ } X(z)z^{n-1}dz$. How can I ...
1
vote
1answer
47 views

why does my integrator perform noise shaping?

I am designing (in matlab) a simple system with a DC signal at 0 Hz and some added noise. I am performing a differentiation of the signal (by simply subtracting the last values from each other), next ...
1
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1answer
59 views

CIC SINC3 Filter with unsigned Integrators

I want to use the AD7401 Sigma delta modulator connected to my FPGA. The Datasheet pg.15 has some demo verilog code which I understand alright except one thing: They use unsigned integers for all of ...
1
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1answer
81 views

y(t) a of an integrator circuit

I have a signal $$ x(t)= \frac{1}{T} e^{-\frac{t}{T}} u(t) - \frac{1}{T} e^{\frac{t}{T}} u(-t) $$ and I know that it transits in a integrator circuit and I have to find y(t) in time and frequency ...
1
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1answer
175 views

Integral over power spectral density

The wikipedia entry on PSD has one confusing line: Summation or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process) But ...
1
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1answer
92 views

How to differentiate a time domain signal in the complex transfer function?

I have an input-output data set where the input is current and the output velocity. I am interested in the transfer function from current to acceleration though. So suppose: $H(s) = \frac{I(s)}{V(s)}$ ...
1
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2answers
2k views

How to calculate the displacement from the measured acceleration data?

I have an acceleration data which is collected from a vehicle. I do also have a displacement measurement. I would like to calcuate the displacement vector from the measured acceleration data, but no ...
1
vote
1answer
846 views

Eigenfilters in Matalb [closed]

I have to implement an eigenfilter for an arbitrary frequency response in MATLAB. I have this algorithm: $N$ - order of the filter $M = N/2$ $c(\omega) ...
1
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0answers
55 views

Rocketry : Combine two accelerometers to reduce noise?

I am designing an IMU for an experimental rocket. I'll be using the BN055 9DOF that has sensor fusion - orientation quaternion - and gravity compensation. My main goal is speed computation, and I was ...
1
vote
0answers
27 views

Estimating a discrete summer with constrained input bandwidth

I have a discrete-time system which can be described as: $$ Y_m = \sum_{r=-N_g}^{R-1+N_g} c_r x[R(m-1) + r] $$ The unknowns are $c_k$ but I know that they have the following approximate behavior: $$...
1
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0answers
244 views

How use FIR filter for simulating an integral implemented using the trapezoidal integration?

Suppose I have this equation $$ \phi = \frac{60}{T^5} \int_0^T \left( T^2 - 6T \tau + 6\tau^2\right) y(t-\tau) d\tau - \frac{30 \alpha}{T^5} \int(T-\tau)^2\tau^2 u(t-\tau) d\tau $$ and I want to use ...
1
vote
1answer
139 views

Verlet integration first step

I'm trying to implement color-to-grayscale method from this paper. And they use Verlet's integration as: $$L^*(t+\Delta t)=\frac{F(t)}{m}\Delta t^2+2L^*(t)-L^*(t-\Delta t),$$ for computing new ...
1
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1answer
87 views

Integrating over an image?

I am currently trying to implement the method described in this paper. In short we have a system of the form $a=B\times c$. Where $$a_i = \int d^3r \space w(r)f_i(r)t(r)\quad\text{and}\quad B_{ij} = ...
1
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1answer
799 views

The essential bandwidth of a rectangular pulse

The problem is to estimate the essential bandwidth of a rectangular pulse \begin{equation} g(t) = \Pi(t/T), \end{equation} Where the essential bandwidth must contain at least $90\%$ of the pulse ...
1
vote
1answer
504 views

Filter wrap-around to avoid dealing with settling time? Using a Butterworth

I've been using a Butterworth high pass filter to correct the low frequency distortion of an acceleration waveform integrated to a velocity waveform. I throw out a small number of the first data ...
1
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0answers
313 views

Why does my overlap-add integrator in frequency domain result in distorted signals?

With help from the dsp guide and DSP-Related, I tried to implement a leaky integrator in octave/matlab. It seems to work in general, but there are a few problems. So that's what I do: Calculate the ...
1
vote
0answers
265 views

One of the best ways to numerically integrate a signal?

I need to get position $x$ (integration) from velocity $v$. One could use 1st order Euler integration as $x_{t+1} = x_t + \Delta v_t$ However, doing so leads to errors proportional to sampling time ...
0
votes
2answers
848 views

BIBO stability of $y(t)=\int_{-\infty}^{t}{x(\tau)d\tau}$

How can I prove that the LTI system with (output $y(t)$, input $x(t)$) $$y(t)=\int_{-\infty}^{t}{x(\tau)d\tau}$$ is BIBO (bounded-input/bounded output) stable?
0
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2answers
154 views

Reconstructing band-limited signal's integral from discrete samples [closed]

I'm thinking about building an e-bike consumed power meter. I'm planning on sampling a low-pass filtered current sensing and voltage analog signals on the battery to e-bike controller connection, ...
0
votes
1answer
988 views

Difference between these two integrators

I am an electrical engineering student but new to digital signal processing. However while searching references for my senior project I came across following two different discrete integrator blocks ...
0
votes
1answer
330 views

Absolute and Square Summability & Integration of sinc function

Is the sinc function both absolutely summable (L1 norm for Continuous time signals and l1 norm for Discrete time signals) and square summable (L2 norm for Continuous time signals and l2 norm for ...
0
votes
1answer
950 views

How to model state space for complex valued system correctly in SIMULINK (MATLAB)?

When trying to use the default state-space model block, if there is a complex number valued in the matrices, there will be an error To resolve that, firstly I need to look at pseudo reference model ...
0
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1answer
41 views

The computation procedure problem of calculating the BER

Here is the processing of calculating the BER average \begin{align} P_e &=\int^\infty_0Q(\sqrt{2uSNR})e^{-u}du \\ & =\frac{1}{\sqrt{2\pi}}\int^{\infty}_0\int^\infty_{\sqrt{2uSNR}}e^{-\frac{t^...
0
votes
1answer
66 views

Finding the output of a system where the input is a sum of complex exponentials

So, I have to find $H\{ x(t)\})$ (which is an LTI system), where $$x(t) = \sum_{k=0}^{\infty} a_ke^{ \ jw_kt}$$ and where the impulse response of the system is given by: $$h(t) = \frac{\delta(t+\tau)-\...
0
votes
1answer
2k views

Cancel Drift after numerical integration

I am trying to integrate angular acceleration obtained from a set of accelerometers positioned specifically at opposite corners of a cube, based on the paper EcoIMU: A Dual Triaxial-Accelerometer ...