Questions tagged [continuous-signals]

A continuous signal or a continuous-time signal is a varying quantity (a signal) whose domain, which is often time, is a continuum.

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The plot of instantaneous power of the Dirac function

I am very confused. I have tried researching this question for the last two weeks and I cannot get a conclusive answer. I was wondering how would I go about plotting the instantaneous power in the ...
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How do I check for linearity for the following piecewise-defined system?

The problem at hand: Where I'm currently stuck: I'm not entirely sure about how to move on from this point, I'm trying to find the superposition of the responses of the two individual signals so I ...
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Is there processing gain for FMCW using heterodyne-style receiver as opposed to matched filter?

Beat signal of a single target will be a sinusoid in the idealized world, so theoretically the signal processing gain of an FMCW pulse correlated with Tx waveform in this way should be analogous to ...
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Chebyshev Filter Transfer Function

I'm trying to derive the transfer function for Chebyshev filter. $$|H(\Omega)|^2=\frac{1}{\sqrt{(1+\epsilon^2T_n^2(\frac{\Omega}{\Omega_c})}}$$ where $$T_n(x)=\cos(N\cos^{-1}(x)) \forall x \le 1$$ $$...
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Following a smoothing filter of a biological signal

)Following a smoothing filter of a biological signal. y[n] = Ay[n − 1] + Bx[n − 2] Please find the transfer function in the z-domain and find the impulse response function of this filter.
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Why is the range of frequency for discrete time Fourier transform $-\pi<\omega<\pi$? [duplicate]

In my class we are taught that the range for the frequency is $-\pi<\omega<\pi$ for discrete time Fourier transform, however for continuous time the limit is $-\infty<\omega<\infty$ why is ...
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question with chirped signal

I have some difficulty understanding the following question. I have written a code to plot continuous and discrete version of the chirped signal. ...
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1answer
68 views

Update: Fourier Transform of a shifted and scaled $\operatorname{sinc}$ signal

Let $x_N$ be the function given by $$x_N(t)=A\frac{\sin(M\pi(t-N))}{\pi(t-N)}$$ The Fourier Transform of $x_N$ is $$\begin{align} X_N(j\omega)&=\mathscr{F}\{x_N\}(j\omega)\\\\ &=\int_{-\infty}^...
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3D (time, scale, amplitude) plot in Continuous Wavelet Transform

I will be extremely grateful if someone could please answer this basic question. How can one plot a 3D (translation, scale, amplitude) plot from the Continuous wavelet transform (CWT) coefficients? ...
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Random function covariance

I was studying signal processing and I was frequently asked to verify if a certain covariance is possible for a given random function. I tried to check by verifying the property to prove it: ${γ}_{xx}(...
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How to simulate a continuous signal passing through a capacitor (simple coupling capacitor)

the differential equation for the current flowing through a serial capacitor (see for example https://www.allaboutcircuits.com/textbook/direct-current/chpt-13/capacitors-and-calculus/) indicates that ...
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Which step response matches the system transfer function

A system has the following open loop bode plot: - Which one of the plots below describe the closed loop step response for the entire system? My attempt My initial thought was to look at the static ...
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Finding Interval of Integration

If we let : $$ x(t)=\begin{cases} 1&\text{if $0<t<1$}\\ 0&\text{if otherwise} \end{cases} $$ and $$ h(t)=x(t/a)=\begin{cases} 1&\text{if $0<t<a$}\\ 0&\text{if otherwise}\...
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A. V. Oppenheim Problem $(1.14)$

Problem 1.14) Consider a periodic signal : $$ x(t):=\left\{\begin{array}{ll} 1, & 0 \leq t \leq 1 \\ -2, & 1<t<2 \end{array}\right. $$ with period $T=2$. The derivative of this signal is ...
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Removing white noise by taking the mean of many samples

I'm learning about signal processing, and I am attempting to remove white noise from an output of a blackbox system. As you can see from the above image (left is the input sin(t), right is the output)...
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Time domain representation of a modulated signal when symbol rate >> carrier frequency?

I need to understand the relationship between carrier vs. bits with PSK modulation, and what the signal in the time domain would look like if my symbol rate is far greater than my carrier frequency. ...
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UPDATE : How to continue Computing the Convolution

$$ x(t):=\begin{cases} 1&\text{if $0<t<T$}\\ \\ 0&\text{if otherwise} \end{cases} \qquad\text{and}\qquad h(t):=\begin{cases} t&\text{if $0<t<2T$}\\ \\ 0&\text{if otherwise} ...
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1answer
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Initial rest condition applied on $x(t)$ vs $h(t)$

Define the LTI system $\mathcal{H} : x\mapsto y$ Define the convolution for continuous-time system : $$ (x*h)(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\;\text{d}\tau $$ The initial rest condition ...
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I am looking for an analytic description of a continuous-time Butteworth High-pass filter in the time domain (=impulse response)

having derived the Butterworth Lowpass Time domain response, I am now struggling to find a similar function for a Butterworth Highpass filter. I understand you need to replace s by 1/s. But this leads ...
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Linkwitz-Riley Crossover Sum as Allpass Filter

I was curious about crossover filter design, so I did some reading on Linkwitz-Riley filters. Seems to me that the general idea is that if you add HP and LP filters and they are properly designed, ...
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1answer
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Do we assume modulating and carrier signal uncorrelated in AM modulation?

I am new to communication and studying amplitude modulation. Let us assume an amplitude modulated wave given by $$\big(m(t)+ A\big) \cos(2\pi f_c t)$$ Now we have formula for efficiency as $$ \eta=\...
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Separation of many signals all containing 1 common component

I have been struggling with a research question and would appreciate any pointers (or full solutions). I have a data set of many (+/- 50k) observed - correlated - signals. These signals are known to ...
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Advice on generating stable FMCW waveform

I am trying to derive parameters for a triangular FMCW waveform such that the phase of the signal has consistency from one period to the next. Perhaps this is arbitrary and feel free to tell me so, ...
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From transfer function to differential equation

I have the below detailed solution (boxed in blue) that I don't understand completely: I can reconstitute the differential equation from: $$ (1+Ts) X(s) = K_v U(s) $$ $$ x(t) + T\dot x(t) = K_v u(t) $...
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Find the autocorrelation function of signal $x(t) = u(t) - u(t-1)$

I have used the energy-type signal autocorrelation function: $$\mathcal{R}_{xx}(\tau)=\int_{-\infty}^{\infty}x(t)x^*(t+\tau)dt$$ I have rewritten the equation as: $$\begin{align} \int_{-\infty}^{\...
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How do I find the Energy Density Function of $g(t)$ if i am not given an input or impulse response?

$$g(t)=\frac{12a}{t^2+a^2}$$ I need to find the Energy Density Function of the signal, but everywhere I look has an input and an impulse response. Does anyone know how to solve this. Would I just take ...
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205 views

Difficulty with a Fourier Transform

What would be the best way to take the Fourier transform of $$ f(t)\cdot \cos\big(\pi(t-1)\big) $$ I'm aware that when you take the Fourier Transform of $\cos(kt)$ you get two impulse at the location ...
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How to find inverse Fourier transform of summ of delta functions?

I am practicing for my exam that I have this semester and I stumbled upon this one. How can i find inverse Fourier transform given: $$ X(j\omega) = \sum_{k=-\infty}^{\infty}\delta(\omega-2k+1) $$
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Solution verification for this Fourier series problem

We have a signal with period $T = 2$ We want to find the continuous time fourier series for this signal. Since $T = 2$, $\omega = \pi$. All we have to do know is find the frequency domain. $$x(t) = \...
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Looking for a Mathematical Derivation for the Energy Formula in Continuous-time Domain

I have just started my signal and system course and I would like to know how we derive the corresponding formula for the energy of a continuous-time signal $x(t)$ over an interval $[t_{1},t_{2}]$ : $$...
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Convergence of a finite series on MATLAB

If I have output values of a signal $y$ (or series) stored in a $1\times N$ matrix, where $N$ is finite. Is there any MATLAB code or function I can use to determine if the signal converges or diverges?...
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1answer
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Represent DFT coefficients with respect to Continuous time-Fourier series coefficients

Does anyone know how to represent the Discrete Fourier transform (DFT) coefficient, $X[k]$, with respect to the Continuous time-Fourier series (CT-FT) coefficient, $X_k$? I come to the conclusion as $...
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1answer
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How can we prove the correctness of the integration property of the Laplace transform?

I was going through an Electrical Engineering textbook for understanding the Laplace transform and came across the following proof for one of the properties of the Unilateral Laplace transform. ...
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1answer
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Relation between continuous time transfer function and sampled approximation

Suppose I have some continuous time system and associated transfer function: $$ y(n)=x(n)+x(n-1)$$ $$ H(j \omega) = 1+e^{j \omega (-T_s)} $$ Now suppose I create a discrete-time approximation of this ...
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1answer
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MIT 6.003 HW#8 Problem 4 - Fourier Coefficients of Triangle Wave

In the mentioned homework, part of the solution involves finding the Fourier coefficients of the triangle wave. The solution mentions that we can express this function as follows: What does that ...
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What is impulse response in simple AWGN channel?

Assume a sender and receiver communicate through an AWGN channel. Let $x(t)$ be a transmitted signal and $y(t)$ be a received signal and $z(t)$ be the Gaussian noise. It is known that $y(t) = kx(t-\...
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1answer
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Transformation of random variables vs shift of functions

I am a beginner to random variables and I am understanding the concept of the transformations of a random variable. Consider a random variable $X$ to be Gaussian distributed with $a_x = 1.6$ and $\...
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2answers
167 views

Find power of a sum of sinusoids

We got this question to solve: Calculate the power of the signal: $$s(t) = 8\cos\left(20\pi t-\frac \pi4\right) + 4\sin(15\pi t)$$ Now, I thought of two approaches : Use Parseval theorem, so first ...
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195 views

Autocorrelation function of a triangular wave

I am interested to find the analytical expression for the autocorrelation function of a signal that comprises triangular pulses. I have followed the derivation in "Statistical Theory of ...
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62 views

Fractional Bandwidth of a Gaussian Amplitude Modulated Signal

a gaussian modulated sinusoidal signal may be expressed as $$x(t)=A\cdot e^{(j2\pi ft)}\cdot e^{\left[-\frac{1}{2\sigma^{2}}\cdot(t-t_{0})^{2}\right]}$$ Let's consider the case in which the gaussian ...
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1answer
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Ending points of the root locus

Let $$D(s) + KN(s) = 0 \tag{1}$$where $D(s)$ and $N(s)$ are polynomials of $s \in \mathbb{C}$ such that $\text{Deg}(D) = n, \ \text{Deg}(N) = m$ and $n\ge m$. The root locus method tells us how the ...
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1answer
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Inverse Fourier transform of $\frac{\frac{2}{1+2i\pi f}-\frac{2}{4+2i\pi f}}{\frac{1}{1+2i\pi f}+\frac{1}{3+2i\pi f}}$

I wanted to calculated the inverse fourier transform of the transfer function : \begin{align} H(f) &= \frac{\frac{2}{1+2i\pi f}-\frac{2}{4+2i\pi f}}{\frac{1}{1+2i\pi f}+\frac{1}{3+2i\pi f}}\\ &...
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What equation predicts the amplitudes of harmonics from a square/triangle/sawtooth/pulse oscillator?

I have seen pictures like this which depict the shapes of amplitudes from the various common types of audio oscillators: Similar pictures of spectra are shown here. I am attempting recreating these ...
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Converting Audacity Filter Curve EQ into transfer function and applying it to a signal via python

First of I am very new to Signal Processing and to python in general. I am trying to write a script where I would feed a voice recording into it, internally apply an eq and have the modified signal ...
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1answer
113 views

Recovering a signal after nonuniform sampling

Let $x(t)$ be a bandlimited signal such that $X(j\omega) =0 $ when $|\omega|>M$. Also $p(t) = p_1(t) - p_1(t-\Delta)$ is a nonuniformly spaced periodic pulse train where $$p_1(t) = \sum_{k = -\...
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1answer
45 views

How to describe voltage fluctuations

I am looking for a way to describe the properties of voltage fluctuations - is the RMS amplitude a suitable measure? Attached is a screenshot showing the random current fluctuations (in the case of ...
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Sampling interval $T$ as a multiplier in digital processing of a continuous-time signals

[from: Discrete-Time Signal Processing, Oppenheim and Schafer, p.224] Q: Why do we have $T$ as multiplier in $TY_a(j\Omega)$ in Eq.155?
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Why not use the same “standard” exponentials for both continuous and discrete time

In continuous time the standard exponential signal is usually defined as $$ e^{st}, \quad\text{with}\quad s = \sigma+j \omega $$ In discrete time the standard exponential signal is usually defined as ...
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Quantify the amplitude of fluctuations

I consider the potential of a nerve cell, which is -65mV. If I introduce a random noise source, for example, this leads to fluctuations (see attached picture). I would like to know what is a good ...
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1answer
51 views

If the impulse response of a causal LTIC system is $h(t) = \delta(t) + \sin(t)u(t)$, is it marginally stable or unstable (BIBO)?

If you take the $H(s) = \mathcal{L}[h(t)]$ the poles are on the imaginary axis so the system should be marginally stable, but is it?

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