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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Mixed - Discrete and Continuous system Laplace domain stability - Effect of Sampler and DAC

I have a system whose the plant transfer function is continuous and the compensation is discrete. I have an ADC which allows to measure the output of the system and a DAC which allows to control the ...
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Meaning of Rect and Train of Rect Spectra

The Fourier transform of $x(t)=\operatorname{rect}(t)$ is $X(f)=\operatorname{sinc}(f)$ The Fourier transform of a periodic train of rectangular pulses $x(t)=\sum\limits_{n=-\infty}^{\infty}\...
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A system having impulse response $ h(t)=u(t) $ stable or not?

I know that for a system to be BIBO stable its impulse response must be absolutely integrable and the impulse response $ h(t)= u(t)$ integrates to approach infinity (i guess) I proceeded as$$ \int_{-\...
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Fourier Transform of $u(t)$ [duplicate]

I am just unable to find the correct Fourier transform of these signals (unit step, sine and cosine functions) which are containing delta functions in their Fourier transform. For unit step function, ...
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Is it possible that the value of a continuous mother wavelet at origin is zero, i.e. $\psi(t=0)=0$?

According to Fourier transform, a continuous wavelet could be written as $$ \psi(t)=\frac{1}{2\pi}\int\hat\psi(k)\text{e}^{-ikt}\text{d}k $$ From the equation above, we know that $\psi(t=0)$ is $$ \...
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Meaning of frequency and bandwidth of a signal, despite the fact that we do not know the signal

First of all, I am completely new to the domain of signal processing. As far as I know, a signal can be represented with an infinite integral of infinitesimal complex exponentials, which is known as a ...
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What's the point of defining the signal over the whole time domain?

This question is classic for anyone starting with some signal processing course, suppose $y(t)=x(t/2)$ then the system is noncausal because we have that the output at t=-6 depends on the input at t=-3 ...
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Finding the wiggles pattern in the original dataset. (Wiggles appear after performing division by another dataset)

I have multiple measurements regarding scientific observations. The problem is that there is a subtle noise pattern caused by the instrument - the wiggles. These wiggles are invisible when looking at ...
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Finding the integral of a signal

I'm trying to find the integral of the following signal: $x(t)=A, 0 \le t \le T$ $x(t)=0, otherwise$ The integral is defined as $y(t)=\int_0^t{x(\tau)d\tau}$ For $y(t)$, I'm getting $AT$ when $t \gt T$...
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Linear system stability criteria

Suppose we have a closed-loop system $H(s)=\frac{A(s)}{1+A(s)f(s)}=\frac{A(s)}{1+T(s)}$. I've seen the stability of the system stated a couple ways: If $H(s)$ has any poles in the RHP, then it is ...
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Periodicity of complex exponential in continuous and discrete time (Eq 1.51, Signals and Systems by Oppenheim & Wilsky)

Hi All: This is very basic but I've always wondered about it and now I see it in print in a textbook so I may as well ask. In Signals and Systems on page 26, it says $$e^{j(\omega_0 + 2\pi)n} = e^{j2\...
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Best temperature compensation equation?

I'm looking for the correct temperature compensation equation to use on our project. We are measuring the output of a detector who's signal is very sensitive to temperature drift. Any external ...
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FFT of square wave - what does output represent?

I am really new to FFT and signal processing. I am doing an analysis of square waves with FFT and I am trying to understand why the FFT output on the frequency domain has a downward slope for square ...
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Why we only can transmit a real signal? [duplicate]

i just wondering, why we always transmit a real signal but when we deal with a baseband signal we use a complex signal ? are this is related to up and down conversion of the signal (Because complex ...
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Can complex envelope be writen in the form of quadrature components when it has symmetric spectrum?

I am reading a chapter on VCO noise in "Design of CMOS phase-locked loops from circuit level to architecture level by Behzad Razavi";I am confused when the upconverted noise is writen as $...
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Does $\cos(bt)\cdot u(t)$ have a Fourier Transform?

If it does, $$\int_{-\infty}^{\infty} \cos(bt)\,u(t)e^{-j\omega t} dt = \int_{0}^{\infty} \cos(bt)\,e^{-j\omega t} dt = \int_{0}^{\infty} \frac{e^{jbt} + e^{-jbt}}{2}\,e^{-j\omega t} dt$$ Then how do ...
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Confused about the Fourier Transform of $e^{at}u(t)$

This is the problem at hand: I'm unaware of why we didn't have to say anything about $\omega$ like that it should be also greater than $0$, I know it's variable...but it's multiplied by $t$ ...
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MATLAB Plot of FT(Cos) Displays Weird Impulse

It is known that: $$ \mathcal{F}\{\cos(2\pi t)\}=\frac{\delta(f-1)+\delta(f+1)}{2} $$ However, on MATLAB, I used F=fftshift(fft(x))/N; to obtain the FT of $\cos(2\...
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Under what conditions is the convolution of an input signal with the system's impulse response periodic?

I'm currently solving the following convolution problem from Oppenheimer's book: In the solution, it was stated that "$x(t)$ periodic implies $y(t)$ is periodic" So I wondered if it's ...
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$\int_{-\infty}^{+\infty} |G(f)| \,e^{j2\pi ft}df=|g(t)|$?

Given the absolute value of the Fourier transform of a signal $g(t)$: $|G(f)|$ If I compute the inverse Fourier transform of $|G(f)|$, $$\int_{-\infty}^{+\infty} |G(f)|\, e^{j2\pi ft}df$$ do I obtain ...
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Application of the spherical Radon transform property in tomography

Let function $f$ be even and continuous on the unit sphere $S^n$. Let $R$-be a spherical Radon transform. There is a known property: $R(f^n)=f$ whenever $f$=constant. What would this property mean in ...
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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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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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1answer
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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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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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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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209 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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