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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Impulse response of ideal filters

I am aware that an ideal low-pass filter in both continuous time and discrete time has a $\mathrm{sinc}$ impulse response. What would the impulse response of an ideal high-pass or band-pass filter ...
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2answers
547 views

BIBO Stability and the convergence of the frequency response of a system

It is my understanding that an LTI system is BIBO stable if and only if its impulse response $h(t)$ is absolutely integrable. This also happens to be one of the Dirichlet conditions for the ...
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581 views

Fourier transform of even/odd parts of a complex signal

Why does Oppenheim state the following properties: \begin{align} \mathcal F\big\{x_e (t) \big\} &= \Re\big\{ X(j\omega) \big\}\\ \mathcal F\big\{x_o (t) \big\} &= j \Im\big\{ X(j\omega) \big\...
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128 views

Uniqueness of Fourier Series Representation and the Fourier Transform of Periodic Signals

If we are given a signal of the form $$x(t) = \sum_{k = -\infty}^{+\infty} a_k e^{j k \omega_0 t},$$ can we call it a Fourier Series representation of $x(t)$ right away? Suppose we are given the ...
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3answers
614 views

Periodicity of the discrete-time Fourier Transform

The DTFT of a sequence $x[n]$ can be written as $$X(e^{j\omega}) = \sum_{n = -\infty}^{\infty} x[n] e^{-j\omega n}.$$ Is the smallest (fundamental) period in frequency of the DTFT always $2\pi$? Or ...
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1answer
51 views

Echo effect on sinusoidal input: Intuition

I am currently reading about adding echo effects on signals. I was wondering if my understanding is correct. In the attached figure, the blue line represents the original signal and the red line is ...
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3answers
143 views

Causality as applied to capacitors

This question stems from a point of confusion that I still have about the causality, linearity, and time-invariance in LCCDEs. I wanted to use the capacitor as an example. Consider a capacitor with ...
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2answers
169 views

Duality of the continuous-time Fourier transform - derivation and notation

Suppose we have the Fourier transform pair $x(t)$ and $X(\omega)$ such that $$X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} \mathrm{d}t$$ The duality property states that $X(t)$ and $2\pi ...
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1answer
58 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 ...
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57 views

Measured signal from tachometer has ripples, why and how to minimize?

I don't know if it would be straightforward enough to chalk it up to this being a real system. Thus there's friction and physical mechanics of gears and teeth pulling.
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1answer
506 views

Fourier coefficients of product of two periodic signals

question: If $x(t)$ and $y(t)$ are two periodic signals(both with period T) with Fourier coefficients $c_{n}$ and $d_{n}$ respectively then, Fourier coefficient of $z(t)=x(t)\cdot y(t)$ is: (a) $\...
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515 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\...
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4answers
202 views

Very basic question about how we define frequency in signal processing

When talking about general periodic continuous-time signals for which $$x(t + T_0) = x(t)$$ where $T_0$ is the fundamental period we define the fundamental frequency $\omega_0$ as $\omega_0 = 2\pi/T_0$...
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1answer
69 views

Generay way to find Stability of system

I am trying to find a general way to find system stability.I have applied these methods and struck in confusion For eg: If $$y(t)= \int_{-\infty}^{t} x(\tau) \sin(4\tau) d\tau$$ then find whether the ...
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2answers
437 views

Linear Constant Coefficient Differential Equations: Zero-Input and Zero-State responses

The solution to a linear constant coefficient differential equation of the form $$\sum_{k = 0}^{N} a_k y^{(k)} (t) = \sum_{k = 0}^{M} b_k x^{(k)} (t)$$ can be written as $y(t) = y_{ZI} (t) + y_{ZS} (t)...
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Finding the fundamental frequency of a periodic signal

Suppose we have the signal $$x(t) = e^{j\omega_1 t} + e^{j\omega_2 t} + e^{j\omega_3 t},$$ where all the frequencies are rationally related (that is, the ratio of any pair of frequencies is a rational ...
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56 views

Rationally related frequencies and the Fourier Series representation

Suppose that we have the signal $$x(t) = e^{j\omega t} + e^{j\frac{3}{2} \omega t},$$ and we want to find a Fourier Series representation for that signal. Is this possible? According to my ...
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1answer
3k views

Calculating output of a system given impulse response and input

I'm trying to calculate the output given an impulse response $h(t)=e^{-t}u(t)$ and input $x(t)=\cos(2\pi t)$. I know I need to use the convolution but given that cosine is periodic, I don't see how ...
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1answer
325 views

Power or Energy Signal of $x(t) = t \cdot u(t)$ [closed]

Is $x(t)=t \cdot u(t)$ a power signal or an energy signal? Please "show" me why - preferably using equations.
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46 views

What will be the code for this following signal in MATLAB?

I was trying to generate a signal like this: I have done this so far like this: ...
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1answer
137 views

Find Fourier Transform of signal that satisfies this equation

I have attempted it this way: $G(j\omega)$ is a rectangular pulse in frequency domain. Using results, inverse Fourier transform of $G(\omega)$ will be $$g(t)=\frac{\sin(2t)}{\pi t}$$ Now given that $...
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2answers
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how to evaluate derivative of convolution integral?

If the signal $x(t)$ has ordinary first derivative $\dot x(t)$, then $\dfrac{d}{dt}\big(x(t)\star y(t)\big)$ is: (a) $\dot x(t)y(t)$ (b) $x(t)\dot y(t)$ (c) $\dot x(t)\star y(t)$ (d) $\...
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475 views

Discrete vs Continuous Signal

Why do the same signals in the continuous-time and discrete-time differ in their frequency? For example, if $x(t)$ is a continuous-time signal taking values only at $t= 1,2,3,4$ and $x(n)$ takes the ...
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157 views

Does the Nyquist Sampling Theorem hold for a triangular wave?

Does the Nyquist Sampling theorem hold for a triangular wave produced by a function generator? For example, a triangular wave with a frequency of 15 Hz sampled at 60 Hz, 180 Hz and 15000 Hz.
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130 views

How to determine the poles from a graph

From my knowledge of stability, I understand that if the function approaches a finite number then the system will be stable. Thus if a system is stable its poles will be on the left of the $j\omega$-...
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1answer
36 views

Convolution including $\delta(t-5)$

I know the two properties of convolution that are related to my question $\quad x(t)*\delta(t)=x(t) $ $\quad x(t)*\delta(t-t_0)=x(t-t_0)$ But my question is, how can I use those two to calculate $$ ...
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353 views

why total energy of a finite duration continuous signal becomes infinite after sampling

If I have a continuous-time energy signal and multiply it with an impulse train, the frequency domain representation has infinite replicas. The total energy if we sum them up becomes infinite. What ...
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2answers
706 views

Minimum possible sampling frequency for continuous time signal

A continuous-time signal $x(t)$ has the magnitude spectrum $\lvert X(F)\rvert$ shown in Figure 6. The signal is sampled to obtain $x(n)$ using the sampling frequency $F_s$. Determine the minimum ...
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1answer
899 views

What is the product of Dirac Delta function and $t$? [closed]

$t\cdot \delta (t)$ Does it equal to zero? If so, how can we prove it?
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2answers
160 views

Problem identifying the analytic expression of such determined signal

I came across this problem I am supposed to find the Fourier transform of $g(t)$, but I am not able to find the analytical expression of such signal. The teacher suggests that I should consider ...
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1answer
191 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 ...
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1answer
231 views

Reconstruction of a signal using an ideal DAC

I think I was able to do part (a) correctly, but now am stuck on part (b). I don't know how to approach this part or what is the theory behind it. My MATLAB code is shown below. Please let me know if ...
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1answer
252 views

Impulse response rest conditions of LTI systems

Why the impulse response (initial) rest conditions are taken like $h(t)=0$ and $h'(t) =1$, if the order of differential equation is $2$.
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1answer
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How to generate colored Gaussian noise and adding it to a ODE system - Do I need Euler-Maruyama method?

In the tutorial, when white noise process is added to ordinary differential equations (ODE), the ODE becomes a stochastic process. Then the stochastic process needs to be solved using Euler Maruyama ...
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3answers
90 views

Derivative of delta function

I've come across the below. Can you please help me understand the proof for this? $$\int_{-\infty}^{\infty}x(t)\delta'(t-2.5)dt=-\frac{dx(t)}{dt}{\Big |}_{t=2.5}$$
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1answer
529 views

How to Generate White Gaussian Noise with Known PSD in MATLAB

I want to generate white Gaussian noise with known PSD. Using wgn I can generate noise with specific power (dBW) but I want to generate a white noise with constant or know PSD. Can anyone give me ...
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1answer
550 views

Question on convolution

I am stuck at question number 2.8 This is how I have gone about solving it: I have calculated $y(t)$ by convolving $x(t)$ with $h(t)$ using the fact that $x(t)$ convolved with an impulse at $t=t_0$ ...
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1answer
119 views

Time invariance of a System

I have this small question about the time invariance of a system. Which is: If the current output is multiplied by the current input (see both are variables) will the system be time variant or time ...
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2answers
83 views

whether the system is linear or not for the given problem

Given the system: $$y(t)=x(t+1)+x(t−1)$$ is the system linear? For a system to be a linear first it should satisfy zero input and zero output. How can we calculate output at 0 input if the system ...
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1answer
57 views

Smoothing a Signal which contain White Noise

I'm trying to smooth a signal in order to reduce white noise as a result of changes in amplitude across sequential frames of an audio signal. I've been investigating using a moving average or ...
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2answers
1k views

Different definitions of unit step signal

While I was learning signal theory, I have come across different definitions for the unit step signal. For example. $u(t)=\begin{cases} 1 & t\geq0\\ 0 & t<0\\ \end{cases}$ $u(t)=\begin{...
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1answer
140 views

Overshoot computations not matching

Take the following transfer function of a 3rd order system: $$H(s)=\dfrac{2.302~s+0.3548}{s^3+0.739~s^2+3.223~s+0.3548}$$ with poles: ...
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1answer
274 views

Frame overlapping

I 'm reading the following article from NVIDIA which involves some audio processing a the beginning : link just here (part 3.2) I don't understand the following part : In practice, we use 520ms ...
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1answer
68 views

Trouble understanding an article about MFCC

I'm willing to apply the same audio preparation that this article page 3. However, I don't understand the final part : ...
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1answer
566 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 ...
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0answers
217 views

Software based synchronizing of multiple data streams/sensors with slightly differing sampling rates

Situation: I try to synchronize the data streams of multiple sensors (ADXL372) of the same type but with different data output rates. The data sheet states all sensors should have the same sampling ...
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1answer
568 views

Measure the amount of drift in video and audio frames

I setup a stereo system using 2 GoPro Hero Session Cameras, since I need a larger baseline than the one of the Dual Hero System. This unfortunately means the two cameras are not ...
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1answer
92 views

What is special about the frequency $\omega_0=\pi$ that suddenly causes rate of oscillation decrease?

In Alan Oppenheim's book Signals and Systems a comparison is made between the properties of discrete-time and continuous-time complex exponential signals in section 1.3 pg. 26. Specifically it says: ...
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1answer
166 views

Confusion about subtle difference between discrete-time and continuous-time

In Alan Oppenheim's book Signals and Systems a comparison is made between the properties of discrete-time and continuous-time complex exponential signals in section 1.3 pg. 26. Specifically it says: ...
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1answer
52 views

Is my answer incomplete? Checking the stability of a system

Yesterday, during my exam, I had the following exercise: Given $$H(s) = \frac{1}{s^2+2s+4}$$ check if it's stable. which was supposed to be the hardest (since it was the last one). From my ...