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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questions
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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
23 views
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
104 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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1answer
78 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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2answers
47 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
106 views
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
53 views
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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1answer
35 views
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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2answers
52 views
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
106 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 $$p_1(t) = \sum_{k = -\...
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1answer
41 views
How to describe voltage fluctuations
I am looking for a way to describe the properties of voltage fluctuations - is the RMS amplitude a suitable measure? Attached is a screenshot showing the random current fluctuations (in the case of ...
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1answer
36 views
Sampling interval $T$ as a multiplier in digital processing of a continuous-time signals
[from: Discrete-Time Signal Processing, Oppenheim and Schafer, p.224]
Q: Why do we have $T$ as multiplier in $TY_a(j\Omega)$ in Eq.155?
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1answer
74 views
Why not use the same “standard” exponentials for both continuous and discrete time
In continuous time the standard exponential signal is usually defined as
$$
e^{st}, \quad\text{with}\quad s = \sigma+j \omega
$$
In discrete time the standard exponential signal is usually defined as
...
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0answers
16 views
Quantify the amplitude of fluctuations
I consider the potential of a nerve cell, which is -65mV. If I introduce a random noise source, for example, this leads to fluctuations (see attached picture).
I would like to know what is a good ...
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1answer
41 views
If the impulse response of a causal LTIC system is $h(t) = \delta(t) + \sin(t)u(t)$, is it marginally stable or unstable (BIBO)?
If you take the $H(s) = \mathcal{L}[h(t)]$ the poles are on the imaginary axis so the system should be marginally stable, but is it?
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2answers
119 views
Determining Causality and Time-Invariance of a system
Consider the following system:
$$y(t-1)=\int_{-\infty}^\infty x(š)u(š-t) dš $$
where $u(t)$ is the unit step function, which is zero for $t<0$ and equals $1$ for $t>0$.
$(1)$ Is the system ...
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0answers
27 views
Finding arbitrary/random repetitive patterns in signals (both self and across two signals)
I am trying to figure out a direction for my research. I need to find random repetitive patterns between two signals and on each individual signal. I have read about FFT and time series-motif ...
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0answers
137 views
Wideband FM bandwidth estimation
In the book Modern Digital and Analog Communication Systems by B. P Lathi, there is a section for the estimation of WBFM bandwidth. First, the staircase approximation of $m(t)$ is constructed by the ...
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2answers
100 views
Power of an angle-modulated wave
Let $$\phi(t) = A\cos(\omega_c t + km(t)) \tag{1}$$ be an angle-modulated wave(FM or PM). What's the power of $\phi(t)$? Intuitively, it seems that the answer is $$P = \frac{A^2}{2} \tag{2}$$ since ...
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1answer
39 views
Quantization and Sampling - putting it all together
So after I learned this two topic: quantization and sampling, I'm learning the way to look at both of them and try to optimize the split of a given amount of bit B to N and k, where N is the amount of ...
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40 views
What types of distributions (except the Gaussian) that follow for random jitter in a correlated signal?
For the simple case, let's consider the correlated signal $s$ with jitter (without noise) as follows:
\begin{align}
s(t, \theta_i) = \cos(2\pi f (t + \epsilon_t) + \phi + \theta_i), ~~ i=0,1, ...
\end{...
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1answer
61 views
Relation between power of the baseband signal and the low pass equivalent
Let $x(t)$ be a real bandpass signal and $x_l(t) = (x(t) + j\hat{x}(t))e^{-j2\pi f_0t}$ be the lowpass equivalent. Is there any relation between $P_x$ and $P_{x_l}$ where $P_x$ refers to the average ...
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1answer
38 views
What does the variable $\tau$ mean in frequency modulation?
Frequency modulation formula:
$$
x_{FM}(t) = A_{c}\cos\left(2\pi f_{c} t+ 2\pi k_{f}\displaystyle\int_0^{t}m(\tau)d\tau+\phi_{0}\right)
$$
Its change from $m(t)$ to $m(\tau)$.
What is $\tau$? I wonder ...
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1answer
48 views
Finding impulse response $h(t)$ by impulse matching; problem understanding Example 2.3 in “Linear systems and signals”, Lathi 3rd edition
Before the example it has been stated that in a system described by
$$
Q(D)y(t) = P(D)x(t), \quad (1.) \iff \\
(D^N + a_1D^{N-1} + ... + a_{N-1}D + a_N)y(t) = (b_{N-M}D^M + b_{N-M+1}D^{M-1} + ...+b_{N-...
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2answers
91 views
Magnitude of the analytic signal
It's well known that magnitude of the analytic signal for narrowband signals gives the envelope. For example we can demodulate AM signal by abs(hilbert(s_AM)) in ...
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2answers
74 views
Why this system is linear?
Hi guys i'm studying signals and systems, and my professor told us that
$$y(t) = \int\limits_{ t+T }^{t-T/2} {x(a+T/2)}\mathrm{d} a$$
is a linear system.
But a primitive of $x$ isn't $ x^2$ ? How it'...
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2answers
87 views
Periodic signals in Continuous and discrete time
Is there any signal which is periodic in Continuous Time but not in Discrete Time?
I have this doubt prevailing in me for a long time.
Are all CT periodic signals periodic in DT?
If so, how is sin ...
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1answer
63 views
What is the average of $\DeclareMathOperator{\rect}{rect} \rect(\cos(\pi t/2)) $?
We have this signal:
$$\operatorname{rect}\left(\cos\left(\frac{\pi t} {2}\right)\right) $$
I must find the average power , how can i get there ?
My solution:
I have seen that
$$-\frac 12 < \...
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4answers
167 views
the $L^2$-norm of a signal is also applied as its energy!
I am a newcomer in signal processing. I saw that the $L^2$-norm of a signal is also applied as its energy! How is this concept illustrated for those ones who are working in pure math.
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2answers
225 views
A path to get the foundation of SP and DSP for a pure math
I have a Ph.D. in pure math (interested in Harmonic analysis and operator theory). I am looking forward some proper references to lead me get the foundation of discrete/signal processing more and ...
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1answer
73 views
two dimensional integration of a trigonometric function
I am working on a detection problem which finally, I have to solve the following 2-D integral:
$$\int\limits_{a}^{b} \int\limits_{c}^{d}e^{A\sin(x)\cos(y-B)}\, \mathrm{d}x \, \mathrm{d}y \ ,$$
where $...
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1answer
26 views
Impulse invariance vs. DT representation of a CT system: Where is the inconsistency?
Suppose you have a continuous-time (CT) system $h_c(t)$, bandlimited to $B$. Your goal is to represent the system as a discrete-time (DT) system $h[n]$, sampled at $f_s \leq 2 B$. Clearly $h[n]$ won't ...
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1answer
26 views
Does fading memory mean impulse response with finite support?
Given a continuous-time impulse response $h(t)$, bandlimited to $B$.
The discrete-time $h[n]=h(n/(2B))$ is then a unique and perfect representation of $h(t)$ and a discrete-time system $h[n]$ is then ...
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1answer
45 views
Sinusoidal function
I have a problem. I am confused, I have two function like this:
$\sin(t)$
$\sin(2\pi t)$
what is the difference between of them and how to calculate period and frequency for second?
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0answers
26 views
Generating a sawtooth function with a variable frequency
So I'm trying to write a script that produces a sawtooth function using the equation:
s = 2 a / Ļ atan ( cot ( Ļ f t ) )
Where:
a : amplitude
f : frequency
t : time
It's all working fine when f is ...
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1answer
31 views
elementary one and multidimensional real examples of continuous/discrete LTI systems
By an LTI system, we mean a time-invariant linear map on continuous/discrete-time signals.
What (elementary one and multidimensional) real examples of continuous/discrete LTI systems do you suggest to ...
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1answer
102 views
An invertible system with memory
Suppose $\mathcal{L}$ be invertible system with memory. Does $\mathcal{L}^{-1}$ have memory necessarily?
Intuitively I think the answer is "yes". There are many examples showing that. For ...
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0answers
38 views
Find CT Fourier transform of $ \left[ \frac{ \sin(\pi ~ t) }{\pi ~t} \right] \left[ \frac{ \sin(2\pi ~ (t-1)) }{\pi ~(t-1)} \right] $using properties
Use properties of Fourier Transform to solve the question.
The question is in the imgur link below.
I got $f_t$ of $\frac {sin(pi \cdot t)} {pi \cdot t}$ as rectangular pulse with value $1$ from -pi ...
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1answer
87 views
Multiplying signals in discrete-time vs continuous-time
Given two discrete-time signals $a[n]$, $b[n]$ and its product $c[n]=a[n] b[n]$. The ideally interpolated, continuous-time version of $c[n]$ is
\begin{align}
c_1(t)&=\sum_{n=-\infty}^{\infty} a[n] ...
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1answer
42 views
Bandwidth of a bandpass signal
If the Fourier transform of an aperiodic continuous time signal has signal components between the minimum frequency w1 and the maximum frequency w2, but not all the frequencies between w1 and w2, is ...
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0answers
57 views
What does voices per octave means?
I was studying the description of continuous wavelet transform in Matlab and I came across this term
'**cwt uses 10 voices per octave**'
...
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0answers
22 views
Energy or power signal for continuous signal
I have solved $w(t) = 1dt$ and determined it as power but when $w(t) =\Pi\left(\frac{t}{T_0}\right)$ it is energy.
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37 views
Signal processing on hourly atmospheric temperature, wind speed and humidity data for solar panel maintenance
I am working on a machine learning problem, where i have to predict if a device connected with solar panel at a specific block has been damaged. I have many factors to consider, out of which ...
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0answers
35 views
Problem with implementing DC offset for streaming data
I am new to the DSP world. I'm trying to run a raw input signal through a LPF, remove any DC offset, amplify the signal, and then decode the signal. The problem with my code seems to be the ...
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0answers
21 views
Low pass filter output
What will be the output when a square wave of time period 1sec is applied to the input of LPF of cutoff frequency 1Hz. Please help me, I am not getting the exact expression. What if cutoff of LPF is ...
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2answers
47 views
Conceptualising the continuous time unit impulse function as derivative of unit step
This a very newbie question.
I just watched Lecture 3 of Oppenheim's Signals course and he defines here the continuous time function as the derivative of the unit step function like so:
$$ \delta_\...
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39 views
Finding the Energy & Power of a Composition of Odd and Even Signal
Given two CT signals $x_1(t)$ (even signal) and $x_2(t)$ (odd signal). If $x_1(t)+x_2(t)$ is an even signal then what is the energy and power of $x_2(t)$.
My Attempt
$\int \limits _{-\infty}^{+\infty}$...
1
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1answer
42 views
Time-shifting operation post the time-reversal operation when performing convolution
I'm confused with the time-shifting operation post the time-reversal operation when performing convolution. Let's say we were to convolve $x(t)$ and $h(t)$, so I would have the term $x(k)$ and $h(t-k)$...
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1answer
81 views
Prove a property using shift theorem and duality
I'm reading Lectures on the Fourier Transform and Its Applications and I'm going to prove shift theorem for the inverse Fourier transform using duality. According to the mentioned source, the duality ...
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1answer
128 views
Inverse Hilbert Transform
The Hilbert Transform of a 1D/real-valued vector signal returns the analytic signal, x, from a real data sequence, xr. The analytic signal x = xr + jxi has a real part, xr, which is the original data, ...
