29 votes
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What sampling frequency should I use if Nyquist is not available?

HINT When you sample at below the Nyquist rate, aliasing happens. That means frequencies higher than half the sampling rate get folded back down to below half the sampling rate. Have a read about ...
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What sampling frequency should I use if Nyquist is not available?

As correctly stated in Peter K.'s answer, this question is about aliasing. Since you can't sample at a rate that is sufficiently high to avoid aliasing - i.e., $f_s>50\textrm{ kHz}$ - you have to ...
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13 votes
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FFT vs DFT Run Time Comparison (Complexity Analysis) in MATLAB

Abhinav Jain, Welcome to DSP Community. I build for you a proper testing of the run time comparison. Few tips about timing in MATLAB: Never time in a script. Always call a function to do the heavy ...
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11 votes
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Why is $A\cos(2\pi f_ct)$ a non-stationary process?

A random process is a collection of random variables, one random variable for each time instant. It is best to write the random process as $$\{X(t)\colon -\infty < t < \infty\} \tag{1}$$ where ...
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8 votes

Intuitive interpretation of Laplace transform

Why is the fourier transform a special case of the laplace transform? The Laplace transform produces a 2D surface of complex values, while the Fourier transform produces a 1D line of complex values. ...
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8 votes

Filter design with zero - pole placement method

Here let me show you a simple procedure very similar to pole zero placement which will be helpful for your notch filter design. First, lets analyse the frequency response of a single zero and let $$ ...
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8 votes
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Difference between causality and memorylessness

A causal system does not need to know the future in order to compute its output. A memoryless system computes the output only from the current input. A memoryless system is always causal (as it doesn'...
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8 votes
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Does $\cos(bt)\cdot u(t)$ have a Fourier Transform?

The integral doesn't converge in the conventional sense, so you can't solve it with standard methods. Assuming that you know (or can look up) the Fourier transform of the unit step function $u(t)$, it ...
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7 votes
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Calculate the Inverse DTFT of the DTFT Derivative in Terms of $ x \left[ n \right] $

This is pretty straight forward using the definition of the Discrete Time Fourier Transform (DTFT). The definition of the DTFT: $$ X \left( {e}^{j \omega} \right) = \sum_{m = -\infty}^{\infty} x \...
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6 votes

Filtering $ n \times n $ Images by Separable $ m \times m $ Filters. Computation Time for Filtering Using FFT, 2D Convolution and Two 1D Convolutions

I'm not sure exactly what you're after, but just to try to add data: Using the separability property of the filter is always the right choice. Given it is separable we now have to apply 1D ...
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6 votes
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Properties of DTFT to Infer the Inverse DTFT of Altered Data

The property is the Linearity of the DTFT. Linearity means that if your input is a linear combination of signals the output will be the same linear combination of each input by itself: $$ \...
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6 votes
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Question About Sampling White Noise

A continuous-time white noise process $\{X(t)\colon -\infty < t < \infty\}$ is a hypothetical construct that we can treat (in the simplified versions that we use on dsp.SE) as a zero-mean wide-...
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6 votes
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Is this system causal or not?

Note that in this case you can see that the system is causal only from the given implementation. It's important to understand that you can't see it from the difference equation (if no initial ...
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5 votes

Intuitive interpretation of Laplace transform

The best intuitive description of Laplace transform I've ever seen: At first glance, it would appear that the strategy of the Laplace transform is the same as the Fourier transform: correlate the ...
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5 votes

Filtering without using any transforms

You can achieve this result by using two combs filters : https://en.wikipedia.org/wiki/Comb_filter Put simply, the comb filter consists of adding a delayed version of the signal to itself, causing ...
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5 votes
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how to evaluate derivative of convolution integral?

Note that option (b) is not correct, and that it is also not equal to what you came up with. Option (b) is just the multiplication of $x(t)$ and $y'(t)$, not the convolution. Your solution and option (...
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State space equations

Two principles here: When dealing with a differential equation, you define intermediate state variables so everything is in terms of first derivatives. This system is nonlinear, so the state-space ...
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4 votes

BIBO stability of $1/x(t)$

Hint: define a bound for $|x(t)|$, i.e., $|x(t)|\le A$; now try to find a positive number $B$ such that $|y(t)|\le B$ for any $|x(t)|\le A$ (that's simply the definition of BIBO stability). For the ...
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4 votes

Why is $A\cos(2\pi f_ct)$ a non-stationary process?

In easy words: A process is stationary if its stochastic properties are independent of the time you look at it. Think of it like this: A stochastic process is just a Random Variable (RV) that, ...
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4 votes

Fourier Transform of shifted Impulse

Technically, the impulse $\delta(t)$ is called a distribution, and not a function, but for the purposes of your first course in Fourier transforms, what you need to know is that $\delta(t)$ has the ...
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Stability of a system

For BIBO stability in the case of discrete time, there is a necessary and sufficient condition given by $\sum |h[n]| < \infty$ that is if the impulse response is absolute summable then the system ...
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4 votes

Would anyone be able to help out with the following discrete convolution question?

Hint The simplest way is to use the Z transform property "convolution in time domain is multiplication in z domain". See Z transform convolution $$\mathrm{Z}(x[n]*h[n]) = \mathrm{Z}(x[n]) \times \...
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4 votes
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What will be the filtered output?

First note that: $$ \cos(2\pi 50 t) \longleftrightarrow 0.5 \delta(f+50) + 0.5\delta(f-50) $$ $$\sin(2\pi 150 t) \longleftrightarrow 0.5 j \delta(f+150) -j 0.5\delta(f-150)$$ Hence the baseband ...
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4 votes
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Transmitting over Unknown Fading Channel

The strategy depends heavily your "realistic scenario", e.g. which estimator, which equalizer, which estimation error, etc. In general scenario, you cannot trust the estimate of $h$, it means that ...
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4 votes
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Proof of $E[y(n)]=E[x(n)] \, \sum h(k)$

The standard meaning of white noise includes an insistence (whether implicit or explicit) that the mean is $0$. Thus, what you want to prove is trivially true: since $$Y[n] = \sum_{k=-\infty}^\infty ...
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4 votes

Fourier components of $\cos(2\pi f_1t)$

HINT: Going from your last equation, $$\frac{\sqrt{T}}{2}\bigg(\frac{e^{j2\pi (f_1T-n)}-1}{j2\pi (Tf_1-n)} + \frac{e^{-j2\pi (f_1T+n)}-1}{-j2\pi (Tf_1+n)}\bigg)$$ This can be simplified further down ...
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4 votes
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Calculate the Output of Linear Time Invariant System Given it Impulse Response

The Discrete Delta Function, $ \delta \left[ n \right] $ is the identity operator of Linear Time Invariant Systems. Moreover, since it LTI System we can computer for each element of the filter by ...
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4 votes

Difference between causality and memorylessness

A memoryless system's output is determined by the current input value only, hence, every memoryless system must also be causal (a system is causal if its output does not depend on the future input ...
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4 votes
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Determine whether the system is linear?

In the following, I suggest that, before using the generic $T(\alpha_1 x_1+\alpha_2 x_2)$ versus $\alpha_1 T( x_1)+\alpha_2T( x_2)$, it can be more informative to try with simpler partial tests, or ...
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4 votes

Question About Sampling White Noise

Working with your definitions: $$ v \left( \left( n + 1 \right) {T}_{s} \right) - v \left( n {T}_{s} \right) = \int_{0}^{ \left( n + 1 \right) {T}_{s} } g(u) du - \int_{0}^{ n {T}_{s} } g(u) du = \...
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