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 ...
Peter K.'s user avatar
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21 votes

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 ...
Matt L.'s user avatar
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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 ...
Dilip Sarwate's user avatar
10 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 $$ ...
Fat32's user avatar
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10 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'...
Matt L.'s user avatar
  • 90k
10 votes

How can I find the following Fourier Transform without directly using FT pairs?

The results shown in the two answers provided by Ahsan Yousaf don't agree. This answer is about explaining why the two solutions are different, and how to arrive at the correct solution. Note that ...
Matt L.'s user avatar
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9 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 ...
Royi's user avatar
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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 ...
Matt L.'s user avatar
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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-...
Dilip Sarwate's user avatar
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 ...
Matt L.'s user avatar
  • 90k
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 ...
Albits's user avatar
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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 (...
Matt L.'s user avatar
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5 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 ...
Laurent Duval's user avatar
5 votes
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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 ...
Robert L.'s user avatar
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5 votes
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DFT exercise in the book Understanding digital signal processing 3 Ed

@Henrique Luna. Please forgive me. In Part (b) of the problem the words "sequence time" should "time sequence". Sorry for the confusion! Years ago when I created that Part (b) ...
Richard Lyons's user avatar
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 ...
Matt L.'s user avatar
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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, ...
Marcus Müller's user avatar
4 votes
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Showing linear-phase frequency response when only real part is known

I agree with Maximilian Matthé's answer, but I'd like to show you another route to the solution, which might be a bit more straightforward, and which avoids the explicit application of the Hilbert ...
Matt L.'s user avatar
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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 ...
Dilip Sarwate's user avatar
4 votes
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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 ...
Arka Sadhu's user avatar
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 \...
AlexTP's user avatar
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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 ...
Fat32's user avatar
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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 ...
AlexTP's user avatar
  • 6,595
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 ...
Dilip Sarwate's user avatar
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 ...
Gilles's user avatar
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4 votes
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Expressing 2N point DFT in terms of N point DFT

Consider N-point sequence $x[n]$ whose N-point DFT is $X[k]$. To compute 2N-point DFT, $X_2[k]$, of $x[n]$, one should append N-zeroes to $x[n]$, and make it a 2N-point sequence, denoted $x_2[n]$. ...
Fat32's user avatar
  • 28.2k
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 ...
Fat32's user avatar
  • 28.2k
4 votes

Finite impulse response FIR filters

In order for H(z) to be a linear phase filter, it must zeros both on the inside of the unit circle and at the complementary locations (1/z) which are outside the unit circle. Therefore a linear phase ...
Dan Boschen's user avatar
4 votes
Accepted

Discrete-time Fourier transform of $\frac{\cos(\frac{n\pi} 6)}{(n+3)\pi}$

HINT: $$\frac{\cos(n\pi /6)}{(n+3)\pi}=\frac{\cos[(n+3)\pi/6 -\pi/2]}{(n+3)\pi}=\frac{\sin[(n+3)\pi/6]}{(n+3)\pi}$$
Matt L.'s user avatar
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4 votes

Is this system causal or not?

Purely by inspection of the block diagram the system is causal, because the output is the sum of the current input sample and stuff that's delayed -- there's no $z$ blocks in there to predict the ...
TimWescott's user avatar
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