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21

It is actually not distorted, it is sampled at high enough rate. What fools you is the straight lines drawn between sample points, it gives you a false impression of the waveform. It shows you a linear interpolation of the signal. It does not represent how the signal would actually look like. A sampled signal exists only at the sample points, and to convert ...


9

The actual requirement is to sample at GREATER then twice the bandwidth, not at a rate equal to it... So only your 80Hz same set actually meets the requirement, because the 60Hz case is ambiguous in general, consider if you were sampling sin (2PiFt) instead then you would get a flat line at zero amplitude.... And changing the angle between sin and cos would ...


7

The OP's opening statement is incorrect: $f_s > f_{max}/2$ prevents frequency aliasing for a bandlimited signal, but not amplitude aliasing $f_s > 2 f_{max}$ prevents aliasing. It's as simple as that. There is no such distinction as "amplitude aliasing". Since the OP has stated the signal is band-limited; as long as we can assume that means ...


6

There is no aliasing as 𝑓 = 30 Hz is less than or equal to the folding frequency, 30 Hz and 40 Hz, respectively. Yes and no. There isn't significant aliasing when you're sampling at 80Hz, because the resulting signal has frequency components at 30Hz and 50Hz. The result is thus unambiguous as long as you take that 50Hz signal into account. There is ...


5

No, because this is a sufficient condition (for regularly sampled signals), and not a necessary one. This condition restricts the space of all possible continuous signals to a subspace of discrete sequences that contain the same information. Suppose that you can constrain the signal space, eg limited band-width, positivity, parametric models, sparsity, etc....


4

Looks like a potential application for blue noise also known as Poisson disk sampling, which is random placement of samples but with a guaranteed minimum distance between sample locations. I think that would give more accurate transform results than independently located random samples. Various spherical harmonic transform algorithms with specific sampling ...


4

I'm a little confused about what you are asking and how it relates to the set of spherical basis functions. Those are the same one that electron orbitals are based on, right? I have never understood them to be constrained to the surface of a sphere. Then again, that is not my usual stomping grounds. Nevertheless, I think I can generalize your question ...


4

Yes the OP is correct in that you can implement pulse shaping in less than 2 samples per symbol for exactly the reasons that was outlined. However importantly we must also keep in mind having excess bandwidth to simplify subsequent filtering required (such as after the DAC on the transmitter side). The Nyquist criteria is the sampling rate must be twice the ...


4

The claim is wrong. Sampling of a pure sinusodial whose frequency is below but arbitrarily close to the Nyquist frequency (half the sampling frequency) is a perfectly valid operation, as long as you can create ideal (zero width transition band) brickwall lowpass filters to be used at the reconstruction interpolation of the continuous waveform from its ...


3

Answer: You will see residual images of $X(f)$ at multiples $f_s$, $2f_s$ and $3f_s$, and distorted image of $X(f)$ at non-zero multiples of $4f_s$, when sampling in the manner you explained. Depending on value $e$, the size of residual will change. I have explained how in detail below. Ideally, sampling at $4f_s$ would have completely cancelled those ...


3

Remembering from my 1970 Signal Processing lectures we have ... The crucial thing is the filter used to reconstruct the signal. Let's do the theory first for ideal sampling a perfect sine wave at 2x its frequency and filtering with an ideal low pass filter. The samples are infinitely thin - they are delta functions separated by time t. The filter is an ...


3

From an ADC perspective, it is just taking a sample of the voltage in time. I fail to see how a "misinterpretation" could be made since there is no "turning car wheel" to take pictures of at the wrong time. Do the harmonics alias in such a way that the wave shape is preserved? You can reason this out yourself, in the time domain. Consider a square wave ...


3

I think that considering the DFT from a linear algebraic point of view has some value, so I will attempt to introduce the foundations. We will assume that our signal is a vector of $N$ complex entries. $\mathbb{C}^N$ is the vector space of vectors with $N$ complex entries. Let $\mathbf{u}_0,\mathbf{u}_1,\ldots,\mathbf{u}_{N-1}$ be vectors in $\mathbb{C}^{N}...


3

However when I look at the closed loop transfer function, I would say that this system is unstable for 𝐺𝐻=βˆ’1. In this case the transfer function becomes infinity so a bounded input will result in a unbounded (=infinity) output. This depends on your definition of stability. $GH = -1$ is called marginally stable because depending on how you look at it, it ...


2

I have some very short signals in the range of 8 to 16 samples. These represent a bandlimited signal, sampled at or slightly above the Nyquist rate. Nope. A signal can't be limited in time and in frequency at the same time. If it's very short, than chances are the bandwidth is a lot higher than you think it is and that you've already picked up some ...


2

Since this is a pure sinusoid, it has a bandwidth of 0 Hz. You can multiply it by a carrier signal of the same frequency, pass it through a low pass filter then take only a few samples. What matters is NOT the frequency of the signal, rather the bandwidth. Consider for example a voice signal modulating a 1 GHz carrier. It will be very costly, to sample this ...


2

You may be interested in the simple experiment using matlab. https://poweidsplearningpath.blogspot.com/2020/04/ch4-adcdac-how-to-simulate-adcdac.html Reconstruction is essentially a kind of interpolation or so called digital to analog conversion (DAC). Detail descriptions are introduced in chapter 4.8.3 of the DSP Bible 1. However, we all understand the ...


2

I'd say that this is not only "similar to a cross-domain equivalent to Nyquist's Sampling Theorem", but it simply is the sampling theorem. The sampling theorem does not specify the domains of the signals involved; it is rather a mathematical condition that a function of a continuous variable needs to satisfy such that it is perfectly represented by ...


2

There are a few things I can note about your question. As far as I have always learned, the nyquist stability criterion is taken over the openloop transfer function. if you take the closed loop transfer function, you should count the encirclements of 0 instead (if i recall correctly). The formal definition of stability, as expressed by the Lyapunov's ...


2

Can we sample the Dirac function? Strictly speaking: "sampling" would be taking the instantaneous value. Since the Dirac Delta doesn't have a value at $t=0$ (it is not really a function!), NO. Realistically speaking: an ADC can't measure instantaneous values. That's impossible, because it would require infinite bandwidth of the conversion system (...


1

Answer : What you are considering as $\Omega_{N_x}$ is equal to $\frac{\Omega_N}{2}$ according to question. So, what you are saying is same as what answer mentions given we are considering Baseband Samping of $y_a(t)$. I think you are confused because of the terms Nyquist rate and Nyquist Frequency. Nyquist rate and Nyquist frequency are two different ...


1

Firstly, your book uses the term Nyquist frqeuency as $\Omega_N + 2\Omega_o$, this is incorrect, this is the Nyquist rate (minimum sampling rate) if we consider the signal to be baseband. The maximum frequency content is then $\Omega_o + \frac{\Omega_N}{2}$, since you have defined maximum frequency of $x(t)$ as $\Omega_{N_x}$ this is nothing but $\frac{\...


1

Update: I feel there is no need to bring concept of Bandpass sampling (as mentioned in comments below) because this is specifically regarding a simple misunderstanding that OP has. The text referenced asks about Nyquist Rate (which wrongly mentions it as Nyquist Frequency) but OP asks about highest Frequency in $y(t)$. For the component $\frac{1}{2}X(\Omega-...


1

A good 1d example of this is the foundation of the FFT algorithm in how an $N$ length DFT can be created from two $N/2$ length DFTs. If you look under the hood of this, we are increasing the resolution through multiple copies of a time domain signal each sampled at a different offset, and resulting in each signal containing the low frequency content as well ...


1

does aliasing occur always if i sample a vibration in real world applications? Yes. The aliasing always occurs. The sampling theorem assumes band limited signals, but these strictly band limited signal do not exist in reality (as they would be infinitely long). Of course any signal can be low pass filtered to be reduce the aliasing to an acceptable level ...


1

If I have a vibrations sensor that has a max sample rate of 8kHz -> It can reconstruct signals till 4kHZ perfectly right? Theoretically, yes. Though I would like to add that all the noise signals beyond $+/-4kHz$ will alias back into your sampled signal. But what about frequencies which occur also in the measurements with much higher frequencies? If ...


1

That is why, before sampling, a (steep) lowpass filter with cutoff frequency $f_c \leq \frac{f_s}{2}$ shall be applied. Thus, the amount of aliasing will be insignificant.


1

Sampling operation has its roots in the mathematical interpolation theory which was used to generate certain function values at specified points from the availabe set (the samples) of existing values. These kind of work is summarized as Whittaker interpolation. Lagrange interpolation is also another related concept. Sampling theorem in electrical ...


1

One more possibility, if you have a lot more and longer training data than production data. Attempt to train a machine learning model (DNN, etc.) against shorter segments of the test data to predict the interpolated values in regions where Sinc interpolation alone is too inaccurate. Use data from longer segmenting to validate during training. If you don'...


1

One possibility is to use a "discrete sinc interpolation", which uses a compact support version of a sinc (which is not a truncated sinc). Otherwise there are methods based on the discrete cosine transform (DCT) and discrete sine transform (DST). Another interesting approach is based on "sinc-lets". These are reviewed in this paper. In particular, look at ...


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