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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 ...

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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 ...

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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 ...

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In contrast to common misunderstanding, aliasing does not affect the process of sampling, rather it affects the process of reconstruction. i.e., the samples themselves do not contain error, but their interpretation at the reconstruction (or anything related with it) will be in error. So, if you sample a sine wave $$x(t) = A \sin( 2 \pi f_0 t + \theta)$$ at ...

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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 $\mathbb{C}^{N}... 3 However when I look at the closed loop transfer function, I would say that this system is unstable for 𝐺𝐻=−1. In this case the transfer function becomes infinity so a bounded input will result in a unbounded (=infinity) output. This depends on your definition of stability.$GH = -1$is called marginally stable because depending on how you look at it, it ... 3 Answer: You will see residual images of$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 ... 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 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 SDR is not applying a lowpass (baseband) sampling on its RF input, instead it effectively employs a band-pass signal sampling. According to Shannon-Nyquist baseband sampling criteria, the bandwidth of the signal to be sampled should be less than half the sampling frequency. That's what you are talking about. However for narrowband modulated signals bandbass ... 1 you can successfully recover (or reconstruct) the signal so long as$f_2<F_s/2$Nope. The sampling theorem says that you need 2 samples per Hz of bandwidth, so in this case you'd need $$Fs > 2*(f_2-f_1)$$ For more info on how this works google "bandpass sampling" or just read https://en.wikipedia.org/wiki/Undersampling. The basic idea is that ... 1 Yes; what you describe doesn't mathematically look any different than a single sampler at twice the rate. But they sensors need to sense the same, band-limited thing! Actually, that's how some very high-speed ADCs work. If you're not actually doing GS/s, getting a faster ADC would probably be easier than staggering multiple ADCs. 1 From the definition of sampling, and the definition of periodicity follows that any signal is identical to signal at a frequency higher by an integer amount of sampling frequencies. Since for real signals, negative frequencies aren't distinguishable from positive ones, it directly follows that for every$0.5f_\text{sample}\le f \le f_\text{sample}$, there is ... 1 Yes, you can see that in pictures when the subject has tiny stripes or plaid shirts. This causes a disturbing visual effects, and professional are sometimes warned about that when they go on TV. Images are always of finite size, thus aliasing always exists to a certain amount, but might be attenuated in quantization, or barely noticeable to the eye. In short,... 1 To have exactly zero aliasing, you need to reduce energy >= samplerate/2 to exactly zero (assuming that you are interested in the lowpass part). I believe that is usually impossible, but say that the residue is <-48dB below the desired lowpass signal. Then your aliasing noise is in the ballpark of an 8-bit quantizer, which might be enough for some ... 1 i see aliasing happening when the image is captured with a sensor that cannot capture higher frequencies. Exactly! If i capture a image, is it always aliased? Not when the image content is sufficiently bandlimited. That's often the case because of the physics behind things, or simply because the motive isn't higher in frequency. (Physical effects include ... 1 Well! all signals in this world are made up of sum of different rotations(sinusoidals) - different in three senses: a. how big is the amplitude (A) b. how fast is the rotation ($\omega$) c. where is the starting point of the rotation (phase$\phi$) Fourier made this very clear. How do we measure rapidness of the rotating signals(sinusoidals) : by their ... 1 Both. And more. From just looking at the FFT results, you can't tell if the input samples were samples of a low frequency sinewave (below half the sample rate) reversed in time, or of a high frequency (above half the sample rate and below the sample rate), or any of an infinite multiple of image folding frequencies thereof (both high (under sampled) and/or ... 1 Okay, to within a scaling factor where definitions may vary, this is the DFT and it's inverse: $$X[k] = \sum\limits^{N-1}_{n=0} x[n] \ e^{-j2\pi n k/N}$$ $$x[n] = \frac{1}{N} \sum\limits^{N-1}_{k=0} X[k] \ e^{+j2\pi n k/N}$$ Both the discrete-time function$x[n]$and discrete-frequency function$X[k]$are periodic with period$N$. Even if your ... 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{\...

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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$length DFTs. If you look under the hood of this, we are increasing the resolution through multiple copies of a time domain signal each sampled at a different offset, and resulting in each signal containing the low frequency content as well ... 1 When you add 2 or more sinusoids at the same frequency$f_o$but with different phase shifts you get a sinusoid at same frequency but an additional attenuation term. Mathematically, you will have following: $$cos(2\pi f_ot + \phi)+cos(2\pi f_ot + \theta) = 2cos(\frac{\phi - \theta}{2}).cos(\frac{2\pi f_ot + \phi + 2\pi f_ot + \theta}{2})$$$$= 2cos(\frac{\phi ... 1 Let$x(t)$be an infinite analog signal, and$w$be a rectangular window function non-zero on$[-T,T]$. Let us fix the sampling rate$F_s$and let$N$be the DFT size. The DFT of$x(t)w(t)$limits to the DTFT as$N\rightarrow\infty$i.e. increasing interpolating the DTFT (note since$F_s$is fixed we must be sampling where$w$is zero also). Now let$F_s\...

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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 ...

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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 ...

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