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Approaching The Sampling Theorem as Inner Product Space Preface There are many ways to derive the Nyquist Shannon Sampling Theorem with the constraint on the sampling frequency being 2 times the Nyquist Frequency. The classic derivation uses the summation of sampled series with Poisson Summation Formula. Let's introduce different approach which is more ...


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It appears that your understanding of harmonics is not entirely correct yet. Your example of two sinusoids with frequencies $f_1$ and $f_2$ can actually be explained in terms of harmonics IF $$f_1=n_1f_0\quad\text{and}\quad f_2=n_2f_0,\qquad n_1,n_2\in\mathbb{Z^+}\tag{1}$$ is satisfied for some positive $f_0$. Note that this means that the ratio $f_1/f_2$ is ...


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If padding L-1 zeros were the only thing that's done for upsampling, nobody would actually store the end result with zeros that way, because the zeros would be redundant. In practice, padding with L-1 zeros is always followed by low pass filtering. If that filtering is done with an FIR filter, a dumb implementation would have many coefficients being ...


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At first, I thought harmonics come from the signal being periodic. That's correct. However, we know a sine wave is also periodic but contains a single frequency and no harmonics. This is the only periodic signal that doesn't have harmonics. Or to be precise the amplitudes of all harmonics are zero. No, I'm wondering what's the real root-cause for ...


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The basic idea is that a periodic sound can have a missing fundamental. e.g. If you mix the right multiples of 110 Hz, a human will hear 110 Hz, whether or not there is any non-zero 110 Hz sinusoidal component in the mix (via FT or DFT). So all those multiples are still called the same thing: harmonics. Where do they come from: lots of physical objects/...


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It pretty much doesn’t. One could say a boxcar window is used, but you most likely won’t hear any mention it. The trick with doing FFT convolution is making sure your FFT length is at least equal to the number of signal samples plus the impulse response length minus one. Both your input signal and impulse response need to be padded with zeros appended to the ...


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what role if any does windowing (e.g. Blackman, Hanning etc.) play None. If you do apply a window (other than rectangular) for a regular overlap add/save algorithm, your results will be wrong. Time domain windowing can be helpful if your frequency domain processing is time variant, but that requires more complicated algorithm with partially overlapping and ...


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The theoretical data rate to transmit a signal with bandwidth $W$ Hz and resolution $R$ b/sample is simply $2WR$ b/s. In practice, you will need somewhere between that and $4WR$. One way to reduce the required rate is to preprocess the signal in the SDR's FPGA. This will require delving into Verilog, integrating your hardware with what is already there, and ...


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Suppose that you could write $x_.$ as $x_.(t) = a_1x_1(t-n_1)+a_2x_1(t-n_2)$, then by LTI hypothesis, you can derive $y_.(t) = a_1y_1(t-n_1)+a_2y_1(t-n_2)$ with very little computation. You can get a first insight to that method by looking at the shape of $x_2$ which is self-evident, and a little more imagination gives you the structure for $x_3$. I find ...


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So, writing your $F^b$ and $F^{'b}$ as permutations $p, p^{-1}$ of the DFT matrix $F$ , IDFT $F'$, and the cyclic prefixing and removal also as Matrix operation $P$, $R$ (which boils down to cyclically extended or truncated identity matrices), and the circular convolution with the channel impulse response as circulant matrix $C$, \begin{align} x&=F^{'b}X=...


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Generally, an echo of a signal is not just its delayed and weakened copy, but is characterized by an impulse response of the echo path. The basic model of an echo is a linear one. In order to simulate a linear echo $E$ one will implement the following convolution: $E = S * H, $ where $S$ is your sampled signal and $H$ is an echo path impulse response, both ...


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I think the formula you got the 1m resolution from is actually the one for pulse radars with non-interpolating single pulses. You can do better! Note that in (linear-chirp) FMCW, the beat frequency, i.e. the difference in frequency between the received and transmitted signal is proportional to range. There's nothing in there limiting your resolution! Your ...


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You have two different meanings for ${γ}_{xx}$ when you write stuff such as $${γ}_{xx}(τ) = {γ}_{xx}(t+τ,t)$$ because on the LHS, the function ${γ}_{xx}$ has one argument while on the RHS it has two arguments. Be that as it may, a very important property of what you call the covariance but what most everyone else on dsp.SE would call the autocovariance ...


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The term "Hanning" is illiterate. The window is attributed to Julius von Hann. So it is a Hann window, or von Hann window. If you use overlap add/save fast convolution on von Hann windowed data where the windows are 50% overlapped, you get the same result as rectangular windows of data that are non-overlapped, except for the initial and ending ...


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Below, the result of a simulation where I have the sum of two sine-waves with frequencies 100 Hz and 201 Hz, respectively: $$ x(t) = > \sin(2 \pi 100 t) + \sin(2 \pi 201 t) $$ The signal is periodic (with a period equal to 1 s) but it does not contain harmonics Yes it does. It is a 1 Hz periodic signal that only contains the 100th and 201st harmonics.


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What was previously posted as an answer should have been an extended comment -- sorry for that. There simply was not enough space for what I thought I'd write about how the Octave code seemed to behave. Now that the code is cleared, what should have been the answer in the first place (given the rather obvious signs), is now the answer: spectral leakage. The ...


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The first equation, $ \frac{1}{N_s}\sum |x[n]|^2 $, is correct. Applying the RRC filter kind of smears the energy of the symbol over several samples so that when you add them all back up again you get $E_s$, and of course divide by $N_s$ for multiple symbols. The RRC filter should be normalized as someone in the comments noted. The second equation I can't ...


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There are two meanings of harmonics that may be confused. One is the mathematical, where in a Fourier expansion of any periodic signal (actually one satisfying Dirichlet conditions, as was pointed out), any components in the expansion with frequency an integer multiple of the frequency of repitition of the signal, might be understood to be a harmonic of the ...


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I think that maybe there is a mis understanding in your question. The harmonics doesn't appear in every signal. Usually they appear because the main frequency excite a natural frequency (resonance) or because there is any other physical phenomena involved in its generation.


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We can also imagine the sum of two different frequencies, one is not necessarily the harmonic of the other, and get a periodic signal but no harmonics, just the two frequencies No this is not true. If you eventually have a periodic signal (which meets Dirichlet Conditions) from their sum, then those two sine waves must be harmonically related, at the period ...


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