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The Impulse Invariant method does not promise to represent the frequency response of the continuous-time system as a lowpass version. What the Impulse Invariant method does is frequency-alias the frequency response by sliding it by multiples of the sampling frequency and adding up all of the translated copies.


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Here is an improved version starting from the example suggested by Engineer (thanks!). I did use the sinc function of Octave, which is defined in zero (not getting warning messages and not introducing that small error due to wrong calculation). Moreover, I did show a step by step plotting to see how further samples change the signal and how the errors at the ...


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Direct response from me, the OP, to everyone else: you're right. But so am I, and the responders could've handled this much better. Explanation follows. Where I agree If the actual, continuous/analog signal has no continuous Fourier transform frequency components above $f_s / 2$, then there can be no aliasing. This is explained in detail in Dan's answer. ...


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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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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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Assuming $\frac{f_s}{f} \neq N$ where N is an integer. There are 2 easy ways that I can see to compute the RMS value 1st method : You could try to accumulate several periods of data. For example with fs = 1 kHz, f = 61 Hz, you could compute the RMS of 82 points which gives about 5.002 signal periods. The error introduced by the non-integer number of periods ...


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That's coherent sampling. Assuming this is a AC power application: it's only coherent sampling if you derive the sampling clock directly from the AC frequency using a PLL (Phase Locked Loop) or equivalent. But does this account for frequency variations? Depends on how you derive the sampling clock. If you use a PLL it will track the AC frequency, if you ...


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When referring to the SNR achieved by oversampling, we have to be careful in using the term "SNR". There are essentially two SNR's to consider: The SNR that is the signal-to-quantization-noise ratio. The SNR that is your signal-to-noise ratio. Here the noise is produced by your system. For simplicity this can be modeled by the kTB relation that ...


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OP clarified that the question in the comments as follows: If we ignore any modulation for now and assume that we are receiving pure tones plus the band limited noise and we try to improve the SNR in post processing how much improvement can we expect by oversampling and is there a limit to it? My original question was about this aspect. First consider the ...


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You've made an error in this statement: "I know by sampling in time domain we get shifted replicas of the original spectrum which is two delta functions at $f_0$ and $−f_0$ but how this corresponds to $f_s$ and aliasing at $f_0+kf_s$?" The two delta functions at $-f_0$ and $f_0$ are the result of taking the Fourier transform of a sinusoid $x(t)$: $$...


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Can't be done. Next question? First, the author does not define $x[n]$. Let's assume, then, that $x[n] = x(nT_s)$, where $T_s$ is the sampling interval; i.e. $f_s = 1/T_s$. So assume that $x_1(t) = x_1[n]\ \delta \left(t - \left(n + \epsilon\right) T_s \right)$, where $\epsilon$ is any number where $0 < |\epsilon| < 1$, and $x_2(t) = x_2[n] \left( u \...


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Note the trig identity: -sin(x) == sin(-x) This means that a negative aliased frequency just means the aliased samples happen to be inverted in phase from that of the fundamental signal. You can't tell, just by looking at these aliased samples, whether the harmonic was an inverted sinewave, or a negative frequency sinewave. So you can call that set of ...


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Aliasing is more like circular wrapping around rather than just mirroring. Frequencies wrap circularly between -Fs/2 to Fs/2, an 800 Hz range from -400 to 400 at a sample rate of 800 sps. 120 is inside -Fs/2 to Fs/2, so doesn't need to wrap around at all. But 480 wraps 80 past 400, and shows up at -320, which looks like 320 if you don't care about phase. ...


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First, about "why we have negative frequencies and which one is copied"... In a real signal, there is no difference between a negative frequency and a positive frequency in this sampling context. Consider that you have captured 1000 samples that you know contains a single sine wave. You analyze it and find it contains exactly ten cycles of a sine ...


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You reached a puzzling conclusion about $c_1(t) = c_2(t)$, and wonder whether you made a mistake in deriving them, or if the equality is indeed correct then how to explain it, perhaps by explicitly deriving one from the other. I cannot tell whether it's possible to explicitly manipulate the double summation in $c_2(t)$ so as to convert it into the single ...


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There's multiple ways you can approach this, and as what's inside frontends is a business secret, I'm not sure anyone can say for certain what 4G devices do in general, but it's safe to assume they do a combination of a couple measures. Among these to cancel a Sampling Clock Offset (SCO) will be: Ignoring the problem Your frames are of limited length. ...


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There are many advantages, but the most obvious to me Advantage 1 : Oversampling followed by decimation allows you use to simpler and smaller anti-aliasing filters. These filters cost less and take up less space on a PCboard. Advantage 2 : In multi-channels application, the tolerance and variation of the analog components of your anti-aliasing filters can ...


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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 (...


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Actually in OFDM, it is very easy. You can check Van De Beek algorithm for compensation of carrier frequency offset.


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And additional consideration not mentioned that comes up in radio design is in the decision to use quadrature sampling of a baseband signal (as in "Zero-IF receivers") over a "Digital-IF" receiver that is achievable when the signal can be sampled at a much higher rate as a real signal. The Digital IF signal avoids the quadrature imbalance ...


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With downsampling you have complete control over the process and it comes down to what compromise of processing complexity, delay, aliasing and loss of passband you can accept. With a lower rate A/D you are pretty much at the mercy of someone elses spectral trade-offs and in addition you get the quantization/noise of one analog pass. -k


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