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Just think of the value of $t$ where $x(0)$ appears. For $x(t+t_0)$ it is at $t=-t_0$, which corresponds to a left shift if $t_0>0$. However, for $x(-t+t_0)$ the value $x(0)$ occurs at $t=t_0$, which is to the right of its original position if $t_0>0$. You can think of deriving $x(-t+t_0)$ from $x(t)$ in two different ways: invert the time axis: $x(-t)...


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If $X(f)=\mathcal{F}\{x(t)\}$ you have, in analogy with Eq. $(7)$, $$x(f)\Longleftrightarrow X(-t)\tag{1}$$ where we assume that functions with the independent variable $f$ are Fourier transforms of the corresponding functions with independent variable $t$. Applying $(1)$ to $$x(t-b)\Longleftrightarrow e^{-2\pi jbf} X(f)\tag{2}$$ gives $$x(f-b)\...


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I think the confusion comes from the fact that the command hilbert in Scipy (and also in Matlab/Octave) does not just compute the Hilbert transform, but its output is the analytic signal. So if $x(t)$ is the (real-valued) input to such a function, its (complex-valued) output is $$y(t)=x(t)+j\mathcal{H}\{x(t)\}\tag{1}$$ Clearly, if you want to obtain $x(t)$ ...


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The autocorrelation of $x(t)$ is $$r_x(t)=x(t)\star x(-t)\tag{1}$$ where $\star$ denotes convolution. Taking the Fourier transform of $(1)$ gives $$S_x(\omega)=X(\omega)X^*(\omega)=|X(\omega)|^2\tag{2}$$ $S_x(\omega)$ is the energy density of $x(t)$, and according to $(2)$ it equals the squared magnitude of the Fourier transform of $x(t)$. So if $x(t)$ is ...


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Let me suppose that by Nyquist, you mean that there is some frequency $F_s$ such that there is a existence theorem that says: if one samples any continuous $s(t)$ at $F_s$ (or above), $F_s$ denoting an extremal frequency or a extremal bandwidth, to get a discrete signal $s[k]$, then one can recover $s(t)$ by some algorithm. Case 0: neither oversampling ...


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Nyqusit Frequency simply means "half the sample rate" Undersampling means your signal bandwidth is higher than the Nyquist Frequency. You get aliasing Oversampling means your signal bandwidth is lower than the Nyquist Frequency. You don't get aliasing Most practical applications choose a Nyquist Frequency that's slightly higher than the highest ...


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