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4

$x(t)$ is a real-valued finite energy signal with Fourier transform $X(f)$. Its autocorrelation function is \begin{align} R_x(\tau) &= \int_{-\infty}^{\infty}x(t)x(t-\tau) \,\mathrm dt\\ &= \int_{-\infty}^{\infty}x(t)y(\tau-t) \,\mathrm dt & {\scriptstyle{\text{Define}~y(t)~\text{as the time-reversal}~x(-t)~\text{of}~x(t)}}\\ &= x\star y\big|...


3

$$ \mathscr{F} \Big\{ x(t) \Big\} \triangleq X(f) \triangleq \int\limits_{-\infty}^{\infty} x(t) \, e^{-j2 \pi f t} \ \mathrm{d}t $$ $$ \mathscr{F}^{-1} \Big\{ X(f) \Big\} \triangleq x(t) = \int\limits_{-\infty}^{\infty} X(f) \, e^{j2 \pi f t} \ \mathrm{d}f $$ Assuming $x(t)$ is real (which means that $R_x(\tau)$ is also real). $$\begin{align} S_x(f) &...


0

I would say most of your confusion arises from conflating the continuous case with the discrete case of the FT. The are similar, and related, but not equivalent. "Area" is a continuous concept. "Sum" is discrete. I normalize my DFTs by 1/N principally because it makes the bin values essentially sample count independent. Other reasons too. For the sum ...


3

I would simply say that it doesn't matter so much. The Fourier Transformation and its inverse are a pair and the two formulas are entangled and require a normalization factor, which is up to conventions. See eg enter link description here If your theory/model does not require a special convention, it is basically a degree of freedom that you can choose to ...


1

Considering real valued WSS processes whose auto-correlation and auto-covariance functions are defined as $$ r_x(\tau) = E[ x(t)x(t+\tau) ] $$ $$ c_x(\tau) = E[ (x(t)-\mu_x)(x(t+\tau)-\mu_x) ] = r_x(\tau) - \mu_x^2 $$ Then the following are just basic observations. The total power of a WSS random process is given by $$ P_x = E[ x^2(t) ] = E[ x(t)x(t) ] = ...


2

Synthetic Method: If $\{\hat X(t)\}$ and $\{\hat Y(t)\}$ are zero-mean uncorrelated low-pass WSS processes with identical autocorrelation function $R(\tau)$ and identical power spectral density $S(f)$ enjoying the property that $S(f) = 0$ for $|f|>B$, then $$\hat{N}(t) = \hat X(t)\cos(2\pi f_ct) - \hat Y(t)\sin(2\pi f)ct$$ is a band-pass process whose ...


3

I think you are making this more complicated than it needs to be. The Fourier and autocorrelation integrals are defined from $-\infty$ to $+\infty$. That's always correct and the saftest way to write them. If you are integrating over a function with limited support on $[0,T]$ than you can adjust the integration interval to $[0,T]$ too. The areas outside ...


0

Energy spectral density is mostly applied to time-limited signals. In such cases, energy is finite, and average power (throughout all time) is zero. These signals are not stationary. Normally you work with (or model your signals as) stationary signals. These have constant power (and infinite energy). Power spectral density applies to stationary signals, and ...


2

Your first steps seem OK to me. Even if I'd put absolute values in the time domain as well (for generality), if the first three integrals exist, the fourth is valid. Integration is linear. The weak point resides in the reverse inference. The equality of energy integrals does not mean that the integrands are related. For one, PDSs are real and positive ...


2

A personal rule: in general, it can be useful to perform non-linear operations before linear ones. One reason behind that is that a lot of practical concerns are related to outliers or suspect behavior, which can easily be smoothed by linear filters. Let me reformulate. If $f_i$ denote filters, and $s_i$ signals, should one do $f_0 \ast(s_1 .s_2)$ or $(...


1

In general you will need to multiply first and then low pass filter. You also have to make sure that your sample rate is high enough so the multiply doesn't create aliasing. Let's look at a simple example: feed a 1kHz signal into a loudspeaker and measure current and voltage to determine the average (thermal) power with maybe a 100ms time constant. The ...


1

Assuming that the process is wide-sense-stationary, the total power in a random process with autocorrelation function $R(\tau)$ and power spectral density $S(f) = \mathcal F\{R(\tau)\}$ is given by $$\text{Total Power} = \int_{-\infty}^\infty S(f) \,\mathrm df = R(0)$$ while the "DC power" (also equal to the squared mean of the process) is given by $$\text{...


1

PSD (power spectral density) : distribution of power (of a WSS random process) along frequency... Power Spectrum is the same thing. For zero mean discrete-time processes, PSD is defined as the DTFT of the ACS: $$ \boxed{ S_x(e^{j\omega}) = \sum_{m=-\infty}^{\infty} r_{x}[k] e^{-j \omega m} }$$ Where $r_{x}[m]$ is the ACS (auto-correlation sequence) of the ...


1

You are right, the derivation is full of typos. The first equation below Eq. $(8.39)$ should read $$\int_{-\infty}^{\infty}x(t+\tau)e^{\color{red}{-}j\omega\tau}d\tau=X(\omega)e^{j\omega \color{red}{t}}\tag{1}$$ Substituting into $(8.39)$ gives $$\begin{align}\mathcal{F}\big\{R(\tau)\big\}&=\int_{-\infty}^{\infty}x(t)X(\omega)e^{j\omega t}dt\\&=X(\...


0

Ok. Thank you for this explanation. I'll try to explain better my problem, because I tried to simplify it, but maybe (since I am not an expert) I have worsen the situation. I had to do Power spectral density of a random signal (the velocity of a cfd simulation). Firstly I did the fft of the autocorrelation: v is my signal r =xcorr(v,v) PSD = abs(...


1

Regarding what we discussed in the comments, here is an explanation of the different concepts I was referring to: The concrete usage of terms may vary, but my interpretation is this: For a stochastic process $x(t)$, the autocorrelation function is defined as $$r(\tau) = \mathbb{E}\{x(t) \cdot x(t+\tau)\}.$$ Note that the process needs to be stationary, ...


-1

$$R_{n}(\tau) = E(n(t)n(t-\tau))$$ $$R_{n_i}(\tau) = \cos(2\pi f_ct)\cos(2\pi f_c(t-\tau))E(n(t)n(t-\tau))=\cos(2\pi f_ct)\cos(2\pi f_c(t-\tau))R_{n}(\tau) $$ Using $\cos(a-b) = \cos a \cos b + \sin a \sin b$, we get $$R_{n_i}(\tau) = \cos(2\pi f_ct)[ \cos(2\pi f_ct ) \cos(2\pi f_c\tau)+ \sin(2\pi f_ct) \sin(2\pi f_c\tau)]R_{n}(\tau) $$ or $$R_n(\tau) = \...


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