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I think $$R_{XX}(\tau) \triangleq \int_{-\infty}^\infty X(t)X(t+\tau)dt \tag{2}$$ Is just for a continuous signal $X$ that isn't a stochastic process. When it becomes a stochastic process, the expected value arrives to compute the ensemble average. In this case, that is the definition of a finite duration real signal This can be rewritten as: $$R_{XX}(\tau) ...


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Possibly off-topic but in order to provide context, i.e. Parseval's identity: I think a more general outlook should be pointed out. It's applicable in "reality" because we believe that Energy is conserved irrespective of description and there are equivalent similar relations for any of the linear transforms/representations; Laplace, Mellin, ...


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From physics, energy is a term often used as a quantitative property. In other words, energy is a quantity that is preserved under some actions, transformations, etc. In signal processing (where physics vanish), this often takes the shape of a sum or an integral of a squared quantity for reals, or its modulus for complex data. We can write it symbolically ...


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Yes, the square of the $L_2$ norm of a signal is also by definition its energy $\mathcal{E}_x$. There's nothing surprising, unbelievable, or mysterious in that though? The concept of signal energy : $$ \mathcal{E}_x = \int_{-\infty}^{ \infty } x(t)^2 dt\tag{1} $$ is fundamentally based on the concept of energy (or work) in physics as the Kinetic Energy of a ...


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How is this concept illustrated for those ones who are working in pure math. I've never seen a pure mathematician need an illustration for a definition! Really, the energy is defined as the sum of squares (discrete time) or the integral of squared (continuous time) signal. At that point, it's not concept you have to apply, just a definition. When leaving ...


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You have already read those Oppenheim's Signals & Systems, and Discrete-Time Signal Processing books. I'm not sure what you mean by foundations but in some sense these two are also the foundations on signal processing. In other words, there are no (popular & successful) graduate level DSP books that discuss at an advanced level the same topics that ...


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I found these books to be very good in their respective field: J.R. Ohm - Multimedia Communication Technology This has focus on representation and transmission of signals. It follows a practical approach hands down and features very good, informative illustrations. In general, Ohm's books are recommendable. His newest one is about feature extraction, but I ...


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Since, the Taylor series expansion for $e^z$ about $z=0$ is $$e^z = \sum_{n=0}^{\infty}\dfrac{z^n}{n!}$$ then, ignoring the question of convergence, you can say $$\begin{align*} \int_a^b \int_c^d e^{A\sin(x)\cos(y-B)}dx dy &= \int_a^b \int_c^d \sum_{n=0}^{\infty}\dfrac{(A\cos(y-B))^n}{n!}\sin^n x \; dx \, dy\\ \\ &= \int_a^b \sum_{n=0}^{\infty}\...


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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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What you want is theoretically impossible: if $h(t) \neq 0$ only on a finite interval, then it is not bandlimited. Fortunately, all real-world signals and systems have the property that $H(f) \rightarrow 0$ as $|f| \rightarrow \infty$. This allows you to approximate $h(t)$ very closely with only a finite number of samples. For engineering purposes, this is ...


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We call a signal $f(t)$ periodic with $T>0$ if $f(t) = f(t+T)$ for all $t$. Note that if a function is $T$-periodic, it is also $2T$, $3T$, ... periodic. Therefore, typically, the (fundamental) period is associated with the smallest $T>0$ for which $f(t) = f(t+T)$ for all $t$. The function $\sin(t)$ has a fundamental period of $2\pi$, i.e., $\sin(t) = \...


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If you care for multidimensional signals, let us use "shift-invariant" maps. To analyze such systems, Fourier transforms are tools (for engineers) of choice. I often say that Fourier transforms are to convolutions what logarithms are to multiplications: a tool that helps explaining what is less than obvious (diagonalization of the convolution ...


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For time-based systems, I understand that it is difficult to imagine a memory of the future. But for general systems, $-t$ and $t$ are just left and right (think of a spatial system). Other discussions are in LTI system $y(t)=x(t−T)$ with or without memory, What is a memory less system?, or A question about the concept of the time. By definition of ...


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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 are 2 types of digital here. Think of OFDM using 16QAM. You have a bit stream that you are taking and correlating it to discrete amplitudes of 2 4PAM signals, one in phase and one in quadrature. This is a digital signal right, due to the discrete amplitude part. But how do you transmit it? With a digital sample to analogue converter on the output of ...


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