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Coherent demodulation\detection is usually done via matched filtering in RF signal processing No, that's not generally true: Matched filtering only is an optimal technique if you want to maximize SNR in an AW(G)N channel In many, if not most, systems that do matched filtering, that matched filter is realized in baseband, i.e. after coherent demodulation. ...

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The secret to proving the limit is to not convert the $sin$ to $sinc$. This leaves the $T$ in the argument and not outside. No need to worry about the complex conjugate as the values are real. The $sin$ values are bounded by -1 and 1. At that point it becomes:  \begin{align} 0 \le S_{xx}(f) &= \lim_{T\rightarrow\infty}\frac{1}{T}\left|X_{T}(f) \...

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You don't need to make it so complicated as to use two variables $t$ and $t^\prime$. \begin{align} X_T(f) &= \int_{-T/2}^{T/2} x(t)\exp(-j2\pi ft)\,\mathrm dt\\ &= \int_{-T/2}^{T/2}\frac{\exp(j2\pi f_0t)+\exp(-j2\pi f_0t)}{2}\exp(-j2\pi ft)\,\mathrm dt\\ &= \frac 12\int_{-T/2}^{T/2} \exp(j2\pi (f_0-f)t)+\exp(-j2\pi (f_0+f)t)\,\mathrm dt\\ &= ...

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