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The condition $$x(nT)=\delta[n]\tag{1}$$ is called the Nyquist criterion for zero intersymbol interference (ISI). It is important for the design of transmit pulses in digital communication systems. Condition $(1)$ can be expressed in the frequency domain as $$\frac{1}{T}\sum_{k=-\infty}^{\infty}X\left(\omega-\frac{2\pi k}{T}\right)=1\tag{2}$$ where $X(\omega)...


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I don’t think this is necessarily additionally introduced ISI (beyond the ISI of the pulse shaping filter itself, which is zero when there is no timing offset) but may in fact be the result of timing offset error, given all the points within a small offset from the ideal sampling location appear to be selected, and the overall pattern of the full ...


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Short Answer In order to do a complete Zero-IF transmit and receive without inversion, the operations must be done as follows (or otherwise have the signs equally flipped on Q for both transmit and receive): Zero-IF (Direct Conversion) Transmit: $$I_{RF}(t) = I_s(t)\cos(\omega_{LO}t) - Q_s(t)\sin(\omega_{LO}t)$$ Zero-IF Receive: $$I(t) = \text{LPF}\{I_{RF}(t)...


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Since Discrete Fourier Basis Vectors are $2\pi$-Periodic, hence, negative frequencies $-\frac{2\pi k}{N}$ are conventionally represented as $(2\pi - \frac{2\pi k}{N} = \frac{2\pi}{N} (N-k))$. It has nothing to do with Butterfly Structure of FFT Algorithm. The above statement basically means that $-k^{th}$ tone is represented as $(N-k)^{th}$ tone in DFT/FFT. ...


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