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To ensure smooth continuity we generate $$x_0 = \cos(\omega_0 t), \ t_0 \leq t < t_1 \\ x_1 = \cos(\omega_1 t + t_1 \cdot (\omega_0 / \omega_1)), \ t_1 \leq t < t_2 \\$$ where $\omega_0 t$ is input phase for $x_0$. Example It cannot be perfectly smooth for any arbitrary $t_1$ per discretization limitations (but $t_1$ can be chosen to maximize ...

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Generate the second sine wave as $$x(t) = sin(\omega_2 t + \omega_1t_0)$$ where $t_0$ is the time when the frequency changes from $\omega_1$ to $\omega_2$ .

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Easy ones first: Ramp: Position when moving through space at a constant rate. Or modulo position from a closest 1 foot (or 1 mile, or 1 inch, or 1 meter etc) boundary when moving through space at a constant rate for a repeating ramp. Sine wave: The horizontal displacement of a spinning wheel at a fixed location. The shadow of a stick spinning in the air with ...

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The key here is that A signal that's discrete in one domain is periodic in the other (and vice versa) The opposite holds as well: i.e. a signal that continuous in one domain is aperiodic in the other. That gives a total of four possible types of signals and hence there are four different flavors of the Fourier Transform Time: continuous & aperiodic + ...

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A full answer would require half a textbook, but here are some pointers about aspects not covered in the other answers. The diagram shown in your question is known as a "train" (or sequence) of orthogonal pulses, and can be written as $$\sum_k a_k g(t-kT),$$ where $a_k \in \{0, A\}$ is the pulse amplitude (which carries the actual information), $T$...

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Although a sample is often described as representing the signal value at a single point in time, it's important to remember that this is not entirely realistic, and why it's a perfectly reasonable simplification. If you actually want to sample any weighted distribution of points around each sample time, instead of a single value at the exactly sample time, ...

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Constructing a constellation diagram is done in the transmitter (generation of the signal) while the OP's questions have to do with sampling a constellation already constructed (receiver). Perhaps the question is mistitled? (1) Now for the phase modulation case, do we do the same in practice? Do we extract the phase at a single point at the center of the ...

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