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As the OP has described, the noise as shown is AM noise only as the constellation points are not changing in phase at all. A double sided spectrum with symmetric sidebands in phase with the carrier is amplitude noise: Consider a reference carrier as 1 at angle 0 degrees for which we do the following to create sinusoidal sideband noise by adding to it a small ...


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Let the complex frequency response of an LTI system be $$H(\omega)=M(\omega)e^{j\phi(\omega)}\tag{1}$$ with magnitude $M(\omega)$ and phase $\phi(\omega)$. If the input to such a system is $x(t)=A\sin(\omega_0t+\theta)$, then its output is given by $$y(t)=AM(\omega_0)\sin\big(\omega_0t+\theta+\phi(\omega_0)\big)$$ The quantity $\phi(\omega_0)$ is called the ...


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Phase shift is often confused with time delay, although related these are two different quantities. Phase shift specifically refers to the rotation of a complex number. For example given the complex number $x = Ae^{j\theta}$ which through Euler's formula has real and imaginary components given as $A\cos(\theta)+jA\sin(\theta) = I + jQ$, a phase shift of $\...


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This post provides many details on implementing carrier recovery for a higher order QAM system (or even QPSK). High modulation index PSK - carrier recovery Using a PI loop filter where in addition to a simple accumulator, a direct (proportional) path is also summed in results in a 2nd order Type 2 tracking loop as required to track phase offsets to zero ...


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First, you want to find the peaks in this 2D plot. In your case, you know that there will be 2 peaks in your FFT (because you added two sines). Your plot shows 4 peaks because the FFT magnitude is symmetric, since your signal is real. You can discard negative wave numbers and look for peaks among the positive wave numbers. The wave numbers corresponding to ...


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Subspace-based spectrum estimation methods give you a high resolution pseudospectrum that is used for frequency detection only. The complex amplitude (that includes the phase) can be calculated with linear least-squares once your frequency estimates are available. In MUSIC and root-MUSIC, You calculate the autocorrelation matrix of the sensor signals. ...


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The basis function for the Fourier Transform is the complex exponential $e^{j\omega t}$ Since $$sin(\omega t) = \frac{1}{2j}(e^{j\omega t} - e^{-j\omega t}) $$ the Fourier Transform of a sine wave a phase of -90 degrees at the positive frequency and a phase of +90 degrees at the negative frequency. For a cosine, the phase would be 0 for both the positive and ...


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Simple Fourier Transform propery for differentiation is $$\mathcal{F}\{\frac{\partial}{\partial t}x(t)\}=j\omega \cdot\mathcal{F}\{x(t)\}$$ Differentiation in time corresponds to multiplication with $j\omega$ in frequency. Hence the +6dB/octave slope. The phase shift is a constant 90 degrees for all frequencies.


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Its frequency response can be decomposed into a real part and an imaginary part as $$ H(e^{j\omega}) = 1-e^{-j\omega} = 1-(\cos\omega - j\sin\omega) = (1-\cos\omega) + j\sin\omega $$ The magnitude response is the modulus of the complex frequency response $$ |H(e^{j\omega})| = \sqrt{\mathcal{Re}\{H(e^{j\omega})\}^2 + \mathcal{Im}\{H(e^{j\omega})\}^2} = \sqrt{(...


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There are many effects that take part in forming the signal distribution. The multitude of references are popped up when searching with keywords "signal phase uniform distribution", and many are about the communication channel effects -- for the most part, about the Rayleigh fading in the channel. I think the channel impulse response is the subject ...


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Do an FFTShift (rotate the data halfway) before doing an FFT for phase analysis. That will re-reference the measured phase to the center of your data, not to potential circular discontinuities at the ends. Any discontinuities near the phase reference point (due to any non-integer-periodic-in-aperture frequencies) will corrupt phase interpolation. Do not &...


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What other "modern" methods exist for accurate phase extraction? Unless the frequencies in the signal are phase locked to your sampling clock an FFT is not a great way to determine either frequency or phase of a sinusoidal component. In many cases a Phase Locked Loop (PLL) or Delay Lock Loop works much better. You can use an FFT to quickly ...


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A zero mean complex white noise process has a uniform phase distribution. The uniform phase distribution would occur for any complex noise process whose I and Q (real and imaginary) components are zero-mean, independent and identically distributed. The thermal noise floor common to radio receivers for receiving signals without other specific interference is ...


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Using an FFT to measure phase for just two tones results in a lot more processing that doing the following alternate approach that can be either streamed or processed in blocks. No windowing is needed: Apply the received signal as the input to two multipliers. Apply a normalized local copy of one tone as the second input to one of the multipliers and a ...


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