New answers tagged phase
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Equalizing a channel by inverting its frequency response is NOT recommended. This is because only systems that have a minimum phase response (all zeros inside the unit circle) have a stable causal inverse. For example, any channel with leading and trailing echos (most wireless channels) are a mixed phase system so therefore cannot simply be inverted.
The ...
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I actually don't see how the following diagram can be possibly used for phase correction only as it will have an unavoidable amplitude correction as well. An all pass filter by definition will pass all magnitude without modification and only change the phase. The Time Delay block will similarly pass all magnitude without modification with a linear phase ...
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As explained in the comments for previous versions of this question, a matched filter alone (using the cross-correlation specifically which can be done efficiently with FFT's) should be used with caution for purposes of estimating time delay. The reason why has been detailed in this other post linked below along with a robust least-square solution for ...
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For a constant frequency signal, with no frequency offset between transmitter and receiver, the phase of a complex correlation, at the correlation peak, is the phase difference between the received signal and the reference signal used for the matched filter. It need not be 0. And that is for the actual correlation peak, which may have to be obtained by ...
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As @hotpaw2 says, the FFT assumes periodic boundary conditions, that is the next point on the right should be equal to the first point on the left. It's not the case in your code because the time $t=10$ is included in your time array. The next point on the right is thus $t=10+dt$ and you get the discontinuity because $y(10+dt) \neq y(0)$. You can fix this by ...
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FIR filters that have coefficients symmetric about their center coefficient(s) are linear phase. Digital pulse filters commonly in use, including a Root-Raised-Cosine filter, are FIR and have this property, so they will have will have a linear phase response.
For all digital modulations, I would avoid non-linear phase response filters.
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The principle behind FMCW is that you transmit a chirped signal and receive a time delayed version of it after reflecting from a target. After mixing and filtering, the resulting signal is a sinusoid at a frequency that is a function of the target's range. This frequency is known as the "beat" frequency $f_b$. Thus, the dechirped signal will have a form of
$...
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If there are phase discontinuities after all $2\pi$-jumps have been removed, then these discontinuities are usually caused by zeros of the frequency response. The phase jumps by $\pi$ at these frequencies, and the group delay doesn't exist, or, if one prefers, is non-finite. Note that the group delay is meaningless anyway at frequencies where the frequency ...
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First of all you see that the phase is a piecewise linear function, so it's a linear phase FIR filter. There's a phase jump at half the Nyquist frequency, which shows that the filter has a zero at that frequency. Note that the phase jumps by $\pi$ corresponding to sign inversion.
You can also see what type of linear phase FIR filter it is. Since the phase ...
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You would need to recover some timing information from the transmitted signal.
A common way is to transmit some sort of preamble before the rest of the data in your frame that is transmitted. The preamble would contain known symbols, so it could be used for multiple purposes, including channel estimation and timing recovery (also known as synchronization).
...
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You need to synchronize phase first, otherwise QAM decision is not possible.
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The ratio
$$\frac{\textrm{Im}\{H(u,v)I(u,v)\}}{\textrm{Re}\{H(u,v)R(u,v)\}}\tag{1}$$
is only independent of $H(u,v)$ if $H(u,v)$ is real-valued, which is exactly the condition for a zero-phase filter. The phase can only be zero if the frequency response is real-valued, as you've already noted.
For real-valued $H(u,v)$ we have
$$\frac{\textrm{Im}\{H(u,v)I(...
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Filters that affect the real and imaginary parts equally, and thus
have no effect on the phase, are appropriately called zero-phase-shift
filters."
The filter itself is zero-phase, but the input (and therefore the output) waveform can be complex. In this case the real and imaginary components of the input would be effected equally since the filter is ...
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