I read some sentence that said "It is usually assumed that the multi-path channel has a constant gain and linear phase within the coherence bandwidth". I can't understand what it mean " linear phase ". To my best of knowledge, difference path delay would caused difference phase shift in the receiving signal, but it's would be rarely that different path delay is linear, or it should be said that there is almost no such situation. Therefore, I am very curious about what it mean about "linear phase". Can someone please explain it? Thanks in advanced.
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3 Answers
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For the classical baseband channel model for multipath, $h(t)=\sum_ka_k\delta(t-\tau_k)$, the frequency response $H(f)=\sum_k a_ke^{-j2\pi f\tau_k}$. If $K=1$, single tap the phase would have been linear $\phi(f)=2\pi f\tau_k$. But for different path incident on the receiver, there is no guarantee that phase is linear. But let us see how phase changes between first path $k$ and last path $p$. For these incident paths, $2\pi f\tau_k-2\pi f\tau_p \approx \pi$ would cause fading. So if your frequency $f$ changes from $f_0$ to $f_0 +\frac{1}{2(\tau_k-\tau_p)}$, you will encounter fading. Hence $f_w=\frac{1}{2(\tau_k-\tau_p)}$ is called Coherence Bandwidth. It is the amount of bandwidth for which frequency response is 'flat' or constant. If the delay spread $\Delta \tau = (\tau_k-\tau_p)$ was less, $H(f)$ would be varying less over frequencies. If the difference in delay of paths was much much smaller than symbol time, that is, $\Delta \tau << T$, the width of the channel would be much much greater than width of symbols. That is $\frac{1}{\Delta\tau} >> \frac{1}{T}$. So your transmission data would see a 'flat' or constant channel in terms of frequency response. A constant channel in frequency corresponds to a single tap as explained earlier. It's phase response is linear $\phi(f)=2\pi f\tau_k$.
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When we talk about linear phase in an LTI system we are talking about the phase of the FFT of the time domain system response. Thus the linearity is with respect to the frequency of the signal and not time shifts. This is where I think you have the confusion.
The phase is linear in $\omega$, the frequency.
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Path delay is not linear but phase is linear. Path delay is negative derivative of phase. So, if phase is linear then delay is constant because derivative is constant.
This is expected within Coherence Bandwidth, because we expect channel to be flat within Coherence Bandwidth. Flat channel means no distortion of the input signal, which means, the channel just results into some attenuation and a constant group delay. Basically, if you confine your input signal within the Coherence Bandwidth, then you input will be suffering only from Flat Fading, but no distortion due to Multi-path Fading.
This also means that the channel will behave as a Single Tap channel, of the form $\alpha \delta[n-n_o] $ in time domain, and $\alpha e^{-j\omega n_o}$ in Frequency domain. $\alpha$ is the magnitude attenuation given by channel and $n_o$ samples delay in Time domain.
So, in frequency domain, check that the phase is linear, i.e. $-\omega.n_o$, and negative derivative of this phase is actually the constant delay of $n_o$ samples in time domain.
