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I read from material that coherent integration can be provided by DFT. But it doesn't give further explanation on this remark. As far as I know, coherent integration is the summation of discrete samples with phase considered and it is usually used in time domain. How to understand that DFT's capability of providing coherent integration?

Thanks.

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Rick's answer concerns trying to combine the results of two (or more) DFT's coherently. I'm interpreting your question differently - does a single DFT provide coherent integration? If you are trying to detect a sinusoid then yes the DFT provides Coherent Integration. Consider a signal consisting of a complex sinusoid of amplitude 1 and zero mean white Gaussian Noise with a variance of 1. The input signal is given by $$x(n) = \exp(j2\pi mn/N) +w(n),$$ where $w(n)$ is the noise and $m$ is an integer and therefore the frequency will correspond exactly with one of the basis vectors of the FFT or equivalently this means the frequency is exactly on one of the FFT frequency bins. Looking at the SNR of the input signal, the signal power is equal to 1 and the noise power is equal to 1 (by the given conditions). There for the SNR = 1.

Given the definition of the FFT as $$X(k)= \sum_{n=0}^{N-1}x(n)e^{-j2\pi kn/N}$$ we can examine the resulting SNR in a single FFT bin. For the signal only case $X(k)=0$ except when $m=k$ then we have $$ X(m)=\sum_{n=0}^{N-1}e^{j2\pi mn/N}e^{-j2\pi mn/N}=\sum_{n=0}^{N-1}1=N$$ then the signal power is $|(X(m))|^2=N^2$.

For the noise power in a FFT bin we use the expected value or variance i.e. then letting $W(k)$ denote the FFT of the noise only signal - then since W(k) is a linear combination of zero mean Gaussian variables then $W(k)$ is also a zero mean Gaussian variable and the variance is the sum of the individual variances in each term.

$$var[W(K)] = \sum_{n=0}^{N-1} var \left[w(n)e^{-j2\pi kn/N} \right] =var[w(n)] \sum_{n=0}^{N-1} |e^{-j2\pi kn/N}|^2= N \cdot var [w(n)] = N$$

where we have used the scaling properties of the variance and the fact that the noise is zero mean. The SNR in the FFT bin is $N^2/N=N$. This is the coherent gain of the FFT when the sinusoid lies exactly on an FFT bin. You can also view FFT as a bank of Matched Filters for a sinusoidal input and look at the theory of Matched Filters.

Note - the analysis can be generalized for any zero mean and independent noise process, since the scaling property still holds for the variance. It's just a bit quicker and easier to do for a Gaussian noise process.

In the case when the signal is not exactly on a FFT bin there is a processing loss, this is commonly referred to as Scalloping Loss. You can read more about this in the classic paper on Windows by Fred Harris.

Notice that if you average the magnitude of sequential FFTs there is no coherent gain, instead you are reducing the variance of the estimate caused by the noise. For white noise if you look at the magnitude of a single FFT then there is quite a variation across all the frequency bins. Once you start averaging the magnitude or magnitude squared then the amplitude will show reduced variation and will approach a constant value. This is referred to as non-coherent processing because by using the Amplitude you are losing the phase information required for coherent processing.

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@ecook: Your time-domain notion of coherent integration is correct. (Where multiple periodic sequences must be time-synchronized before the averaging operation.)

The only way I could justify the remark that "coherent integration can be provided by DFT" is in the following scenario: let's say your DFT's input is an $x1(n)$ sinusoidal sequence and your DFT result is the complex valued $X1(m)$ sequence. Then some moments later you collect a new $x2(n)$ time sequence of the same sinusoidal signal and produce a new complex-valued DFT result of $X2(m)$. To average (integrate) those $X1(m)$ and $X2(m)$ sequences you would add their magnitudes. That's a form of incoherent averaging (incoherent integration), also called "scalar averaging." (Averaging $X1(m)$'s and $X2(m)$'s real and imaginary parts individually may lead to spectral energy cancellation and you don't want that to happen.)

If, on the other hand, your $x2(n)$ time sequence was time-synchronized with the previous $x1(n)$ time sequence you can add $X1(m)$'s and $X2(m)$'s real and imaginary parts individually. That's a form of coherent averaging (coherent integration), also called "time-synchronous averaging."

Know this however, as Peter Kootsookos points out, averaging phase angles is tricky. Have a look at Peter's explanation at: http://www.dsprelated.com/showarticle/57.php

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