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Let $x[n]$ be a discrete signal of 2 samples. We know that its DFT with N=4 is $X[k]=[0, 1+j, 2, 1-j]$. Without calculating $x[n]$, how can we know the DFT with N=2?

I have tried to use the relation between DFT and DTFT. Specifically, that the DFT is the result of sampling the DTFT in the frequency domain every $\frac{2\pi k}{N}$ samples. However, I don't know how to continue. I would really appreciate some help.

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$$X_4[0] = x[0] + x[1]$$ $$X_4[2] = x[0] - x[1]$$

Now $$X_2[0] = x[0] + x[1]$$ $$X_2[1] = x[0] - x[1]$$

Therefore $$X_2[0] = X_4[0]$$ $$X_2[1] = X_4[2]$$

An alternative explanation is the following, if we were to decimate in the frequency domain by 2 (the 4 point FFT), you would incur circularly shifted Aliasing in discrete time domain. Since we started with a two point sequence then zero padded it to get a 4 point sequence, this decimation will result in Aliasing at those zero stuffed places leaving your original sequence non aliased. So say if we were to decimate more than 2 the sequence would incur Aliasing in time domain. So the fact that we are still able to have the original 2 point sequence non aliased after the decimation, therefore the FFT values of this 4 point FFT still carries the information about the 2 point FFT

Ex: if say we would have a 4 point sequence and you zero pad to get a 16 point sequence and take the FFT, now if you decimate the FFT by a factor of 2 you would get Aliasing in samples 8 to 15 of the time domain sequence.

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Simple explanation:

Zero padding in one domain is simply interpolation in the other. Doing a 4-point DFT over a 2-point sequence can be interpreted as zero padding to 4 points. Hence the 4-point DFT is just an up-sampled version of the 2-point DFT and simple decimation will yield the 2-point DFT.

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The N-point DFT corresponds to sampling the DTFT at frequencies $0$, $\frac{2\pi}{N}$, ..., $\frac{2\pi N-1}{N}$. Thus the 4-point DFT in question corresponds to the values of DTFT at frequencies $0$, $\frac{2\pi}{4}$, $\pi$ and $\frac{3\pi}{4}$. Now we are interested in the 2-point DFT which are simply the values of DTFT at frequencies $0$, $\frac{2\pi}{2} = \pi$. Hence, the 2-point DFT values are 0, 2

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