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The conventional definition of the DFT for a length $N$ signal (without zero-padding) is $$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\tag{1}$$ So there is no scaling involved. Scaling is applied to th …

For a) you're correct. For b), $x_1$ is a length $2N$ signal, and its DFT is given by $$X_1[k]=\sum_{n=0}^{2N-1}x_1[n]e^{-j2\pi kn/2N}=\sum_{n=0}^{2N-1}x_1[n]e^{-j\pi kn/N}\tag{1}$$ With $x_1=x[n]+x …

This is just about obtaining a symmetric signal after zero-padding. Take a symmetric signal (w.r.t. to time index $n=0$) and append zeros. Due to the implicit periodicity of the time signal used as in …

All effects you see have to do with windowing. Your signal can be seen as a truncated (i.e., rectangularly windowed) sinusoid. If $s[n]$ is your signal, and $w[n]$ is the window, the signal you analyz …

The clue is that without zero-padding, the circulary shifted sequence is just a shifted version of the periodic continuation of the original sequence. That's why without zero-padding the magnitudes of …

One easy way to understand interpolation in the time domain by zero-padding in the frequency domain is to realize that all interpolated sequences can be derived from sampling a single periodic continu …

The obvious reason why the DFTs of the x and y signals are different is because the signals themselves are different. Different versions of zero-padding result in different DFTs. What you probably wan …