2 corrected formula symbol
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The Fourier transform of a sine signal is a Dirac impulse in frequency domain. But for the sampled sine signal you won't get the expected Dirac impulse in most cases (1) - even if you don't violate the sampling theorem. This is because your sampled time sequence must be finite in length, which means that the original sine signal is implicitly multiplicated with a rectangular impulse in time domain. This corresponds to a convolution with a $sin(x)/x$ function in frequency domain and is referred to as windowing effect.

This observation is reflected in the constraint to the sampling theorem: the original signal can be reconstructed perfectly from its sampled version if the sampling frequency is higher than double the highest frequency in the band-limited analog signal and if an inifinite number of samples is drawn from the analog signal.

(1) Let $x(n) = sin(\alpha n)$ be the sampled sine signal. The discrete Fourier transform will be a Dirac impulse if $\alpha = k \Delta \omega$, where $\Delta \omega = \frac{2\pi}{N}$ and $N$ is the number of samples. This is because in this case it happens that you sample the $sin(x)/x$ shaped frequency spectrum at its zero crossings (except for the fundamental frequency). If you use the discrete time Fourier transform, however, you will always see the $sin(x)/x$ spectrum.

The Fourier transform of a sine signal is a Dirac impulse in frequency domain. But for the sampled sine signal you won't get the expected Dirac impulse in most cases (1) - even if you don't violate the sampling theorem. This is because your sampled time sequence must be finite in length, which means that the original sine signal is implicitly multiplicated with a rectangular impulse in time domain. This corresponds to a convolution with a $sin(x)/x$ function in frequency domain.

This observation is reflected in the constraint to the sampling theorem: the original signal can be reconstructed from its sampled version if the sampling frequency is higher than double the highest frequency in the analog signal and if an inifinite number of samples is drawn from the analog signal.

The Fourier transform of a sine signal is a Dirac impulse in frequency domain. But for the sampled sine signal you won't get the expected Dirac impulse in most cases (1) - even if you don't violate the sampling theorem. This is because your sampled time sequence must be finite in length, which means that the original sine signal is implicitly multiplicated with a rectangular impulse in time domain. This corresponds to a convolution with a $sin(x)/x$ function in frequency domain and is referred to as windowing effect.

This observation is reflected in the constraint to the sampling theorem: the original signal can be reconstructed perfectly from its sampled version if the sampling frequency is higher than double the highest frequency in the band-limited analog signal and if an inifinite number of samples is drawn from the analog signal.

(1) Let $x(n) = sin(\alpha n)$ be the sampled sine signal. The discrete Fourier transform will be a Dirac impulse if $\alpha = k \Delta \omega$, where $\Delta \omega = \frac{2\pi}{N}$ and $N$ is the number of samples. This is because in this case it happens that you sample the $sin(x)/x$ shaped frequency spectrum at its zero crossings (except for the fundamental frequency). If you use the discrete time Fourier transform, however, you will always see the $sin(x)/x$ spectrum.

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The Fourier transform of a sine signal is a Dirac impulse in frequency domain. But for the sampled sine signal you won't get the expected Dirac impulse in most cases (1) - even if you don't violate the sampling theorem. This is because your sampled time sequence must be finite in length, which means that the original sine signal is implicitly multiplicated with a rectangular impulse in time domain. This corresponds to a convolution with a $sin(x)/x$ function in frequency domain.

This observation is reflected in the constraint to the sampling theorem: the original signal can be reconstructed from its sampled version if the sampling frequency is higher than double the highest frequency in the analog signal and if an inifinite number of samples is drawn from the analog signal.