The NyquistShannon sampling theorem concerns signals with continuous time Fourier transforms that are only nonzero on the interval (B,B)(B,B) for some constant BB. Such a function is said to be bandlimited to (B,B)(B,B). Essentially, the sampling theorem has already been implicitly introduced in the previous module concerning sampling. Given a continuous time signals xx with continuous time Fourier transform XX, recall that the spectrum XsXs of sampled signal xsxs with sampling period TsTs is given by
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(1)It had previously been noted that if xx is bandlimited to (π/Ts,π/Ts)(π/Ts,π/Ts), the period of XsXs centered about the origin has the same form as XX scaled in frequency since no aliasing occurs. This is illustrated in Figure 1. Hence, if any two (π/Ts,π/Ts)(π/Ts,π/Ts) bandlimited continuous time signals sampled to the same signal, they would have the same continuous time Fourier transform and thus be identical. Thus, for each discrete time signal there is a unique (π/Ts,π/Ts)(π/Ts,π/Ts) bandlimited continuous time signal that samples to the discrete time signal with sampling period TsTs. Therefore, this (π/Ts,π/Ts)(π/Ts,π/Ts) bandlimited signal can be found from the samples by inverting this bijection.
This is the essence of the sampling theorem. More formally, the sampling theorem states the following. If a signal xx is bandlimited to (B,B)(B,B), it is completely determined by its samples with sampling rate ωs=2Bωs=2B. That is to say, xx can be reconstructed exactly from its samples xsxs with sampling rate ωs=2Bωs=2B. The angular frequency 2B2B is often called the angular Nyquist rate. Equivalently, this can be stated in terms of the sampling period Ts=2π/ωsTs=2π/ωs. If a signal xx is bandlimited to (B,B)(B,B), it is completely determined by its samples with sampling period Ts=π/BTs=π/B. That is to say, xx can be reconstructed exactly from its samples xsxs with sampling period TsTs.
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