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Sampling Theory

The dual of the Poisson Summation Formula is the continuous-time aliasing theorem, which lies at the foundation of elementary sampling theory [264, Appendix G]. If $ x(t)$ denotes a continuous-time signal, its sampled version $ x(nT)$ , $ n\in{\bf Z}$ , is associated with the continuous-time signal

$\displaystyle x_d(t) \isdefs x(t)\psi_T(t) \isdefs x(t)\sum_m\delta(t-mT).$ (B.60)

where $ T$ denotes the (fixed) sampling interval in seconds. The sampled signal values $ x(nT)$ are thus treated mathematically as coefficients of impulses at the sampling instants. Taking the Fourier transform gives

\begin{eqnarray*} X_d(f) &=& \hbox{\sc FT}_f(x\cdot\psi_T) \eqsp X\ast \Psi_T\\ &=& \frac{1}{T}X\ast \psi_{1/T} \eqsp f_s\sum_{k=-\infty}^{\infty}X(f-kf_s) \end{eqnarray*}

where $ f_s\isdef 1 f_s\isdef 1/T$ denotes the sampling rate in radians per second. Note that $ X_d(f)$ is periodic with period $ f_s$ . We see that if $ X(f)$ is bandlimited to less than $ f_s$ radians per second, i.e., if $ X(f)=0$ for all $ \vert f\vert\geq f_s/2$ , then only the $ k=0$ term will be nonzero in the summation over $ k$ , and this means there is no aliasing. The terms $ X(f-kf_s)$ for $ k\neq 0$ are all aliasing terms.

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Poisson Summation Formula