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TheA very nice overview paper is Unser: Sampling - 50 years after Shannon. Your problem arises from the fact that pure, infinite sine signals are not covered by the Shannon sampling theorem. The applicable theorem for periodic signals is the earlier Nyquist sampling theorem.


The Shannon sampling theorem applies to functions that can be represented as $$ x(t)=\int_{-W}^WX(f)e^{i2\pi ft}\,df $$ where

\$ x(t)=\int\limits_{-W}^WX(f)e^{i2\pi ft}\,df \$

where X is a square-integrable function. Then this signal can be exactly represented from discrete samples as

\$ x(t)=\sum\limits_{k=-\infty}^\infty x(k\frac{T}2)\frac{\sin(\pi W(t-k\frac{T}2))}{\pi W(t-k\frac{T}2)} \$

with \$T=\frac1{W}\$ a "period". Note that the perfect reconstruction depends on samples from arbitrarily large times in the future and past. As their influence only falls as \$\frac1t\$, truncating the sum has to include a rather large number of terms to reduce errors.

A pure sine function is not contained in that class, as its Fourier transform is composed of Dirac-delta distributions.


The earlier Nyquist sampling theoremNyquist sampling theorem states (or re-interprets an earlier insight) that if the signal is periodic with period T and highest frequency W=N/T, then it is a trigonometric polynomial $$ x(t)=\sum_{n=-N}^NX_ne^{i2\pi\frac{n}{T}t} $$ with

\$ x(t)=\sum\limits_{n=-N}^NX_ne^{i2\pi\frac{n}{T}t} \$

with 2N+1 (non-trivial) coefficients and these coefficients can be reconstructed (by linear algebra) from 2N+1 samples in the period.

The case of a pure sine function falls in this class. It promises perfect reconstruction if 2N+1 samples over a time NT are taken.

The Shannon sampling theorem applies to functions that can be represented as $$ x(t)=\int_{-W}^WX(f)e^{i2\pi ft}\,df $$ where X is a square-integrable function. Note that a pure sine function is not contained in that class.


The earlier Nyquist sampling theorem states (or re-interprets an earlier insight) that if the signal is periodic with period T and highest frequency W=N/T, then it is a trigonometric polynomial $$ x(t)=\sum_{n=-N}^NX_ne^{i2\pi\frac{n}{T}t} $$ with 2N+1 (non-trivial) coefficients and these coefficients can be reconstructed (by linear algebra) from 2N+1 samples in the period.

The case of a pure sine function falls in this class. It promises perfect reconstruction if 2N+1 samples over a time NT are taken.

A very nice overview paper is Unser: Sampling - 50 years after Shannon. Your problem arises from the fact that pure, infinite sine signals are not covered by the Shannon sampling theorem. The applicable theorem for periodic signals is the earlier Nyquist sampling theorem.


The Shannon sampling theorem applies to functions that can be represented as

\$ x(t)=\int\limits_{-W}^WX(f)e^{i2\pi ft}\,df \$

where X is a square-integrable function. Then this signal can be exactly represented from discrete samples as

\$ x(t)=\sum\limits_{k=-\infty}^\infty x(k\frac{T}2)\frac{\sin(\pi W(t-k\frac{T}2))}{\pi W(t-k\frac{T}2)} \$

with \$T=\frac1{W}\$ a "period". Note that the perfect reconstruction depends on samples from arbitrarily large times in the future and past. As their influence only falls as \$\frac1t\$, truncating the sum has to include a rather large number of terms to reduce errors.

A pure sine function is not contained in that class, as its Fourier transform is composed of Dirac-delta distributions.


The earlier Nyquist sampling theorem states (or re-interprets an earlier insight) that if the signal is periodic with period T and highest frequency W=N/T, then it is a trigonometric polynomial

\$ x(t)=\sum\limits_{n=-N}^NX_ne^{i2\pi\frac{n}{T}t} \$

with 2N+1 (non-trivial) coefficients and these coefficients can be reconstructed (by linear algebra) from 2N+1 samples in the period.

The case of a pure sine function falls in this class. It promises perfect reconstruction if 2N+1 samples over a time NT are taken.

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The Shannon sampling theorem applies to functions that can be represented as $$ x(t)=\int_{-W}^WX(f)e^{i2\pi ft}\,df $$ where X is a square-integrable function. Note that a pure sine function is not contained in that class.


The earlier Nyquist sampling theorem states (or re-interprets an earlier insight) that if the signal is periodic with period T and highest frequency W=N/T, then it is a trigonometric polynomial $$ x(t)=\sum_{n=-N}^NX_ne^{i2\pi\frac{n}{T}t} $$ with 2N+1 (non-trivial) coefficients and these coefficients can be reconstructed (by linear algebra) from 2N+1 samples in the period.

The case of a pure sine function falls in this class. It promises perfect reconstruction if 2N+1 samples over a time NT are taken.