Sampling and the Sampling Theorem

date2026-07-14document_iddoc_39a00152c4e334961250e32dbdb8c3c5descriptionインパルス列による標本化、スペクトルの反復、標本化定理、エイリアシング、アンチエイリアスフィルタを説明する。prerequisitesフーリエ変換の基本 / 通信工学の基本type講義statusactive

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1Introduction

This lecture represents uniform sampling of a continuous-time signal as multiplication by an impulse train and derives reconstruction conditions and aliasing from the resulting spectral replicas.

2Uniform Sampling

SamplingSampling obtains values of a continuous-time signal at discrete times. Let $T_{s} > 0$ be the sampling period and let the sampling frequencySampling frequency be $f_{s} = 1 / T_{s}$ . The sample sequence is

x [n] = x (n T_{s}), n \in Z .

$T_{s}$ is measured in seconds and $f_{s}$ in hertz.

The Dirac deltaDirac delta $δ (t - t_{0})$ is a distribution rather than an ordinary function. For every smooth, compactly supported test function $φ$ , it has the sifting property

\int_{- \infty}^{\infty} φ (t) δ (t - t_{0}) d t = φ (t_{0}) .

Using this idealization, represent uniform sampling in continuous time by the impulse train of period $T_{s}$ ,

p (t) = \sum_{n = - \infty}^{\infty} δ (t - n T_{s}) .

The sampled signal is

x_{s} (t) = x (t) p (t) = \sum_{n = - \infty}^{\infty} x (n T_{s}) δ (t - n T_{s}) .

This is an ideal model. Finite acquisition time and quantization in a physical analog-to-digital converter are separate effects.

3Spectral Replicas

The Fourier transform of the impulse train is

P (f) = f_{s} \sum_{k = - \infty}^{\infty} δ (f - k f_{s}) .

Because multiplication in time corresponds to convolution in frequency,

X_{s} (f) = X (f) * P (f) = f_{s} \sum_{k = - \infty}^{\infty} X (f - k f_{s}) .

Copies of $X (f)$ therefore repeat at intervals of $f_{s}$ . The spacing of the replicas is $f_{s}$ , while the Nyquist frequencyNyquist frequency $f_{s} / 2$ is the conventional boundary of the baseband. It is distinct from the Nyquist rateNyquist rate $2 B$ associated with a band limit $B$ . Uniqueness at this boundary requires care, as discussed below.

4Sampling Theorem

Assume that $x (t)$ is a continuous finite-energy signal that is band-limited and satisfies

X (f) = 0 (| f | > B) .

If, in addition, $f_{s} > 2 B$ , adjacent spectral replicas remain separated by $f_{s} - 2 B > 0$ and do not overlap. An ideal low-pass filter can select the central replica and apply gain $T_{s}$ , recovering $X (f)$ and hence $x (t)$ uniquely.

Thus, for a strictly band-limited continuous-time signal,

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is a sufficient condition for perfect reconstruction from ideal samples. This statement assumes an infinite-duration signal, exact sample values, and an ideal reconstruction filter.

Equivalently, with $sinc (u) = \sin (π u) / (π u)$ , reconstruction can be written as the sinc interpolationSinc interpolation formula

x (t) = \sum_{n = - \infty}^{\infty} x (n T_{s}) sinc (\frac{t - n T_{s}}{T_{s}}) .

This infinite sum expresses the same ideal low-pass reconstruction.

5The Boundary $f_{s} = 2 B$

At $f_{s} = 2 B$ , adjacent spectral replicas touch at $f = \pm B$ . For the finite-energy signals considered in the preceding section, values at isolated endpoints do not change a signal as an $L^{2}$ function, so this contact alone does not establish non-uniqueness. The boundary nevertheless requires care in engineering signal models that also include power signals. For example, $x (t) = \sin (2 π B t)$ lies outside the finite-energy assumption; if it is sampled at $f_{s} = 2 B$ with every sampling instant at a zero crossing, every sample is zero. This lecture therefore adopts $f_{s} > 2 B$ as a conservative engineering guarantee that avoids ambiguity at the endpoints.

6Aliasing and the Anti-Aliasing Filter

When $f_{s} < 2 B$ , or when the input contains components above $f_{s} / 2$ , spectral replicas overlap. A component exactly at $f_{s} / 2$ is treated as the boundary case described in the preceding section. AliasingAliasing is the phenomenon in which this overlap makes distinct continuous-time frequencies appear as the same sample sequence. Sinusoids at $f$ and $f + k f_{s}$ , for example, cannot be distinguished at the sampling instants.

Once spectra overlap after sampling, their original components cannot generally be separated. An analog anti-aliasing filterAnti-aliasing filter is therefore placed before analog-to-digital conversion to attenuate components at and above $f_{s} / 2$ sufficiently. Because a physical filter has a transition band, practical systems leave a guard band between the signal bandwidth and $f_{s} / 2$ .

7Key Points

Sampling with an impulse train produces spectral replicas spaced by $f_{s}$ in the frequency domain.
Under $X (f) = 0$ for $| f | > B$ and $f_{s} > 2 B$ , ideal perfect reconstruction is possible.
Treatment of $f_{s} = 2 B$ depends on the signal class and endpoint conditions; a conservative engineering guarantee uses a strict inequality.
Aliasing cannot generally be removed after sampling, so pre-sampling filtering and a sufficiently high sampling frequency are necessary.

8Next Lecture

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