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Sampling: review

Let us consider a signal x ( t ) which is uniformly sampled meaning that the samples are x n = x ( n τ ) ( n = ... - 2 , - 1 , 0 , 1 , 2 , ... ). The sampling frequency f e = 1 / τ is the number of samples per time unit.

In practical applications, the signal will be sampled only over an interval, say of length L and into K samples. Then τ = L / K as for discrete signals.

Spectral copies (also called spectral repetitions or images)

The sampled signal can be written as the Dirac comb of step τ modulated by x ( t )

x e ( t ) = x ( t ) · τ Δ τ ( t ) = n = - τ x ( n τ ) · δ ( t - n τ )

The factor τ is added to obtain a Fourier transform that does not depend on τ and to preserve an averaging property (the integral of x e provides a good approximation of the integral of x since x e ( t ) d t = n = - τ x ( n τ ) is the Riemann sum approximating the x ( t ) d t as a sum of rectangles of width τ .)

Using [link] we may write

x e ( t ) = x ( t ) · τ Δ τ ( t ) = k = - x ( t ) · e j 2 π k t / τ

Note, that [link] is not a Fourier representation of x e ( t ) (the “coefficients” x ( t ) of the sinusoids depend on t instead of k ). Computing the Fourier transform of the sampled signal x e ( t ) once more, using [link] , we find

X e ( f ) = k = - F x ( t ) e j 2 π k t / τ ( f ) = k = - X ( f - k / τ )

We observe that the Fourier transform, and also the spectrum of the sampled signal is periodic with period f e = 1 / τ , the sample rate (see [link] right). However, we see now that X e is composed of copies of X , shifted by a multiple of the period, i.e., shifted by k / τ = k · f e . These are called spectral copies .

Sampling and the discrete Fourier transform

Finite energy signals: Using [link] again we find the Fourier transform of the sampled signal x e ( t )

X e ( f ) = n = - τ x ( n τ ) F δ ( t - n τ ) ( f ) = τ n = - x n · e - j 2 π f n τ

Setting now f = k / L and recalling τ = L / K we get approximatively

finite energy: X e ( k / L ) τ n = - K / 2 K / 2 x n · e - j 2 π ( k / L ) n ( L / K ) = τ x ˆ k .

This agrees with [link] for finite energy signals and is actually valid for all k , since both sides are periodic with period K . However, it is only useful if x is of finite energy, similar to [link] .

For periodic signals we could write X and X e as sums of Dirac delta functions as in [link] . Doing so, [link] confirms that also periodic signals possess spectral copies when sampled.

For periodic functions, the relation [link] shows that the terms τ x ˆ k will attempt to approximate the Dirac-shape of X e , meaning that the values of τ x ˆ k will be huge for k / L close to a peak location but very small otherwise (see [link] , [link] , and [link] , compare discussion after [link] ). In order to identify the amplitude X k of the Dirac delta functions better, one uses the earlier approximation

periodic: X k 1 K x ˆ k .
Sampling at 300 dpi appears to be sufficient to keep the quality of the image.
Sampling at 300 dpi appears to be sufficient to keep the quality of the image.
Sampling at 50 dpi introduces aliasing.
Sampling at 50 dpi introduces aliasing.
Sampling of a detail at 500dpi reveals high-frequency content.
Sampling of a detail at 500dpi reveals high-frequency content. These frequencies will leak from the spectral copies into the low frequencies of the main spectral copy when sampled at too low a frequency.

Aliasing

Assume that the continuous-time signal x ( t ) is bandlimited , meaning there is some B > 0 such that X ( f ) = 0 for | f | > B . If the sample rate had been too small, namely

f e < 2 B

then this copies would overlap and the original signal can not be recovered. The sampled signalshows artifacts called aliasing (recouvrement); these artifacts manifest is erroneouslow frequency content. Such content spills or leaks from the spectral copies into the main spectral period (see [link] ).

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Source:  OpenStax, Sampling rate conversion. OpenStax CNX. Sep 05, 2013 Download for free at http://legacy.cnx.org/content/col11529/1.2
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