0.6 Sampling: review

Sampling: review

Let us consider a signal $x\left(t\right)$ which is uniformly sampled meaning that the samples are $x_n=x\left(n\tau \right)$ ( $n=...-2,-1,0,1,2,...$ ). The sampling frequency $f_e=1/\tau$ 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 $\tau =L/K$ as for discrete signals.

Spectral copies (also called spectral repetitions or images)

The sampled signal can be written as the Dirac comb of step $\tau$ modulated by $x\left(t\right)$

The factor $\tau$ is added to obtain a Fourier transform that does not depend on $\tau$ and to preserve an averaging property (the integral of $x_e$ provides a good approximation of the integral of $x$ since $\int x_e\left(t\right)dt={\sum }_{n=-\infty }^{\infty }\tau x\left(n\tau \right)$ is the Riemann sum approximating the $\int x\left(t\right)dt$ as a sum of rectangles of width $\tau$ .)

Using [link] we may write

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

We observe that the Fourier transform, and also the spectrum of the sampled signal is periodic with period $f_e=1/\tau$ , 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 $k/\tau =k·f_e$ . 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 $x_e\left(t\right)$

Setting now $f=k/L$ and recalling $\tau =L/K$ we get approximatively

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 $\tau {\hat{x}}_k$ will attempt to approximate the Dirac-shape of $X_e$ , meaning that the values of $\tau {\hat{x}}_k$ 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

Aliasing

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

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] ).