Ideal reconstruction

In order to reconstruct a function we again take a look at what happens during sampling in frequency domain. As stated in Sec. 3.6, sampling can be seen as multiplying a function with a comb function which means that the frequency response of the function gets convolved with the frequency response of the comb function, which is another comb function but with reciprocal spacing.

This again means that the frequency response of the function gets replicated with reciprocal spacing of the sampling rate between the replicas. Lets assume that these replicas do not overlap (and now the importance of the function being band-limited as well as the Nyquist frequency becomes clear, because the replicas do not overlap if and only if the function is band-limited and sampled above the Nyquist frequency). Then the frequency response of the original function can be perfectly reconstructed by multiplying the frequency response of the sampled function with a box function which half-width is equal to the bandwidth of the original function. The box function is defined by

$$\Pi_a(x) = \left\{ \begin{array}{ll} 1 & \textrm{if $\vert... ...extrm{if $\vert x\vert > \frac{a}{2}$}\\ \end{array} \right.$$ (4.1)

where a denotes the half-width of the box function (in this case we would set $a=\omega_N$, the Nyquist frequency or twice the bandwidth of the function). So far so good, the sampled function must be multiplied by a box function in frequency domain. From Theorem 2 one can infer that this means that the samples are convolved with the Fourier transform of the box-function. The box function can quite easily be transformed to frequency domain analytically:

$$\begin{eqnarray*} \mathcal{F}(\Pi_a)(\omega) &=& \int_{-\infty}^{\infty} \Pi_a(x... ... \pi a\omega}{\pi a\omega}\\ &=& a \cdot \mathrm{sinc}(a\omega) \end{eqnarray*}$$

The sinc function is a quite important function and will be often used in later chapters:

Definition 3 (sinc function)  

$$\mathrm{sinc}(x) = \left\{ \begin{array}{cl} \frac{\sin \p... ...x \neq 0$}\\ 1 & \textrm{if $x = 0$}\\ \end{array} \right.$$ (4.2)

and the following duality holds (both functions are depicted in Fig. 4.1):

$$\mathrm{sinc}(ax) \rightleftharpoons \Pi_a(x)$$ (4.3)

Figure 4.1: The sinc function, which is the ideal reconstruction filter, on the left and its frequency response, the box function, on the right

A band-limited and properly sampled function can be perfectly reconstructed by convolving the set of samples with the sinc function. This is the second part of the Sampling theorem, which can now be given in its complete form:

Theorem 5 (Sampling Theorem, complete)   A function f(x) that is

• band-limited and
• sampled above the Nyquist frequency

is completely determined by its samples and can be perfectly reconstructed by convolving the set of samples with the sinc function.

Figure 4.2: When a function is sampled below its Nyquist frequency, the original frequency response will overlap with nearby replicas of itself. Even if the ideal reconstruction filter is used, an aliased spectrum is reconstructed.

However, there are a few problems which arise in practice. The first problem arises when the function is not band-limited or it is sampled below the Nyquist frequency. Then it happens that the replicas overlap with the original spectrum of our function and when we multiply in frequency domain with the box function we get an aliased, i.e., modified, spectrum, that is a spectrum which has aliases of higher frequencies (therefore this phenomenon is called aliasing). Fig 4.2 depicts this concept graphically. The real bad thing about this is that this problem depends only on the sampling rate and whether the function is band-limited or not. There is no way to exactly reconstruct an improperly sampled function. What happens when a filter different from the ideal filter is used for reconstruction will be discussed in the next section.