What is Ideal Reconstruction Filter?

What is Data Reconstruction?

Data reconstruction is defined as the process of obtaining the analog signal $x\mathrm{\left(\mathrm{t}\right)}$ from the sampled signal $\mathrm{x_{s}(t)}$. The data reconstruction is also known as interpolation.

The sampled signal is given by,

$$\mathrm{x_{s}(t)\:=\:x(t)\sum_{n=-\infty}^{\infty}\:\delta(t\:-\:nT)}$$

$$\mathrm{\Rightarrow\: x_{s}(t)\:=\:\sum_{n=-\infty}^{\infty}\:x(nT)\delta(t \:-\: nT)}$$

Where, $\mathrm{\delta(t\:-\:nT)}$ is zero except at the instants t = nT. A reconstruction filter which is assumed to be linear and time invariant has unit impulse response $\mathrm{h(t)}$. The output of the reconstruction filter is given by the convolution as,

$$\mathrm{y(t)\:=\:\int_{-\infty}^{\infty}\:\sum_{n=-\infty}^{\infty}\:x(nT)\delta(k\:-\:nT)h(t\:-\:k)dk}$$

By rearranging the order of integration and summation, we get,

$$\mathrm{y(t)\:=\:\sum_{n=-\infty}^{\infty}\:x(nT)\int_{-\infty}^{\infty}\:\delta(k\:-\:nT)h(t\:-\:k)dk}$$

$$\mathrm{\therefore\:y(t)\:=\:\sum_{n=-\infty}^{\infty}\:x(nT)h(t\:-\:nT)}$$

Ideal Reconstruction Filter

An ideal reconstruction filter is used to construct a smooth analog signal from a sampled signal. If a signal $\mathrm{x(t)}$ is sampled at a frequency greater than the Nyquist rate and the sampled signal $\mathrm{x_{s}(t)}$ is then passed through an ideal reconstruction filter (or ideal low pass filter), with bandwidth greater than $\mathrm{f_{m}}$ (which is maximum frequency present in the signal) but less than $\mathrm{(f_{s}\:-\:f_{m})}$ and a band amplitude response of T, then output of the filter is $\mathrm{x(t)}$. The bandwidth of the ideal reconstruction filter is taken equal to 0.5 $\mathrm{f_{s}}$.

Therefore, the transfer function of the ideal reconstruction filter is given by,

$$\mathrm{H(f)\:=\:\begin{cases} T \:;\: \text{ for } \:\left|f \right|\:\lt\:0.5 f_{s} \\\\ 0 \:;\: \text{ otherwise} \end{cases}}$$

The block diagram of an ideal reconstruction filter is shown in the figure.

The impulse response of the ideal reconstruction filter is given by,

$$\mathrm{h(t)\:=\:\int_{-0.5 f_{s}}^{0.5 f_{s}}T\:e^{j2\pi ft}\:df}$$

$$\mathrm{\Rightarrow\: h(t)\:=\:T\left[\frac{e^{j2\pi ft}}{j2\pi t}\right ]^{0.5 f_{s}}_{-0.5f_{s}}\:=\:\frac{T}{j2\pi t}\left(e^{j\pi f_{s}t}\:-\:e^{-j\pi f_{s}t} \right)}$$

$$\mathrm{\Rightarrow\: h(t)\:=\:\frac{1}{\pi f_{s}t}\left(\frac{e^{j\pi f_{s}t}\:-\:e^{-j\pi f_{s}t}}{2j}\right )\:=\: \frac{sin\:\pi f_{s}t}{\pi f_{s}t}}$$

$$\mathrm{\therefore\: h(t)\:=\:sinc(f_{s}t)}$$

By substituting value of the impulse response in the expression for the output of the reconstruction filter, we have,

$$\mathrm{y(t)\:=\:x(t)\:=\:\sum_{n=-\infty}^{\infty}\:x(nT)\:sinc\:f_{s}\left(t\:-\:nT \right)}$$

$$\mathrm{\therefore\: x(t)\:=\:\sum_{n=-\infty}^{\infty}\:x(nT)\:sinc\:\left ( \frac{t}{T}\:-\:n \right )}$$

Thus, it is clear that the original signal can be reconstructed by weighing each sample by a sinc function centred at the sample time and summing. The ideal reconstruction filter is non-causal and its impulse response is not limited. Therefore, it cannot be used for real-time applications.