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What is Ideal Reconstruction Filter?
What is Data Reconstruction?
Data reconstruction is defined as the process of obtaining the analog signal $x\mathrm{\left(\mathit{t}\right)}$ from the sampled signal $x_{\mathit{s}}\mathrm{\left ( \mathit{t}\right)}$. The data reconstruction is also known as interpolation.
The sampled signal is given by,
$$\mathrm{\mathit{x}_{\mathit{s}}\mathrm{\left ( \mathit{t}\right)}\:\mathrm{=}\:\mathit{x}\mathrm{\left(\mathit{t}\right)}\sum_{\mathit{n}=-\infty}^{\infty}\:\delta \mathrm{\left ( \mathit{t-nT} \right )}}$$
$$\mathrm{\Rightarrow \mathit{x}_{\mathit{s}}\mathrm{\left ( \mathit{t}\right)}\:\mathrm{=}\sum_{\mathit{n}=-\infty}^{\infty}\:\mathit{x}\mathrm{\left(\mathit{nT}\right )}\delta\mathrm{\left(\mathit{t-nT}\right)}}$$
Where, $\mathit{\delta}\mathrm{\left(\mathit{t-nT} \right)}$ is zero except at the instants t = nT. A reconstruction filter which is assumed to be linear and time invariant has unit impulse response $\mathit{h\mathrm{\left({\mathit{t}}\right)}}$. The output of the reconstruction filter is given by the convolution as,
$$\mathrm{\mathit{y\mathrm{\left({\mathit{t}}\right)}}\:\mathrm{=}\:\int_{-\infty}^{\infty}\sum_{\mathit{n}=-\infty}^{\infty}\:\mathit{x}\mathrm{\left(\mathit{nT} \right )}\delta\mathrm{\left(\mathit{k-nT} \right)}\mathit{h}\mathrm{\left ( \mathit{t-k} \right )}\mathit{dk}}$$
By rearranging the order of integration and summation, we get,
$$\mathrm{\mathit{y\mathrm{\left({\mathit{t}}\right)}}\:\mathrm{=}\:\sum_{\mathit{n}=-\infty}^{\infty}\:\mathit{x}\mathrm{\left(\mathit{nT} \right )}\int_{-\infty}^{\infty}\delta\mathrm{\left(\mathit{k-nT} \right)}\mathit{h}\mathrm{\left ( \mathit{t-k} \right )}\mathit{dk}}$$
$$\mathrm{\therefore \mathit{y\mathrm{\left({\mathit{t}}\right)}}\:\mathrm{=}\:\sum_{\mathit{n}=-\infty}^{\infty}\mathit{x}\:\mathrm{\left(\mathit{nT} \right )}\mathit{h}\mathrm{\left ( \mathit{t-nT} \right )}}$$
Ideal Reconstruction Filter
An ideal reconstruction filter is used to construct a smooth analog signal from a sampled signal. If a signal $x\mathrm{\left(\mathit{t}\right)}$ is sampled at a frequency greater than the Nyquist rate and the sampled signal $x_{\mathit{s}}\mathrm{\left ( \mathit{t}\right)}$ is then passed through an ideal reconstruction filter (or ideal low pass filter), with bandwidth greater than $\mathit{f_{m}}$(which is maximum frequency present in the signal) but less than $\mathrm{\left(\mathit{f_{s}-f_{m}}\right )}$ and a band amplitude response of T, then output of the filter is $x\mathrm{\left(\mathit{t}\right)}$. The bandwidth of the ideal reconstruction filter is taken equal to 0.5 $\mathit{f_{s}}$.
Therefore, the transfer function of the ideal reconstruction filter is given by,
$$\mathrm{\mathit{H\mathrm{\left(\mathit{f}\right)}}\:\mathrm{=}\:\begin{cases} T & \text{ for } \left|f \right|<\:0.5f_{s} \ 0 & \text{ otherwise} \end{cases}}$$
The block diagram of an ideal reconstruction filter is shown in the figure.
The impulse response of the ideal reconstruction filter is given by,
$$\mathrm{\mathit{h}\mathrm{\left(\mathit{t}\right)}\mathrm{=}\int_{-0.5\mathit{f_{s}}}^{0.5\mathit{f_{s}}}\mathit{\mathit{T}\:e^{j\mathrm{2}\pi ft}\:df}}$$
$$\mathrm{\Rightarrow \mathit{h}\mathrm{\left(\mathit{t} \right)}\:\mathrm{=}\:\mathit{T}\mathrm{\left[\frac{\mathit{e^{j\mathrm{2}\pi ft}}}{\mathit{j}\mathrm{2}\mathit{\pi t}}\right ]^{0.5\mathit{f}_{s}}_{-0.5\mathit{f}_{s}}}\:\mathrm{=}\:\frac{\mathit{T}}{\mathit{j}\mathrm{2}\mathit{\pi t}}\mathrm{\left(\mathit{e^{j\pi f_{s}t}-e^{-j\pi f_{s}t}} \right)}}$$
$$\mathrm{\Rightarrow \mathit{h}\mathrm{\left(\mathit{t}\right)}\:\mathrm{=}\frac{1}{\mathit{\pi f_{s}t}}\mathrm{\left(\frac{\mathit{e^{j\pi f_{s}t}-e^{-j\pi f_{s}t}}}{2\mathit{j}}\right )}\:\mathrm{=}\:\frac{\mathrm{sin}\:\mathit{\pi f_{s}t}}{\mathit{\pi f_{s}t}}}$$
$$\mathrm{\therefore \mathit{h\mathrm{\left(\mathit{t}\right)}}\:\mathrm{=}\:\mathrm{sinc}\mathrm{\left(\mathit{f_{s}t}\right)}}$$
By substituting value of the impulse response in the expression for the output of the reconstruction filter, we have,
$$\mathrm{\mathit{y\mathrm{\left({\mathit{t}}\right)}}\:\mathrm{=}\:\mathit{x\mathrm{\left({\mathit{t}}\right)}}\:\mathrm{=}\:\sum_{\mathit{n}=-\infty}^{\infty}\:\mathit{x}\mathrm{\left(\mathit{nT} \right )}\:\mathrm{sinc\:\mathit{f_{s}}}\mathrm{\left ( \mathit{t-nT} \right )}}$$
$$\mathrm{\therefore \mathit{x\mathrm{\left({\mathit{t}}\right)}}\:\mathrm{=}\:\sum_{\mathit{n}=-\infty}^{\infty}\:\mathit{x}\mathrm{\left(\mathit{nT} \right )}\:\mathrm{sinc\:\mathrm{\left ( \frac{\mathit{t}}{\mathit{T}}-\mathit{n} \right )}}}$$
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.
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