# Reconstruction filter(Low Pass Filter)

Reconstruction filter (Low Pass Filter) Procedure to reconstruct actual signal from sampled signal:-

Low Pass Filter is used to recover original signal from it’s samples. This is also known as interpolation filter.

An LPF is that type of filter which passes only low frequencies up to cut-off frequency and rejects all other frequencies above cut-off frequency.

For an ideal LPF, there is a sharp change in the response at cut-off frequency as shown in the figure.

i.e, Amplitude response becomes suddenly zero at cut-off frequency which is not possible practically that means an ideal LPF is not physically realizable.

i.e, in place of an  ideal LPF a practical filter is used.

In case of a practical filter, the amplitude response decreases slowly to zero (this is one of the reason why we choose  $f_{s}>2f_{m}$)

This means that there exists a transition band in case of practical Low Pass Filter in the reconstruction of original signal from its samples.

Signal Reconstruction (Interpolation function):-

The process of reconstructing a Continuous Time signal x(t) from it’s samples is known as interpolation.

Interpolation gives either approximate (or) exact reconstruction (or) recovery of CT signal.

One of the simplest interpolation procedures is known as zero-order hold.

Another procedure is linear interpolation. In linear interpolation the adjacent samples (or) sample points are connected by straight lines.

We may also use higher order interpolation formula for reconstructing the CT signal from its sample values.

If we use the above process (Higher order interpolation) the sample points are connected by higher order polynomials (or) other mathematical functions.

For a Band limited signal, if the sampling instants are sufficiently large then the signal may be reconstructed exactly by using a LPF.

In this case an exact interpolation can be carried out between sample points.

Mathematical analysis:-

A Band limited signal x(t) can be reconstructed completely from its samples, which has higher frequency component fm Hz.

If we pass the sampled signal through a LPF having cut-off frequency of  fm  Hz.

From sampling theorem

$g(t)&space;=&space;x(t).\delta&space;_{T_{s}}(t)$.

$g(t)=\frac{1}{T_{s}}\left&space;\{&space;1+2\cos&space;\omega&space;_{s}t+2\cos&space;2\omega&space;_{s}t+2\cos&space;3\omega&space;_{s}t+.....&space;\right&space;\}$.

g(t)     has a multiplication factor  $\frac{1}{T_{s}}$. To reconstruct  x(t)  (or)  X(f) , the sampled signal must be passed through an ideal LPF of Band Width of  $f_{m}$   Hz and gain  $T_{s}$ .

$\left&space;|&space;H(\omega&space;)&space;\right&space;|=T_{s}&space;\&space;for&space;\&space;-\omega&space;_{m}\leq&space;\omega&space;\leq&space;\omega&space;_{m}$.

$h(t)&space;=&space;\frac{1}{2\pi&space;}&space;\int_{-\omega&space;_{m}}^{\omega&space;_{m}}T_{s}e^{j\omega&space;t}\&space;d\omega$.

$h(t)&space;=&space;2f_{m}T_{s}&space;\&space;sinc(2\pi&space;f_{m}t)$.

If sampling is done at Nyquist rate , then Nyquist interval is  $T_{s}&space;=&space;\frac{1}{2f_{m}}$ .

therefore  $h(t)&space;=&space;\&space;sinc(2\pi&space;f_{m}t)$ .

h(t) = 0.      at all Nyquist instants  $t=&space;\pm&space;\frac{n}{2f_{m}}$  , when    g(t)    is applied at the input to this filter the output will be  x(t)  .

Each sample in g(t)  results a sinc pulse having amplitude equal to the strength of sample. If we add all these sinc pulses that gives the original signal  x(t) .

$g(t)&space;=&space;x(kT_{s})\delta&space;(t-kT_{s})$ .

$x(t)&space;=\sum_{k}&space;x(kT_{s})\&space;h&space;(t-kT_{s})$ .

$x(t)&space;=\sum_{k}&space;x(kT_{s})\&space;sinc(2\pi&space;f_{m}&space;(t-kT_{s}))$.

$x(t)&space;=\sum_{k}&space;x(kT_{s})\&space;sinc(2\pi&space;f_{m}t-k\pi&space;)$ .

This is known as interpolation formula

It is assumed that the signal  x(t) is strictly band limited but in general an information signal may contain a wide range of frequencies and can not be strictly band limited this means that the maximum frequency in the signal can not be predictable.

then it is not possible to select suitable sampling frequency  fs  .

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