NYQUIST SAMPLING THEOREM

                       555 Timer

The 555 timer IC is  one of the most important and widely used single ICs in history. The design has remained unchanged for over 40 years, which makes it one of the longest running IC designs. It’s been used in everything from toys to spacecraft.

This IC consists of 23 transistors, 2 diodes and 16 resistors. It is basically a  monolithic timing circuit that produces accurate and highly stable time delays or oscillation. When compared to the applications of an op-amp in the same areas, the 555IC is also equally reliable and is cheap in cost. Apart from its applications as a monostable multivibrator and astable multivibrator a 555 timer can also be used in dc-dc convertors digital logic probes,waveform generators analog frequency meters and tachometers,temperature measurements and control devices, voltage regulators etc. The timer IC is setup to work in either of the two modes – one-shot or monostable or as a free-running or astable multivibrator.The  555 can be used for temperature ranges between – 55°C to 125°

All the above applications are basics of electronics circuit design,which is build using ne555, hence its so important.

 

     NYQUIST SAMPLING THEOREM

Sampling is a process of converting a signal (for example, a function of continuous time and/or space) into a numeric sequence (a function of discrete time and/or space). Shannon’s version of the theorem states: 

  If a function x(t) contains no frequencies higher than B hertz,
  it is completely determined by giving its ordinates at a series 
  of points spaced 1/(2B) seconds apart.

A sufficient sample-rate is therefore 2B samples/second, or anything larger. Equivalently, for a given sample rate fs, perfect reconstruction is guaranteed possible for a bandlimit B < fs/2.

When the bandlimit is too high (or there is no bandlimit), the reconstruction exhibits imperfections known as aliasing. Modern statements of the theorem are sometimes careful to explicitly state that x(t) must contain no sinusoidal component at exactly frequency B, or that B must be strictly less than ½ the sample rate. The two thresholds, 2B and fs/2 are respectively called the Nyquist rate and Nyquist frequency. And respectively, they are attributes of x(t) and of the sampling equipment. The condition described by these inequalities is called the Nyquist criterion, or sometimes the Raabe condition. The theorem is also applicable to functions of other domains, such as space, in the case of a digitized image. The only change, in the case of other domains, is the units of measure applied to t, fs, and B.

The normalized sinc function: sin(πx) / (πx) … showing the central peak at x= 0, and zero-crossings at the other integer values of x.

The symbol T = 1/fs is customarily used to represent the interval between samples and is called the sample period or sampling interval. And the samples of function x(t) are commonly denoted by x[n] = x(nT) (alternatively “xn” in older signal processing literature), for all integer values of n. A mathematically ideal way to interpolate the sequence involves the use of sinc functions. Each sample in the sequence is replaced by a sinc function, centered on the time axis at the original location of the sample, nT, with the amplitude of the sinc function scaled to the sample value, x[n]. Subsequently, the sinc functions are summed into a continuous function. A mathematically equivalent method is to convolve one sinc function with a series of Dirac delta pulses, weighted by the sample values. Neither method is numerically practical. Instead, some type of approximation of the sinc functions, finite in length, is used. The imperfections attributable to the approximation are known as interpolation error.

Shannon’s original proof 

Poisson shows that the Fourier series  produces the periodic summation of X(f), regardless of fs and B. Shannon, however, only derives the series coefficients for the case fs = 2B. Virtually quoting Shannon’s original paper:

Let  \scriptstyle X(\omega ) be the spectrum of  \scriptstyle x(t).  Then
 x(t)\,  ={1 \over 2\pi }\int _{-\infty }^{\infty }X(\omega )e^{i\omega t}\;{\rm {d}}\omega \
 ={1 \over 2\pi }\int _{-2\pi B}^{2\pi B}X(\omega )e^{i\omega t}\;{\rm {d}}\omega \
since  \scriptstyle X(\omega ) is assumed to be zero outside the band  \scriptstyle |{\frac {\omega }{2\pi }}|<B. If we let
 t={n \over {2B}}\,
where n is any positive or negative integer, we obtain
 x\left({\tfrac {n}{2B}}\right)={1 \over 2\pi }\int _{-2\pi B}^{2\pi B}X(\omega )e^{i\omega {n \over {2B}}}\;{\rm {d}}\omega .
On the left are values of  \scriptstyle x(t) at the sampling points. The integral on the right will be recognized as essentially  the nth coefficient in a Fourier-series expansion of the function  \scriptstyle X(\omega ), taking the interval –B to B as a fundamental period. This means that the values of the samples  \scriptstyle x(n/2B) determine the Fourier coefficients in the series expansion of  \scriptstyle X(\omega ).  Thus they determine  \scriptstyle X(\omega ), since  \scriptstyle X(\omega ) is zero for frequencies greater than B, and for lower frequencies  \scriptstyle X(\omega ) is determined if its Fourier coefficients are determined. But \scriptstyle X(\omega )determines the original function  \scriptstyle x(t) completely, since a function is determined if its spectrum is known. Therefore the original samples determine the function  \scriptstyle x(t) completely.

 

Let   \scriptstyle x_{n} be the nth sample. Then the function  \scriptstyle x(t) is represented by:
 x(t)=\sum _{n=-\infty }^{\infty }x_{n}{\sin \pi (2Bt-n) \over \pi (2Bt-n)}.\,