Switches-Circuit Switching

Switches are used in Circuit-Switched and Packet-Switched Networks.  The switches are used are different depending up on the structure and usage.

Circuit Switches (or) Structure of Circuit Switch:-

The Switches used in Circuit Switching are called Circuit-Switches

Space-Division Switch:-

  • The paths are separated spatially from one switch to other.
  • These were originally designed for analog circuits but currently used for both analog and digital Networks.

Cross-bar Switch:-

In this type of Switch we connect n inputs and m outputs using micro switches (Transistors) at each cross point to form a cross-bar switch of size n X m.

The number of cross points required = n X m.

As n and m increases, cross points required also increases, for example n=1000 and m=1000  requires n X m= 1000 X 1000 cross points. A cross-bar with these many number of cross points is impractical and statics show that 25% of the cross points are in use at any given time.

Multi stage Switch:-

The solution to Cross-bar Switch is Multi stage switching. Multi stage switching is preferred over cross-bar switches to reduce the number of cross points.  Here number of cross-bar switches are combined in several stages.

Suppose an N X N cross-bar Switch can be made into 3 stage Multi bar switch as follows.

  1.  N is divided into groups , that is N/n Cross-bars with n-input lines and k-output lines forms n X k cross points.
  2. The second stage consists of k Cross-bar switches with each cross-bar switch size as (N/n) X (N/n).
  3. The third stage consists of N/n cross-bar switches with each switch size as k X n.

The total number of cross points = 2kN + k (\frac{N}{n})^{2}, so the number of cross points required are less than single-stage cross-bar Switch = N^{2}.

for example k=2 and n=3 and N=9 then a Multi-stage switch looks like as follows.

The problem in Multi-stage switching is Blocking during periods of heavy traffic, the idea behind Multi stage switch is to share intermediate cross-bars. Blocking means times when one input line can not be connected to an output because there is no path available (all possible switches are occupied). Blocking generally occurs in tele phone systems and this blocking is due to intermediate switches.

Clos criteria gives a condition for a non-blocking Multi stage switch 

n = \sqrt{\frac{N}{2}}k\geq (2n-1)  and  Total no.of Cross points \geq 4N(\sqrt{2N}-1).

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GATE problems in EMT

pb. An Electro Magnetic Wave Propagates through a loss less insulator with a velocity 1.5 X 10^{10} Cm/sec . Calculate the electric magnetic properties of the insulator if its intrinsic impedance is 90\pi Ohms.

Ans:- Given loss less insulator  \Rightarrow \alpha =0

v_{p} =1.5 X 10^{10} Cm/Sec

v_{p} = \frac{1}{\sqrt{\mu _{o}\epsilon _{o}\mu _{r}\epsilon _{r}}}

1.5 X 10^{10}= \frac{1}{\sqrt{\mu _{o}\epsilon _{o}\mu _{r}\epsilon _{r}}}

1.5 X 10^{10}= \frac{3 X 10^{10}}{\sqrt{\mu _{r}\epsilon _{r}}}

{\sqrt{\mu _{r}\epsilon _{r}}} = 2 ------------EQNI

from \eta =\sqrt{\frac{\mu _{o}\mu _{r }}{\epsilon _{o}\epsilon _{r }}}

90\pi = \sqrt{\frac{\mu _{o}\mu _{r }}{\epsilon _{o}\epsilon _{r }}}

90\pi = \sqrt{\frac{\mu _{r }}{\epsilon _{r }}} . 120 \pi

\sqrt{\frac{\mu _{r }}{\epsilon _{r }}} =\frac{3}{4}--------EQNII

from Equations I and II \mu _{r} = \sqrt{\frac{\mu _{r }}{\epsilon _{r }}} \sqrt{\mu _{r }\epsilon _{r}}

\mu _{r} = 2 X \frac{3}{4}

\mu _{r} = 1.5

\sqrt{\mu _{r }\epsilon _{r }} = 2

\sqrt{1.5 \epsilon _{r }} = 2

\epsilon _{r} = 2.66

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Solved Example problems in Electro Magnetic Theory

  1. Convert Points P(1,3,5)  from Cartesian to Cylindrical and Spherical Co-ordinate system.

Ans. Given P(1,3,5) \fn_cm \Rightarrow x=1,y=3, z=5

Cylindrical :- \fn_cm \phi =tan^{-1}(\frac{y}{x})

                              \fn_cm =tan^{-1}(\frac{3}{1})

                                \fn_cm =75^{o}

Similarly \fn_cm \rho = \sqrt{x^{2}+ y^{2}} \Rightarrow \sqrt{1^{2}+ 3^{2}} = \sqrt{10}=3.16

\fn_cm P(\rho ,\phi ,z)= P(3.16,71.5^{o},5)





Spherical :- \fn_cm r= \sqrt{x^{2}+y^{2}+z^{2}}

                        \fn_cm r= \sqrt{1^{2}+3^{2}+5^{2}}

                      \fn_cm r= \sqrt{35}

                    \fn_cm r=5.91

\fn_cm \theta =tan^{-1}(\frac{\sqrt{x^{2}+y^{2}}}{z})

\fn_cm \theta =tan^{-1}(\frac{\sqrt{1^{2}+3^{2}}}{5})

\fn_cm \theta =32.31^{o}

\fn_cm \phi =tan^{-1}(\frac{y}{x})

\fn_cm \phi =tan^{-1}(\frac{3}{1})

\fn_cm \phi = 75^{o}

\fn_cm P(r,\theta ,\phi ) = P(5.91,32.31^{o},71.5^{o})

 

Phase Locked Loop (PLL)

Demodulation of an FM signal using PLL:-

Let the input to PLL is an FM signal S(t) = A_{c} \sin (2 \pi f_{c}t+2\pi k_{f} \int_{0}^{t}m(t)dt)

let  \Phi _{1} (t) = 2\pi k_{f} \int_{0}^{t}m(t)dt ------Equation (I)

 Now the signal at the output of VCO is FM signal (another FM signal, which is different from input FM signal) Since Voltage Controlled Oscillator is an FM generator.

\therefore b(t) = A_{v} \cos (2 \pi f_{c}t+2\pi k_{v} \int_{0}^{t}v(t)dt)

the corresponding phase    \Phi _{2} (t) = 2\pi k_{v} \int_{0}^{t}v(t)dt ------Equation (II)

It is observed that S(t) and b(t) are out of phase by 90^{o}. Now these signals are applied to a phase detector , which is basically a multiplier

\therefore the error signal e(t) =S(t) .b(t)

e(t) =A_{c} \sin (2 \pi f_{c}t+2\pi k_{f} \int_{0}^{t}m(t)dt). A_{v} \cos (2 \pi f_{c}t+2\pi k_{v} \int_{0}^{t}v(t)dt)

e(t) =A_{c}A_{v} \sin (2 \pi f_{c}t+\phi _{1}(t)). \cos (2 \pi f_{c}t+\phi _{2}(t))

on further simplification , the product yields a higher frequency term (Sum) and a lower frequency term (difference)

e(t) =A_{c}A_{v}k_{m} \sin (4 \pi f_{c}t+\phi _{1}(t)+\phi _{2}(t))- A_{c}A_{v}k_{m}\sin (\phi _{1}(t)-\phi _{2}(t))

e(t) =A_{c}A_{v}k_{m} \sin (2 \omega _{c}t+\phi _{1}(t)+\phi _{2}(t))- A_{c}A_{v}k_{m}\sin (\phi _{1}(t)-\phi _{2}(t))

This product e(t) is given to a loop filter , Since the loop filter is a LPF it allows the difference and term and rejects the higher frequency term.

the over all output of a loop filter is 

 

Frequency domain representation of a Wide Band FM

To obtain the frequency-domain representation of Wide Band FM signal for the condition \beta > > 1 one must express the FM signal in complex representation (or) Phasor Notation (or) in the exponential form

i.e, Single-tone FM signal is S_{FM}(t)=A_{c}cos(2\pi f_{c}t+\beta sin 2\pi f_{m}t).

Now by expressing the above signal in terms of  Phasor notation (\because \beta > > 1 , None of the terms can be neglected)

S_{FM}(t) \simeq Re(A_{c}e^{j(2\pi f_{c}t+\beta sin 2\pi f_{m}t)})

S_{FM}(t) \simeq Re(A_{c}e^{j2\pi f_{c}t}e^{j\beta sin 2\pi f_{m}t})

S_{FM}(t) \simeq Re(e^{j2\pi f_{c}t} A_{c}e^{j\beta sin 2\pi f_{m}t})-------Equation(I)

Let    \widetilde{s(t)} =A_{c}e^{j\beta sin 2\pi f_{m}t}      is the complex envelope of FM signal.

\widetilde{s(t)} is a periodic function with period \frac{1}{f_{m}} . This \widetilde{s(t)} can be expressed in it’s Complex Fourier Series expansion.

i.e, \widetilde{S(t)} = \sum_{n=-\infty }^{\infty }C_{n} e^{jn\omega _{m}t}  this approximation is valid over [-\frac{1}{2f_{m}},\frac{1}{2f_{m}}] . Now the Fourier Coefficient  C_{n} = \frac{1}{T} \int_{\frac{-T}{2}}^{\frac{T}{2}} \widetilde{S(t)} e^{-jn2\pi f_{m}t}dt

T= \frac{1}{f_{m}}

C_{n} = \frac{1}{\frac{1}{f_{m}}} \int_{\frac{-1}{2f_{m}}}^{\frac{1}{2f_{m}}} \widetilde{S(t)} e^{-jn2\pi f_{m}t}dt

C_{n} = f_{m} \int_{\frac{-1}{2f_{m}}}^{\frac{1}{2f_{m}}} A_{c}e^{j\beta sin 2\pi f_{m}t} e^{-jn2\pi f_{m}t}dt

C_{n} = f_{m} \int_{\frac{-1}{2f_{m}}}^{\frac{1}{2f_{m}}} A_{c}e^{{j\beta sin 2\pi f_{m}t-jn2\pi f_{m}t}}dt

C_{n} = f_{m} \int_{\frac{-1}{2f_{m}}}^{\frac{1}{2f_{m}}} A_{c}e^{j({\beta sin 2\pi f_{m}t-n2\pi f_{m}t})}dt

let x=2\pi f_{m}t       implies   dx=2\pi f_{m}dt

as x\rightarrow \frac{-1}{2f_{m}} \Rightarrow t\rightarrow -\pi     and    x\rightarrow \frac{1}{2f_{m}} \Rightarrow t\rightarrow \pi

C_{n} = \frac{A_{c}}{2\pi } \int_{-\pi }^{\pi } e^{j({\beta sin x-nx})}dx

let J_{n}(\beta ) = \frac{1}{2\pi } \int_{-\pi }^{\pi } e^{j({\beta sin x-nx})}dx   as    n^{th}  order Bessel Function of first kind then   C_{n} = A_{c} J_{n}(\beta ).

Continuous Fourier Series  expansion of  

\widetilde{S(t)} = \sum_{n=-\infty }^{\infty }C_{n} e^{jn\omega _{m}t}

\widetilde{S(t)} = \sum_{n=-\infty }^{\infty }A_{c} J_{n} (\beta )e^{jn\omega _{m}t} 

Now substituting this in the Equation (I)

S_{WBFM}(t) \simeq Re(e^{j2\pi f_{c}t} \sum_{n=-\infty }^{\infty }A_{c} J_{n} (\beta )e^{jn\omega _{m}t})

S_{WBFM}(t) \simeq A_{c} Re( \sum_{n=-\infty }^{\infty }J_{n} (\beta ) e^{j2\pi f_{c}t} e^{jn\omega _{m}t})

S_{WBFM}(t) \simeq A_{c} Re( \sum_{n=-\infty }^{\infty }J_{n} (\beta ) e^{j2\pi (f_{c}+nf _{m}t)})

\therefore S_{WBFM}(t) \simeq A_{c} \sum_{n=-\infty }^{\infty }J_{n} (\beta ) cos 2\pi (f_{c}+nf _{m}t)

The  Frequency spectrum  can be obtained by taking Fourier Transform 

S_{WBFM}(f) = \frac{A_{c}}{2}\sum_{n=-\infty }^{\infty }J_{n}(\beta ) [\delta (f-(f_{c}+nf_{m}))+\delta (f+(f_{c}-nf_{m}))]

n value wide Band FM signal
0 S_{WBFM}(f) = \frac{A_{c}}{2}\sum_{n=-\infty }^{\infty }J_{0}(\beta ) [\delta (f-f_{c})+\delta (f+f_{c})]
1 S_{WBFM}(f) = \frac{A_{c}}{2}\sum_{n=-\infty }^{\infty }J_{1}(\beta ) [\delta (f-(f_{c}+f_{m}))+\delta (f+(f_{c}+f_{m}))]
-1 S_{WBFM}(f) = \frac{A_{c}}{2}\sum_{n=-\infty }^{\infty }J_{-1}(\beta ) [\delta (f-(f_{c}-f_{m}))+\delta (f+(f_{c}-f_{m}))]
….

From the above Equation it is clear that 

  • FM signal has infinite number of side bands at frequencies (f_{c}\pm nf_{m})for n values changing from -\infty to  \infty.
  • The relative amplitudes of all the side bands depends on the value of  J_{n}(\beta ).
  • The number of significant side bands depends on the modulation index \beta.
  • The average power of FM wave is P=\frac{A_{c}^{2}}{2} Watts.

 

 

 

 

 

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Matched Filter, impulse response h(t)

Matched Filter can be considered as a special case of Optimum Filter. Optimum Filter can be treated as Matched Filter when the noise at the input of the receiver is White Gaussian Noise.

Transfer Function of Matched Filter:-

Transfer Function of Optimum filter is H(f)=\frac{k X^{*}(f)e^{-j2\pi fT}}{S_{ni}(f)}

if input noise is white noise , its Power spectral density (Psd) is S_{ni}(f)=\frac{N_{o}}{2}.

then H(f) becomes H(f)=\frac{k X^{*}(f)e^{-j2\pi fT}}{\frac{N_{o}}{2}}

H(f)=\frac{2k}{N_{o}}X^{*}(f)e^{-j2\pi fT}-----Equation(I)

From the properties of Fourier Transforms , by Conjugate Symmetry property  X^{*}(f) = X(-f)

Equation (I) becomes

H(f)=\frac{2k}{N_{o}}X(-f)e^{-j2\pi fT}------Equation(II)

From Time-shifting property of Fourier Transforms

x(t)\leftrightarrow X(f)

From Time-Reversal Property  x(-t)\leftrightarrow X(-f)

By Shifting the signal x(-t) by T Seconds in positive direction(time) ,the Fourier Transform is given by  x(T-t)\leftrightarrow X(-f)e^{-j2\pi ft}

Now the inverse Fourier Transform of the signal from the Equation(II) is

F^{-1}[H(f)]=F^{-1}[\frac{2k}{N_{o}}X(-f)e^{-j2\pi fT}]

h(t)=\frac{2k}{N_{o}}x(T-t)

Let the constant \frac{2k}{N_{o}} is set to 1, then the impulse response of Matched Filter will become h(t) = x(T-t).

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Mutual Information I(X ; Y) Properties

Property 1:- Mutual Information is Non-Negative

Mutual Information is given by equation I(X ; Y) =\sum_{i=1}^{m}\sum_{j=1}^{n}P(x_{i}, y_{j})\log _{2} \frac{P(\frac{x_{i}}{y_{j}})}{P(x_{i})}---------Equation(I)

we know that P(\frac{x_{i}}{y_{j}})=\frac{P(x_{i}, y_{j})}{P(y_{j})}-------Equation(II)

Substitute Equation (II) in Equation (I)

I(X ; Y) =\sum_{i=1}^{m}\sum_{j=1}^{n}P(x_{i}, y_{j})\log _{2}\frac{P(x_{i}, y_{j})}{P(x_{i})P(y_{j})}

The above Equation can be written as 

I(X ; Y) =-\sum_{i=1}^{m}\sum_{j=1}^{n}P(x_{i}, y_{j})\log _{2}\frac{P(x_{i})P(y_{j})}{P(x_{i}, y_{j})}

-I(X ; Y) =\sum_{i=1}^{m}\sum_{j=1}^{n}P(x_{i}, y_{j})\log _{2}\frac{P(x_{i})P(y_{j})}{P(x_{i}, y_{j})}------Equation(III)

we knew that \sum_{k=1}^{m} p_{k}\log _{2}(\frac{q_{k}}{p_{k}})\leq 0---Equation(IV)

This result can be applied to Mutual Information I(X ; Y) , If p_{k} = P(x_{i}, y_{j}) and q_{k} be P(x_{i}) P( y_{j}), Both p_{k} and q_{k} are two probability distributions on same alphabet , then Equation (III) becomes 

-I(X ; Y) \leq 0

i.e, I(X ; Y) \geq 0  , Which implies that Mutual Information is always Non-negative (Positive).

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Example Problems in Electro Magnetic Theory Wave propagation

  1. A medium like Copper conductor which is characterized by the parameters \bg_black \sigma = 5.8 X 10^{7} Mho's/meter and \epsilon _{r}=1,\mu _{r}=1 uniform plane wave of frequency 50 Hz. Find \alpha ,\beta ,v,\eta  and \lambda.

Ans.  Given \bg_black \bg_black \sigma = 5.8 X 10^{7} Mho's/meter  ,     \bg_black \epsilon _{r}=1,\mu _{r}=1    and \bg_white f= 50 Hz

\bg_white \alpha =? ,\beta =? ,v = ?,\eta =? and \bg_white \lambda =?

Find the Loss tangent \bg_white \frac{\sigma }{\omega \epsilon } = \frac{5.8X 10^{7}}{2 \pi X50X\epsilon _{o}\epsilon _{r}}

                                              \bg_white \bg_white \frac{\sigma }{\omega \epsilon } = \frac{5.8X 10^{7}}{100\pi X\epsilon _{o}}

                                            \bg_white \bg_white \frac{\sigma }{\omega \epsilon } = 2.08 X 10 ^{16}> > 1

So given medium is a Conductor (Copper)

then \bg_white \alpha (or) \beta =\sqrt{\frac{\omega \mu \sigma }{2}}

                         \bg_white =\sqrt{\frac{5.8X10^{7}X2\pi X 50X\mu _{o}}{2}}

                      \bg_white \alpha = 106.99  , \bg_white \beta =106.99.

\bg_white v_{p}=\frac{\omega }{\beta }  \bg_white =\frac{2\pi X50}{106.99}\bg_white =2.936 meters/Sec.

\bg_white \lambda =\frac{2\pi }{\beta }=\frac{2\pi }{106.99}=0.0587 meters.

\bg_white \eta =\sqrt{\frac{j\omega \mu }{(\sigma +j\omega \epsilon )}}

    \bg_white =\sqrt{\frac{jX2\pi X50X\mu _{o}}{(5.8X10^{7}+j2\pi X50X\epsilon _{o})}}

    \bg_white = \sqrt{\frac{j 3.947 X10^{-4}}{(5.8X10^{7}+j 2.78 X10^{-9})}}

    \bg_white = \sqrt{\frac{ 3.947 X10^{-4}\angle 90^{o}}{(5.8X10^{7}\angle -2.74 X 10^{-15})}}

    \bg_white = \sqrt{0.68 X10 ^{-11}}\angle \frac{90-(2.74 X 10^{-5})}{2}

\eta = 2.6 X 10^{-6}\angle 45^{o}.

2. If \bg_white \epsilon _{r}=9,\mu =\mu _{o} for a medium in which a wave with a frequency of \bg_white f= 0.3 GHz is propagating . Determine the propagation constant and intrinsic impedance of the medium when \bg_white \sigma =0.

Ans: Given \bg_white \epsilon _{r}=9,  \bg_white \mu =\mu _{o} , \bg_white f=0.3GHz and \bg_white \sigma =0.

\bg_white \gamma =?,\eta =?

Since \bg_white \sigma =0, the given medium is a lossless Di-electric.

which implies \bg_white \alpha = \frac{\sigma }{2}\sqrt{\frac{\mu }{\epsilon }} =0.

\bg_white \beta = \omega \sqrt{\mu \epsilon }

    \bg_white =2\pi X o.3X10^{9}\sqrt{\mu _{o}X9\epsilon _{o}}

    \bg_white = 18.86.

\bg_white \eta = \sqrt{\frac{\mu }{\epsilon }}

\bg_white \eta = \sqrt{\frac{\mu_{o}\mu _{r} }{\epsilon_{o}\epsilon _{r} }}

\bg_white \eta = \sqrt{\frac{\mu_{o} }{9\epsilon_{o} }}

\bg_white \eta = \frac{120\pi }{3}

\bg_white \eta = 40 Ω.

 

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Delta modulation and Demodulation

DM is done on an over sampled message signal in its basic form, DM provides a stair case approximated signal to the over sampled version of message signal.

i.e, Delta Modulation (DM) is a Modulation scheme in which an incoming  message signal is over sampled (i.e, at a rate much higher than the Nyquist rate f_{s}> 2f_{m}) to purposely increase the correlation between adjacent samples of the signal. Over sampling is done to permit the use of a sample Quantizing strategy for constructing the encoded signal.

Signaling rate and Transmission Band Width are quite large in PCM. DM is used to overcome these problems in PCM . 

DM transmits one bit per sample. 

The process of approximation in Delta Modulation is as follows:-

The difference between the input (x[nT_{s}]) and the approximation (x[(n-1)T_{s}]) is quantized into only two levels \pm \Delta corresponding to Positive and negative differences.

i.e, If the approximation  (x[(n-1)T_{s}]) falls below the signal (x[nT_{s}])at any sampling epoch(the beginning of a period)output signal level is increased by \Delta.

On the other hand the approximation   (x[(n-1)T_{s}]) lies above the signal (x[nT_{s}]) , output signal level is diminished by \Delta provided that the input signal does not change too rapidly from sample to sample.

it is observed that the  change in stair case approximation lies with in  \pm \Delta .

This process can be illustrated in the following figure

Delta Modulated System:- The DM system consists of Delta Modulator and Delta Demodulator.

Delta Modulator:- 

Mathematical equations involved in DM Transmitter are 

error signal: e[nT_{s}]=x[nT_{s}]-x_{q}[(n-1)T_{s}]

Present sample of the (input) sampled signal: x[nT_{s}]

last sample approximation of stair case signal: x_{q}[(n-1)T_{s}]

Quantized  error signal( output of one-bit Quantizer): e_{q}[nT_{s}]

if         x[nT_{s}]\geq x_{q}[(n-1)T_{s}] \Rightarrow e_{q}[nT_{s}] = \Delta.

and  x[nT_{s}]< x_{q}[(n-1)T_{s}] \Rightarrow e_{q}[nT_{s}] = -\Delta.

encoding has to be done on the after Quantization that is when the output level is increased by \Delta from its previous quantized level, bit ‘1’ is transmitted .

similarly when output is diminished by \Delta from the previous level  a ‘0’ is transmitted.

from the accumulator x_{q}[nT_{s}]=x_{q}[(n-1)T_{s}]+ e_{q}[nT_{s}]

                                               e_{q}[nT_{s}]=e[nT_{s}]+ q[nT_{s}]

where q[nT_{s}] is the Quantization error.

 

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Fourier Series and it’s applications

The starting point of Fourier Series is the development of representation of signals as linear combination (sum of) of a set of basic signals.

f(t)\approx C_{1}x_{1}(t)+C_{2}x_{2}(t)+.......+C_{n}x_{n}(t)+....

The alternative representation if  a set of complex exponentials are used,

f(t)\approx C_{1}e^{j\omega _{o}t}+C_{2}e^{2j\omega _{o}t}+.......+C_{n}e^{jn\omega _{o}t}+....

The resulting representations are known as Fourier Series in Continuous-Time [Fourier Transform in the case of Non-Periodic signal]. Here we focus on representation of Continuous-Time and Discrete-Time periodic signals in terms of basic signals as Fourier Series and extend the analysis to the Fourier Transform representation of broad classes of aperiodic, finite energy signals.

These Fourier Series & Fourier Transform representations are most powerful tools used

  1. In the analyzation of signals and LTI systems.
  2. Designing of Signals & Systems.
  3. Gives insight to S&S.

The development of Fourier series analysis has a long history involving a great many individuals and the investigation of many different physical phenomena.

The concept of using “Trigonometric Sums”, that is sum of harmonically related sines and cosines (or) periodic complex exponentials are used to predict astronomical events.

Similarly, if we consider the vertical deflection f(t,x) of the string at time t and at a distance x along the string then for any fixed instant of time, the normal modes are harmonically related sinusoidal functions of x.

The scientist Fourier’s work, which motivated him physically was the phenomenon of heat propagation and diffusion. So he found that the temperature distribution through a body can be represented by using harmonically related sinusoidal signals.

In addition to this he said that any periodic signal could be represented by such a series.

Fourier obtained a representation for aperiodic (or) non-periodic signals not as weighted sum of harmonically related sinusoidals but as weighted integrals of Sinusoids that are not harmonically related, which is known as Fourier Integral (or) Fourier Transform.

In mathematics, we use the analysis of Fourier Series and Integrals in 

  1. The theory of Integration.
  2. Point-set topology.
  3. and in the eigen function expansions.

In addition to the original studies of vibration and heat diffusion, there are numerous other problems in science and Engineering in which sinusoidal signals arise naturally, and therefore Fourier Series and Fourier T/F’s plays an important role.

for example, Sine signals arise naturally in describing the motion of the planets and the periodic behavior of the earth’s climate.

A.C current sources generate sinusoidal signals as voltages and currents. As we will see the tools of Fourier analysis enable us to analyze the response of an LTI system such as a circuit to such Sine inputs.

Waves in the ocean consists of the linear combination of sine waves with different spatial periods (or) wave lengths.

Signals transmitted by radio and T.V stations are sinusoidal in nature as well.

The problems of mathematical physics focus on phenomena in Continuous Time, the tools of Fourier analysis for DT signals and systems have their own distinct historical roots and equally rich set of applications.

In particular, DT concepts and methods are fundamental to the discipline of numerical analysis , formulas for the processing of discrete sets of data points to produce numerical approximations for interpolation and differentiation were being investigated.

FFT known as Fast Fourier Transform algorithm was developed, which suited perfectly for efficient digital implementation and it reduced the time required to compute transform by orders of magnitude (which utilizes the DTFS and DTFT practically).

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Block Diagram of Digital Communication System/Elements of DCS

A General Communication System can be viewed as a Transmitting unit and a Receiving Unit connected by a medium(Channel). Obviously Transmitter and Receiver consists of various sub systems (or) blocks.

Our basic aim is to understand the various modules and sub systems in the system. If we are trying to understand the design and various features of DCS , it is plus imperative that we have to understand how we should design a transmitter and we must understand how to design a very good quality Receiver. Therefore one must know the features of the channel to design a good Transmitter as well as receiver that is the channel and it’s contribution will come repeatedly in digital Communications. 

Source:- the primary block (or) the starting point of a DCS is an information source, it may be an analog/digital source , for example the signal considered is analog in nature, then the signal generated by the source is some kind of electrical signal which is random in nature. if the signal is a speech signal (not an electrical signal) that has to be converted into electrical signal by means of a Transducer, which can be considered as a part of source itself.

Sampling & Quantization:- the secondary block involves the conversion of analog to discrete signal 

this involves the following steps

Sampling:- it is the process that involves in the conversion of Continuous Amplitude Continuous Time (CACT) signal into Continuous Amplitude Discrete Time (CADT) signal.

Quantization:- it is the process that involves in the conversion of Continuous Amplitude Discrete Time (CADT) signal into Discrete Amplitude Discrete Time (DADT) signal.

Source Encoder:-  An important problem in  Digital Communications is the efficient representation of data generated by a Discrete Source, this is accomplished by source encoder.

” The process of representation of incoming data  from a Discrete source into a more suitable form required for Transmission is known as source encoding”

 Note:-The blocks Sampler, Quantizer followed by an Encoder constructs ADC (Analog to Digital Converter).

∴ the output of Source encoder is a Digital Signal, the advantages of Source coding are

  • It reduces the Redundancy.
  • Minimizes the average bit rate.

Channel encoder:-Channel coding is also known as error control coding. Channel coding is a technique which reduces the probability of error P_{e} by reducing Signal to Noise Ratio at the expense of Transmission Band Width.The device that performs the channel coding is known as Channel encoder.

Channel encoding increases the redundancy of incoming data , this also involves error detection and error correction  along with the channel decoder at the receiver.

Spreading Techniques:- Spread Spectrum techniques are the methods by which a signal generated with a particular Band Width is deliberately spread in the frequency domain, resulting in a signal with a wider Band width.

There are two types of spreading techniques available

  1. Direct Sequence Spread Spectrum Techniques.
  2. Frequency Hopping Spread Spectrum Techniques.

The output of a spreaded signal is very much larger than incoming sequence. Spreading increases the BW required for transmission, which is a disadvantage even though spreading is done for high security of data.

SS techniques are used in Military applications.

Modulator:- Spreaded sequence is modulated by using digital modulation schemes like ASK, PSK, FSK etc depending up on the requirement, now the transmitting antenna transmits the modulated data into the channel.

Receiver:- Once you understood the process involved in transmitter Block. One should perform reverse operations in the receiver block. 

i.e the input of the demodulator is demodulated after that de-spreaded and then the channel decoder removes the redundancy added by the channel encoder ,the output of channel decoder is then source decoded and is given to Destination.

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Why Digital Communication is preferred over Analog Communication?

Introduction:-

Communication is the process of establishing Connection (or) link between two points (which are separated by some distance) and transporting information between those two points. The electronic equipment used for communication purpose is called Communication equipment. The equipment when assembled together forms a communication system.

Examples of different types of communications are

  • Line Telephony & Telegraphy.
  • Radio Broadcasting.
  • Point-to-Point Communication.
  • Mobile Communication.
  • TV Broadcasting.
  • Radar and Satellite Communications etc.

Why Digital?

A General Communication system has two devices and a medium (channel) connecting those two devices. This can be understood that a Transmitter and Receiver are separated by a medium called as Communication channel. To transport an information-bearing signal from one point to another point over a communication channel either Analog or digital modulation techniques are used.

Now Coming to the point, Why Digital communication is preferred over analog Communication?

Why are communication systems, military and commercial alike, going digital?

  1. There are many reasons; the primary advantage is the ease with which digital signals compared with analog signals are generated. That is the generation of digital signals is much easier compared to analog signals.
  2. Propagation of Digital pulse through a Transmission line:-

When an ideal binary digital pulse propagating along a Transmission line. The shape of the waveform is affected by two mechanisms

  • Distortion caused on the ideal pulse because all Transmission lines and Circuits have some Non-ideal frequency Transfer function.
  • Unwanted electrical noise (or) other interference further distorts the pulse wave form.

Both of these mechanisms cause the pulse shape to degrade as a function of line length. During the time that the transmitted pulse can still be reliably identified (i.e. before it is degraded to an ambiguous state). The pulse is amplified by a digital amplifier that recovers its original ideal shape. The pulse is thus “re-born” (or) regenerated.

Circuits that perform this function at regular intervals along Transmission system are called “regenerative repeaters’. This is one of the reasons why Digital is preferred over Analog.

3.Digital Circuits Vs Analog Circuits:-

Digital Circuits are less subject to distortion and Interference than are analog circuits because binary digital circuits operate in one of two states FULLY ON (or) FULLY OFF to be meaningful, a disturbance must be large enough to change the circuit operating point from one state to another. Such two state operation facilitates signal representation and thus prevents noise and other disturbances from accumulating in transmission.

However, analog signals are not two-state signals, they can take an infinite variety of shapes with analog circuits and even a small disturbance can render the reproduced wave form unacceptably distorted. Once the analog signal is distorted, the distortion cannot be removed by amplification because accumulated noise is irrecoverably bound to analog signals, they cannot be perfectly generated.

 4. With digital techniques, extremely low error rates and high signal fidelity is possible through error detection and correction but similar procedures are not available with analog techniques.

5.  Digital circuits are more reliable and can be produced at a lower cost than analog circuits also; digital hardware lends itself to more flexible implementation than analog hardware.

Ex: – Microprocessors, Digital switching and large scale Integrated circuits.

6. The combining of Digital signals using Time Division Multiplexing (TDM) is simpler than the combining of analog signals using Frequency Division Multiplexing (FDM).

7. Digital techniques lend themselves naturally to signal processing functions that protect against interference and jamming (or) that provide encryption and privacy and also much data communication is from computer to computer (or) from digital instruments (or) terminal to computer, such digital terminations are normally best served by Digital Communication links.

8. Digital systems tend to be very signal-processing intensive compared with analog systems.

Apart from pros there exists a con in Digital Communications that is non-graceful degradation when the SNR drops below a certain threshold, the quality of service can change suddenly from very good to very poor. In contrast most analog Communication Systems degrade more gracefully.

 

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Maxwell’s Equations in Point (or Differential form) and Integral form

Maxwell’s Equations for time-varying fields in point and Integral form are:
  1. \overrightarrow{\bigtriangledown }X\overrightarrow{H}=\overrightarrow{J}+\frac{\partial \overrightarrow{D}}{\partial t}      \Rightarrow \oint_{l}\overrightarrow{H}.\overrightarrow{dl}=\oint_{s}\overline{J}.\overrightarrow{ds}+\int_{s}\frac{\partial \overrightarrow{D}}{\partial t}.\overrightarrow{ds}.
  2. \overrightarrow{\bigtriangledown }X\overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}       \Rightarrow \oint_{l}\overrightarrow{E}.\overrightarrow{dl}=-\int_{s}\frac{\partial \overrightarrow{B}}{\partial t}.\overrightarrow{ds} . 
  3. \overline{\bigtriangledown }.\overrightarrow{D}=\rho _{v}            \Rightarrow \oint_{s}\overrightarrow{D}.\overrightarrow{ds}=\int_{v}\rho _{v}dv.
  4. \overrightarrow{\bigtriangledown }.\overrightarrow{B}=0      \Rightarrow \oint_{s}\overrightarrow{B}.\overrightarrow{ds}=0.

The 4 Equations above are known as Maxwell’s Equations. Since Maxwell contributed to their development and establishes them as a self-consistent set.  Each differential Equation has its integral part. One form may be derived from the other with the help of Stoke’s theorem (or) Divergence theorem.

word statements of the field Equations:-

A word statement of the field Equations is readily obtained from their mathematical statement in the integral form.

1.\overrightarrow{\bigtriangledown }X\overrightarrow{H}=\overrightarrow{J}+\frac{\partial \overrightarrow{D}}{\partial t} \Rightarrow \oint_{l}\overrightarrow{H}.\overrightarrow{dl}=\oint_{s}\overline{J}.\overrightarrow{ds}+\int_{s}\frac{\partial \overrightarrow{D}}{\partial t}.\overrightarrow{ds}.

i.e, The magneto motive force (\because \oint_{l}\overrightarrow{H}.\overrightarrow{dl}\rightarrow is m.m.f)around a closed path is equal to the conduction current plus the time derivative of the electric displacement through any surface bounded by the path.

 2. \overrightarrow{\bigtriangledown }X\overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}\Rightarrow \oint_{l}\overrightarrow{E}.\overrightarrow{dl}=-\int_{s}\frac{\partial \overrightarrow{B}}{\partial t}.\overrightarrow{ds}.

The electro motive force (\because \oint_{l}\overrightarrow{E}.\overrightarrow{dl}\rightarrow is e.m.f)around a closed path is equal to the time derivative of the magnetic displacement through any surface bounded by the path.

3.\overrightarrow{\bigtriangledown }.\overrightarrow{D}=\rho _{v}  \Rightarrow \oint_{s}\overrightarrow{D}.\overrightarrow{ds}=\int_{v}\rho _{v}dv.

The total electric displacement through the surface enclosing a volume is equal to the total charge within the volume.

4.    \overrightarrow{\bigtriangledown }.\overrightarrow{B}=0  \Rightarrow \oint_{s}\overrightarrow{B}.\overrightarrow{ds}=0.

The net magnetic flux emerging through any close surface is zero.

the time-derivative of electric displacement is called displacement current. The term electric current is then to include both conduction current and displacement current. If the time-derivative of electric displacement is called an electric current, similarly \frac{\partial \overrightarrow{B}}{\partial t} is known as magnetic current, e.m.f as electric voltage and m.m.f as magnetic voltage.

the first two Maxwell’s Equations can be stated as 

  1. The magnetic voltage around a closed path is equal to the electric current through the path.
  2. The electric voltage around a closed path is equal to the magnetic current through the path.
Maxwell’s Equations for static fields in point and Integral form are:

Maxwell’s Equations of static-fields in differential form and integral form are:

  1. \overrightarrow{\bigtriangledown } X\overrightarrow{H}=\overrightarrow{J}        \Rightarrow \oint_{l}\overrightarrow{H}.\overrightarrow{dl}=\oint_{s}\overline{J}.\overrightarrow{ds}.
  2. \overline{\bigtriangledown } X\overrightarrow{E}=0           \Rightarrow \oint_{l}\overrightarrow{E}.\overrightarrow{dl}=0.
  3. \overline{\bigtriangledown }.\overrightarrow{D} = \rho _{v}            \Rightarrow \oint_{s}\overrightarrow{D}.\overrightarrow{ds}=\int_{v}\rho _{v}dv.
  4. \overline{\bigtriangledown }.\overrightarrow{B} = 0             \Rightarrow \oint_{s}\overrightarrow{B}.\overrightarrow{ds}=0.

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Inconsistensy in Ampere’s law (or) Displacement Current density

Faraday’s experimental law has been used to obtain one of Maxwell’s equations in differential form \overrightarrow{\bigtriangledown }X \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t} , which shows that a time-varying Magnetic field produces an Electric field.

From Ampere’s  Circuital law which is applicable to Steady Magnetic fields

\overrightarrow{\bigtriangledown } X \overrightarrow{H}=\overrightarrow{J}

By taking divergence of Ampere’s law the Ampere’s law is not consistent with time-varying fields

\overrightarrow{\bigtriangledown }.(\overrightarrow{\bigtriangledown } X \overrightarrow{H})=\overrightarrow{\bigtriangledown }.\overrightarrow{J}

\overrightarrow{\bigtriangledown } . \overrightarrow{J}=0 ,since \overrightarrow{\bigtriangledown }.(\overrightarrow{\bigtriangledown } X \overrightarrow{H})=0---------Equation(1)

the divergence of the curl is identically zero which implies \overrightarrow{\bigtriangledown }.\overrightarrow{J}=0------Equation(2), but from the continuity equation \overrightarrow{\bigtriangledown }.\overrightarrow{J} = -\frac{\partial \rho _{v}}{\partial t}-------Equation(3) which is not equal to zero, as \frac{\partial \rho _{v}}{\partial t}\neq 0 is an unrealistic limitation(i.e we can not assume \frac{\partial \rho _{v}}{\partial t} as zero) .

\therefore to make a compromise between the above two situations we must add an unknown term \overrightarrow{G} to Ampere’s Circuital law

i.e, \overrightarrow{\bigtriangledown }X\overrightarrow{H} = \overrightarrow{J}+\overrightarrow{G}

then by taking the Divergence of the above equation

\overrightarrow{\bigtriangledown }.(\overrightarrow{\bigtriangledown }X\overrightarrow{H}) = \overrightarrow{\bigtriangledown }.\overrightarrow{J}+\overrightarrow{\bigtriangledown }.\overrightarrow{G}------Equation(4)

from Equation(1),Equation(4) becomes     \overrightarrow{\bigtriangledown }.\overrightarrow{J}+\overrightarrow{\bigtriangledown }.\overrightarrow{G}=0

\overrightarrow{\bigtriangledown }.\overrightarrow{G}=-\overrightarrow{\bigtriangledown }.\overrightarrow{J}

thus \overrightarrow{\bigtriangledown }.\overrightarrow{G} = \frac{\partial \rho _{v}}{\partial t}---------Equation(5)

from Maxwell’s first Equation \overrightarrow{\bigtriangledown }.\overrightarrow{D}=\rho _{v} 

then Equation (5) becomes \overrightarrow{\bigtriangledown }.\overrightarrow{J} = \frac{\partial }{\partial t} (\overrightarrow{\bigtriangledown }.\overrightarrow{D})

\overrightarrow{\bigtriangledown }.\overrightarrow{G} =\overrightarrow{\bigtriangledown }. \frac{\partial \overrightarrow{D}}{\partial t}

then   \overrightarrow{G} = \frac{\partial \overrightarrow{D}}{\partial t}

\overrightarrow{\bigtriangledown }X\overrightarrow{H} = \overrightarrow{J}+\frac{\partial \overrightarrow{D}}{\partial t}

This is the equation obtained which does not disagree with the continuity equation. It is also consistent with all other results. This is a second Maxwell’s Equation is time-varying fields so the term \frac{\partial \overrightarrow{D}}{\partial t} has the dimensions of current density Amperes/Square-meter. Since it results from a time-varying electric flux density (\overrightarrow{D} ) , Maxwell termed it as displacement current density \overrightarrow{J_{D}}.

\overrightarrow{\bigtriangledown }X\overrightarrow{H} = \overrightarrow{J}+\frac{\partial \overrightarrow{D}}{\partial t}

\overrightarrow{\bigtriangledown }X\overrightarrow{H} = \overrightarrow{J}+\overrightarrow{J_{D}}

\overrightarrow{J_{D}}=\frac{\partial \overrightarrow{D}}{\partial t}

up to this point three current densities are there \overrightarrow{J}=\sigma \overrightarrow{E} , \overrightarrow{J}=\rho _{v} \overrightarrow{v} and \overrightarrow{J_{D}}= \frac{\partial\overrightarrow{D} }{\partial t}.

when the medium is Non-conducting medium \overrightarrow{\bigtriangledown }X \overrightarrow{H}=\frac{\partial\overrightarrow{D} }{\partial t}

the total displacement current crossing any given surface is expressed by the surface integral I_{d} = \oint_{s} \overrightarrow{J_{D}}.\overrightarrow{ds}

I_{d} = \oint_{s} \frac{\partial \overrightarrow{D}}{\partial t}.\overrightarrow{ds}

from Ampere’s law \oint_{s}(\overrightarrow{\bigtriangledown }X \overrightarrow{H}).\overrightarrow{ds}=\int_{s} \overrightarrow{J}.\overrightarrow{ds} +\oint_{s} \frac{\partial \overrightarrow{D}}{\partial t}.\overrightarrow{ds}

\oint_{l}\overrightarrow{H}.\overrightarrow{dl}=\int_{s} \overrightarrow{J}.\overrightarrow{ds} +\oint_{s} \frac{\partial \overrightarrow{D}}{\partial t}.\overrightarrow{ds}

\oint_{l}\overrightarrow{H}.\overrightarrow{dl}=I +\oint_{s} \frac{\partial \overrightarrow{D}}{\partial t}.\overrightarrow{ds}

\oint_{l}\overrightarrow{H}.\overrightarrow{dl}=I +I_{d}

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Correlation Receiver(Special case of Optimum Receiver)

we know that one form of Optimum filter is Matched filter, we will now derive another form of Optimum filter that is different from Matched filter Let the input to the Optimum filter is v(t) which is a noisy input that is v(t)=x(t)+n(t)

from the figure output of the filter after sampling at t=T_{b} seconds is v_{o}(T_{b})

v_{o}(t)=v(t)*h(t)

v_{o}(t) = \int_{-\infty }^{T_{b}}v(\tau ) h(t-\tau )d\tau

at t= T_{b} output becomes  v_{o}(T_{b}) = \int_{-\infty }^{T_{b}}v(\tau ) h(T_{b}-\tau )d\tau -----Equation(1)

Now by substituting h(\tau )=x_{2}(T_{b}-\tau )-x_{1}(T_{b}-\tau )

h(T_{b}-\tau )=x_{2}(T_{b}-T_{b}+\tau )-x_{1}(T_{b}-T_{b}+\tau )

h(T_{b}-\tau )=x_{2}(\tau )-x_{1}(\tau )------Equation(2)

by substituting the  Equation(2)  in Equation(1) over the limits [0,T_{b}]

v_{o}(T_{b}) = \int_{0 }^{T_{b}}v(\tau ) (x_{2}(\tau )-x_{1}(\tau ))d\tau

v_{o}(T_{b}) = \int_{0 }^{T_{b}}v(\tau ) x_{2}(\tau ) d\tau-\int_{0}^{T_{b}}v(\tau )x_{1}(\tau )d\tau

Now by replacing \tau with t the above equation becomes 

v_{o}(T_{b}) = \int_{0 }^{T_{b}}v(t ) x_{2}(t ) dt-\int_{0}^{T_{b}}v(t )x_{1}(t )dt-----Equation(3)

The equation (3) suggests that the Optimum Receiver can be implemented as shown in the figure, this form of the Receiver is called  as correlation Receiver. This receiver requires the integration operation be ideal with zero initial conditions. Correlation Receiver performs coherent-detection.

in general Correlation Receiver can be approximated with Integrate and dump filter.

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