The post Switches-Circuit Switching appeared first on ecenotes.

]]>**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.

- 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.
- The second stage consists of k Cross-bar switches with each cross-bar switch size as (N/n) X (N/n).
- The third stage consists of N/n cross-bar switches with each switch size as k X n.

The total number of cross points = , so the number of cross points required are less than single-stage cross-bar Switch = .

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

, and Total no.of Cross points .

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]]>The post GATE problems in EMT appeared first on ecenotes.

]]>Ans:- Given loss less insulator

Cm/Sec

from

from Equations I and II

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]]>The post Solved Example problems in Electro Magnetic Theory appeared first on ecenotes.

]]>Ans. Given P(1,3,5)

Cylindrical :-

Similarly

Spherical :-

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]]>The post Phase Locked Loop (PLL) appeared first on ecenotes.

]]>Let the input to PLL is an FM signal

let

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.

the corresponding phase

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

the error signal

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

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

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]]>The post Frequency domain representation of a Wide Band FM appeared first on ecenotes.

]]>i.e, Single-tone FM signal is

Now by expressing the above signal in terms of Phasor notation ( , None of the terms can be neglected)

Let is the complex envelope of FM signal.

is a periodic function with period . This can be expressed in it’s Complex Fourier Series expansion.

i.e, this approximation is valid over . Now the Fourier Coefficient

let implies

as and

let as order Bessel Function of first kind then .

Continuous Fourier Series expansion of

Now substituting this in the Equation (I)

The Frequency spectrum can be obtained by taking Fourier Transform

n value | wide Band FM signal |

0 | |

1 | |

-1 | |

… | …. |

From the above Equation it is clear that

- FM signal has infinite number of side bands at frequencies for n values changing from to .
- The relative amplitudes of all the side bands depends on the value of .
- The number of significant side bands depends on the modulation index .
- The average power of FM wave is Watts.

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]]>The post Matched Filter, impulse response h(t) appeared first on ecenotes.

]]>**Transfer Function of Matched Filter:-**

Transfer Function of Optimum filter is

if input noise is white noise , its Power spectral density (Psd) is .

then H(f) becomes

From the properties of Fourier Transforms , by Conjugate Symmetry property

Equation (I) becomes

From Time-shifting property of Fourier Transforms

From Time-Reversal Property

By Shifting the signal by T Seconds in positive direction(time) ,the Fourier Transform is given by

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

Let the constant is set to 1, then the impulse response of Matched Filter will become .

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]]>The post Mutual Information I(X ; Y) Properties appeared first on ecenotes.

]]>Mutual Information is given by equation

we know that

Substitute Equation (II) in Equation (I)

The above Equation can be written as

we knew that

This result can be applied to Mutual Information , If and be , Both and are two probability distributions on same alphabet , then Equation (III) becomes

i.e, , Which implies that Mutual Information is always Non-negative (Positive).

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]]>The post Example Problems in Electro Magnetic Theory Wave propagation appeared first on ecenotes.

]]>Ans. Given , and

and

Find the Loss tangent

So given medium is a Conductor (Copper)

then

, .

.

.

2. If for a medium in which a wave with a frequency of is propagating . Determine the propagation constant and intrinsic impedance of the medium when

Ans: Given , , and .

Since , the given medium is a lossless Di-electric.

which implies

.

Ω.

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]]>The post Delta modulation and Demodulation appeared first on ecenotes.

]]>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 ) 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 () and the approximation () is quantized into only two levels corresponding to Positive and negative differences.

i.e, If the approximation () falls below the signal ()at any sampling epoch(the beginning of a period)output signal level is increased by .

On the other hand the approximation () lies above the signal () , output signal level is diminished by 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 .

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:

Present sample of the (input) sampled signal:

last sample approximation of stair case signal:

Quantized error signal( output of one-bit Quantizer):

if .

and .

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

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

from the accumulator

where is the Quantization error.

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]]>The post Fourier Series and it’s applications appeared first on ecenotes.

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

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

- In the analyzation of signals and LTI systems.
- Designing of Signals & Systems.
- 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 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

- The theory of Integration.
- Point-set topology.
- 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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]]>