## Current equation of the diode

The diode under Forward bias is as follows

The current equation related to the voltage V and current I is given by

$I=I_{o}(e^{\frac{V}{\eta&space;v_{T}}}-1)&space;Amperes$

I- Diode current

Io– Reverse saturation current of the diode at room temperature.

V-applied External voltage

$\eta$– constant   = 1  For Ge

= 2 For Si.

$v_{T}=\frac{kT}{q}$ – volt equivalent temperature 26 mV at room temp.

where k-Boltzmann constant = 1.38 X 10-23 J/K.

T-Temperature of Diode in kelvin   oK = o C + 273.

q- charge of electron  = 1.6X10-19 C.

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Reverse Saturation Current (Io) of PN-Diode:-

## Pulse Position Modulation(PPM)

Pulse Position modulation is another type of Pulse Time modulation technique that is in PPM the position of the pulse carrier is varied in accordance with the instantaneous values of the message signal, where as the amplitude and width of the pulse remains constant. here message lies in the position(OFF periods) of the PPM signal.

PPM demodulator:-

The PPM Demodulator consists of a Transistor T1  which acts as a switch followed by a second order Low pass filter circuit( using OP-AMP).

As the  input to the demodulator is a PPM signal, the gaps between pulses contains the information in PPM signal. Let us consider a PPM signal with OFF and ON periods marked from A to F.

Here Transistor T1 acts as a switch  as follows

• input to the base of T1 is low  —–> Transistor T1 is in cut-off region.
• input to the base of T1 is high  —–> Transistor T1 is in Saturation region.

during  the time inerval AB, the input to the base of T1 is low and transistor T1 moves into cut-off region in this condition capacitor C charges to a vlotage proportional to length of time duaration AB that is the height of the ramp is equals to duration AB.

During the time interval BC, the input to T1 is high and T1 moves into Saturation region in this case Capacitor ‘C’ discharges through T1 , this discharge is rapid and the collector voltage remains low over the duartion BC.

This process continues and results a saw-tooth wave form at the output of transistor T1 , by applying this signal to a second order LPFn Demodulated signal has been obtained as the final output.

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This is  the most commonly used Receiver and it uses “hetero dyning” principle which is used almost in all types of receivers like TR Receiver and Radar Receiver etc. The word hetero(≈different) dyne(≈mixing) means mixing  different frequencies using a Mixer. Hence the name given as super hetero dyne Receiver.

The block diagram consists of  a receiving antenna followed by an RF stage as the primary block , the receiving signal has been fed to RF stage through the antenna.

In a Super hetero dyne Receiver the incoming RF signal frequency ($f_{s}$) is combined with local oscillator frequency($f_{o}$) through a mixer and converts a signal of a lower fixed frequency (IF) this lower fixed frequency is called as Intermediate Frequency ($f_{i}$ or $f_{IF}$). A constant frequency difference is maintained between the Local Oscillator and incoming RF signal. This is provided through Capacitance tuning that is all capacitors are ganged together and operated by a common control knob.

$\therefore$ incoming RF is  down translated to IF using a mixer now this IF is given as input to the secondary stage of the block diagram that is IF amplifier. IF amplifier consists of number of transformers each consisting of a pair of mutually tuned circuits thus with a large number of double tuned circuits, operating at a specially chosen frequency the IF amplifier provides most of the gain.

Thus IF stage full fills most of the gain (sensitivity) and Band width(selectivity) requirements of the Receiver. For a Super hetero dyne receiver Sensitivity and selectivity are quite uniform throughout it’s tuning range this is one of the advantage over TRF Receiver.

The amplified IF signal is given as an input to the Detector. The Detector or the demodulator demodulates the signal and down translates the IF signal to AF(Audio Frequency) signal.

The AF signal is amplified by Audio amplifier and further by power amplifier. The last stage of the receiver is a Loud speaker , which receives AF signal. Loud speaker is in general a transducer which converts electrical signal into a voice (or) Audio.

• It provides high gain through IF amplifier that is more sensitivity is being provided by it.
• Improved selectivity over TRF receiver.
• BW remains constant over the entire operating range.
• Selectivity and Sensitivity are uniform throughout it’s tuning range.
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The block diagram consists of  a receiving antenna followed by an RF stage as the primary block , the receiving signal has been fed to RF stage through the antenna. This RF stage consists of two (or) three RF Amplifiers, these amplifiers are tuned RF Amplifiers.i.e they have variable tuned circuits at input and output sides.

The received signal has been amplified by the RF amplifiers and the amplified signal is being given as an input to the Detector. The Detector or the demodulator demodulates the signal and down converts the RF signal to AF(Audio Frequency) signal.

The AF signal is amplified by Audio amplifier and further by power amplifier. The last stage of the receiver is a Loud speaker , which receives AF signal. Loud speaker is in general a transducer which converts electrical signal into a voice (or) Audio.

1. Selectivity of TRF Receiver is poor. This is because achieving sufficient selectivity at high frequencies is difficult due to enforced use of single-tuned Circuits.
2. Instability:-(RF Stage)  The TRF Receiver suffers from a tendency to oscillate at a higher frequencies (i.e, instability), this is because multi-stage RF amplifiers has to provide high gain at high frequencies. RF amplifiers provides high gain which results in positive feed back leads to oscillations and then causes instability of the circuit. This positive feedback (caused by the leakage of output of RF stage back to it’s input) could result from power supply coupling through any other element common to input and output stages.
3. Variation of band width over tuning range:- One more draw back in TRF receiver is the BW variation over the tuning range i.e the BW of TRF receiver varies with the incoming frequency.

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## Sampling Theorem

Sampling of signals is the fundamental operation in signal processing, a Continuous Time (CT) signal can be converted into a Discrete Time (DT) signal using Sampling process. Sampling is required since the advancement in both signals and systems which are digitized i.e, Digital systems operates only on digital signals only.

Sampling Theorem:-

A CT signal is first converted into DT signal by Sampling process. The sufficient number of samples must be taken so that the original signal is represented in it’s samples completely, and also the signal is represented from it’s samples, these two conditions representation and reconstruction depends on the sampling process ‘fs‘ Hz.

Sampling theorem can be given into two parts

i. A band limited signal of finite energy, which has no frequency component higher than ‘fm‘ Hz, is completely described by it’s sample values at uniform intervals less than (or) equal to 1/2fm seconds apart.

i.e, $T_{s}\leq&space;\frac{1}{2f_{m}}$  Seconds.

ii. A Band limited signal of finite energy, which has no frequency component higher than fm Hz may be completely recovered from the knowledge of it’s samples if samples are taken at the rate of 2fm samples/second.

i.e, $f_{s}\geq&space;2f_{m}$ Hz.

Statement:- A Continuous Time signal can be completely represented in it’s samples and recovered from it’s samples if the sampling frequency $f_{s}\geq&space;2f_{m}Hz.$

where $f_{s}$ is the sampling frequency.

$f_{m}$ is the highest frequency present in the original signal / Band width of the signal.

proof of Sampling theorem:-

Let us consider a CT signal x(t), which is a band limited to $f_{m}$ Hz as shown

To prove Sampling theorem, it should be shown a signal whose spectrum is band limited to fm Hz can be reconstructed exactly without any error from it’s samples taken uniformly at a rate of $f_{s}>&space;2f_{m}$ Hz.

The circuit shows the sampler

Now sampling of x(t) at a rate of fs may be achieved by multiplying x(t) with a train of impulses  $\delta&space;T_{s}(t)$ with a period ‘Ts‘ seconds.

The sampling signal is an ideal (or) instantaneous signal. This is also known as ideal (or) instantaneous sampling.

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

As $\delta&space;T_{s}(t)$ is a periodic impulse train it can be expressed in it’s Fourier Series expansion as follows

Exponential Fourier Series is

$\delta&space;T_{s}(t)&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}F_{n}e^{jnw_{s}t}$

$F_{n}=&space;\frac{1}{T_{s}}\int_{\frac{-T_{s}]}{2}}^{\frac{T_{s}}{2}}\delta&space;T_{s}(t)e^{-jn\omega&space;_{s}t}dt$

$F_{n}=\frac{1}{T_{s}}$

$F_{n}=f_{s}$

∴ Exponential Fourier Series is $\delta&space;T_{s}(t)=\sum_{n=-\infty&space;}^{\infty&space;}f_{s}e^{jn\omega&space;_{s}t}$

now the sampled signal $g(t)&space;=&space;x(t).\delta&space;T_{s}(t)$

$g(t)=x(t)\sum_{n=&space;-\infty&space;}^{\infty&space;}f_{s}e^{jn\omega&space;_{s}t}$

$g(t)=\sum_{n=&space;-\infty&space;}^{\infty&space;}f_{s}x(t)e^{jn\omega&space;_{s}t}$

By finding Fourier Transform of g(t) is G(f)

$G(f)=\sum_{n=&space;-\infty&space;}^{\infty&space;}f_{s}X(f-nf_{s})$

Now the frequency spectrum of the sampled signal G(f) is of the form

From G(f) spectrum the original spectrum of X(f) has been shifted to different center frequencies

i.e, when n=0  center frequency is 0.

n=1  center frequency is fs

n=-1 center frequency is -fs etc

Some important conclusions from frequency spectrum of sampled signal:-

1. The spectrum of sampled signal G(f)/G(w) will repeat periodically if $f_{s}>&space;2f_{m}$ without any overlapping.
2. G(f) is extending up to infinity and the Band width is infinity as well, out of G(f) , X(f) need to be recovered , which is band limited to fm Hz.
3. X(f) is centered at f=0 and has fm as the highest frequency, X(f) may be recovered by passing it through a Loe Pass filter with cutoff frequency approximately equals to fm  Hz.
4. to reconstruct x(t) from g(t) the condition that must be satisfied is  $f_{s}\geq&space;2f_{m}$.

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