# Modulating Wave To Be Tone Signal Computer Science Essay

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Modulating index is defined as the ratio of the amplitude of message signal to the amplitude of carrier signal. is the % modulation. To avoid envelope distortion

Comparing (3) & (4)

By dividing (5) by (6) we get

## SQUARE LAW MODULATOR:

## SWITCHING MODULATOR:

In switching modulator the diode is used as a switching element. As it acts as a device with time-varying characteristic, generating the desired AM signals when it is used in the circuit shown in figure (a)

(a) Switching modulator

(b) Switching characteristic of the diode-load combination

Figure

The (ideal) transfer characteristics of the diode-load combination is shown at (b) in Fig. 4.35. This is explained as follows. We have,

Where m(t) is the message signal and c(t) is the carrier signal. Since [m(t)]max << Ac, the behaviour of the diode is governed by c(t) and can be approximated as

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That is the diode offers infinite impedance when reverse biased and has zero impedance when forward biased. Hence whether V1(t) is switched or not switched to the output depends on the carrier cycle. V2(t) can be expressed as

Where xp(t) - periodic rectangular pulse train of frequency fc and duty cycle 50%

Fourier series of Xp(t) is given by,

Substituting (1) and (4) in (3) we get

The first term in the RHS of (5) is the desired AM wave with the carrier amplitude Ac/2 and the amplitude sensitivity ka = 4/Ac

V2(t) is composed of two components. They are

The desired AM wave :

The undesired terms with

impulses in spectra at f=0,+2fc,+4fc,â€¦â€¦â€¦.

Spectral lobes of width 2w centered at 0,+3fc,+5fc.,,,,,

The undesired terms of v2(t) can be filtered by using properly designed filter. The required AM signal available at the output of the filter is given by

## ADVANTAGES:

Generated AM signals can have larger power levels.

Filtering requirements are less stringent because we can separate the desired AM signal if fc>2w

## DISADVANTAGES:

The percentage modulation has to be low inorder that the switching characteristics of the diode are controlled only by the carrier.

## DEMODULATION:

Demodulation is the process of recovering the original baseband signal from an incoming modulated wave.

Demodulation is the reverse process of modulation

DEMODULATION OF AM WAVES:

There are two devices used for the demodulation of AM waves. They are

Square law detector

Envelope detector

## (1)SQUARE LAW DETECTOR:

Square law detector is obtained by using a square law modulator for the purpose of demodulation. When a non-linear element such as semiconductor diode is properly biased and operated in a restricted portion of the characteristic curve has a non-linear V-I characteristic.

For small variations of applied signal around a suitable operating point, the transfer characteristic of the diode can be represented by square law i.e.

V2(t) = a1v1(t) + a2v12(t) +a3v13(t)+â€¦â€¦â€¦â€¦

Since the signal is weak, the higher terms are neglected

V2(t) = a1v1(t) + a2v12(t) â€¦â€¦â€¦â€¦.(1)

Where a1 , a2 - constants

The last term in the (3) is the desired base band signal. This can be extracted from (3) by using a low pass filter. The output of LPF is given by

## ENVELOPE DETECTOR:

When AM signal not over modulated allows the recovery of m(t) from its envelope. A good approximation to the ideal envelope detector can be realized with a simple electronic circuit shown in fig(a).

Let us assume the diode D to be ideal. When it is forward biased it acts as a short circuit and thereby making the capacitor C charge through the source resistance Rs. When it is reverse biased, it acts as an open circuit and C discharges through the load resistance RL. The operation of the detector depends upon the charge and discharge of C.

If the time constants RsC and RLC are properly chosen, v1(t) follows the envelope of s(t) are fairly closely. During the conduction cycle of diode, C quickly charges to the peak value of the carrier at the time instant. It will discharge a little during the next off cycle of the diode.

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The time constants of the circuit will control the ripple about the actual envelope. Cb is a blocking capacitor and the final v0(t) will be proportional to m(t) as shown in fig(b).

A small high frequency ripple at the carrier frequency could be present on v0(t). For audio transmission, this would not cause any problem as fc is generally much higher than the upper limit of the audio frequency range.

During the charging cycle, the C will be charged to the peak value of the carrier in as short a time as possible. i.e.

RsC << 1/fc

Discharging time constant should be large enough so that C does not discharge too much between the positive peaks of the carrier but small enough to be able follow the maximum rate of change of m(t). This is the maximum rate of change of m(t). This maximum rate depends on W, the highest frequency in M(f). i.e.

1/fc << RLC < 1/W

Too small a value for RLC will make v1(t) somewhat ragged (i.e. sort of saw tooth ripple on the top) whereas with too large value for RLC, envelope detector fails to follow the envelope during the periods when m(t) is decreasing.

LINEAR MODULATION SCHEMES:

Linear modulation is defined by its general form by

This equation is the canonical representation of a narrowband signal. In linear modulation SI(t) and SQ(t) are low-pass signals that are linearly related to the message signal m(t)

Where SI(t) - in-phase component of the modulated wave s(t)

SQ(t) - quadrature component of s(t)

Depending on how these two components of s(t) are defined there are three types of linear modulation involving a single message signal. They are

Double sideband suppressed carrier(DSBSC)

Single sideband modulation (SSB)

Vestigial sideband (VSB) modulation

## DOUBLE SIDE BAND-SUPPRESSED CARRIER (DSBSC) MODULATION:

The DSBSC is the simplest form of all the 3 linear modulation schemes.

DSBSC modulation is the process in which the upper band and lower sidebands are transmitted.

GENERATION OF DSBSC WAVES:

There are methods of generating DSBSC signals. They are

Product Modulator

Ring Modulator

Balanced Modulator

PRODUCT MODULATOR:

This DCBSC modulation involves just the multiplication of the message signal and the carrier, this scheme is also known as product modulation.

Let the carrier signal e(t) be

e(t)=Ec cos(2âˆfct)

and the message signal em(t) with â”‚em(f)â”‚=0 for â”‚fâ”‚>w.

This form of linear modulation is generated by using a product modulator which simply multiplies the message signal em(t) by the carrier wave. i.e

s(t)= e(t) em(t)

s(t)= Ec em(t) cos(2âˆfct)

The modulated signal s(t) undergoes a phase reversal whenever the message signal em(t) crosses zero. Hence the envelope of a DSBSC modulated signal is different from the message signal.

Taking Fourier Transform of s(t) we get

s(f)=1/2 Ec [M(f-fc)+M(f+fc)]

Except for a change in scale factor, the modulation process simply translates the spectrum of the baseband signal by Â±fc . The transmission bandwidth of DSBSC wave is 2w as Am.

RING MODULATOR:

The balanced ring modulator is shown in the fig (A). In ring modulator four are connected in the form of a ring in which all four diode point in the same manner. All the four diodes in ring are controlled by a square wave carrier signal e(t) applied through a center tapped transformer e(t) is much larger than the message signal em(t).

During the positive half cycle of e(t) i.e. when T1 is positive and T2 is positive D2 and D3 are forward biased where as D1 and D4 are reverse biased. The voltage at point a and b are switched to b' and a' and hence -em(t) is switched to the output. The carrier signal e(t) is a periodic signal of frequency fc defined by

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Examples of our workFourier series as

## âˆž

xp(t)=4/âˆ âˆ‘ (-1)n /2n+1 cos[2âˆ(2n+1)]

n=0

and v(t) is given by

v(t) = em(t) xp(t)

## âˆž

v(t) = em(t) [4/âˆ cos(2âˆfct) + 4/âˆ âˆ‘ (-1)n/2n+1 cos[2âˆ(2n+1)fct]]_

n=1

This v(t) is passed through a BPF tuned to fc, the output is the desired DSBSC signal namely

s(t) = 4/âˆ m(t) cos(2âˆfct)

ADVANTAGES:

It's output is stable.

Require no external power source to activate the diodes.

Virtually no maintenance and long life.

3. BALANCED MODULATOR:

The carrier voltage is applied in parallel to the input of the two matched diodes D1 and D2 where as the modulating voltage in push pull

To the same two diodes are shown in the fig (B).

Let the carrier signal ec(t) be

ec(t) = Ec cos 2âˆfct

and the message signal em(t) be

em(t) = em cos(2âˆfmt)

then the voltage applied to the diodes D1 and D2 are

v1(t) = ec(t) + em(t) = ec cos(2âˆfct) + em cos(2fmt)

v2(t) = ec(t) - em(t) = ec cos(2âˆfct) - em(t) cos(2fmt)

According to square law the output voltage of the diodes D1 and D2 are

V3(t) = a1v1 + a2v12 + a3v13+â€¦â€¦.

V4(t) = a1v2 m+ a2v22 + a3v23 +â€¦â€¦.

v3 (t) = a1 Ec cos(2Î fct) + a2 Em cos(2Î fmt) + a2 Ec2

cos2(2Î fct) + a2 Em2 cos2(2Î fmt) + 2a2Ec Em

cos(2Î fct) cos(2Î fmt)

v4(t) = a1 Eccos(2Î fct) - a1 Emcos(2Î fmt) + a2 Ec2

cos2(2Î fct) + a2 Em2cos2(2Î fmt) - 2a2EcEm

cos(2Î fct) cos(2Î fmt)

The output voltage is given by

v0(t) = v3(t) - v4(t)

= 2a1 Em cos(2Î fmt) + 4a2 Em Ec

cos(2Î fct) cos(2Î fmt)

v0(t) = 2a1 em(t) + a2 Ec em(t) cos(2Î fct)

This v(t) is passed through a BPF which is tuned to fc, the output is the desired DSBSC signal.

s(t) = 4a2 Ec em(t) cos(2Î fct)

ADVANTAGES:

Suppression of carrier results in economy of power.

It is commonly used in carrier current telephony system.

It increases the efficiency because the carrier is suppressed.

DEMODULATION OF DSBSC WAVE:

There are many methods to demodulate DSBSC wave. They are

Coherent detection.

Carrier recovery for coherent demodulation.

Costos Receiver.

Squaring loop.

Quadrature carrier multiplexing.

## SINGLE SIDE BAND (SSB) MODULATION:

SSB modulation is defined as the process in which only one sideband either lower side band or upper side band is transmitted.

## GENERATION OF SSB WAVE:

There are two methods to generate SSB wave. They are

Frequency Discrimination method

Phase Discrimination method

## FREQUENCY DISCRIMINATION METHOD:

BPF

PRODUCT

MODULATOR

em(t) s(t)

Em cos2Î fmt SSB wave

Ec cos(2Î fct)

The frequency discrimination method consists of two stages.

The first stage is a product modulator which generates a DSBSC wave.

The second stage is a band pass filter which is designed to pass one of the sidebands of this modulated wave and suppress the other.

The design of BPF in frequency discriminator must satisfy the following 3 requirements.

The desired sideband lies inside the pass band of the filter

The unwanted sideband lies inside the stop band of the filter

The filters transition band which separates the pass band from stop band is twice the lowest frequency component of the message signal

Let em(t) and ec(t) be the message and carrier signals given by

em(t) = Em cos(2Î fmt)

ec(t) = Ec cos(2Î fct)

The output of the product modulator is given by

v(t) = em(t) ec(t)

= EmEc/2 cos2Î fct cos2Î fmt

v(t) = Em Ec/2 [cos[2Î (fc+fm)t]+cos[2Î (fc-fm)t]]

This v(t) signal is applied to the BPF which is used to neglect the lower sideband. The resultant wave at the filter output is the required SSB wave.

s(t) = Em Ec cos[2Î (fc+fm)t]

## 1. PHASE DISCRIMINATION METHOD:

The phase discriminator consists of two product modulator, two

âˆ/2 phase shifters and an adder. One of the phase shifter is a Hilbert Transformer which provides âˆ/2 phase shift for all the components in M(f).

Hilbert Transformer (HT) is not an easy circuit to realize. So instead of wide band phase shifter (HT), it is possible to have two phase shifting networks as shown is fig (2).

Let em(t) and ec(t) be the message and carrier signals and given by

em(t) = Em cos(2Î fmt)

ec(t) = Ec cos(2Î fct)

The output of the phase shift network 1is

v1(t) = Em cos(2Î fmt+Î¸1)

The output of the phase shift network 2 is

v2(t) = Em cos(2Î fmt+Î¸2)

## FREQUECNY DISCRIMINATION METHOD:

The frequency discrimination method consists of two stages.

The first stage is a product modulator which generates a DSBSC wave

The second stage is a band pass filter which is designed to pass one of the sidebands of this modulated wave and suppress the other

The design of BPF in frequency discrimination must satisfy the following 3 requirements

The desired sideband lies inside the pass band of the filter

The unwanted sideband lies inside the stop band of the filter

The filters transition band which separates the pass band from stop band is twice the lowest frequency component of the message signal

Let m(t) and c(t) be the message and carrier signals given by

The output of the product modulator is given by

V(t) = m(t) c(t)

This v(t) signal is applied to the BPF which is used to neglect the lower side band. The resultant wave at the filter output is the required SSB wave

PHASE DISCRIMINATION METHOD:

The phase discriminator consists of two product modulator, two phase shifters and an adder. One of the phase shifter is a Hilbert Transformer which provides phase shift for all the components in m(f).

Hilbert Transformer (HT) is not an easy circuit to realize so instead of the wideband phase shifter(HT), it is possible to have two phase shifting networks as shown in fig(2)

Let m(t) and c(t) be the message and carrier signals and are given by

The output of the phase shift network1 is

The output of the phase shift network2 is

## FREQUENCY TRANSLATION:

If we have a modulated signal s1(t) whose spectrum is centered on a carrier frequency f1 and the requirement is to translate it upward in frequency such that its carrier frequency is changed from f1 to a new value f2. This requirement is accomplished by using a mixer as shown in fig(1). The mixer is a device which consists of a product modulator followed by a BPF.

It is assumed that the mixer input s(t) is an AM signal with carrier frequency f1 and band width 2W. Fig(A) displays the AM spectrum s(f) assuming that f1>W. Fig(B) displays the spectrum s'(f) of the resulting signal s'(t) at the output of the product modulator. The s'(t) is the sum of two modulated components. One component is represented by shaded part and another component is represented by unshaded part of the spectrum shown in fig(B). Depending on whether the incoming carrier frequency f1 is translated upward or downward, there are two different conversions. They are

Up conversion

Down conversion

(1). UP CONVERSION:

In this case the translated carrier frequency f2 is greater than incoming carrier frequency f1.

i.e. f2>f1

f2=f1+fl

The local oscillator frequency is given by

Fl=f2-f1

The unshaded part of the spectrum in fig(B) defines the wanted modulated signal and the shaded part of this spectrum defines the image signal associated with s2(t). The mixer is now referred to as frequency up converter.

FREQUENCY DOWN CONVERTER:

In this case the translated carrier frequency f2 is smaller than the incoming carrier frequency

i.e. f1<f2

f2=f1-fl

The local oscillator frequency is given by

Fl=f1-f2

The shaded part of the spectrum shown in fig(B) defines the modulated signal and the unshaded part of this spectrum defines the image signal associated with s2(t). The mixer is now referred to as frequency down converter.

The purpose of the band pass filter in the mixer of fig(1) is used to pass the wanted modulated signal s2(t) and eliminate the associated image signal.

## FREQUENCY DIVISION MULTIPLEXING:

An important form of signal processing operation is multiplexing whereby a number of independent signals can be combined into a composite signal suitable for transmission over a common channel. To transmit a number of these signals over the same channel, the signal must be kept apart so that they do not interfere each other and thus they can be separated at the receiving end. This is accomplished by separating the signals either in frequency or in time.

The technique of separating the signals in frequency is called Frequency Division Multiplexing (FDM). The technique of separating the signals in time is called time division multiplexing(TDM).

The block diagram of FDM system is shown in fig(1). The incoming message signals are assumed to be of the low pass type but their spectra do not necessarily have non zero values all the way down to zero frequency.

Following each input signal, we have shown a LPF which is designed to remove high frequency components that do not contribute significantly to signal representation but are capable of disturbing other message signals that are the common channel. These LPF may be omitted only if the input signals are band limited initially.

The filtered signals are applied to modulators that shift the frequency ranges of the signals so as to occupy mutually exclusive frequency intervals. The necessary carrier frequency needed to perform frequency translation are obtained from a carrier supply. The most widely used method of modulation in FDM is SSB modulation which in the case of voice signals, requires the bandwidth that is approximately equal to that of original voice signals.

The band pass filters following the modulators are used to restrict the band of each modulated wave to its prescribed range. The resulting BPf outputs are next combined in parallel to form the input to the common channel.

At the receiving terminal a bank of BPFs, with their inputs are connected in parallel is used to separate the message signals on a frequency occupancy basis.

Finally the original message signals are recovered by individual demodulators.

## SUPER HETERODYNE RECEIVER:

The receiver not only demodulates the modulated wave but also performs some system functions given by

Carrier frequency tuning the purpose of which is to select the desired signal.

Filtering which is required to separate the desired signal from other modulated signals that may be picked up along the way.

Amplification which is intended to compensate for the loss of signal power incurred in the course of transmission.

The super heterodyne receiver or superhet is a special type of receiver that fulfills all the three functions. It overcomes the difficulty of having to built a tunable highly selective and variable filter.

The incoming AM wave is picked up by the receiving antenna and amplified in the RF section that is tuned to the carrier frequency of the incoming wave. The combination of mixer and local oscillator provides a heterodying function whereby the incoming signal is converted to a predetermined fixed intermediate frequency lower than the incoming carrier frequency. This translation is achieved without disturbing the relation of the sidebands to the carrier. The result of hetrodying is to produce and intermediate frequency carrier defined by

FIF=fLO-fRFâ€¦â€¦â€¦â€¦.(1)

where fLO - frequency of local oscillator

fRF - carrier frequency of incoming RF signal

fIf - Intermediate frequency

The mixer - local oscillator combination is called the first detector and the demodulators is called the second detector.