MAYURBHANJ SCHOOL OF ENGINEERING � LAXMIPOSI ,BARIPADA,757107

• DEPARTMENT- E&TC ENGG.

SEMISTAR- 5^TH

• SUBJECT-A & D Communication
• NAME OF TOPIC –PULSE MODULATION SYSTEM
• PREPARED BY - U S Panda

Pulse Modulation

Analog Pulse Modulation

Digital Pulse Modulation

Pulse Amplitude Modulation (PAM)

Pulse Width Modulation (PWM)

Pulse Code Modulation (PCM)

Delta Modulation (DM)

Pulse Position Modulation (PPM)

​

Advantage of Pulse modulation: (i) Transmitted power is no longer continuous as in CW

Modulation, but pulsed in nature

(ii) Vacant time between pulse occurrence filled by interleaving/multiplexing pulse waveforms of some other Message (TDM)

Sampling Theorem

This provides a mechanism for representing a continuous time signal by a discrete time signal , taking sufficient number of samples of signal so that original signal is represented in its samples completely. It can be stated as:

𝟏

(i) A band-limited signal of finite energy with no frequency component higher than f_m Hz, is completely described by its sample values which are at uniform intervals less than or equal to 1/2f_m

seconds apart. [T_s=_𝟐𝒇𝒎 ]where T_s is sampling time.

(ii) Sampling frequency must be equal to or higher than 2f_m Hz. [fs ≥ 2fm]

​

A continuous time signal may be completely represented in samples and recovered back, if f_s≥2f_m, where f_s is sampling frequency and f_m is maximum frequency component of message signal

Proof of sampling theorem

• Sampling of input signal x(t) can be obtained by multiplying x(t) with an impulse train δ(t) of period T_s.
• The output of multiplier is a discrete signal called sampled signal which is represented with y(t) in the diagrams,
• y(t)=x(t).δ(t)......(1)

​

The Fourier series representation of δ(t) :

• δ(t)=a₀+Σ^∞ (a_ncosnω_st + b_nsinnω_st)......(2)

n=1

0

𝑠

where a = ¹ ∫ δ(t) dt =

1

δ(0) = ¹

𝑇𝑠 𝑇

𝑠

𝑇

n s

a =² ∫δ(t)cosnω dt =

𝑇

𝑠

s

² δ(0)cosnω 0= ²

𝑇

𝑆

2

b_n= ∫δ(t)sinnω_st dt

^𝑇 -T/2

=² δ(0)sinnω 0=0

s

T/2

^𝑠 -T/2

δ(t)= ¹ +Σ ( ² cosnω t+0)

𝑆

𝑇 𝑇

𝑆

s

1

𝑇

2

𝑇

𝑆 𝑆

s

∴δ(t) = +Σ ( cosnω t+0)

Substitute δ(t) in equation 1.

→y(t)=x(t).δ(t)

𝑇_𝑠

1 2

𝑇_𝑠

= x(t)[ +Σ ( cosnω_st+0)]

1

𝑇_𝑆

= [x(t)+2Σ(cosnω_st)x(t)]

𝑻

𝑺

s s s

y(t)= ^𝟏 [x(t)+2cosω t.x(t)+2cos2ω t.x(t)+2cos3ω t.x(t)......]

Take Fourier transform on both sides.

𝑇

𝑆

s s s s s

Y(ω) = ¹ [X(ω)+ X(ω-ω ) +X(ω+ω )+X(ω-2ω )+X(ω+2ω )+ X(ω+3ω )+………..]

n=1

n=∞

n=1

n=∞

n=1

n=∞

n=∞

Y(ω)=^{𝟏 +∞} 𝑿(𝝎 − 𝒏𝝎 )

𝑻

𝑺

−∞

𝒔

To reconstruct x(t), one has to recover input signal spectrum X(ω) from sampled signal spectrum Y(ω), which is possible when there is no overlapping between the cycles of Y(ω) which is possible if

f_s≥2f_m

​

For f_s=2f_m, is known as Nyquist rate.

s

T =

𝟏

𝟐𝒇𝒎

is known as Nyquist interval

Aliasing Effect

The overlapped region in case of under sampling

represents Aliasing effect. It can be termed as “the phenomenon of a high-frequency component in the spectrum of a signal, taking on the identity of a lower-frequency component in the spectrum of its sampled version.

This effect can be removed by considering

1. fs >2f_m or
2. by using anti aliasing filters which are low pass filters and eliminate high frequency components

Three types of sampling techniques:

​

• Impulse sampling: Obtained by multiplying input signal x(t) with impulse train of period 'T_s.

Also called ideal sampling. Practically not used because pulse width cannot be zero and the generation of impulse train not possible.

Natural sampling

• This type of sampling similar to ideal sampling except for the fact that instead of delta function, now we use rectangular train of period T_s. i.e. multiply input signal x(t) to pulse train
• An electronic switch is used to periodically shift between the two contacts at a rate of f_s = (1/T_s ) Hz, staying on the input contact for C seconds and on the grounded contact for the remainder of each sampling
• The output x_s(t) of the sampler consists of segments of x(t) and hence X_s(t) can be considered as the product of x(t) and sampling function s(t).
• X_s(t)= x(t)×s(t)

Using Fourier series, we can rewrite the signal S(t) as:

0 ^𝑛=1 𝑠

S(t)= C + ^∞ 2𝐶𝑛𝑐𝑜𝑠(𝑛ω 𝑡)

𝑻

𝒔

0 n s

s

Where the Fourier coefficients C = ^𝝉 and C =f τsinc(nf τ)

Therefore: x (t)=x(t)[C + ^∞ 2𝐶𝑛(𝑐𝑜𝑠𝑛𝜔 𝑡)]

s 0 ^𝑛=1 𝑠

x_s(t)=C₀x(t)+2C₁x(t) cos(ω_st)+2C₂x(t) cos(2ω_st)+……

Applying Fourier Transform for the above equation

​

Using x(t)↔X(f)

𝟐

0 0

0

x(t) cos(2πf t)↔ ^𝟏[X(f-f )+X(f+f )]

X_s(f)=C₀X(f)+C₁[X(f-f₀)+X(f+f₀)]+C₂[X(f-f₀)+X(f+f₀)]+…………

s 0

𝒏=−∞

𝒏

X (f)= C X(f)+ ^∞ 𝑪 𝑿(𝒇 − 𝒏𝒇𝒔)

X_s(f) = Aτ/ Ts .[ Σ sin c(n fs.τ) X(f-n fs)]

​

The signal X_s(t) has the spectrum which consists of message spectrum and repetition of message spectrum periodically in the frequency domain with a period of fs. But the message term is scaled by ‘Co”( sinc function) which is not the case in instantaneous sampling.

• Flat Top sampling: During transmission, noise is introduced at top of the transmission pulse which can be easily removed if the pulse is in the form of flat top.
• Here, the top of the samples are flat i.e. they have constant amplitude and is equal to the instantaneous value of the baseband signal x(t) at the start of sampling. Hence, it is called as flat top sampling or practical sampling.
• Flat top sampling makes use of sample and hold circuit
• Theoretically, the sampled signal can be obtained by convolution of rectangular pulse h(t) with

ideally sampled signal ,s_δ(t) g(t)= s(t) ⊗ h(t)

δ(t)

t

h(t)

⊗

=

0 τ

f(t) ⊗ δ(t) = f(t); property of delta function Applying a modified form; s(t) in place of δ(t)

On convolution of s(t) and h(t), we get a pulse whose duration is equal to h(t) only but amplitude

defined by s(t).

Train of impulses given by:

Ts

𝒏=−∞

δ (t) = ^∞ 𝜹(𝒕 − 𝒏𝑻𝒔)

Signal s(t) obtained by multiplication of message signal x(t) and δ_Ts(t)

​

Thus, s(t) = x(t). δ_Ts(t)

𝒏=−∞

𝒔

s(t)= ^∞ 𝒙(𝒏𝑻𝒔)𝜹(𝒕−𝒏𝑻 )

Now sampled signal g(t) given as:

g(t)=s(t) ⊗ h(t)

=

−∞

∞

𝑠 τ ℎ 𝑡 − τ 𝑑τ

G(f)=S(f) H(f)

S(f)=fs 𝑋(𝑓 − 𝑛𝑓𝑠)

g(t) =

−∞

∞

𝑛=−∞

∞

s

𝑥(𝑛𝑇𝑠)δ(τ-nT ) h(t-τ)dτ

−∞

g(t)= ^∞

𝒙(𝒏𝑻𝒔)

−∞

∞

𝒔

𝜹(𝝉 − 𝒏𝑻 ) 𝒉(𝒕 − 𝝉)𝒅𝝉

Using shifting property of delta function:

−∞

∞

0

𝑓(𝑡)δ(𝑡 − 𝑡𝑜)=f(t )

−∞

g(t)= ^∞

𝒙 𝒏𝑻

𝒔

𝒉(𝒕 − 𝒏𝑻𝒔)

s

−∞

G(f)=f ^∞

𝑿 𝒇 − 𝒏𝒇𝒔 𝑯(𝒇) Spectrum of flat top samples

Aperture Effect: Spectrum of flat topped sample is given by;

​

G(f)=f_s ∑〖𝑿(𝒇−𝒏𝒇_𝒔)𝑯(𝒇)〗 , where H(f)= τ.sin c(f_s.t)𝒆^(−𝒋𝝅𝒇𝝉)

​

This equation shows that signal g(t) is obtained by passing the signal s(t) through a filter having transfer function H(f).

Figure(a) shows one pulse of rectangular pulse train and each sample of x(t) i.e. s(t) is

convolved with this pulse

Figure (b) shows the spectrum of this pulse. Thus, flat top sampling introduces an amplitude distortion in reconstructed signal x(t) from g(t). There is a high frequency roll off making H(f) act like a LPF, thus attenuating the upper portion of message signal spectrum. This is known as aperture effect

How to minimize aperture effect?? An equalizer at the receiver end is needed to compensate aperture effect. The receiver contains low pass reconstruction Filter with cut off slightly higher than f_m Hz.

Reconstruction Filter

​

Equalizer

PAM

Signal g(t)

Message signal x(t)

Equalizer in cascade with reconstruction filter has the effect of decreasing the in band loss of reconstruction filter, frequency increases in such away so as to compensate aperture effect.

eq

𝑯(𝒇)

−𝒋𝟐𝝅𝒇𝒕𝒅

H (f)=^𝑲.𝒆 ,

where t_d is time delay introduced by LPF being equal to τ/2

H_eq(f) =

𝑲

𝝉𝒔𝒊𝒏 𝒄(𝒇𝝉)

Pulse Amplitude Modulation (PAM)

• Amplitude of the pulse carrier varies proportional to the instantaneous amplitude of the message

signal.

• The width and positions of the pulses are constant in this modulation.
• PAM could be:
1. Single polarity PAM: A suitable fixed DC bias is added to the signal to ensure that all the pulses are positive.
2. Double polarity PAM: In this the pulses are both positive and negative.

• Depending on type of sampling PAM can be:
1. Ideal Sampling PAM, (ii) Natural sampling PAM and (iii) Flat top PAM.

​

• The advantage of this modulation is the generation and detection is easy in this modulation and

also allows multiplexing.

​

• The disadvantage is large band width of transmitted signal.

​

BPF characteristics

• For a PAM signal produced with natural sampling, the sampled signal follows the waveform of

the input signal during the time that each sample is taken.

• A PAM signal is generated by using a pulse train, called the sampling signal (or clock signal) to operate an electronic switch or "chopper". This produces samples of the analog message signal.
• The switch is closed for the duration of each pulse, allowing the message signal at that sampling time to become part of the output.
• The switch is open for the remainder of each sampling period making the output zero. This is

known as Natural PAM.

​

​

In simplest form PAM can be visualized as o/p of an AND gate whose two inputs are message signal x(t) and pulses at sampling rate

.

• For flat-top sampling, a sample-and-hold circuit is used in conjunction with the chopper to hold

the amplitude of each pulse at a constant level during the sampling time,

• Flat-top sampling, produces pulses whose amplitude remains fixed during the sampling time. The amplitude value of the pulse depends on the amplitude of the input signal at the time of sampling.
• Aperture Effect seen in this type of PAM. Equalizers used at receiver end

​

Transmission Bandwidth in PAM

​

𝝉 ≪ T_s

f ≥ 𝟐𝒇𝒎 ; 𝑻𝒔 ≤

𝟏

s

𝝉≪ Ts≤

𝟐𝒇𝒎

𝟏

𝟐𝒇𝒎

max

₌ 𝟏

𝟐𝝉

BW≥ 𝒇𝒎𝒂𝒙; 𝑩𝑾 ≥

If on and off time of PAM pulse is same then f

𝟏

𝟐𝝉

𝟐𝝉

𝑩𝑾 ≥ ^𝟏 ≫ fm

Transmission of PAM signals

​

• For PAM signals to be transmitted through space using antennas, they must be amplitude/ frequency/ phase modulated by a high frequency carrier and only then they can be transmitted. Thus the overall system is PAM-AM. PAM-FM or PAM-PM and at receiving end, AM/ FM/PM detection is first employed to get the PAM signal and then message signal is recovered.

​

Drawbacks of PAM

• Bandwidth required for transmission of PAM signal is very large in comparison to maximum frequency present in modulating signal.
• Since amplitude of PAM pulses varies in accordance with modulating signal so interference of noise is maximum in PAM
• Variation of the peak power required by transmitter

Demodulation of PAM

• PAM signal sampled at Nyquist rate can be reconstructed at the receiver end , by passing it through an efficient Low Pass Filter (LPF) with exact cut off frequency of fs/2. This is known as Reconstruction or Interpolation Filter.
• The low pass filter eliminates the high-frequency ripples and generates the demodulated signal. This signal is then applied to the inverting amplifier to amplify its signal level to have the demodulated output with almost equal amplitude with the modulating signal

• For a flat topped PAM, a holding circuit followed by a LPF gives demodulated signal

Holding circuit

Received PAM signal

C

Zero order Holding Circuit

• Switch S closes after the arrival of pulse and opens at the end of pulse.
• Capacitor C charges to pulse amplitude value and holds this value during interval between two pulses.
• The sampled values are shown in fig.
• Holding circuit o/p smoothened in LPF.
• Known as zero order holding circuit, which considers only the previous sample to decide value between two

pulses

• First order holding circuit considers previous two samples, second order holding circuit considers previous three samples.

PAM

signal

Pulse Time modulation

• In PTM, amplitude of pulse is constant while position or width of pulse is made proportional to the amplitude of the signal at the sampling instant.
• It can be PWM and PPM
• In both the cases amplitude constant and does not carry information so amplitude limiters can be used ( like in FM) providing good noise immunity

​

Generation of PTM signals can be either by:

1. Indirect Method: Firstly PAM signals are generated, Synchronized is generated during each pulse interval. These two signals are added and the sum is applied to a comparator whose reference level is suitably chosen. The second crossing of comparator level used for PPM
2. Direct method: PTM waveforms generated without using PAM waveforms

​

Pulse Width modulation

The pulse width modulation is the modulation of signals by varying the width of pulses. The amplitude and

positions of the pulses are constant in this modulation

Generation of PWM and PPM by Direct Method

• The non inverting input of the comparator is fed by the input message or modulating signal x(t)

and the other input by a saw-tooth signal which operates at carrier frequency.

• The comparator compares the two signals together to generate the PWM signal at its output. Its o/p is high only when the instantaneous value of x(t) is higher than sawtooth waveform.
• The rising edges of the PWM signal occurs at the fixed time period (kT_s) while trailing edge depends on amplitude of message signal x(t).
• When saw-tooth voltage waveform greater than x(t), o/p of comparator is zero, trailing edge is modulated
• If saw-tooth. waveform is reversed, trailing edge is fixed while leading edge is modulated.
• Replacing saw-tooth waveform by triangular, both leading and trailing edge modulated. (symmetrical PWM)
• The amplitude of PDM/PWM will be positive saturation of the comparator shown as ‘A’, being same

for all pulses,

Three types of pulse-width modulation (PWM) are possible:

​

• The leading edge of the pulse being constant, the trailing edge varies according to the message signal.
• The trailing edge of the pulse being constant, the leading edge varies according to the message signal
• The center of the pulse being constant, the leading edge and the trailing edge varies according

to the message signal (Symmetrical PWM)

Indirect Method:

Modulating signal (A) applied to i/p of PAM circuit [s(t) pulse train] and PAM signal generated(B). S(t) also is i/p to Ramp generator(Integrator circuit), all having equal slopes, amplitude and generation(D). These ramp pulses added to PAM pulses to produce varying height samples. These varying height ramp gates a S.T ckt to generate varying width rectangular pulses of PWM.

PWM

​

Summer

Schmitt Trigger

Ramp Generator

x(t)

A

PAM

generator

B

C

D

E

F

PWM detector

Schmitt

Trigger

Ramp Generator

Synchronization

Pulse generator

Adder

Level Shifter

Rectifier

LPF

1

2

3

4

5

6

• Received PWM signal applied to ST circuit to remove noise
• Regenerated PWM applied to Ramp generator and synchronization pulse.
• Heights of Ramp proportional to width of pulses.
• Pulse generator produces reference pulses with constant

amplitude and width but delayed by specific amount.

• Delayed reference pulses added to o/p of ramp generator
• The o/p given to level shifter, negative offset shifts waveform. Then clipped by rectifier followed by LPF to give message signal.

Noisy PWM

Pulse position modulation

• (PPM) is an analog modulating scheme in which the amplitude and width of the pulses are kept constant, while the position of each pulse, with reference to the position of a reference pulse varies according to the instantaneous sampled value of the message signal.
• The transmitter has to send synchronizing pulses (or simply sync pulses) to keep the transmitter and receiver in synchronism. These sync pulses help maintain the position of the pulses.
• PPM is done in accordance with the PWM signal.
• PWM signal is used as the trigger input to a monostable multivibrator.
• Its o/p remains zero until it is triggered on the trailing edge of PWM
• O/P of monostable MV switches to positive saturation value A and remains high for fixed period then goes low
• Hence, the position of these pulses is proportional

to the width of the PWM pulses.

Advantage As the amplitude and width are constant

the power handled is constant

Disadvantage: Synchronization between Transmitter and receiver is a necessity

• The PDM is differentiated, and then rectified and shaped.
• PPM carries exactly the same information as long as the position of the clock pulses (leading edge) is well defined in the received signal.
• PPM is superior to PDM for message transmission, since the wide pulses of PDM require more energy than PPM when transmitted
• PPM is suited for communication in the presence of noise.
• Very high peak narrow pulses can be transmitted and the pulse position can be determined even when the noise level is high,
• However, transmitting very narrow pulses requires a large band

width

• When light is used as the media for transmitting analog signals, PPM or PCM are the most suitable types of modulation because the maximum power output in the modulated light source, such as LED or LASER is achieved when it is pulsed at a very low duty cycle.
• In PPM, necessary to transmit a series of sync pulses at a much lower repetition rate than the sampling pulses, to avoid interference with original signal and/or minimise the number of pulses transmitted in order to conserve transmission power

Transmission BW of PWM and PPM

• Both PWM and PPM have DC value.
• Both need a shrp rise time and fall time to preserve the message information
• Rise time be very less than Ts i.e. t_r≪ Ts

T

• Transmission BW: B ≥

𝟏

𝟐𝒕𝒓

• BW higher than PAM

PAM

• The amplitude of the pulse is proportional to the amplitude of modulating the signal.
• Band width of transmitting channel depends on the width of the pulse
• Instantaneous power of transmitter varies. Noise interference is high
• Complex system. Similar to A.M.

PWM

• Width of pulse is proportional to amplitude of modulating signal.
• The Bandwidth of transmitting channel depends on rise time of the pulse.
• Instantaneous power of transmitter varies. Noise interference is minimum.
• Simple to implement Similar to F.M.

PPM

• Relative position of pulse is proportional to amplitude of modulating signal.
• The bandwidth of transmitting channel depends on the rise time of the pulse.
• Instantaneous power remains constant. Noise interference is minimum.
• Simple to implement. Similar to P.M.

 Difference Between PAM, PWM, and PPM

Parameter

• Type of Carrier:

PAM

Train of Pulses

Amplitude

Low

Low

PWM

Train of Pulses Width

High

High

• Variable Characteristic :
• Bandwidth Requirement:
• Noise Immunity :
• Information Contained in: Amplitude Variations

• Power efficiency (SNR)
• Transmitted Power

Low Varies

• Need to transmit synchronizing pulses Not needed

Width Variations Moderate Varies

Not needed

rise time of the pulse

PPM

Train of Pulses Position

High

High

Position Variations High

Remains Constant Necessary

rise time of the pulse

Instantaneous power varies with Constant width of the pulses

• Bandwidth depends on width of the pulse
• Transmitter power Inst. power varies

with amplitude of pulses

• Complexity of generation and detection

Complex Easy

 12 Similarity with other Modulation Systems Similar to AM

Similar to FM

Complex Similar to PM

Q. For a PAM transmission of voice signal with f_m=3kHz, calculate the transmission BW. Given that f_s=8kHz and the pulse duration τ=0.1T_s

Soln: Ts= ¹ = 125µs

𝑓𝑠

τ=0.1Ts=0.1×125=12.5µs

𝟏

𝟐𝝉

BW≥ ≥ 40 kHz

Q. For the above signal if rise time is 1% of pulse width, find minimum Tx BW for PWM and PPM? Soln: t_r=τ×0,01= 1.25x 10^-7

T

B ≥

1

2𝑡

𝑟

≥ 4MHz

Thus BW of PWM/PPM much higher than PAM

Multiplexing

• Multiplexing refers to the combination of information streams from multiple sources for

transmission over a shared medium .

• Multiplexor is a mechanism that implements the concept. It permits hundreds or even thousands of signals to be combined and transmitted over a single medium. De-multiplexing refers to the separation of a combination, back into separate information streams .

Principle used

• Each sender communicates with a single receiver
• All pairs share a single transmission medium
• Multiplexor combines information from the senders for transmission in such a way that the de multiplexer can separate the information for receivers.
• Cost savings obtained using single channel to send Multiple signals.

Four basic types of multiplexing

• Frequency Division Multiplexing (FDM)
• Wavelength Division Multiplexing (WDM)

•Time Division Multiplexing (TDM) • Code Division Multiplexing (CDM)

Frequency Division Multiplex (FDM): Separation of spectrum into smaller frequency.

• Channel gets band of the spectrum for the whole time. Each signal allocated different frequency band i.e, Multiple carriers used
• Each message signal is limited to f_m Hz.

Example: Multiplexing of telephonic signals from n subscribers

• Telephonic message (BW=3kHz) and broadcast signal limited to 5kHz. Without multiplexing if n channels transmitted,

Interference and no useful information.

• In FDM, each baseband signal translated by Analog Modulation (AM/Angle) to different carrier frequencies.
• Each carrier separated from neighbouring by at least 2fm
• Multiplexed signals can be transmitted over a common channel without interference.
• At receiver, various carrier frequencies selected using BPF tuned to appropriate carrier frequencies and demodulated by separate detector.

FDM System

Advantages: No dynamic coordination needed and works also for analog signals

​

Disadvantages: Waste of bandwidth if traffic distributed uneven; inflexible;

Time Division Multiplexing

• Time Division Multiplexing (TDM) is the time interleaving of samples from several sources so that the information from these sources can be transmitted serially over a single communication channel.
• At the Transmitter :Simultaneous transmission of several signals on a time-sharing basis.

Each signal occupies its own distinct time slot, using all frequencies, for the duration of the transmission. Slots may be permanently assigned on demand

• At the Receiver : Decommutator (sampler) has to be synchronized with the incoming waveform
• In Pulse modulation techniques, there is a free space between any two consecutive pulses of a signal. This free space between pulses can be occupied by pulses from other channel. This is Time Division Multiplexing (TDM) and makes maximum utilization of transmission channel.
• Applications of TDM: Digital Telephony, Data communications, Satellite Access, Cellular radio

Block diagram of TDM and PAM-TDM

• The system shows TDM of ‘N’ PAM channels.
• Each channel to be transmitted is passed through LPF to band limit its frequency to f_m Hz ( W Hz)
• Outputs of LPF are connected to the rotating sampling switch or commutator.
• It takes sample from each channel per revolution and rotates at the rate of fs.
• Function of commutator is two fold: (i) Taking narrow samples of each of N input messages at

rate 1/T_s (ii) To sequentially interleave N samples Inside a sampling interval T_s

• The single signal composed due to multiplexing of input channels given to Transmission channel
• If W or f_m is highest signal frequency in the message signal: fs ≥ 2f_mor 2W; Ts or 𝟏/𝐟𝐬≤𝟏/𝟐𝐟𝐦
• Thus time interval Ts contains one sample from each input. It is called frame. If N input channels multiplexed, each frame will have one sample from each of N channel's input
• Spacing between two samples: 𝐓𝐬/𝐍
• No. of pulses/sec=1/ spacing between two pulses=1/(𝑇𝑠/𝑁) =𝑁/𝑇𝑠
• Ts=1/fs; No. of pulses per second=Nfs
• The no. of pulses transmitted per second is called signalling rate of TDM ‘r’
• r=Nfs; fs ≥2fm; r ≥2Nf_m or r ≥2NW
• Pulsed TDM passed through LPF to convert it to baseband signal whose BW given by half signalling rate
• B.W=𝒓/𝟐 = Nf_m=NW; Minimum Transmission Bandwidth
• At the receiver, decommutator separates the time multiplexed input channels which then passed through reconstruction filter

Synchronization in TDM system

• The time division multiplexing (TDM) needs synchronization between multiplexer and demultiplexer. If synchronization is not there between multiplexer and demultiplexer, a bit going to one channel may be received by the wrong channel.
• Because of this reason, one or more synchronization bits are usually added to the beginning of each

frame called Markers (highest amplitude)

• These bits are called framing bits (Marker pulse), allows the demultiplexer to synchronize with the incoming stream so that that it can separate time slot accurately.
• Because of the marker pulse, no of channels to be multiplexed reduced by 1

Marker Pulse

x

1

x₂

^xN-1

Crosstalk and Guard Times

𝟎.𝟓𝟓

• RF transmission of TDM needs further modulation
• TDM signal converted to smooth modulating waveform by passing through a baseband filter
• This filtering gives rise to inter-channel crosstalk which means individual signal sample amplitude

interfere with each other. Thus interference between adjacent TDM channels is crosstalk

• This interference can be reduced by increasing distance between individual signal amplitudes
• The minimum distance between individual signal samples to avoid crosstalk is called guard time.
• Ideally communication channel over which TDM signal is transmitted should be infinite but in

practise has a finite BW, known as band limited channels

• Whenever signal passed through band limited channel, shape of signal will change.
• Whenever a PAM-TDM signal transmitted over band limited channel, signals corresponding to x₁(t)

get mixed with x₂(t) and this overlap causes crosstalk.

• To keep cross talk below -30dB, T_g≥ _𝐁𝐖

• Thus, guard time required to avoid cross talk decreases with increase in BW

Transmission Bandwidth for ‘N’ PAM-TDM channels: Nf_m Where f_m is the maximum frequency of baseband signal Advantages:

• Full available channel BW can be utilized.
• TDM circuitry not very complex
• Problem of cross talk not very severe

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Disadvantages

• Synchronization for proper operation

Application of PWM

• Although PWM is also used in communications, its main purpose is actually to control the power that is supplied to various types of electrical devices, most especially to inertial loads such as AC/DC motors.
• Pulse-width modulation (PWM) is used for controlling the amplitude of digital signals in order to control devices and applications requiring power or electricity. It essentially controls the amount of power, in the perspective of the voltage component, that is given to a device by cycling the on- and-off phases of a digital signal quickly and varying the width of the "on" phase or duty cycle. To the device, this would appear as a steady power input with an average voltage value, which is the result of the percentage of the on time. The duty cycle is expressed as the percentage of being fully (100%) on.

 A very powerful benefit of PWM is that power loss is very minimal. Compared to regulating power levels using an analog potentiometer to limit the power output by essentially choking the electrical pathway, thereby resulting in power loss as heat, PWM actually turns off the power output rather than limits it. Applications range from controlling DC motors and light dimming to heating elements.

This simple circuit based around the familiar NE555 or 7555 timer chip is used to produced the required pulse width modulation signal at a fixed frequency output. The timing capacitor C is charged and discharged by current flowing through the timing networks RA and RB as we looked at in the 555 Timer tutorial.

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The output signal at pin 3 of the 555 is equal to the supply voltage switching the transistors fully “ON”. The time taken for C to charge or discharge depends upon the values of RA, RB.

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The capacitor charges up through the network RA but is diverted around the resistive network RB and through diode D1. As soon as the capacitor is charged, it is immediately discharged through diode D2 and network RB into pin 7. During the discharging process the output at pin 3 is at 0 V and the transistor is switched “OFF”.

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Then the time taken for capacitor, C to go through one complete charge- discharge cycle depends on the values of RA, RB and C with the time T for one complete cycle being given as:

 The time, TH, for which the output is “ON” is: TH = 0.693(RA).C

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 The time, TL, for which the output is “OFF” is: TL = 0.693(RB).C

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 Total “ON”-“OFF” cycle time given as: T = TH + TL with the output frequency being ƒ = 1/T

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 With the component values shown, the duty cycle of the waveform can be adjusted from about 8.3% (0.5V) to about 91.7% (5.5V) using a 6.0V power supply. The Astable frequency is constant at about 256 Hz and the motor is switched “ON” and “OFF” at this rate.

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 Resistor R1 plus the “top” part of the potentiometer, VR1 represent the resistive network of RA. While the “bottom” part of the potentiometer plus R2 represent the resistive network of RB above.

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 These values can be changed to suite different applications and DC motors but providing that the 555 Astable circuit runs fast enough at a few hundred Hertz minimum, there should be no jerkiness in the rotation of the motor.

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 Diode D3 is our old favourite the flywheel diode used to protect the electronic circuit from the inductive loading of the motor. Also if the motor load is high put a heatsink on the switching transistor or MOSFET.

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 Pulse width modulation is a great method of controlling the amount of power delivered to a load without dissipating any wasted power. The above circuit can also be used to control the speed of a fan or to dim the brightness of DC lamps or LED’s. If you need to control it, then use Pulse Width Modulation to do it.