MAYURBHANJ SCHOOL OF ENGINEERING � LAXMIPOSI ,BARIPADA,757107

• DEPARTMENT- E&TC ENGG.
• SEMISTAR- 5^TH
• SUBJECT-A & D Communication
• TOPIC – 5 – ANLAOG TO DIGITAL CONVERSION & PULSE MODULATION
• NAME OF TOPIC – PULSE MODULATION, SAMPLING
• PREPARED BY - U S Panda (Sr. Lect. E & TC Engineering)
• AY – 2021-2022, WINTER-2021

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)

THANK YOU