The document discusses intersymbol interference (ISI) in digital communication systems. It defines ISI as signal distortion caused by one symbol interfering with subsequent symbols. Nyquist's criterion for zero ISI is introduced, which states that the pulse shaping function p(t) must satisfy p(iTb-kTb) = 1 for i = k and 0 for i ≠ k. This ensures each sample only contains the desired symbol. The ideal solution to achieve zero ISI is for p(t) to be a sinc function with a bandwidth of 1/2Tb, known as the Nyquist bandwidth. This results in distinct non-overlapping pulses that can be correctly sampled without interference between symbols.

1. Dr. AMBEDKAR INSTITUTE OF TECHNOLOGY
(An Autonomous Institution, Affiliated to Visveswaraya Technological University, Belagavi)
Near Jnana Bharathi Campus, Mallathahalli, Bengaluru-560056
DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING
Group Activity Report
on
“INTERSYMBOL INTERFERENCE,NYQUIST’S CRITERION FOR
BASEBAND,IDEAL SOLUTION FOR ZERO (ISI)”
Submitted in partial fulfillment of the curriculum
DIGITAL COMMUNICATION-18EC53
In
ELECTRONICS AND COMMUNICATION ENGINEERING
For
BACHELOR’S IN ENGINEERING
Submitted By:
AKSHATHA B R 1DA20EC009
ANUSHA CHOWDARY D 1DA20EC016
ARADHANA B 1DA20EC017
CHINMAY D 1DA20EC032
DARSHAN D 1DA21EC403

2. Dr. AMBEDKAR INSTITUTE OF TECHNOLOGY
(An Autonomous Institution, Affiliated to Visvesvaraya Technological University, Belagavi)
Near Jnana Bharathi Campus, Mallathahalli, Bengaluru-560056
DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING
CERTIFICATE
Certified that the group activity entitled “INETRSYMBOL INTERFERENCE,NYQUIST”S CRITERRION FOR
BASEBAND,IDEAL SOLUTION FOR ZERO(ISI)” carried out by Ms. ,AKSHATHA B R bearing USN
1DA20EC009, Ms.,ANUSHA CHOWDARY D bearing USN 1DA20EC016, Ms.,ARADHANA B
bearing USN 1DA20EC017, Mr. CHINMAY D, bearing USN 1DA20EC032, Mr.DARSHAN bearing
USN 1DA21EC403 a bonafide student of Dr. Ambedkar Institute of Technology, Bengaluru-560056 in
partial fulfillment of curriculum in “DIGITAL COMMUNICATION” in Bachelor of Engineering in
Electronics and Communication Engineering of Dr. Ambedkar Institute of Technology during the year
2022-23.
The intergroup Activity report has been approved as it satisfies the academic requirements in respect
of the subject prescribed for the said subject.
Signature of Co-ordinator
SPOORTHI P A
Assistant professor

3. CONTENTS :
1). INTERSYMBOL INTERFERNCE (ISI):
2).NYQUIST CRITERION FOR DISTORTIONLESS BASEBAND
(FOR BINARY TRANSMISSION):
3).IDEAL SOLUTION OR NYQUIST SDOLUTION FOR ZERO
ISI:

4. 1).Intersymbol Interfernce (ISI):
Intersymbol Interferance:
 Inter symbol Interference is a form of a distortion of a signal, in
which one or more symbols interfere with subsequent signals,
causing noise or delivering a poor output.
Block diagram:
Fig: Block Diagram Of Intersymbol Interferance
Let us assume that the channel is free from noise. Let the input to the
transmitting filter be represented in time domain form as
x(t)= ∑ 𝐴𝑘𝑔(𝑡 − 𝑘𝑇𝑏)
∞
𝑘=−∞ --> (1)
where:
 g(t) is a rectangular pulse.
 A discrete random variable taking the value of +a for symbol 1
and -a for symbol 0.
 T is the time allocated for one bit and is known as the bit
duration.

5. Hence, the fourier transform of equation gives
X(f)=∑ 𝐴𝑘𝐺(𝑓)𝑒−𝑗2𝜋𝑓𝑘𝑇𝑏
∞
𝑘=−∞ --> (2)
Let us denote the output of the receiving filter by
Y(t)=∑ 𝜇𝐴𝑘𝑝(𝑡 − 𝑘𝑇𝑏)
∞
𝑘=−∞ --> (3)
Where:
p(t) is the pulse shaping function of y(t).
𝜇 is the scaling factor.
let the FT of y(t) is
Y(f)=∑ 𝜇𝐴𝑘𝑃(𝑓)𝑒−𝑗2𝜋𝑓𝑘𝑇𝑏
∞
𝑘=−∞ --> (4)
As a second step, the output of the receiving filter in frequency domain
is given by
Y(f)=X(f)Ht(f)Hc(f)Hr(f) -->(5)
Substituting equations 2 and 4 in equation 5, we get
∑ 𝜇𝐴𝑘𝑃(𝑓)𝑒−𝑗2𝜋𝑓𝑘𝑇𝑏
∞
𝑘=−∞ =∑ 𝐴𝑘𝐺(𝑓)𝑒−𝑗2𝜋𝑓𝑘𝑇𝑏
∞
𝑘=−∞ Ht(f)Hc(f)Hr(f)
𝜇P(f) = G(f)Ht(f)Hc(f)Hr(f) -->(6)

6. Taking inverse FT on both sides of equation 6 ,we can determine the
shape of p(t).
y(t) = ∑ 𝜇𝐴𝑘𝑝(𝑡 − 𝑘𝑇𝑏)
∞
𝑘=−∞
Let t=iTb,
Where
i=0, +1 or -1, +2 or -2
y(iTb)=∑ 𝜇𝐴𝑘𝑝(𝑖𝑇𝑏 − 𝑘𝑇𝑏)
∞
𝑘=−∞
=∑ 𝜇𝐴𝑘𝑝[(𝑖 − 𝑘)𝑇𝑏]
∞
𝑘=−∞
Therefore,
y(iTb)=𝜇Aip(0) + ∑ 𝜇𝐴𝑘𝑝[(𝑖 − 𝑘)𝑇𝑏]
∞
𝑘=−∞
𝑘≠𝑖
-->(7)
Since p(0)=1,
y(iTb)=𝜇Ai + ∑ 𝜇𝐴𝑘𝑝[(𝑖 − 𝑘)𝑇𝑏]
∞
𝑘=−∞
𝑘≠𝑖
--> (8)
Specification:
 The first term on the right-hand side of equation 8 represents the
contribution of ith transmitted symbol.
 The second term represents the unwanted contribution of all other
transmitted bits on the detection of ith transmitted bit.
 This unwanted contribution is called intersymbol interference.

7. 2).NYQUIST’SCRITERION FOR DISTORTIONLESS BASEBAND BINARY
TRANSMISSION (OR ZERO ISI):
 The pulse shaping function p(t) with Fourier transform given by
P(f),
∑ 𝑃(𝑓 − 𝑛𝑅𝑏)
∞
𝑘=−∞ =Tb
has,
p(iTb-Ktb)={
1, 𝑖 = 𝑘
0, 𝑖 ≠ 𝑘
This condition is known as Nyquist criterion for zero ISI.
Proof:
>>> Let us sample p(t) by using a Dirac comb with a period equal to
Tb. The process of ideal sampling may be defined mathematically by
P𝛿(t)=p(t)S𝛿(𝑡)
Applying FT on both sides,
P𝛿(f)=P𝛿(f)*S𝛿(f)
P𝛿(f)=P(f)*fs∑ 𝛿(𝑓 − 𝑛𝑓𝑠)
∞
𝑛=−∞
Let fs = (1/T) = Rb
then above equation

8. therefore,
P𝛿(f)=P(f)*Rb∑ 𝛿(𝑓 − 𝑛𝑅𝑏)
∞
𝑛=−∞
Applying the convolution property of an impulse function,
P𝛿(f)=Rb∑ 𝑃(𝑓 − 𝑛𝑅𝑏)
∞
𝑛=−∞ -->(1)
Using the defining equation of FT,
P𝛿(f)=Rb∫ 𝑝𝛿(𝑡)
∞
−∞
𝑒−𝑗2𝜋𝑓𝑡
dt
Using,
P𝛿(t)=p(t)S𝛿(𝑡)
=𝑝(𝑡) ∑ 𝛿(𝑡 − 𝑚𝑡𝑏)
∞
𝑚=−∞
then,
P𝛿(t)=∑ 𝑝(𝑚𝑇𝑏)𝛿(𝑡 − 𝑚𝑇𝑏)
∞
𝑚=−∞
Then above equation becomes,
P𝛿(f)=∫ ∑ [𝑝(𝑚𝑇𝑏)𝛿(𝑡 − 𝑚𝑇𝑏)]𝑒−𝑗2𝜋𝑓𝑡
∞
𝑚=−∞
∞
−∞
dt -->(2)
Let the integer, m = i-k,
Then,
i = k corresponds to m = 0, and similarly i ≠ k corresponds to m ≠ 0.
Using the condition:
p[(i-kTb)] = p[mTb]

9. then,
p[(i-kTb)] ={
1, 𝑖 = 𝑘
0, 𝑖 ≠ 𝑘
Equation 2 bcomes,
P𝛿(f)=∫ 𝑝(0)𝛿(𝑡)
∞
−∞
𝑒−𝑗2𝜋𝑓𝑡
dt
=p(0) 𝑒−𝑗2𝜋𝑓𝑡
|t=0 (using shifting property)
=p(0)
Since p(0) = 1, we get P𝛿(f)= 1,
 As a consequence of this, equation 1 gives P𝛿(f) = 1 only when
∑ 𝑃(𝑓 − 𝑛𝑅𝑏)
∞
𝑛=−∞ =Tb
Hence the proof,
 Because of the significance of this theorem in baseband
transmission, the above equation or equivalently
p(iTb-Ktb)={
1, 𝑖 = 𝑘
0, 𝑖 ≠ 𝑘
Finally,it constitutes that Nyquist's criterion for distortionless
baseband transmission (zero ISI).

10. 3).IDEAL SOLUTION OR NYQUIST SOLUTION
FOR ZERO ISI:
 The ISI can be minimized by controlling p(t) in time-domain and
P(f) in frequency domain.
One of the functions that gives zero ISI is
>> p(t) = sinc(2Bot) shown in fig 1
And spectrum of same signal is shown in fig 2
Where:
Bo = 1/2Tb is called Nyquist bandwidth.
“Nyquist bandwidth is defined as the minimum transmission bandwidth
for zero ISI”.
The FT of p(t) gives
P(f)={
1
2𝐵𝑜
, |𝑓| < 𝐵𝑜
0, |𝑓| > 𝐵𝑜
 1
 The above equation implies that frequencies of absolute value
greater than half the bit rate are not needed.

11. Equation 1:
 Suggests that P(f) is the frequency response of an ideal low pass
filterand p(t) = sinc(2Bot) is the impulse response of an ideal low
pass filter.
 Since p(t) is a sinc function, it goes through zero at integer
multiples of Tb.
Thus if,
Tb = 1/2Bo.
 Then,it is clear that p(t — kTb) = sinc[2Bo(t— kTb)] for integer
values of k will appear as shown in fig 3.
Also fig 3 implies that if y(t) is sampled at instants of time t= 0, +Tb
or –Tb , +2Tb or -2Tb….., will have zero ISI.
Fig 1: Impulse response of an ideal Low pass filter
Fig 2: Frequency response of an ideal low pass filter

12. Fig 3: Demonstration of Sampling Instants For Zero ISI
But,we know that

13.  The first term on the right–hand side of the above equation gives the
desired symbol.
 The second term represents the ISI caused by timing error ∆(𝑡) due to
inaccurate synchronisation of the clock in receiver sampling circuit.