ECT305 Analog and Digital Communication

MODULE-4 , PART-2

Dr. Susan Dominic

A s s i stant Pro fess or
De p a rtm ent O f E l e ct ronic s a n d C o m m u n i cat ion E n g i n eeri ng
R a ja giri S c h o ol Of E n gi neering A n d Te c hno logy

RAJAGIRI SCHOOL OF ENGINEERING AND TECHNOLOGY, KOCHI

Topics covered
• Matched filter for noiseless reception
• Baseband transmission through ISI channel
• Mathematical model of ISI
• Nyquist criterion for zero ISI
• Signal design for zero ISI- Nyquist pulse, RC pulse
• Partial response signalling- duobinary coding
• Equalization

Reference- Chapter 8, Digital Communication Systems by Simon Haykin

Introduction
• So far channel was assumed to be distortionless except for the AWGN at the
channel output.
• there was no limitation imposed on the channel bandwidth, with the energy
per bit to noise spectral density ratio Eb/N0 being the only factor to affect the
performance of the receiver.
• In reality, however, every physical channel is not only noisy, but also limited to
some finite bandwidth.
• if, for example, a rectangular pulse, representing one bit of information, is
applied to the channel input, the shape of the pulse will be distorted at the
channel output.
• the distorted pulse may consist of a main lobe representing the original bit of
information surrounded by a long sequence of sidelobes on each side of the
main lobe
• The sidelobes represent a new source of channel distortion, referred to as
intersymbol interference, so called because of its degrading influence on the
adjacent bits of information
• There is a fundamental difference between intersymbol interference and channel noise
that could be summarized as follows:
• Channel noise is independent of the transmitted signal; its effect on data transmission
over the band-limited channel shows up at the receiver input, once the data transmission
system is switched on.
• Intersymbol interference, on the other hand, is signal dependent; it disappears only
when the transmitted signal is switched off.
• In the first part of this section, we focus only on ISI and assume that channel is
effectively noiseless
• This can be done if the SNR is high enough to ignore the effect of channel noise.
• This leads us to a “matched filter receiver” which is an optimum receiver which
maximizes the SNR at its output
• If we assume that we have a matched filter receiver at the receiver end, we can
view the channel to be effectively noiseless
Matched –filter Receiver
• Consider a data stream represented by g(t) (where g(t) would be any line
coded representation of the data stream) applied to a noisy channel where
the additive channel noise w(t) is modeled as white and Gaussian with zero
mean and power spectral density 𝑁0 /2
• Consider a receiver as follows
It consists of an LTI filter of IR ℎ(𝑡)
The input to the receiver x(t) consists of a pulse signal 𝑔(𝑡) corrupted by the additive
white noise, 𝑤(𝑡)

𝑥 𝑡 =𝑔 𝑡 +𝑤 𝑡 0 ≤𝑡≤𝑇

where T is an arbitrary observation interval

𝑔(𝑡) – represents an arbitrary 0 or 1
𝑁𝑜
𝑤(𝑡)- sample function of a white noise process of zero mean and PSD
2

Assumption- It is assumed that the receiver has knowledge of the waveform of the pulse
signal 𝑔(𝑡) . The source of uncertainty is in the noise 𝑤(𝑡)
Problem : Function of the receiver is to detect the pulse signal 𝑔(𝑡) in an optimum
manner, given the received signal 𝑥 𝑡

Solution : Optimize the design so as to minimize the effects of noise at the filter output
and enhance the detection of the pulse signal 𝑔(𝑡)

Since the filter is linear, the output can be given as

𝑦 𝑡 = 𝑔0 𝑡 + 𝑛(𝑡)

where 𝑔0 𝑡 = 𝑔 𝑡 * ℎ(𝑡) Signal component

𝑛 𝑡 = 𝑤 𝑡 * ℎ(𝑡) Noise component
To state that the requirement is to have the output signal component 𝑔0 𝑡 to be
considerably greater than the output noise component 𝑛(𝑡) is the same as saying that
the instantaneous power in the output signal component 𝑔0 𝑡 measured at t = T should
be as large as possible compared to the average power of the output noise component
𝑛(𝑡)

This is equivalent to maximizing the peak pulse signal-to-noise ratio, defined as

|𝑔0 𝑇 |2
𝜂= 1
𝐸[𝑛(𝑡)2 ]

where |𝑔0 𝑇 |2 is the instantaneous power in the output signal, E[ ] is the expectation
operator and 𝐸[𝑛(𝑡)2 ] is the average output noise power
The objective is to design the filter so that its impulse response h(t) is such that the
output signal-to-noise ratio in 1 is maximized

Let 𝐺(𝑓) be the FT of 𝑔(𝑡) and 𝐻(𝑓) denote the frequency response of the filter
Then the FT of the output signal component 𝑔0 𝑡 is 𝐻(𝑓)𝐺(𝑓)
So 𝑔0 𝑡 can be written as the IFT of 𝐻(𝑓)𝐺(𝑓)
+∞
𝑔0 𝑡 =‫׬‬−∞ 𝐻 𝑓 𝐺 𝑓 exp 𝑗2𝜋𝑓𝑡 𝑑𝑓

When the filter output is sampled at t = T

+∞ 2

|𝑔0 𝑇 |2 = න 𝐻 𝑓 𝐺 𝑓 exp 𝑗2𝜋𝑓𝑇 𝑑𝑓 2

−∞
Effect of the filter on the noise : The power spectral density 𝑆𝑁 (𝑓) of the output noise
𝑛(𝑡) is equal to
𝑆𝑁 (𝑓) = 𝑆𝑊 (𝑓) |𝐻 𝑓 |2

𝑁𝑜
= |𝐻 𝑓 |2
2

Average power of the output noise 𝑛(𝑡) is

2 +∞
𝐸[𝑛(𝑡) ] = ‫׬‬−∞ 𝑆𝑁 (𝑓) 𝑑𝑓
+∞
𝑁𝑜
= න |𝐻 𝑓 |2 𝑑𝑓 3
2
−∞
Substituting 2 and 3 in the equation for peak pulse signal-to-noise ratio, we have

+∞ 2
‫׬‬−∞ 𝐻 𝑓 𝐺 𝑓 exp 𝑗2𝜋𝑓𝑇 𝑑𝑓 4
𝜂=
𝑁𝑜 +∞ 2 𝑑𝑓
‫׬‬ |𝐻 𝑓 |
2 −∞
Our problem is , for a given 𝐺(𝑓) we have to find the frequency response 𝐻(𝑓) of the
filter that makes 𝜂 a maximum
To solve the above , we use a mathematical result known as Schwartz’s Inequality

Schwartz’s Inequality- If we have two complex functions 𝜙1 (𝑥) and 𝜙2 (𝑥) in the real
variable 𝑥 , satisfying the conditions
+∞ +∞

න |𝜙1 (𝑥)|2 𝑑𝑥 <∞ න |𝜙2 (𝑥)|2 𝑑𝑥 <∞

and and
−∞ −∞
Then
+∞ 2 +∞ +∞
‫׬‬−∞ 𝜙1 𝑥 𝜙2 𝑥 𝑑𝑥 ≤ ‫׬‬−∞ |𝜙1 (𝑥)|2 𝑑𝑥 ‫׬‬−∞ |𝜙2 (𝑥)|2 𝑑𝑥

The above holds with equality if

𝜙1 (𝑥) = k 𝜙2 (𝑥)∗
where k is and arbitrary constant and * denotes complex conjugation

If we set 𝜙1 𝑥 = 𝐻(𝑓) and 𝜙2 𝑥 = 𝐺 𝑓 exp 𝑗2𝜋𝑓𝑇 , we can rewrite 4 as

+∞ RHS does not depend on

2 the frequency response
𝜂≤ න |𝐺 𝑓 |2 𝑑𝑓 𝐻(𝑓) but on the signal
𝑁𝑜
−∞ energy and noise PSD
The peak pulse signal-to-noise ratio 𝜂 will be maximum when 𝐻(𝑓) is chosen such that
the equality holds, i.e.
+∞
2
𝜂𝑚𝑎𝑥 = න |𝐺 𝑓 |2 𝑑𝑓
𝑁𝑜
−∞

From Schwartz’s inequality, we know that the equality holds when

We have denoted the
𝐻𝑜𝑝𝑡 𝑓 = 𝑘 𝐺 ∗ 𝑓 exp −𝑗2𝜋𝑓𝑇 value of 𝐻(𝑓) which
gives 𝜂𝑚𝑎𝑥 as 𝐻𝑜𝑝𝑡 𝑓

where 𝐺 ∗ 𝑓 is the complex conjugate of 𝐺 𝑓

“ Except for a factor of k exp −𝑗2𝜋𝑓𝑡 the frequency response of the optimum filter
is the same as the complex conjugate of the Fourier transform of the input signal”
In time domain
+∞

ℎ𝑜𝑝𝑡 𝑡 = 𝑘 න 𝐺 ∗ 𝑓 exp −𝑗2𝜋𝑓𝑇 exp 𝑗2𝜋𝑓𝑡 𝑑𝑓

−∞
+∞

= 𝑘 න 𝐺 ∗ 𝑓 exp −𝑗2𝜋𝑓(𝑇 − 𝑡) 𝑑𝑓
−∞

For a real signal 𝐺 ∗ 𝑓 = G(−f)

+∞

ℎ𝑜𝑝𝑡 𝑡 = 𝑘 න G(−f ) exp −𝑗2𝜋𝑓(𝑇 − 𝑡) 𝑑𝑓

−∞
Changing variable −𝑓 = 𝑓
+∞

ℎ𝑜𝑝𝑡 𝑡 = 𝑘 න G(f ) exp 𝑗2𝜋𝑓(𝑇 − 𝑡) 𝑑𝑓

−∞

= 𝑘 𝑔(𝑇 − 𝑡)

Therefore, IR of the optimum filter, except for a scaling factor if k is a time reversed and
delayed version of the input signal 𝑔(𝑡) i.e., it is “matched” to the input signal

An LTI filter defined in such a way is called as a matched filter

Only assumption for deriving the matched filter is that the input noise w(t) is stationary
𝑁𝑜
and white with zero mean and PSD of
2
Properties of Matched Filters

A filter which is matched to a pulse signal 𝑔(𝑡) of duration T is characterized by an IR

that is a time reversed and delayed version of the input 𝑔(𝑡)

ℎ𝑜𝑝𝑡 𝑡 = 𝑘 𝑔(𝑇 − 𝑡)

In frequency domain
𝐻𝑜𝑝𝑡 𝑓 = 𝑘 𝐺 ∗ 𝑓 exp −𝑗2𝜋𝑓𝑇
Property : The peak pulse signal-to-noise ratio of a matched filter depends only on the ratio
of the signal energy to the PSD of the white noise at the filter input

Proof : Consider a filter matched to a known signal 𝑔(𝑡)

The FT of the resulting matched filter output 𝑔0 𝑡 is

𝐺𝑜 𝑓 = 𝐻𝑜𝑝𝑡 𝑓 𝐺(𝑓)

= 𝑘 𝐺 ∗ 𝑓 𝐺(𝑓) exp −𝑗2𝜋𝑓𝑇

= 𝑘|𝐺 𝑓 |2 exp −𝑗2𝜋𝑓𝑇

+∞ +∞

𝑔𝑜 𝑇 = න 𝐺𝑜 𝑓 exp 𝑗2𝜋𝑓𝑇 𝑑𝑓 = 𝑘 න |𝐺 𝑓 |2 𝑑𝑓
−∞ −∞
From Rayleigh’s energy theorem, the integral of the squared magnitude spectrum of a pulse
signal with respect to frequency, is equal to the signal energy 𝐸
+∞ +∞
i.e., 𝐸 = න 𝑔2 𝑡 𝑑𝑡 = න |𝐺 𝑓 |2 𝑑𝑓
−∞ −∞

∴ 𝑔𝑜 𝑇 = k E
+∞
2
𝑁𝑜
𝐸𝑛 𝑡 = න |𝐻𝑜𝑝𝑡 𝑓 |2 𝑑𝑓
2
−∞
+∞ +∞
𝑁𝑜 2 2
𝑁𝑜 2 2 𝑁𝑜 2
= න 𝑘 |𝐺 𝑓 | 𝑑𝑓 = 𝑘 න |𝐺 𝑓 | 𝑑𝑓 = 𝑘 𝐸
2 2 2
−∞ −∞
 𝑘2𝐸2 2E
∴ 𝜂𝑚𝑎𝑥 = =
𝑁𝑜 2 𝑁𝑜
𝑘 𝐸
2

the peak pulse signal-to-noise ratio depends on the input signal energy and power spectral
density of the noise, not on the particular shape of the waveform that is used.
Error Rate Due to Channel Noise in a Matched-Filter Receiver

• Consider a binary data stream is applied to a noisy channel where the additive
channel noise w(t) is modeled as white and Gaussian with zero mean and power
spectral density N0/2
• The data stream is based on polar NRZ signaling, in which symbols 1 and 0 are
represented by positive and negative rectangular pulses of amplitude A and
duration Tb
• In the bit interval 0 ≤ 𝑡 ≤ 𝑇𝑏 , the received signal is represented as
• The receiver operates synchronously with the transmitter, which means that the

matched filter at the front end of the receiver has knowledge of the starting and

ending times of each transmitted pulse.

• The matched filter is followed by a sampler, and then finally a decision device.

• it is assumed that the symbols 1 and 0 are equally likely; the threshold in the

decision device, namely , 𝜆 may then be set equal to zero.

• If this threshold is exceeded, the receiver decides in favor of symbol 1; if not, it

decides in favor of symbol 0. A random choice is made in the case of a tie.

• Based on the signal space representation of BPSK, the transmitted signal

constellation consists of a pair of message points located at + 𝐸𝑏 and − 𝐸𝑏

where the energy per bit

• The only basis function is a rectangular pulse of the form

• Because of its equivalence to BPSK, the average probability of symbol error is given
by
𝟏 𝐄𝐛
𝐏𝐞 = 𝐞𝐫𝐟𝐜
𝟐 𝐍𝐨
 𝑬𝒃 /𝑵𝟎 (in dB) BER
0 7 x 10^(-2)
2 4 x 10^(-2)
4 1.2 x 10^(-2)
6 2.4x 10^(-3)
8 2x 10^(-4)
10 4 x 10^(-6)

RAJAGIRI SCHOOL OF ENGINEERING AND TECHNOLOGY, KOCHI 27

 The matched-filter receiver exhibits an exponential improvement in the average
probability of symbol error Pe with the increase in Eb/N0.

• For example, expressing Eb/N0 in decibels we see that Pe is on the order of 10−6
when Eb/N0 = 10 dB.
• Such a value of Pe is small enough to say that the effect of the channel noise is
ignorable.
• In further discussion we assume that the SNR is high enough to ignore channel
noise and hence ISI is the only source of interference
Intersymbol Interference (ISI)

• To proceed with a mathematical study of intersymbol interference, consider a baseband

binary PAM system
• “baseband” refers to an information-bearing signal whose spectrum extends from (or
near) zero up to some finite value for positive frequencies
• The incoming binary sequence 𝑏𝑘 consists of symbols 1 and 0 each of duration 𝑇𝑏

• The pulse-amplitude modulator modifies this binary sequence into a new sequence

of short pulses (approximating a unit impulse) , whose amplitude 𝑎𝑘 is represented

in the polar form

• This sequence of short pulses is applied to a transmit filter of impulse response

𝑔 𝑡 , producing the transmitted signal
• The signal 𝑠(𝑡) is modified as a result of transmission through the channel of IR ℎ(𝑡)

• In addition, the channel adds random noise to the signal at the receiver input

• The noisy signal 𝑥(𝑡) is then passed through a receive filter of impulse response 𝑐(𝑡)

• The resulting filter output 𝑦(𝑡) is sampled synchronously with the transmitter , with

the sampling instants being determined by a clock or timing signal

• The sequence of samples 𝑦(𝑡𝑖 ) thus obtained is used to reconstruct the original data

sequence by means of a decision device which compares the sample values to a

threshold 𝜆
• If the threshold 𝜆 is exceeded, a decision is made in favor of symbol 1 else 0

• If the sample value is exactly equal to the threshold, the flip of a coin will determine the

symbol

• The receive filter output can be written as

𝑦 𝑡 = 𝜇 ෍ 𝑎𝑘 𝑝(𝑡 − 𝑘𝑇𝑏 ) + 𝑛(𝑡)

𝑘
where 𝜇 is a scaling factor

• The scaled pulse 𝜇 𝑝(𝑡) is obtained by a double convolution involving the IR 𝑔 𝑡 of the

transmit filter, IR h 𝑡 of the channel and IR 𝑐 𝑡 of the receive filter

 𝜇 𝑝 𝑡 = 𝑔 𝑡 ∗ ℎ 𝑡 ∗ 𝑐(𝑡)

• Assume that the pulse 𝑝 𝑡 is normalised by setting 𝑝 0 = 1

• Hence 𝜇 is used to account for amplitude changes that occur during the signal transmission

through the system

• Taking the FT, we have

𝜇 𝑃 𝑓 = 𝐺 𝑓 𝐻 𝑓 𝐶(𝑓)

• 𝑛(𝑡) in the expression for 𝑦(𝑡) is the noise produced at the output of the receive filter due

to channel noise 𝑤(𝑡) . 𝑤(𝑡) is assumed to be white Gaussian noise of zero mean
 • The receive filter output 𝑦(𝑡) is sampled at time 𝑡𝑖 = 𝑖𝑇𝑏 ( with 𝑖 taking on integer
values) , yielding
𝑦 𝑡𝑖 = 𝜇 ෍ 𝑎𝑘 𝑝[(𝑖 − 𝑘)𝑇𝑏 ] + 𝑛(𝑡𝑖 )
𝑘
+∞

= 𝜇𝑎𝑖 + 𝜇 ෍ 𝑎𝑘 𝑝[ 𝑖 − 𝑘 𝑇𝑏 ] + 𝑛(𝑡𝑖 )
𝑘=−∞
𝑘≠𝑖

• In the above equation, the term 𝜇𝑎𝑖 represents the contribution of the 𝑖𝑡ℎ transmitted
bit. The second term represents the residual effect of all other transmitted bits on the
decoding of the 𝑖𝑡ℎ bit
• This residual effect due to occurrence of pulses before and after the sampling instant 𝑡𝑖 is
called intersymbol interference (ISI)
• The term 𝑛(𝑡𝑖 ) represents the noise sample at time 𝑡𝑖

• In the absence of both ISI and noise, we have

𝑦 𝑡𝑖 = 𝜇𝑎𝑖

This shows that under ideal conditions, the 𝑖𝑡ℎ transmitted bit is decoded correctly

• The unavoidable presence of ISI and noise in the system, however, introduces errors
in the decision device at the receiver output
• The objective of the design of the transmit and receive filters would be to minimize
the effects of noise and ISI and deliver the data with the smallest error rate possible
• When SNR is high, the noise is negligible and the system is limited only by ISI

• Hence we can ignore the noise 𝑛(𝑡𝑖 )

• Assumed that this condition holds in further discussions

• The issue is to determine the pulse waveform 𝑝 𝑡 for which the ISI is completely

eliminated
Nyquist Criterion for Distortionless Baseband Binary Transmission

• The frequency response of the channel is fixed, the problem is to determine the

frequency response of the transmit and receive filters so as to reconstruct the original

binary sequence 𝑏𝑘

• The overlapping pulses are to be designed in such a way that at the receiver output

they do not interfere with each other at the sampling times 𝑡𝑖 = 𝑖𝑇𝑏

• i.e., the weighted pulse contribution 𝑎𝑘 𝑝 𝑖𝑇𝑏 − 𝑘𝑇𝑏 =0 for all 𝑘 except for 𝑘 = 𝑖 to

make the transmission ISI free (during the 𝑖𝑡ℎ sampling instant)
• Hence the overall pulse 𝑝(𝑡) must be designed such that

where 𝑝(0) is set to 1 according to the normalisation condition

• If 𝑝(𝑡) satisfies the conditions in 1, the receiver output is (ignoring the noise term)

𝑦 𝑡𝑖 = 𝜇𝑎𝑖

which implies zero intersymbol interference. The conditions in 1 ensure perfect reception in
the absence of noise
• A pulse 𝑝(𝑡) that satisfies the two part condition of 1 is called a Nyquist pulse and the
condition itself is referred to as the Nyquist’s criterion for distortionless baseband binary
transmission
• Let’s look at the frequency domain equivalent of 1

Consider the sequence { 𝑝 𝑛𝑇𝑏 } where n=0, ± 1, ±2,… . We know that sampling in

time domain produces periodicity in frequency domain

∞

𝑝𝛿 𝑡 = ෍ 𝑝 𝑛𝑇𝑏 𝛿(𝑡 − 𝑛𝑇𝑏 )

𝑛=−∞

∞
1 1
𝑃𝛿 𝑓 = ෍ 𝑝 𝑓−𝑛
𝑇𝑏 𝑇𝑏
𝑛=−∞

1
= where 𝑅𝑏 = is the
𝑇𝑏
bit rate
Since 𝑃𝛿 𝑓 is the FT of 𝑝𝛿 𝑡

Let the integer 𝑚 = (𝑖 – 𝑘), then i = k corresponds to 𝑚 = 0 and 𝑖𝑘 corresponds to

𝑚  0.

Imposing the conditions in 1 in the above equation

 Hence the condition for zero ISI is satisfied, provided that

𝑃𝛿 𝑓 =

=𝑝 0 =1
Nyquist criterion for distortionless
i.e., baseband transmission in frequency
domain
2

The Nyquist’s criterion for distortionless transmission in the absence of noise can be stated as :
The frequency function 𝑃(𝑓) eliminates intersymbol interference for samples taken at intervals
𝑇𝑏 provided that it satisfies equation 2

Note that P(f) refers to overall system consisting of transmit filter, channel and receive filter
Ideal Nyquist Channel

Simplest way of satisfying equation 2 is to specify the frequency function P(f) to be in

the form of a rectangular function

3
 The overall system bandwidth W is defined by

The signal that produces zero ISI is defined by the sinc function

The special value of the bit rate 𝑅𝑏 = 2𝑊 is called as the Nyquist rate and W itself is
called Nyquist bandwidth

The ideal baseband pulse transmission system described by 3 in frequency domain or

equivalently 4 in time domain is called the ideal Nyquist channel
• It can be seen that the function 𝑝(𝑡) has its peak value at the origin and goes through zeros

at integer multiples of bit duration 𝑇𝑏

• Hence if the received waveform 𝑦(𝑡) is sampled at the instant of time

then the pulses defined by will not interfere with each other
Although ideal Nyquist channel ensures zero ISI, there are two practical difficulties
which make it undesirable
1. It requires the magnitude characteristic of P(f) to be flat from –W to +W Hz and zero
elsewhere. This is physically unrealizable because of the sharp transition at the band
edges ±W
1
2. The function p(t) decreases as for large |t|, resulting in a slow rate of decay.
|𝑡|

Accordingly there is practically no margin of error in sampling times in the receiver

(small errors will bring in ISI due to significant tails of nearby pulses)
To evaluate the effect of “timing error” , consider the sample y(t) at t = ∆𝑡 (instead of
t=0), where ∆𝑡 is the timing error

𝑦 ∆𝑡 = 𝜇 ෍ 𝑎𝑘 𝑝[∆𝑡 − 𝑘𝑇𝑏 ]
𝑘

ISI caused by timing error ∆𝑡 in sampling y(t)

𝜇 𝜇

desired symbol
Raised Cosine Spectrum

• It is desired to have pulses which have faster decay of tails and are physically realizable

while satisfying the Nyquist criterion for distortionless baseband transmission

• A commonly used pulse is a raised cosine spectrum

• Here, instead of a sudden fall at ±𝑊, the spectrum of the pulse is made to gradually fall

between W and 2W ( Hence now the bandwidth is greater than the minimum possible

𝑅𝑏
value of 𝑊 = )
2
 Nyquist criterion for distortionless
baseband transmission

If we restrict the frequency band of interest to [−𝑊 , 𝑊], then according to the Nyquist’s
condition stated above,

One of the 𝑃(𝑓)s which satisfies the above condition is the raised cosine (RC) spectrum

The frequency response consists of a flat portion and a roll-off portion that has a
sinusoidal form
where the parameters 𝑓1 , the dimensionless parameter 𝛼 and 𝑊 are related as

where 𝛼 is called as the roll-off factor and indicates the excess bandwidth over the ideal
solution of 𝑊
The BW occupied beyond the Nyquist bandwidth 𝑊 is called as excess bandwidth

The new transmission bandwidth , 𝐵𝑇 is given by

It can be observed that for 𝛼=0.5 , 1, the function 𝑃(𝑓) falls gradually compared to the

ideal Nyquist pulse (𝛼=0) and are hence easier to implement in practice

The time response 𝑝(𝑡) is the IFT of 𝑃(𝑓)

Decreases as
1
|𝑡|2

Characterizes
Ideal Nyquist
Pulse
• The first factor ensures zero crossings at sampling instants 𝑡𝑖 = 𝑖𝑇𝑏

• The second factor reduces the tails of the pulse considerably below those obtained

for the ideal Nyquist pulse so that transmission becomes insensitive to sampling

timing errors

For 𝛼= 1, the roll-off is most gradual and the tails of 𝑝(𝑡) have smallest amplitude. Thus

amount of ISI due to timing error reduces as 𝛼 is increased from 0 to 1.

The special case of 𝛼= 1 (i.e., 𝑓1 =0 ) is called as full-cosine roll-off characteristic for

which 𝑃(𝑓) becomes

and correspondingly
Correlative Coding or partial-response signaling schemes

• The practically realizable RC spectrum removes ISI but results in increase in the bandwidth

required

• Can we achieve a trade-off between bandwidth and ISI ?

• Can we allow a little ISI and reduce the bandwidth requirement ?

• Can be done using correlative coding or partial-response signaling schemes

• Uses the concept of controlled ISI, i.e. ISI is added to the transmitted signal in a controlled

manner to achieve a rate of 2W symbols/sec in a channel of bandwidth W Hz

• Since ISI introduced into the transmitted signal is known, its effect can be understood

at the receiver in a deterministic way

• Hence correlative coding is a practical way of achieving a signaling rate of 2W

symbols/sec in BW of W Hz using realizable filters

• The basic idea of correlative coding is illustrated by considering the specific example

of duobinary signaling
Duobinary Signaling

• Also called as Class-I partial response

• Consider a binary sequence 𝑏𝑘 consisting of uncorrelated symbols 1 and 0 each of
duration 𝑇𝑏
• As before, this sequence is applied to a pulse-amplitude modulator modifies this binary
sequence into a new sequence of short pulses (approximating a unit impulse) , whose
amplitude 𝑎𝑘 is represented in the polar form
• Then this signal is applied to a duobinary encoder that converts it into a three level
output, namely -2, 0, +2 ( as shown below)
• The two-level sequence is first passed through a simple filter involving a single delay

element and summer ( Let 𝐻𝐼 (𝑓) denote the filter’s IR)

• Every input symbol 𝑎𝑘 gets added to the previous symbol 𝑎𝑘−1 in the filter

• The duobinary coder output 𝑐𝑘 can be expressed as the sum of present input pulse 𝑎𝑘 and

its previous value 𝑎𝑘−1 ,

𝑐𝑘 = 𝑎𝑘 + 𝑎𝑘−1
• Now the input sequence {𝑎𝑘 } of uncorrelated two-level pulses is transformed to a
sequence {𝑐𝑘 } of correlated three-level pulses this correlation can be seen as
introducing ISI/intersymbol interference into the transmitted signal in an artificial manner
• IR is the output of the filter when the input is an impulse

ℎ𝐼 𝑡 = 𝛿 𝑡 + 𝛿 𝑡 − 𝑇𝑏

𝐻𝐼 𝑓 = 1 + 𝑒 −𝑗2𝜋𝑓𝑇𝑏

• Overall frequency response of the system is,

𝐻 𝑓 = 𝐻𝐼 𝑓 𝐻𝑁𝑦𝑞𝑢𝑖𝑠𝑡 𝑓

= 𝐻𝑁𝑦𝑞𝑢𝑖𝑠𝑡 𝑓 1 + 𝑒 −𝑗2𝜋𝑓𝑇𝑏

= 𝐻𝑁𝑦𝑞𝑢𝑖𝑠𝑡 𝑓 [𝑒 𝑗𝜋𝑓𝑇𝑏 + 𝑒 −𝑗𝜋𝑓𝑇𝑏 ]𝑒 −𝑗𝜋𝑓𝑇𝑏

= 2 𝐻𝑁𝑦𝑞𝑢𝑖𝑠𝑡 𝑓 𝐶𝑜𝑠 𝜋𝑓𝑇𝑏 𝑒 −𝑗𝜋𝑓𝑇𝑏

 1
For an ideal Nyquist channel of BW , 𝑊 = Hz,
2𝑇𝑏

1 Ignoring the scaling

1, |𝑓| ≤
𝐻𝑁𝑦𝑞𝑢𝑖𝑠𝑡 𝑓 = ቐ 2𝑇𝑏 factor 𝑇𝑏
0, 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

The overall frequency response of the duobinary signaling scheme has the form

1
2 𝐶𝑜𝑠 𝜋𝑓𝑇𝑏 𝑒 −𝑗𝜋𝑓𝑇𝑏 , |𝑓| ≤
𝐻 𝑓 =ቐ 2𝑇𝑏
0, 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Magnitude response Phase response

𝐻 𝑓 = 2|𝐶𝑜𝑠 𝜋𝑓𝑇𝑏 | ∠ 𝐻 𝑓 = −𝜋𝑓𝑇𝑏
 • Advantage of this frequency response is that it can be practically implemented ( no
sudden transitions as in ideal pulse) and the BW is still W Hz
• But it comes at the cost of controlled ISI

+∞

ℎ 𝑡 = න 𝐻(𝑓)𝑒 𝑗2𝜋𝑓𝑡 𝑑𝑓
−∞

𝜋𝑡
𝑆𝑖𝑛 𝑇𝑏 2
𝑇𝑏
=
𝜋𝑡(𝑇𝑏 − 𝑡)
• ℎ 𝑡 has value 1 at t=0 and t= 𝑇𝑏 and is zero at all the other sampling instants
It is also called as partial-response signaling since the response to an input is spread over
more than one signaling interval or we can say that the response in any signaling interval
is “partial”
 1 1
• Tails of ℎ(𝑡) decay as , which is faster than the decay rate of in the ideal Nyquist
|𝑡|2 |𝑡|
channel

To detect {𝑎𝑘 } from {𝑐𝑘 }

By using the equation,

𝑐𝑘 = 𝑎𝑘 + 𝑎𝑘−1

Let 𝑎ෞ𝑘 be the estimate of the original pulse 𝑎𝑘 as estimated by the receiver at the
time t= k 𝑇𝑏
Subtracting the previous estimate 𝑎ෟ
𝑘−1
from 𝑐𝑘 ,

𝑎ෞ𝑘 = 𝑐𝑘 − 𝑎ෟ
𝑘−1
If 𝑐𝑘 is received without error and also if the previous estimate 𝑎ෟ
𝑘−1 at the time t= (k-1) 𝑇𝑏

corresponds to correct decision, then the estimate 𝑎ෞ𝑘 will be correct too . This technique of
using stored estimate of previous symbol is called as decision feedback

• Detection procedure is essentially an inverse of the simple delay-line filter at the

transmitter
• A major drawback is that once errors are made, they tend to propagate through the output
because decision on the current input 𝑎𝑘 depends on the correctness of decision made on
previous input symbol 𝑎𝑘−1
• A practical way of avoiding the error-propagation is to use precoding before duobinary
coding
• Precoding is preformed on the input binary sequence {𝑏𝑘 } which converts it into another
binary sequence {𝑑𝑘 } defined by

𝑑𝑘 = 𝑏𝑘 ⊕ 𝑑𝑘−1 1

where ⊕ stands for modulo-2 addition of binary digits 𝑏𝑘 and 𝑑𝑘−1 equivalent to
two- input EXOR

𝑠𝑦𝑚𝑏𝑜𝑙 1, 𝑖𝑓 𝑒𝑖𝑡ℎ𝑒𝑟 𝑏𝑘 𝑜𝑟 𝑑𝑘−1 𝑛𝑜𝑡 𝑏𝑜𝑡ℎ 𝑖𝑠 1

𝑑𝑘 =ቐ
𝑠𝑦𝑚𝑏𝑜𝑙 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
This precoded binary sequence {𝑑𝑘 } is applied to a PAM, producing a corresponding
sequence {𝑎𝑘 }= ±1 as before. This is then applied to duobinary coder which
produces {𝑐𝑘 } as
𝑐𝑘 = 𝑎𝑘 + 𝑎𝑘−1 2

Combining 1 and 2 0, 𝑖𝑓 𝑑𝑎𝑡𝑎𝑠𝑦𝑚𝑏𝑜𝑙 𝑏𝑘 𝑖𝑠 1

𝑐𝑘 =ቐ
±2, 𝑖𝑓 𝑑𝑎𝑡𝑎𝑠𝑦𝑚𝑏𝑜𝑙 𝑏𝑘 𝑖𝑠 0
The decision rule for detecting the original binary sequence {𝑏𝑘 } from {𝑐𝑘 } is

If | 𝑐𝑘 |= 1, make a random guess in favor of 1 or 0

Detector does not require any knowledge other than the present sample , hence there
will be no error-propagation