Course: BEng (Hons) in Electrical and Electronic Engineering

1. EEEC6440315 COMMUNICATION SYSTEMS
Inter Symbol Interference And Pulse Shaping
FACULTY OF ENGINEERING AND COMPUTER TECHNOLOGY
BENG (HONS) IN ELECTRICALAND ELECTRONIC ENGINEERING
Ravandran Muttiah BEng (Hons) MSc MIET

2. Band Limited Channel And Inter Symbol Interference
Rectangular pulses are suitable for infinite bandwidth channels
(practically - wideband).
Practical channels are band limited ⇒ Pulses spread in time and are
smeared into adjacent slots. This is Inter Symbol Interference (ISI)
1

3. 2
1 0 0 0
1 0 1 1
0
0
0
0
0
0
t
t
t t
t
t
Intersymbol Interference
Sampling Points
(Transmitter Clock)
Sampling Points
(Receiver Clock)
Sampling Points
(Receiver Clock)
Input Binary Waveform Individual Pulse Response Received Waveform
Figure 1: Inter Symbol Interference.
𝑇s 𝑇s
𝑇s

4. 3
Figure 2: Channel noise.
Eye Diagram
Convenient way to observe the effect of ISI and channel noise on an oscilloscope.
Best Sampling Time
0 1 1 0 0 1
Eye Pattern
Waveform
Ideal
Filtering
Filtering
With ISI
Noise Plus
ISI
𝑇b
𝑇b
Maximum
Distortion
Noise
Margin
t
t
t

5. 4
Peak distortion Noise Margin
Distortion of
Zero Crossings
Sensitivity to
Timing Error
Optimum
Sampling Time
Figure 3: Eye diagram.
Oscilloscope presentations of a signal with multiple sweeps (triggered by a
clock signal), each is slightly larger than symbol interval.
Quality of a received signal may be estimated.
Normal operating conditions (no ISI, no noise) ⇒ Eye is open.
Large ISI or noise ⇒ Eye is closed.

6. 5
Timing error allowed ⇒ width of the eye, called eye opening
(preferred sampling time – at the largest vertical eye opening).
Sensitivity to timing error ⇒ Slope of the open eye evaluated at
the zero crossing point.
Noise margin ⇒ The height of the eye opening.

7. 6
Transmitting
Filter
𝐻T 𝑓
Channel (Filter)
Characteristics
𝐻C 𝑓
Receiver
Filter
𝐻R 𝑓
Transmission Over A Band - Limited Channel
Flat Top
Pulses
Recovered
Rounded
Pulse
(to sampling
and decoding
circuits)
𝑥 𝑡 y 𝑡
Figure 4: Baseband transmission system.
Input PAM signal, 𝑥 𝑡 = 𝑛 𝑎𝑛𝑠T 𝑡 − 𝑛𝑇
Output signal, y 𝑡 = 𝑛 𝑎𝑛𝑠 𝑡 − 𝑛𝑇 ,
𝑠 𝑡 = 𝑠T 𝑡 ∗ ℎT 𝑡 ∗ ℎC 𝑡 ∗ ℎR 𝑡 = 𝑠T 𝑡 ∗ ℎ 𝑡
Sampled output at 𝑡 = 𝑚𝑇, 𝑦𝑚 = 𝑛 𝑎𝑛𝑠𝑚−𝑛 = 𝑎𝑚𝑠0 + 𝑛≠𝑚 𝑎𝑛𝑠𝑚−𝑛
= 𝑦 𝑚𝑇
𝑠𝑚−𝑛 = 𝑠 𝑚𝑇 − 𝑛𝑇 transmitted symbol inter symbol interference

8. 7
Pulse Shaping To Eliminate ISI
In 1928 Nyquist discovered 3 methods to eliminate ISI:
• Zero ISI pulse shaping.
• Controlled ISI (Eliminated later on by an Equalizer).
• Zero average ISI (negative and positive areas under the pulse in
an adjacent interval are equal).
• Zero ISI pulse shaping, 𝑠 𝑛𝑇 =
𝑠0 𝑛 = 0
0 𝑛 ≠ 0
𝑦𝑚 = 𝑎𝑚𝑠0 + 𝑛≠𝑚 𝑎𝑛𝑠𝑚−𝑛 ⇒ 𝑦𝑚 = 𝑎𝑚𝑠0

9. 8
Example 1:
𝑠 𝑡 = sinc 𝑓0𝑡
𝑠 𝑛𝑇 = sinc 𝑛 = 0, 𝑛 ≠ 0
Hence, 𝑠𝑖𝑛c pulses allows to eliminate ISI at sampling instants.
Figure 5: Zero ISI 𝑠𝑖𝑛c pulse.
-3 -2 -1 -0 1 2 3
-0.3
0
0.3
0.6
0.9

10. 9
Nyquist Criterion For Zero ISI
Provides a generic solution to the zero ISI problem.
A necessary and sufficient condition for s 𝑡 to satisfy,
𝑠 𝑛𝑇 =
𝑠0 𝑛 = 0
0 𝑛 ≠ 0
⇔ 𝑚=−∞
∞
𝑆s 𝑓 +
𝑚
𝑇
= 𝑇
Consider a channel band-limited to 0, 𝐹max for 3 cases.
•
1
𝑇
= 𝑓0 > 2𝐹max ⇒ No way to eliminate ISI.
• 𝑓0 = 2𝐹max ⇒ Only sinc 𝑓0𝑡 eliminates ISI. The highest possible
transmission rate 𝑓0 for transmission with zero ISI is 2𝐹max, not 𝐹max.
• 𝑓0 < 2𝐹max ⇒ Many signals may eliminated ISI in this case.
Note: 𝑓0 =
1
𝑇
is the transmission rate (symbols/second)

11. 10
Figure 6: 𝑓0 > 2𝐹max, 𝑓0 = 2𝐹max and 𝑓0 < 2𝐹max.
1
𝑇
= 𝑓0 > 2𝐹
max
𝑓0 = 2𝐹
max
𝑓0 < 2𝐹
max
𝑚=−∞
∞
𝑆s 𝑓 +
𝑚
𝑇
= 𝑍 𝑓
𝑚=−∞
∞
𝑆s 𝑓 +
𝑚
𝑇
= 𝑍 𝑓
𝑚=−∞
∞
𝑆s 𝑓 +
𝑚
𝑇
= 𝑍 𝑓
𝑊 = 𝐹
max
−
1
𝑇
−
1
𝑇
+ 𝑊
1
𝑇
− 𝑊
−𝑊 0
0
𝑊 1
𝑇
1
𝑇
+ 𝑊
−
1
𝑇
1
𝑇
𝑊 =
1
2𝑇
−
1
𝑇
1
𝑇
−
1
𝑇
+ 𝑊
−𝑊 𝑊
1
𝑇
− 𝑊
Case 1:
Case 2:
Case 3:
⋯
⋯
⋯ ⋯

12. 11
Raised Cosine Pulse
When 𝑓0 < 2𝐹max, raised cosine pulse is widely used.
The raised cosine pulse has Fourier transform, which its spectrum is,
𝐻 𝑓 =
𝑇𝜑, 0 ≤ 𝑓 ≤
1−𝛼
2𝑇
𝑇
2
1 + cos
π𝑇
𝛼
𝑓 −
1−𝛼
2𝑇
,
1−𝛼
2𝑇
≤ 𝑓 ≤
1+𝛼
2𝑇
0, 𝑓 >
1+𝛼
2𝑇
where 𝛼 is roll-off factor, 0 ≤ 𝛼 ≤ 1.
The bandwidth above the Nyquist frequency
𝑓0
2
=
1
2𝑇
is called excess
bandwidth. Where, 𝑓0 =
1
𝑇
Time-domain waveform of the raised cosine pulse, ℎ 𝑡 =
sin
π𝑡
𝑇
π𝑡
𝑇
cos
𝛼π𝑡
𝑇
1−
2𝛼𝑡
𝑇
2

13. 12
When 𝛼 = 0, pulse reduces to sinc 𝑡𝑓0 = sinc
𝑡
𝑇
and 𝑓0 = 2𝐹max.
𝛼 = 1, symbol rate 𝑓0 = 𝐹max.
Tails decay as 𝑡3
, mistiming is not a big problem.
Smooth shape of the spectrum, easier to design filters.
1
0.5
0 0
0.5
1
1
0
-1
-2
-3
Figure 7: Raised Cosine Spectrum. Figure 8: Time Domain Pulse.
𝛼 = 1
𝛼 = 0.5
𝛼 = 0
-0.5
-1 0 0.5 1
2 3

14. 13
Example 2:
Wideband Code-Division Multiple-Access (WCDMA) is the main
technology behind 3rd Generation (3G) cellular systems implements a
Square-Root Raised Cosine pulse shape with excess bandwidth 𝛼 =
0.22, which translates the signaling rate of 3.84 MHz to a bandwidth
of 0.5 × 1 + 𝛼 × 2 × 3.84 = 4.68 MHz (the factor 2 arises for the
RF bandwidth). Accounting for the guard-bands to minimise interference
between neighboring channels, the signal bandwidth in WCDMA
systems is 5 MHz.

15. (1) Leon W. Couch II, Digital and Analogue Communication Systems, Prentice
Hall, 1999.
References
14