Example
What is the minimum bandwidth for transmitting data at a rate of 33.6 kbps without ISI?
Answer:
The minimum bandwidth is equal to the Nyquist bandwidth. Therefore, (BW)min = W = Rb/2 = 33.6/2 = 16.8 kHz
• Note: If a 100% roll-off characteristic is used, bandwidth = W(1+α) = 33.6 kHz
BT.67

Example
Bandwidth requirement of the T1 system
T1 system
– multiplex 24 voice inputs, based on an 8-bit PCM word.
– bandwidth of each voice input (B) = 3.1 kHz
For converting the voice signal into binary sequence,
• The minimum sampling rate = 2B = 6.2 kHz
• Sampling rate used in telephone system =8 kHz

BT.68

1

Example
With a sampling rate of 8 kHz, each frame of the multiplexed signal occupies a period of 125µs.

:
:

8 bit from
1st input

No. of bits = 8 ·24+1=193

8 bit from
2nd input

….

8 bit from
1 bit for
24th input Synchronization

125 µs

BT.69

Example
Correspondingly, the bit duration is 125 µs/193 = 0.647 µs.
For eliminating ISI, the minimum transmission bandwidth is
1 / 2Tb = 772kHz

BT.70

2

Eye diagrams
This is a simple way to give a measure of how severe the ISI
(as well as noise) is.
This pattern is generated by overlapping the incoming signal elements. Example: bipolar NRZ PAM
1

0

1

1

0

0

Tb

BT.71

Eye diagrams
Eye pattern is often used to monitoring the performance of baseband signal.
– The best time to sample the received waveform is when the eye opening is largest.
Effects of noise are ignored

1

BT.72

3

Eye diagrams
The maximum distortion and ISI are indicated by the vertical width of the two branches at sampling time.

2

BT.73

Eye diagrams
The noise margin or immunity to noise is proportional to the width of the eye opening.

3

BT.74

4

Eye diagrams
The sensitivity of the system to timing errors is determined by the rate of closure of the eye as the sampling time is varied. 4

BT.75

Equalization
In preceding sections, raised-cosine filters were used to eliminate ISI. In many systems, however, either the channel characteristics are not known or they vary.
Example
The characteristics of a telephone channel may vary as a function of a particular connection and line used.
It is advantageous in such systems to include a filter that can be adjusted to compensate for imperfect channel transmission characteristics, these filters are called equalizers. BT.76

5

Before equalization

After equalization

BT.77

Transversal filter (zero-forcing equalizer)

xk

T is the bit duration.
BT.78

6

Equalization
The problem of minimizing ISI is simplified by considering only those signals at correct sample times.
The sampled input to the transversal equalizer is x(kT ) = x k

x0

The output is

x2

y (kT ) = y k
For zero ISI, we require that
1 k = 0 yk = 
0 k ≠ 0 …(*)

x1
BT.79

aN xk − N

The output can be expressed as yk =

N

∑ an xk −n

xk

n=− N

−N ≤k≤N

a0 xk a− N xk + N

There are 2N+1 independent equations in terms of an . This limits us to 2N+1 constraints, and therefore (*) must be modified to
1
k =0 yk = 
0 k = ±1,±2,...,± N

BT.80

7

Equalization
The 2N+1 equations becomes
 xo x−1 L x− N L x− 2 N −1

x0
L x− N +1 L x− 2 N
 x1
 M
M
 xN −1 L x0 L x− N −1
 xN
 M
M

 x2 N −1 x2 N − 2 L xN −1 L x− 2
x
x2 N −1 L xN x−1 L
 2N

x− 2 N   a − N   0 

   x− 2 N +1  a− N +1  0
 M   M 

   x− N   a0  = 1
 M   M 

   x−1   a N −1  0 x0   a N  0

  

BT.81

Example
Determine the tap weights of a three-tap, zero-forcing equalizer for the input where x− 2 = 0.0, x−1 = 0.2, x0 = 1.0, x1 = −0.3, x2 = 0.1 , xk = 0 for k > 2
The three equations are
+ 0.2a0 a−1 − 0.3a−1
+ a0
+ 0.2a1
0.1a−1

− 0.3a0

+ a1

N=1

=0
=1
=0

Solving, we obtain a−1 = −0.1779, a1 = 0.2847, a0 =