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 =…