School of Computer and Communication Engineering,
Amir Razif B. Jamil Abdullah
EKT 431: Digital Communications


Transforming the information source to a form compatible with a digital system

Sampling

Aliasing

Quantization

Uniform and non-uniform

Baseband modulation

Binary pulse modulation

M-ary pulse modulation

M-PAM (M-ary Pulse amplitude modulation)

Digital info.
Bit stream
(Data bits)
Pulse waveforms
(baseband signals) source
Textual info.
Format
Analog info.
Sample Quantize Encode
Pulse modulate
Sampling at rate f s

1 / T s
(sampling time=Ts)
Encoding each q. value to l
 log
2
L bits
(Data bit duration Tb=Ts/l )

Quantizing each sampled value to one of the
L levels in quantizer.
m
 log M symbol out of M symbols and transmitting a baseband waveform with duration T
Information (data) rate:

Symbol rate :
R b


For real time transmission: R b
T [bits/sec] b
 mR

• Assuming real time transmission and equal energy per transmission data bit for binary-PAM and 4-ary PAM:
• 4-ary: T=2T b and Binary: T=T b
•
A
2 
10 B
2
Binary PAM
(rectangular pulse)
4-ary PAM
(rectangular pulse)
3B
-A.
‘0’
T
A.
‘1’
T
-3B
‘10’
T
-B
‘00’
T
B
‘01’
T
‘11’
T

0 T s
2T s
2.2762 V 1.3657 V
0 T b
2T b
3T b
4T b
5T b
6T b
1 1 0 1 0 1 s
R b
=1/T b
=3/T
R=1/T=1/T b s
=3/T
0 T 2T 3T 4T 5T 6T
0 T 2T 3T
R b
=1/T b
=3/T s
R=1/T=1/2T b
=3/2T s
=1.5/T s





Receiver structure

Demodulation (and sampling)

Detection
Another source of error:

Inter-symbol interference (ISI)
Nyquist theorem
The techniques to reduce ISI


Pulse shaping
Equalization

•
•
•
The baseband pulse received at receiver not in ideal pulse shape, need to recover the pulse shape. Filtering & channel caused ISI.
The goal of the demodulator receiving signal
~ is to recover a baseband pulse with the best SNR and free of any ISI .
Equalization consists of sophisticated set of signal processing techniques. It is used to compensate for channel-induced interference.

•
•
Two primary causes of error-performance degradation
(1) Effect of filtering at transmitter, channel and receiver
(2) Electrical noise interference produce by galaxy, atmospheric noise, switching transients, intermodulation noise, interference noise from other sources.
Noise and interference can be reduce the intensity or eliminate. But the thermal motion of electrons cannot be eliminated.

• Demodulation
 recovery of waveform to an undistorted baseband pulse.
• Detection
 the decision making process of selecting a digital of the waveform.
• If error-correcting coding is not present
 the detector output consists of estimates of message symbols (or bits), m i
(hard decision).
• If error-correcting coding is present
 the detector output consists of estimates of channel symbols (coded bits) ui, which can take the form of hard or soft decisions.
• Frequency down-conversion block;
 performs frequency translation for bandpass signals operating at some radio frequency (RF)

Format m i
Pulse modulate g i
( t ) transmitted symbol
Bandpass modulate s i
( t ) channel h c
( t )
M-ary modulation i

1 , , M estimated symbol n ( t )

Format
ˆ i
Detect
Major sources of errors:
 z ( T )
Demod.
& sample r ( t )
Thermal noise (AWGN)

 disturbs the signal in an additive fashion (Additive) has flat spectral density for all frequencies of interest (White)

 is modeled by Gaussian random process (Gaussian Noise)
Inter-Symbol Interference (ISI)

Due to the filtering effect of transmitter, channel and receiver, symbols are “smeared”.

h c
)
 
T


Typical demodulation and detection function of a digital receiver r ( t
Step 1 – waveform to sample transformation Step 2 – decision making
Demodulate & Sample Detect
) Frequency down-conversion
Receiving filter
Equalizing filter z ( T )
Threshold comparison
ˆ i
For bandpass signals
Received waveform
Baseband pulse
(possibly distored)
Compensation for channel induced ISI
Baseband pulse
Sample
(test statistic)



Demodulation and sampling:

Waveform recovery and preparing the received signal for detection:

Improving the signal power to the noise power ( SNR ) using matched filter

Reducing ISI using equalizer

Sampling the recovered waveform
Detection:

Estimate the transmitted symbol based on the received sample


Bandpass model of detection process is equivalent to baseband model because:

The received bandpass waveform is first transformed to a baseband waveform.

Equivalence theorem:

Performing bandpass linear signal processing followed by heterodyning the signal to the baseband, yields the same results as heterodyning the bandpass signal to the baseband , followed by a baseband linear signal processing.


Find optimum solution for receiver design with the following goals:

Maximize SNR



Minimize ISI
Steps in design:


Model the received signal
Find separate solutions for each of the goals.
First, we focus on designing a receiver which maximizes the SNR.

Cont’d…
• Receiving Filter;



 or the modulator which perform waveform recovery in preparation of step-detection.
Filtering at transmitter and channel typically caused the received pulse sequence to suffer from ISI and not ready for sampling and detection.
The goal of the receiving filter is to recover a baseband pulse width the best possible SNR, free of ISI.
To accomplish this use matched filter or correlator.
• Frequency down conversion block
 perform frequency translation for bandpass signal operating at some radio frequency.
• Receiving filter
 to recover a baseband pulse with the best possible SNR and free of
ISI
• Optimal equalizing filter
 is only needed for those systems where channel induced ISI can distort the signals.






Find optimum solution for receiver design with the following goals:


Maximize SNR
Minimize ISI
Transmitter

~ the information symbols characterized as impulse or voltage levels, modulate pulses that are then filtered to comply with some bandwidth constraint.
Baseband system

~ the channel has distributed reactance that distorts the pulses .
Bandpass system

~characterized by fading channels that behave like undesirable filters manifesting signal distortion.
Equalizing filter

~ configured to compensate for the distortion caused by transmitter
& channel.

Figure 3.15 Page 136


Equalizing filter or receiving/equalizing filter ~ configure to compensate distortion caused by transmitter & channel.

Baseband system model
  k x
1 x
2
Tx filter h t
( t )

  k
T
H t
( f ) x
3
Equivalent model x
1 x
2
T
Equivalent system h ( t )
T
H ( f ) x
3
T
Channel h c
( t )
H c
( f ) n ( t ) r ( t ) Rx. filter h r
( t )
H r
( f ) t
 kT z k
Detector
  k z ( t ) n
ˆ
( t ) filtered noise t
 kT z k
Detector
  k

Lump filtering effect into ~



H
H
H t
(f) –transmitting filter c
(f) –filtering within the channel r
(f) –equalizing filter
( ) H t
) ( c f H r
)


Cont’d…ISI
ISI in the detection process due to the filtering effects of the system channel induced distortion .

Overall equivalent system transfer function
( ) t
( ) ( c f H r
)

 creates echoes and hence time dispersion causes ISI at sampling time z k
 k
 k
 i i


Nyquist bandwidth constraint:

The theoretical minimum required system bandwidth to detect
[symbols/s] without ISI is rectangular.
R s
/2 [Hz]. Happen when the H[f] is
R s
Ideal Nyquist filter Ideal Nyquist pulse
H ( f )
( )

1 t / T
T
1
2

1

2
0

1
W f
2 T 2 T

2 T

T
0
T 2 T t


W

1
2 T
Single side bandwidth 1/2T  impulse response h(t) =sinc(t/T)
Equivalently, a system with bandwidth W=1/2T=R support a maximum transmission rate of without ISI .
s
/2 [Hz] can
2W=1/T=R s
[symbols/s]

Cont’d…Nyquist Bandwidth constrain

Bandwidth efficiency, R/W [bits/s/Hz]:


An important measure in DCs representing data throughput per hertz of bandwidth.
Showing how efficiently the bandwidth resources are used by signaling techniques.
1
2
R
T s

W





Nyquist pulses (filters):

Pulses (filters) which results in no ISI at the sampling time.
Nyquist filter:

Its transfer function in frequency domain is obtained by convolving a rectangular function with any real evensymmetric frequency function
Nyquist pulse:

Its shape can be represented by a sinc(t/T) function multiply by another time function.
Example of Nyquist filters: Raised-Cosine filter


Goals and trade-off in pulse-shaping




Compress bandwidth of data impulse to small bandwidth greater than Nyquist minimum.
Reduce ISI; right sampling.
Efficient bandwidth utilization
Robustness to timing error (small side lobes)



System operate with small bandwidth



Pulse that spread in time will degrade the system’s error performance due to increase ISI.
Reduce the required system bandwidth.
Compress the bandwidth of the data impulse to some reasonably small bandwidth greater than the Nyquist minimum
Pulse shaping with Nyquist filter

Zero ISI is only when the sampling is performed at exactly the correct sampling time when the tails of pulses are large.

Due to effect of the received pulses can overlap one another

 tail of pulse can smear into adjacent symbol intervals, thereby interfering with the detection process and degrading the error performance ISI.
The effects of filtering & channel induced distortion leads to ISI


Raised-Cosine Filter

One frequently used H(f) transfer function belonging to the Nyquist class ( zero ISI at the sampling time) f
1






| f cos
4
0


|
0
0
| f W h 2
0
2
0
| f
 


0
]
0
2
]




W~ absolute bandwidth
W
0
=1/2T~ minimum Nyquist bandwidth for rectangular spectrum
W-W
0
~ excess bandwidth r =(W-W
0
)/W
0
~ Roll off factor where 0
 r

1

Cont’d…Raise Cosine Filter

Raise cosine characteristics

Roll off value; r . r=0 roll off is Nyquist minimum bandwidth case.

Example of Nyquist filters: Raised-Cosine filter
| ( f ) | | H ( f
RC
) | h ( t )
 h
RC
( t )
1 r

0 1
0.5
r

0 .
5 r

1
0.5
r

1 r

0 .
5 r

0


1

3

1 0
1 3 1
T 4 T 2 T
2 T 4 T T

R
Baseband
2

3 T

2 T

T 0 T
Relationship of bandwidth and symbol transmission rate
2 T 3 T
) s

Relationship of DBS bandwidth W
R s
DBS and symbol transmission rate

Cont’d…Raise Cosine Filter

Pulse shaping filter;





Larger filter roll-off, the shorter the pulse tail (small amplitude)  less sensitive to timing errors & small degradation due to ISI, small excess bandwidth.
Longer pulse tail  larger pulse amplitude & greater sensitivity to timing error.
Nyquist filter
 is one whose frequency transfer function can be represented by a rectangular function convolved with any real even-symmetric frequency function
Nyquist pulse
 is one whose shape can be represented by a sinc (t/T) function multiplied by another time function
Most popular of Nyquist filter
(1) Raised-cosine filter
(2) Root-raised cosine filter


Square-root Raised-Cosine (SRRC) pulse shaping
Amp.
[V]
Baseband tr. Waveform
Third pulse t/T
First pulse
Second pulse
Data symbol


Raised Cosine pulse at the output of matched filter
Amp.
[V]
Baseband received waveform at the matched filter output
(zero ISI) t/T




Eye pattern: Display on an oscilloscope which sweeps the system response to a baseband signal at the rate
1/T
(
T symbol duration)
Eye pattern of binary antipodal signalling.
Optimum sampling time corresponds to maximum eye opening yield greatest protection against noise.
Distortion due to ISI (D
A
)
Noise margin (M
N
)
Sensitivity to timing error (S
T
)

Timing jitter (J
T
)
Eye closed~ ISI is increasing Eye open~ ISI is decreasing.


Perfect channel ( no noise and no ISI )


AWGN ( Eb/N0=20 dB) and no ISI


AWGN ( Eb/N0=10 dB) and no ISI


Non-ideal channel and no noise h c
)






AWGN ( h






AWGN ( h c
)



)

Step 1 – waveform to sample transformation
Demodulate & Sample
Step 2 – decision making
Detect r ( t ) Frequency down-conversion
Receiving filter
Equalizing filter z ( T )
Threshold comparison
ˆ i
For bandpass signals
Received waveform
Baseband pulse
(possibly distored)
Compensation for channel induced ISI
Baseband pulse
Sample
(test statistic)



Equalization~ signal processing or filtering techniques that is designed to eliminate or reduce ISI.
Equalization categories;
(1) Maximum likelihood sequence estimation (MLSE)

Making measurement of h c
(t) & provide means of adjusting the receiver to the transmission environment.

Migittaing technichque of MLSE  adjusting itself so that better deal with the distorted sample; verterbi equalization .
(2) Equalization with filter


Use filter to compensate the distorted pulse.
Popular approach. Linear device (transversal equalizer) or nonlinear device (decision feedback equilizer)

( )
H )
Non-constant amplitude
Amplitude distortion
Non-linear phase
Phase distortion



Overal system transfer function
(
No ISI at the sampling time
 c r
H e f
Transmmit & received filter chosen to match so that; r
H (
 f
Taking care of ISI
 caused by tr. filter
SRRC
H e
( f )

1
H c
( f )
Taking care of ISI caused by channel

Square-Root Raised Cosine (SRRC) filter and Equalizer


Example of a frequency selective, slowly changing (slow fading) channel for a user at 35 km/h


Example of a frequency selective, fast changing (fast fading) channel for a user at 35 km/h


a
1  k

( t

Tx filter h t
( t )
Channel h c
( t ) r ( t ) Equalizer h e
( t )
T a
2 a
3
H t
( f ) H c
( f ) H e
( f ) n ( t )
Rx. filter h r
( t ) z ( t ) t

H r
( f ) kT z k
Detector
  k

a
1
 k

( t
 Equivalent system h ( t ) z ( t )
T a
2 a
3
H ( f )
( ) t
( ) ( c f H r
) n
ˆ
( t ) x ( t )
Equalizer h e
( t )
H e
( f ) z ( t ) t
 kT z k
Detector
ˆ filtered noise n t ) n t ) h r t )
  k


Equalization using

MLSE (Maximum likelihood sequence estimation)

Filtering

Transversal filtering

Zero-forcing equalizer

Minimum mean square error (MSE) equalizer

Decision feedback

Using the past decisions to remove the ISI contributed by them

Adaptive equalizer


Transversal filter:

A weighted tap delayed line that reduces the effect of ISI by proper adjustment of the filter taps .
x ( t ) c

N
N

 k N
    c

N

1 c
N

1 c
N
 z ( t )
Coeff. adjustment


Zero-forcing equalizer:
 The filter taps are adjusted such that the equalizer output is forced to be zero at N sample points on each side:

Adjust
  n
N n
 
N
( )
1
0
Mean Square Error (MSE) equalizer:

0
The filter taps are adjusted such that the MSE of ISI and noise power at the equalizer output is minimized.
Adjust
  n
N n
 
N

( ( )
 k
2
)




2-PAM with SRRQ
Non-ideal channel
Matched filter outputs at the sampling time h
 c
)
 t

 t

One-tap DFE
ISI-no noise,
No equalizer
ISI-no noise,
DFE equalizer
ISI- noise
No equalizer
ISI- noise
DFE equalizer

Example 3.1: Equalization. Pg 331
The tab weight of an equalizing transversal filter are to be determined by transmitting a single pulse as a training signal. Let the equalizer be a three tap one . Given a received distorted set of pulse sample {x(k)}, with voltage values {0.0, 0.1, 1.0, -0.2, 0.1} Use a zero-forcing solution to find the tab weights that reduce the ISI so that the equalized pulse samples
{z(k)} have no ISI. Using these weights, calculate the ISI values of the equalizing pulse at the sample times k=+ 2, + 3. What is the largest magnitude sample contributing to ISI and what is the sum of all the ISI magnitudes?
Solution: