EE456 – Digital Communications · Chapter 9: Signaling Over Bandlimited Channels Introduction to...

Chapter 9: Signaling Over Bandlimited Channels

EE456 – Digital CommunicationsProfessor Ha Nguyen

September 2015

EE456 – Digital Communications 1


Introduction to Signaling Over Bandlimited Channels

We have considered signal design and detection of signals transmitted overchannels of “infinite” bandwidth, or at least sufficient bandwidth to pass most ofthe signal power (say 99% or 99.9%).

Strictly-bandlimited channels are common: twisted-pair wires, coax cables, etc.

Band-limitation depends not only on the channel media but also on the sourcerate (Rs symbols/sec). If one keeps increasing the source rate, any channelbecomes bandlimited.

Band-limitation can also be imposed on a communication system by regulatoryrequirements. In all commercial communications standards, specific spectrummasks are placed on the transmitted signals.

The general effect of band limitation on a transmitted signal of finite timeduration is to disperse (or spread) it out ⇒ Signal transmitted in a particulartime slot interferes with signals in other time slots ⇒ causing inter-symbol

interference (ISI).

In this chapter, we shall consider signal design and demodulation of signals whichare not only corrupted by AWGN, but also by ISI.

Major Approaches to Deal with ISI:

1 Force the ISI effect to zero ⇒ Nyquist’s first criterion.2 Allow some ISI but in a controlled manner ⇒ Partial response signaling.3 Live with the presence of ISI and design the best demodulation for the situation ⇒

Maximum likelihood sequence estimation (Viterbi algorithm).



Baseband Antipodal Communication System Model

��

��

��

��

)( fHC)( fHT) Rate( br

( )

(WGN)

tw

��

)( fH R

bkTt =

��

��

Equivalent system model:

�)( fHC)( fHT )( fHR

bkTt =�

bT0bT2

1 1 0

��

� ��

( )tw

( )tr ( )ty

!



ISI Example: Modulation is NRZ-L and Channel is a Simple LPF

C

R

Lowpass Filter

" # 1( ) expC

th t

RC RC$ %= −& '( )

)(tsAV

bT0t

)(tsB

bT0t

*+, *-,

/1(1 e )bT RCV

RC−−

)(tsA

V

bT0t

bT2bT3 bT4 bT5

)( bA Tts −

)2( bA Tts −−

)3( bA Tts −

)4( bA Tts −−

./0

)(tsB

bT0t

bT2 bT3bT4

bT5

)( bB Tts −

)2( bB Tts −−

)3( bB Tts −

)4( bB Tts −−

123

0 1 2 3During this interval, ( ) ( ) ( ) ( 2 ) ( 3 ) ( )B B b B b B bt s t s t T s t T s t T t= + − + − + − +r b b b b w



Baseband Message Signals with Different Pulse Shaping Filters

2 4 6 8 10 12 14 16 18 20−1

0

1

n

Information bits or amplitude levels

2 4 6 8 10 12 14 16 18 20−2

0

2

t/Tb

Output of the transmit pulse shaping filter − Rectangular

2 4 6 8 10 12 14 16 18 20−2

0

2

t/Tb

Output of the transmit pulse shaping filter − Half−sine

2 4 6 8 10 12 14 16 18 20−2

0

2

t/Tb

Output of the transmit pulse shaping filter − SRRC (β =0.5)

Spectrum of the transmitted signal is much more compact in frequency if theimpulse response of the pulse shaping filter is made longer in time.The signal corresponding to one bit occupies a duration that is longer than onebit interval, hence causing interference to signals of adjacent bits (inter-symbolinterference, or ISI).It is theoretically possible to design the pulse shaping filter so that the effect ofISI at the sampling moments is zero!



Nyquist Criterion for Zero ISI

4)( fHC)( fHT )( fHR

bkTt =5

bT0bT2

1 1 0

678 678

6978

( )tw

( )tr ( )ty

:

y(t) =∞∑

k=−∞

bksR(t − kTb) +wo(t),

where sR(t) = hT (t) ∗ hC(t) ∗ hR(t) is the overall response of the system due to aunit impulse at the input.

bk =

{V if the kth bit is “1”

−V if the kth bit is “0”.

Normalize sR(0) = 1 and look at sampling time t = mTb:

y(mTb) = bm +∞∑

k=−∞k 6=m

bksR(mTb − kTb)

︸︷︷︸

ISI term

+wo(mTb).

What are the conditions on the overall impulse response sR(t), or the overalltransfer function SR(f) = HT (f)HC(f)HR(f) which would make ISI term zero?



Time-Domain Nyquist’s Criterion for Zero ISI

The samples of sR(t) should equal to 1 at t = 0 and zero at all other sampling timeskTb (k 6= 0).

0)()()( fHfHfH RCT ××t )(tsR

)()( tsfS RR ↔

0at

applied pulseIm

=t;<=

;>=

0t

)(tsR

1

bT2− bT2bTbT− bT3 bT4

)( of Samples tsR

?@A



Frequency-Domain Nyquist’s Criterion for Zero ISI

It can be proved that (see textbook) the equivalent Nyquist’s criterion in thefrequency domain is that the sum of sR(f) and all of its delayed copies, delayed byinteger multiples of bit rate, is a constant!

f

B B

0

bT21

bT

1

bT

2

bT23

bT21−

bT

1−

)( fSR CCD

EFFG

H−

bR T

fS1

CCD

EFFG

H−

bR T

fS2

IIJ

KLLM

N+

bR T

fS1

f0

bT2

1

bT2

1−

bTR

k b

kS f

T

∞

=−∞

O P+Q R

S TU

If W < 12Tb

, or rb > 2W ⇒ ISI terms cannot be made zero.

If W = 12Tb

, or rb = 2W ⇒ SR(f) = Tb over f ≤ | 12Tb

|, sR(t) =sin(πt/Tb)(πt/Tb)

.

If W > 12Tb

, or rb < 2W ⇒ Infinite number of SR(f) to achieve zero ISI.



Pulse Shaping when W =1

2Tb, or rb = 2W

f0

bT21

bT21−

bT

)( fSR

−3 −2 −1 0 1 2 3−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t/Tb

s R(t

)

A good approximation of a brick-wall filteris not simple (e.g., requires a very longfilter if implemented digitally).

sR(t) decays as 1/t ⇒ if the sampler isnot perfectly synchronized in time,considerable ISI can be encountered.



Raised Cosine (RC) Pulse Shaping (Industry-Standard)

SR(f) = SRC(f) =

Tb, |f | ≤ 1−β2Tb

Tb cos2[πTb2β

(

|f | − 1−β2Tb

)]

, 1−β2Tb

≤ |f | ≤ 1+β2Tb

0, |f | ≥ 1+β2Tb

.

sR(t) = sRC(t) =sin(πt/Tb)

(πt/Tb)

cos(πβt/Tb)

1− 4β2t2/T 2b

= sinc(t/Tb)cos(πβt/Tb)

1− 4β2t2/T 2b

.

−1 −0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

fTb

S R(f

)

β=0

β=0.5

β=1.0

−3 −2 −1 0 1 2 3−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t/Tb

s R(t

)

β=0

β=0.5

β=1



0 1 2 3 4 5 6 7 8 9 10 11−2

−1

0

1

2

t/Tb

y(t)

/V

With the rectangular spectrum

0 1 2 3 4 5 6 7 8 9 10 11−2

−1

0

1

2

t/Tb

y(t)

/V

With a raised cosine spectrum



Eye Diagrams

Eye diagrams are used to observe and measure (qualitatively) the effect of ISI.

An eye diagram is obtained by overlapping and displaying multiple segments ofthe received signal (after the matched filter) over the duration of a few symbolperiods.

The wider the eye is opened, the better the quality of the samples undermismatched timing and AWGN.

0 1.0 2.0−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

t/Tb

y(t)

/V

0 1.0 2.0−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

t/Tb

y(t)

/V

Left: Ideal lowpass filter; Right: A raised-cosine filter with β = 0.35.



Eye Diagrams under AWGN with SNR= V 2/σ2w = 20 dB

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

t/Tb

y(t)/V

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

t/Tb

y(t)/V

Left: Ideal lowpass filter; Right: A raised-cosine filter with β = 0.35.



Design of Transmitting and Receiving Filters

Have shown how to design SR(f) = HT (f)HC (f)HR(f) to achieve zero ISI.

When HC(f) is fixed, one still has flexibility in the design of HT (f) and HR(f).

Shall design the filters to minimize the probability of error.

bkTt =V VWXYZ[

( ) ( )bk

t V t kTδ∞

=−∞

= ± −\x( )ty( )tr ( )bkTy

( )

Gaussian noise, zero-mean

PSD ( ) watts/Hz

t

S fw

w

)( fHC)( fHT )( fHR

Noise is assumed to be Gaussian (as usual) but does not necessarily have to be

white.



bkTt =] ]^_`ab

( ) ( )bk

t V t kTδ∞

=−∞

= ± −cx( )ty( )tr ( )bkTy

( )


PSD ( ) watts/Hz

t

S fw

w

)( fHC)( fHT )( fHR

Since zero-ISI is achieved, one has

y(mTb) = ±V +wo(mTb),

where wo(mTb) ∼ N (0, σ2w), with σ2

w=

∫∞

−∞Sw(f)|HR(f)|2df .

V− V0

2σ w

( ( ) | )bf y mT V( ( ) | )bf y mT V−

( )by mT

Because P [error] = Q(

Vσw

)

⇒ Need to maximize V 2

σ2w

to minimize P [error].



bkTt =d defghi

( ) ( )bk

t V t kTδ∞

=−∞

= ± −jx( )ty( )tr ( )bkTy

( )


PSD ( ) watts/Hz

t

S fw

w

)( fHC)( fHT )( fHR

1 Compute the average transmitted power: PT = V 2

Tb

∫∞−∞

|HT (f)|2df (watts).

2 Write the inverse of the SNR as

σ2w

V 2=

1

PTTb

[∫ ∞

−∞

|HT (f)|2df] [∫ ∞

−∞

Sw(f)|HR(f)|2df]

=1

PTTb

[∫ ∞

−∞

|SR(f)|2

|HC(f)|2|HR(f)|2df

][∫ ∞

−∞

Sw(f)|HR(f)|2df].

3 Apply the Cauchy-Schwartz inequality:∣∣∣∣

∫∞

−∞

A(f)B∗(f)df

∣∣∣∣2

≤[∫

∞

−∞

|A(f)|2df] [∫

∞

−∞

|B(f)|2df],

which holds with equality if and only if A(f) = KB(f). Identify

|A(f)| =√

Sw(f)|HR(f)|, |B(f)| = |SR(f)|

|HC (f)||HR(f)| . Then

|HR(f)|2 =K|SR(f)|

√Sw(f)|HC(f)|

, |HT (f)|2 =|SR(f)|

√Sw(f)

K|HC(f)|.



Design Under White Gaussian Noise

For white noise (at least flat PSD over the channel bandwidth):

|HR(f)|2 = K1|SR(f)|

|HC(f)|,

|HT (f)|2 = K2|SR(f)|

|HC(f)|=

K2

K1|HR(f)|2,

The above results show that channel equalization (division by |HC(f)|) isoptimally done at both the transmitter and receiver.

Constants K1 and K2 set signal levels at the transmitter and receiver, but themost important design requirement is that the transmit and receive filters are amatched filter pair:

HR(f) = |HR(f)|ej∠HR(f),

HT (f) = K|HR(f)|ej∠−HR(f),

The maximum output SNR is

(V 2

σ2w

)

max

= PTTb

[∫∞

−∞

|SR(f)|√

Sw(f)

|HC(f)|df

]−2

.



Design Under White Gaussian Noise and Ideal Channel

If HC(f) = 1 for |f | ≤ W (ideal channel) and K1 = K2 = 1 then

|HT (f)| = |HR(f)| =√

|SR(f)|

Thus if SR(f) is a raised-cosine (RC) spectrum then both HT (f) and HR(f) aresquare-root raised-cosine (SRRC) spectrum:

HT (f) = HR(f) =

√Tb, |f | ≤ 1−β

2Tb√Tb cos

[πTb2β

(|f | − 1−β

2Tb

)], 1−β

2Tb≤ |f | ≤ 1+β

2Tb

0, |f | ≥ 1+β2Tb

.

hT (t) = hR(t) = sSRRC(t) =(4βt/Tb) cos[π(1 + β)t/Tb] + sin[π(1 − β)t/Tb]

(πt/Tb)[1 − (4βt/Tb)2].

−3 −2 −1 0 1 2 3−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t/Tb

s R(t

)

Raised cosine (RC)Square−root RC



Outputs of the Pulse Shaping Filter and Receive Matched Filter

2 4 6 8 10 12 14 16 18 20−1

0

1

n

Information bits or amplitude levels

2 4 6 8 10 12 14 16 18 20−2

0

2

t/Tb

Output of the transmit pulse shaping filter − SRRC (β =0.5)

2 4 6 8 10 12 14 16 18 20−2

0

2

t/Tb

Output of the receive matched filter − SRRC (β =0.5)

Note that the samples of the output of the pulse shaping filter at t = kTb are notISI-free, but the samples of the output of the receive matched filter are!



Shaping Filters in Bandlimited Passband QAM Systems

De-

multiplexer

Inphase

ASK

decision

,Select I iV

,Select Q iV

)2cos(2

tfT c

s

π

)2sin(2

tfT c

s

π

bits Infor. )(tsi

Multiplexer

Decision

2( ) sin(2 )Q c

s

t f tT

φ π=

2( ) cos(2 )I c

s

t f tT

φ π=)()()( ttst i wr +=

st kT=

st kT=Quadrature

ASK

decision

QAM signal

Transmit

filter

Transmit

filter

Matched

filter

Matched

filter

( )p t

( )p t

( )sp T t−

( )sp T t−



Nyquist’s Criterion for Zero ISI in Passband Channels

st kT=

Matched

filter

( )sp T t−

Inphase

amplitude

)2cos(2

tfT c

s

π

Transmit

filter

( )p t

)2cos(2

tfT c

s

π

f

⋯

0 2

sT

1

sT− 1

2 sT

1R

s

S fT

−

2R

s

S fT

−

( )RS f

1

sT

Passband bandwidth W1

Rs

S fT

+

sT

β

2W

2W− 2

Wcf +2

Wcf − 2

W2

W−0

( )P f

cf

Bandwidth W( )RS f

0

2W

2W−

⋯



Block Diagram of an All-Digital QAM System

1L↑[ ]

IV n

[ ]Q

V n

( )1IFcos 2 sT

Lf nπ

1L↑

( )1

sT

T Lh n

( )1IFsin 2 sT

Lf nπ

( )1

sT

T Lh n

2L↑c

ff

IF[ ]s n

RF( )s t

RFff 2

L↓

( )1IFcos 2 sT

Lf nπ

( )1IFsin 2 sT

Lf nπ

( )1

sT

R Lh n

( )1

sT

R Lh n

1L↓

1L↓

RF( )s t

IF[ ]s n

[ ]IV n

[ ]QV n

( )RF( ) [ ] ( )cos(2 ) [ ] ( )sin(2 )I T s c Q T s c

n

s t V n h t nT f t V n h t nT f tπ π= − + −∑



Sampling a RC (or SRRC) Impulse Response to Obtain a Digital Filter

rc( )H f

0

(Hz)f

1

2 sT

1

2s

T

β+

1

12

transition band

is a raised cosine

rc( )h t

1

2s

T

β−

sT

β

0t

sT

sam

1

4sT T=

rc(e )

jH

ω

0

(rad/sample)ω2π

1

12

π4

π4

π−π−2π−

1

2s

T

β+− 1

2s

T−



Ideal SRRC and RC Responses (Digital)

The frequency response of the ideal Square-Root Raised Cosine (SRRC) filter is:

Hsrrc(ejω) =

1, for |ω| ≤ π(1−β)L√

12

[1 + cos

(π2β

(|ω|Lπ

− 1 + β))]

, for π(1−β)L

< |ω| ≤ π(1+β)L

0, otherwise.

The time-domain SRRC impulse response is

hsrrc[n] =

4βnL

cos(

π(1+β)nL

)+ sin

(π(1−β)n

L

)

πnL

[1 −

(4βnL

)2] , for − ∞ ≤ n ≤ ∞

The frequency response of the ideal overall Raised-Cosine (RC) filter is:

Hrc(ejω

) =

1, for |ω| ≤ π(1−β)L

12

[1 + cos

(π2β

(|ω|Lπ

− 1 + β))]

, for π(1−β)L

< |ω| ≤ π(1+β)L

0, otherwise.

The time-domain ideal RC impulse response is

hrc[n] = hsrrc[n] ∗ hsrrc[n] =sin(πnL

)

πnL

cos(

πβnL

)

1 −(

2βnL

)2, for − ∞ ≤ n ≤ ∞



Implementation with Truncated SRRC Filters

In practice, the implementation of the pulse shaping and matched filter, hT [n]and hR[n], as FIR filters requires truncation of the ideal (infinite-long) SRRCimpulse response.

Any truncation would make the filters non-ideal, hence not exactly satisfying theNyquist criterion, hence causing some amount of ISI.

Truncation also makes the stopband attenuation of the filter not exactly zero. Itis always desired to reduce the stopband attenuation.

The longer the truncated filters (in terms of symbol period Ts), the better theapproximation, hence the less ISI. However, longer filters mean highercomplexity/cost.

EE465 in Term 2 will examine efficient designs of pulse shaping and matchedfilters for a 16-QAM system that meets certain requirements on ISI andout-of-band power.

The next slides show the effects of truncating the ideal SRRC filter.



Truncating the Ideal SRRC Filter

0 5 10 15 20 25 30 35−0.1

0

0.1

0.2

0.3

n

SRRC filter with length 17 (or 4 symbols) and β = 0.2

0 5 10 15 20 25 30 35−0.1

0

0.1

0.2

0.3

n

RC filter by convolving two SRRC filters, β = 0.2

Observe that, due to truncation, the sample values (marked in red) of the RC filterat multiples of symbol period (every 4th sample in this case) are not zero.



Truncating the Ideal SRRC Filter

0 10 20 30 40 50 60−0.1

0

0.1

0.2

0.3

n

SRRC filter with length 33 (or 8 symbols) and β = 0.2

0 10 20 30 40 50 60−0.1

0

0.1

0.2

0.3

n

RC filter by convolving two SRRC filters, β = 0.2

The sample values (marked in red) of the RC filter at multiples of symbol period(every 4th sample in this case) are not exactly 0, but smaller for a longer filter.



Spectra of the Truncated SRRC Filters

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−100

−80

−60

−40

−20

0

20Magnitude Frequency Responses of SRRC filters with β = 0.2

Normalized frequency, ω/2π (cycle/sample)

Gain

(dB)

Length 17 (4 symbols)Length 33 (8 symbols)

Observe that, due to truncation, the stop-band attenuation is not zero. The shorterthe truncation length, the higher the stop-band attenuation.



IQ Plots and Modulation Error Ratio (MER)

−4 −2 0 2 4−4

−2

0

2

4Filter length = 17, MER = 17.17 (dB)

Inphase

Quadrature

−4 −2 0 2 4−4

−2

0

2

4Filter length = 33, MER = 31.31 (dB)

InphaseQuadrature

MER = limN→∞

10 log10

∑N

n=−N(V 2

I [n] + V 2Q[n])

∑Nn=−N

[(VI [n] − V̂I [n])2 + (VQ[n] − V̂Q[n])2

]

,

where VI [n] and VQ[n] are the ideal values and V̂I [n] and V̂Q[n] are the actual values of the

decision variables.



Comparison of harris-Moerder and SRRC Filters: Impulse Responses and ISI

0 10 20 30 40 50 60 70 80 90 100−0.1

0

0.1

0.2

0.3

n

SRRC and hM3 filters, both with length 97 (24 symbols) and β = 0.2

SRRC filterhM3 filter

0 5 10 15 20 25 30 35 40 45 50−3

−2

−1

0

1x 10

−3 ISI after downsampling and removing the peak value of 1

SRRC: MER=47.65dBhM3: MER=60.20

SRRC filter is an industry-standard filter but it is not necessary the best filter. The above figure

and the next two figures show that, for the same length, a filter designed by harris & Moerder

(hM3 filter) performs better than the SRRC filter in terms of both ISI power (or MER) and

stopband attenuation!



Comparison of harris-Moerder and SRRC Filters: IQ Plots and MERs

−4 −2 0 2 4−4

−2

0

2

4SRRC filter, MER = 47.65 (dB)

Inphase

Quadrature

−4 −2 0 2 4−4

−2

0

2

4hM3 filter, MER = 60.20 (dB)

InphaseQuadrature

Note that both SRRC and hM3 filters have the same length of 97 samples, or 24symbols (for an upsampling factor of L1 = 4).



Comparison of harris-Moerder and SRRC Filters: Spectra

0 10 20 30 40 50 60 70 80 90 100−0.1

0

0.1

0.2

0.3

n

SRRC and hM3 filters, both with length 97 (24 symbols) and β = 0.2

SRRC filterhM3 filter

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−150

−100

−50

0

50

Normalized frequency, ω/2π (cycle/sample)

Gain

(dB)

Magnitude frequency responses of SRRC and hM filters

SRRC filterhM filter



Modulation Error Ratio (MER) and Error Vector Magnitude (EVM)

The modulation error ratio (MER) is basically a more general definition of the signal-to-noise ratio (SNR). Here the “noise” termrefers to any unwanted disturbance to the desired signal. For example, this disturbance could be due to thermal noise, ISI, or anyimperfections in the implementation of the transmitter and receiver (such as I/Q imbalance, quadrature error, and distortion).

In the signal space (i.e., the IQ plot for QAM), such disturbance causes the actual constellation points to deviate from the ideallocations.

The MER can be computed from the I/Q decision variables as

MER (dB) = 10 log10

(Psignal

Perror

)= lim

N→∞10 log10

∑Nn=−N

(V 2I

[n] + V 2Q

[n])

∑Nn=−N

[(VI [n] − V̂I [n]

)2+(VQ[n] − V̂Q[n]

)2]

,

where VI [n] and VQ [n] are the ideal values and V̂I [n] and V̂Q [n] are the actual values of the decision variables.

Another common measure is the error vector magnitude (EVM). For a single-carrier QAM, the EVM is conventionally defined asthe ratio between the average error power and the peak signal power (expressed as %):

EVM (%) =

√√√√Perror

Psignal, max

=

√

limN→∞∑N

n=−N

[(VI [n] − V̂I [n]

)2+(VQ [n] − V̂Q[n]

)2]

√V 2I,max

+ V 2Q,max

.

Because the relationship between the peak and average signal powers depends on the constellation geometry, subject to the sameaverage interference power, different constellation types (e.g. 16-QAM and 64-QAM) will report different EVM values.

The MER and EVM are closely related. For example, it can be shown that the relationship between MER and EVM for 64-QAM isas follows:

MER = −

[3.7 + 20 log10

(EVM

100%

)].

The constant 3.7 represents the peak-to-average power ratio (in dB) of the 64-QAM signal. This number can be easily determinedfrom the constellation diagram by computing the average power for all 64 points and dividing by the power of a corner point.


EE456 – Digital Communications · Chapter 9: Signaling Over Bandlimited Channels Introduction to...

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