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Chapter 9: Signaling Over Bandlimited Channels
EE456 – Digital CommunicationsProfessor Ha Nguyen
September 2015
EE456 – Digital Communications 1
Chapter 9: Signaling Over Bandlimited Channels
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).
EE456 – Digital Communications 2
Chapter 9: Signaling Over Bandlimited Channels
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
!
EE456 – Digital Communications 3
Chapter 9: Signaling Over Bandlimited Channels
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
EE456 – Digital Communications 4
Chapter 9: Signaling Over Bandlimited Channels
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!
EE456 – Digital Communications 5
Chapter 9: Signaling Over Bandlimited Channels
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?
EE456 – Digital Communications 6
Chapter 9: Signaling Over Bandlimited Channels
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
EE456 – Digital Communications 7
Chapter 9: Signaling Over Bandlimited Channels
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.
EE456 – Digital Communications 8
Chapter 9: Signaling Over Bandlimited Channels
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.
EE456 – Digital Communications 9
Chapter 9: Signaling Over Bandlimited Channels
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
EE456 – Digital Communications 10
Chapter 9: Signaling Over Bandlimited Channels
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
EE456 – Digital Communications 11
Chapter 9: Signaling Over Bandlimited Channels
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.
EE456 – Digital Communications 12
Chapter 9: Signaling Over Bandlimited Channels
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.
EE456 – Digital Communications 13
Chapter 9: Signaling Over Bandlimited Channels
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.
EE456 – Digital Communications 14
Chapter 9: Signaling Over Bandlimited Channels
bkTt =] ]^_`ab
( ) ( )bk
t V t kTδ∞
=−∞
= ± −cx( )ty( )tr ( )bkTy
( )
Gaussian noise, zero-mean
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].
EE456 – Digital Communications 15
Chapter 9: Signaling Over Bandlimited Channels
bkTt =d defghi
( ) ( )bk
t V t kTδ∞
=−∞
= ± −jx( )ty( )tr ( )bkTy
( )
Gaussian noise, zero-mean
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)|.
EE456 – Digital Communications 16
Chapter 9: Signaling Over Bandlimited Channels
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
.
EE456 – Digital Communications 17
Chapter 9: Signaling Over Bandlimited Channels
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
EE456 – Digital Communications 18
Chapter 9: Signaling Over Bandlimited Channels
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!
EE456 – Digital Communications 19
Chapter 9: Signaling Over Bandlimited Channels
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−
EE456 – Digital Communications 20
Chapter 9: Signaling Over Bandlimited Channels
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−
⋯
EE456 – Digital Communications 21
Chapter 9: Signaling Over Bandlimited Channels
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= − + −∑
EE456 – Digital Communications 22
Chapter 9: Signaling Over Bandlimited Channels
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−
EE456 – Digital Communications 23
Chapter 9: Signaling Over Bandlimited Channels
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 ≤ ∞
EE456 – Digital Communications 24
Chapter 9: Signaling Over Bandlimited Channels
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.
EE456 – Digital Communications 25
Chapter 9: Signaling Over Bandlimited Channels
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.
EE456 – Digital Communications 26
Chapter 9: Signaling Over Bandlimited Channels
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.
EE456 – Digital Communications 27
Chapter 9: Signaling Over Bandlimited Channels
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.
EE456 – Digital Communications 28
Chapter 9: Signaling Over Bandlimited Channels
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.
EE456 – Digital Communications 29
Chapter 9: Signaling Over Bandlimited Channels
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!
EE456 – Digital Communications 30
Chapter 9: Signaling Over Bandlimited Channels
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).
EE456 – Digital Communications 31
Chapter 9: Signaling Over Bandlimited Channels
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
EE456 – Digital Communications 32
Chapter 9: Signaling Over Bandlimited Channels
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 33