EE456 Digital Communications Professor Ha Nguyen September 2015 - - PowerPoint PPT Presentation

SLIDE 1

Chapter 9: Signaling Over Bandlimited Channels

EE456 – Digital Communications

Professor Ha Nguyen September 2015

EE456 – Digital Communications 1

SLIDE 2

Chapter 9: Signaling Over Bandlimited Channels

Introduction to Signaling Over Bandlimited Channels

We have considered signal design and detection of signals transmitted over channels of “infinite” bandwidth, or at least sufficient bandwidth to pass most of the 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 source rate (Rs symbols/sec). If one keeps increasing the source rate, any channel becomes bandlimited. Band-limitation can also be imposed on a communication system by regulatory

• requirements. In all commercial communications standards, specific spectrum

masks are placed on the transmitted signals. The general effect of band limitation on a transmitted signal of finite time duration is to disperse (or spread) it out ⇒ Signal transmitted in a particular time 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 which are 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

SLIDE 3

Chapter 9: Signaling Over Bandlimited Channels

Baseband Antipodal Communication System Model

✁ ✂

• ✆ ✄

☎✝ ✞ ✟ ☛ ✠ ✂ ✡ ☞ ✄ ✌ ✍ ✑ ✄ ✌ ☎ ✍

• ✂

✄ ✝ ✒ ✝ ✠ ✎ ✏ ✡ ✂ ✕ ✡ ✍ ✂ ✟ ✝ ✖ ✠ ✌✍ ✡ ✏ ✡ ✂ ✟ ✌ ✗ ✓

• ✠

✠ ✟ ✍

) ( f HC ) ( f HT ) Rate (

b

r ( ) (WGN) t w

✙ ✟ ✞ ✟ ✡ ✔ ✟ ✕ ✡ ✍ ✂ ✟ ✝

) ( f H R

b

kT t =

✖ ✡ ✠

• ✝

✘ ✁ ✂

• ✁

✂

• ✆

✡ ✠ ✚ ✁ ✟ ✞ ✡ ✎ ✡ ✄ ✠ ✁ ✟ ✔ ✡ ✞ ✟

Equivalent system model:

✛

) ( f HC ) ( f HT ) ( f H R

b

kT t =

✜

b

T

b

T 2 1 1

✢ ✣ ✤ ✢ ✣ ✤ ✢ ✥ ✣ ✤

( ) t w ( ) t r ( ) t y

✦

EE456 – Digital Communications 3

SLIDE 4

Chapter 9: Signaling Over Bandlimited Channels

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

C R Lowpass Filter

✧ ★

1 ( ) exp

C

t h t RC RC

✩ ✪

= −

✫ ✬ ✭ ✮

) (t sA V

b

T t ) (t sB

b

T t

✯ ✰ ✱ ✯ ✲ ✱

/

1 (1 e )

b

T RC

V RC

−

− ) (t sA V

b

T t

b

T 2

b

T 3

b

T 4

b

T 5 ) (

b A

T t s − ) 2 (

b A

T t s − − ) 3 (

b A

T t s − ) 4 (

b A

T t s − −

✳ ✴ ✵

) (t sB

b

T t

b

T 2

b

T 3

b

T 4

b

T 5 ) (

b B

T t s − ) 2 (

b B

T t s − − ) 3 (

b B

T t s − ) 4 (

b B

T t s − −

✶ ✷✸

1 2 3

During this interval, ( ) ( ) ( ) ( 2 ) ( 3 ) ( )

B B b B b B b

t s t s t T s t T s t T t = + − + − + − + r b b b b w

EE456 – Digital Communications 4

SLIDE 5

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 1 n Information bits or amplitude levels 2 4 6 8 10 12 14 16 18 20 −2 2 t/Tb Output of the transmit pulse shaping filter − Rectangular 2 4 6 8 10 12 14 16 18 20 −2 2 t/Tb Output of the transmit pulse shaping filter − Half−sine 2 4 6 8 10 12 14 16 18 20 −2 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 the impulse response of the pulse shaping filter is made longer in time. The signal corresponding to one bit occupies a duration that is longer than one bit interval, hence causing interference to signals of adjacent bits (inter-symbol interference, or ISI). It is theoretically possible to design the pulse shaping filter so that the effect of ISI at the sampling moments is zero!

EE456 – Digital Communications 5

SLIDE 6

Chapter 9: Signaling Over Bandlimited Channels

Nyquist Criterion for Zero ISI

✹

) ( f HC ) ( f HT ) ( f H R

b

kT t =

✺

b

T

b

T 2 1 1

✻ ✼ ✽ ✻ ✼ ✽ ✻ ✾ ✼ ✽

( ) t w ( ) t r ( ) t y

✿

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 a unit 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=m

bksR(mTb − kTb)

• ISI term

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

EE456 – Digital Communications 6

SLIDE 7

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 times kTb (k = 0). ) ( ) ( ) ( f H f H f H

R C T

× × t ) (t sR ) ( ) ( t s f S

R R

↔ at applied pulse Im = t

❀ ❁❂ ❀ ❃ ❂

t ) (t sR 1

b

T 2 −

b

T 2

b

T

b

T −

b

T 3

b

T 4 ) (

• f

Samples t sR

❄ ❅ ❆

EE456 – Digital Communications 7

SLIDE 8

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 the frequency domain is that the sum of sR(f) and all of its delayed copies, delayed by integer multiples of bit rate, is a constant!

f

❇ ❇

b

T 2 1

b

T 1

b

T 2

b

T 2 3

b

T 2 1 −

b

T 1 − ) ( f SR

❈ ❈ ❉ ❊ ❋ ❋

• ❍

−

b R

T f S 1

❈ ❈ ❉ ❊ ❋ ❋

• ❍

−

b R

T f S 2

■ ■ ❏ ❑ ▲ ▲ ▼ ◆

+

b R

T f S 1 f

b

T 2 1

b

T 2 1 −

b

T

R k b

k S f T

∞ =−∞

❖ P

+

◗ ❘ ❙ ❚ ❯

If W <

1 2Tb , or rb > 2W ⇒ ISI terms cannot be made zero.

If W =

1 2Tb , or rb = 2W ⇒ SR(f) = Tb over f ≤ | 1 2Tb |, sR(t) = sin(πt/Tb) (πt/Tb) .

If W >

1 2Tb , or rb < 2W ⇒ Infinite number of SR(f) to achieve zero ISI.

EE456 – Digital Communications 8

SLIDE 9

Chapter 9: Signaling Over Bandlimited Channels

Pulse Shaping when W =

1 2Tb , or rb = 2W f

b

T 2 1

b

T 2 1 −

b

T ) ( f SR

−3 −2 −1 1 2 3 −0.4 −0.2 0.2 0.4 0.6 0.8 1 t/Tb sR(t)

A good approximation of a brick-wall filter is not simple (e.g., requires a very long filter if implemented digitally). sR(t) decays as 1/t ⇒ if the sampler is not perfectly synchronized in time, considerable ISI can be encountered.

EE456 – Digital Communications 9

SLIDE 10

Chapter 9: Signaling Over Bandlimited Channels

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

SR(f) = SRC(f) =        Tb, |f| ≤ 1−β

2Tb

Tb cos2

πTb 2β

• |f| − 1−β

2Tb

• ,

1−β 2Tb ≤ |f| ≤ 1+β 2Tb

0, |f| ≥ 1+β

2Tb

. sR(t) = sRC(t) = sin(πt/Tb) (πt/Tb) cos(πβt/Tb) 1 − 4β2t2/T 2

b

= sinc(t/Tb) cos(πβt/Tb) 1 − 4β2t2/T 2

b

.

−1 −0.5 0.5 1 0.2 0.4 0.6 0.8 1 fTb SR(f) β=0 β=0.5 β=1.0 −3 −2 −1 1 2 3 −0.4 −0.2 0.2 0.4 0.6 0.8 1 t/Tb sR(t) β=0 β=0.5 β=1 EE456 – Digital Communications 10

SLIDE 11

Chapter 9: Signaling Over Bandlimited Channels

1 2 3 4 5 6 7 8 9 10 11 −2 −1 1 2 t/Tb y(t)/V With the rectangular spectrum 1 2 3 4 5 6 7 8 9 10 11 −2 −1 1 2 t/Tb y(t)/V With a raised cosine spectrum

EE456 – Digital Communications 11

SLIDE 12

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 of the received signal (after the matched filter) over the duration of a few symbol periods. The wider the eye is opened, the better the quality of the samples under mismatched timing and AWGN.

1.0 2.0 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 t/Tb y(t)/V 1.0 2.0 −2.5 −2 −1.5 −1 −0.5 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

SLIDE 13

Chapter 9: Signaling Over Bandlimited Channels

Eye Diagrams under AWGN with SNR= V 2/σ2

w = 20 dB

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 0.5 1 1.5 2 2.5 t/Tb y(t)/V 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 0.5 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

SLIDE 14

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.

b

kT t =

❱ ❱ ❲ ❳ ❨ ❩ ❬

( ) ( )

b k

t V t kT δ

∞ =−∞

= ± −

❭

x ( ) t y ( ) t r ( )

b

kT y ( ) Gaussian noise, zero-mean PSD ( ) watts/Hz t S f

w

w ) ( f HC ) ( f HT ) ( f HR

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

EE456 – Digital Communications 14

SLIDE 15

Chapter 9: Signaling Over Bandlimited Channels

b

kT t =

❪ ❪ ❫ ❴ ❵ ❛ ❜

( ) ( )

b k

t V t kT δ

∞ =−∞

= ± −

❝

x ( ) t y ( ) t r ( )

b

kT y ( ) Gaussian noise, zero-mean PSD ( ) watts/Hz t S f

w

w ) ( f HC ) ( f HT ) ( f HR

Since zero-ISI is achieved, one has y(mTb) = ±V + wo(mTb), where wo(mTb) ∼ N (0, σ2

w), with σ2 w =

∞

−∞ Sw(f)|HR(f)|2df.

V − V

2

σ w ( ( ) | )

b

f y mT V ( ( ) | )

b

f y mT V − ( )

b

y mT

Because P [error] = Q

• V

σw

• ⇒ Need to maximize V 2

σ2

w to minimize P [error]. EE456 – Digital Communications 15

SLIDE 16

Chapter 9: Signaling Over Bandlimited Channels

b

kT t =

❞ ❞ ❡ ❢ ❣ ❤ ✐

( ) ( )

b k

t V t kT δ

∞ =−∞

= ± −

❥

x ( ) t y ( ) t r ( )

b

kT y ( ) Gaussian noise, zero-mean PSD ( ) watts/Hz t S f

w

w ) ( f HC ) ( f HT ) ( f HR

1

Compute the average transmitted power: PT = V 2

Tb

∞

−∞ |HT (f)|2df

(watts).

2

Write the inverse of the SNR as σ2

w

V 2 = 1 PT Tb ∞

−∞

|HT (f)|2df ∞

−∞

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

• =

1 PT Tb ∞

−∞

|SR(f)|2 |HC(f)|2|HR(f)|2 df ∞

−∞

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

SLIDE 17

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)|) is

• ptimally done at both the transmitter and receiver.

Constants K1 and K2 set signal levels at the transmitter and receiver, but the most important design requirement is that the transmit and receive filters are a matched filter pair: HR(f) = |HR(f)|ej∠HR(f), HT (f) = K|HR(f)|ej∠−HR(f), The maximum output SNR is V 2 σ2

w

• max

= PT Tb ∞

−∞

|SR(f)|

• Sw(f)

|HC(f)| df −2 .

EE456 – Digital Communications 17

SLIDE 18

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) are square-root raised-cosine (SRRC) spectrum: HT (f) = HR(f) =        √Tb, |f| ≤ 1−β

2Tb

√Tb cos πTb

2β

• |f| − 1−β

2Tb

• ,

1−β 2Tb ≤ |f| ≤ 1+β 2Tb

0, |f| ≥ 1+β

2Tb

. 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 1 2 3 −0.2 0.2 0.4 0.6 0.8 1 1.2 t/Tb sR(t) Raised cosine (RC) Square−root RC

EE456 – Digital Communications 18

SLIDE 19

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 1 n Information bits or amplitude levels 2 4 6 8 10 12 14 16 18 20 −2 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 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 not ISI-free, but the samples of the output of the receive matched filter are!

EE456 – Digital Communications 19

SLIDE 20

Chapter 9: Signaling Over Bandlimited Channels

Shaping Filters in Bandlimited Passband QAM Systems

De- multiplexer Inphase ASK decision

,

Select

I i

V

,

Select

Q i

V ) 2 cos( 2 t f T

c s

π ) 2 sin( 2 t f T

c s

π bits Infor. ) (t si Multiplexer Decision 2 ( ) sin(2 )

Q c s

t f t T φ π = 2 ( ) cos(2 )

I c s

t f t T φ π = ) ( ) ( ) ( t t s t

i

w r + =

s

t kT =

s

t kT = Quadrature ASK decision QAM signal Transmit filter Transmit filter Matched filter Matched filter ( ) p t ( ) p t ( )

s

p T t − ( )

s

p T t −

EE456 – Digital Communications 20

SLIDE 21

Chapter 9: Signaling Over Bandlimited Channels

Nyquist’s Criterion for Zero ISI in Passband Channels

s

t kT = Matched filter ( )

s

p T t − Inphase amplitude ) 2 cos( 2 t f T

c s

π Transmit filter ( ) p t ) 2 cos( 2 t f T

c s

π f

⋯

2

s

T 1

s

T − 1 2

s

T 1

R s

S f T   −     2

R s

S f T   −    

( )

R

S f 1

s

T Passband bandwidth W 1

R s

S f T   +    

s

T β

2 W 2 W

−

2 W c

f +

2 W c

f −

2 W 2 W

− ( ) P f

c

f Bandwidth W ( )

R

S f

2 W 2 W

−

⋯

EE456 – Digital Communications 21

SLIDE 22

Chapter 9: Signaling Over Bandlimited Channels

Block Diagram of an All-Digital QAM System

1

L ↑ [ ]

I

V n [ ]

Q

V n

( )

1

IF

cos 2

s

T L

f n π

1

L ↑

( )

1 s

T T L

h n

( )

1

IF

sin 2

s

T L

f n π

( )

1 s

T T L

h n

2

L ↑

c

f f

IF[ ]

s n

RF( )

s t

RF

f

f

2

L ↓

( )

1

IF

cos 2

s

T L

f n π

( )

1

IF

sin 2

s

T L

f n π

( )

1 s

T R L

h n

( )

1 s

T R L

h n

1

L ↓

1

L ↓

RF( )

s t

IF[ ]

s n [ ]

I

V n [ ]

Q

V 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 t π π = − + −

∑

EE456 – Digital Communications 22

SLIDE 23

Chapter 9: Signaling Over Bandlimited Channels

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

rc( )

H f (Hz) f 1 2

s

T 1 2

s

T β + 1

1 2

transition band is a raised cosine

rc( )

h t 1 2

s

T β −

s

T β t

s

T

sam

1 4

s

T T =

rc(e

)

j

H

ω

(rad/sample) ω 2π 1

1 2

π 4 π 4 π − π − 2π − 1 2

s

T β + − 1 2

s

T − EE456 – Digital Communications 23

SLIDE 24

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

• 1

2

• 1 + cos
• π

2β

• |ω|L

π

− 1 + β

• ,

for π(1−β)

L

< |ω| ≤ π(1+β)

L

0,

• therwise.

The time-domain SRRC impulse response is hsrrc[n] =

4βn L

cos

• π(1+β)n

L

• + sin
• π(1−β)n

L

• πn

L

• 1 −
• 4βn

L

2 , for − ∞ ≤ n ≤ ∞ The frequency response of the ideal overall Raised-Cosine (RC) filter is: Hrc(ejω) =      1, for |ω| ≤ π(1−β)

L 1 2

• 1 + cos
• π

2β

• |ω|L

π

− 1 + β

• ,

for π(1−β)

L

< |ω| ≤ π(1+β)

L

0,

• therwise.

The time-domain ideal RC impulse response is hrc[n] = hsrrc[n] ∗ hsrrc[n] = sin πn

L

• πn

L

cos

• πβn

L

• 1 −
• 2βn

L

2 , for − ∞ ≤ n ≤ ∞

EE456 – Digital Communications 24

SLIDE 25

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) SRRC impulse response. Any truncation would make the filters non-ideal, hence not exactly satisfying the Nyquist criterion, hence causing some amount of ISI. Truncation also makes the stopband attenuation of the filter not exactly zero. It is always desired to reduce the stopband attenuation. The longer the truncated filters (in terms of symbol period Ts), the better the approximation, hence the less ISI. However, longer filters mean higher complexity/cost. EE465 in Term 2 will examine efficient designs of pulse shaping and matched filters for a 16-QAM system that meets certain requirements on ISI and

• ut-of-band power.

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

EE456 – Digital Communications 25

SLIDE 26

Chapter 9: Signaling Over Bandlimited Channels

Truncating the Ideal SRRC Filter

5 10 15 20 25 30 35 −0.1 0.1 0.2 0.3 n SRRC filter with length 17 (or 4 symbols) and β = 0.2 5 10 15 20 25 30 35 −0.1 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 filter at multiples of symbol period (every 4th sample in this case) are not zero.

EE456 – Digital Communications 26

SLIDE 27

Chapter 9: Signaling Over Bandlimited Channels

Truncating the Ideal SRRC Filter

10 20 30 40 50 60 −0.1 0.1 0.2 0.3 n SRRC filter with length 33 (or 8 symbols) and β = 0.2 10 20 30 40 50 60 −0.1 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

SLIDE 28

Chapter 9: Signaling Over Bandlimited Channels

Spectra of the Truncated SRRC Filters

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −100 −80 −60 −40 −20 20 Magnitude 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 shorter the truncation length, the higher the stop-band attenuation.

EE456 – Digital Communications 28

SLIDE 29

Chapter 9: Signaling Over Bandlimited Channels

IQ Plots and Modulation Error Ratio (MER)

−4 −2 2 4 −4 −2 2 4 Filter length = 17, MER = 17.17 (dB) Inphase Quadrature −4 −2 2 4 −4 −2 2 4 Filter length = 33, MER = 31.31 (dB) Inphase Quadrature

MER = lim

N→∞ 10 log10

  N

n=−N(V 2 I [n] + V 2 Q[n])

N

n=−N

• (VI[n] −

VI[n])2 + (VQ[n] − VQ[n])2

• 

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

EE456 – Digital Communications 29

SLIDE 30

Chapter 9: Signaling Over Bandlimited Channels

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

10 20 30 40 50 60 70 80 90 100 −0.1 0.1 0.2 0.3 n SRRC and hM3 filters, both with length 97 (24 symbols) and β = 0.2 SRRC filter hM3 filter 5 10 15 20 25 30 35 40 45 50 −3 −2 −1 1 x 10

−3

ISI after downsampling and removing the peak value of 1 SRRC: MER=47.65dB hM3: 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

SLIDE 31

Chapter 9: Signaling Over Bandlimited Channels

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

−4 −2 2 4 −4 −2 2 4 SRRC filter, MER = 47.65 (dB) Inphase Quadrature −4 −2 2 4 −4 −2 2 4 hM3 filter, MER = 60.20 (dB) Inphase Quadrature Note that both SRRC and hM3 filters have the same length of 97 samples, or 24 symbols (for an upsampling factor of L1 = 4).

EE456 – Digital Communications 31

SLIDE 32

Chapter 9: Signaling Over Bandlimited Channels

Comparison of harris-Moerder and SRRC Filters: Spectra

10 20 30 40 50 60 70 80 90 100 −0.1 0.1 0.2 0.3 n SRRC and hM3 filters, both with length 97 (24 symbols) and β = 0.2 SRRC filter hM3 filter −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −150 −100 −50 50 Normalized frequency, ω/2π (cycle/sample) Gain (dB) Magnitude frequency responses of SRRC and hM filters SRRC filter hM filter

EE456 – Digital Communications 32

SLIDE 33

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” term refers to any unwanted disturbance to the desired signal. For example, this disturbance could be due to thermal noise, ISI, or any imperfections 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 ideal locations. The MER can be computed from the I/Q decision variables as MER (dB) = 10 log10 Psignal Perror

• =

lim N→∞ 10 log10     N n=−N

• V 2

I [n] + V 2 Q[n]

• N

n=−N

• VI [n] −

VI [n] 2 +

• VQ[n] −

VQ[n] 2     , where VI [n] and VQ[n] are the ideal values and VI [n] and VQ[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 as the ratio between the average error power and the peak signal power (expressed as %): EVM (%) =

• Perror

Psignal, max =

• limN→∞

N n=−N

• VI [n] −

VI [n] 2 +

• VQ[n] −

VQ[n] 2

• V 2

I,max + V 2 Q,max . Because the relationship between the peak and average signal powers depends on the constellation geometry, subject to the same average 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 is as 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 determined from 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