Chapter 3 Pulse Modulation - Website of Professor Po-Ning...

1

Chapter 3 Pulse Modulation

We migrate from analog modulation (continuous in both time and value) to digital modulation (discrete in both time and value) through pulse modulation (discrete in time but could be continuous in value).

© Po-Ning [email protected] Chapter 3-2

3.1 Pulse Modulation

o  Families of pulse modulation n  Analog pulse modulation

o  A periodic pulse train is used as carriers (similar to sinusoidal carriers)

o  Some characteristic feature of each pulse, such as amplitude, duration, or position, is varied in a continuous matter in accordance with the sampled message signal.

n  Digital pulse modulation o  Some characteristic feature of carriers is varied in a

digital manner in accordance with the sampled, digitized message signal.

2


3.2 Sampling Theorem

∑∞

−∞=

−=n

ss nTtnTgtg )()()( δδ

( ) ( )∑∑ ∫∞

−∞=

∞

−∞=

∞

∞−−=−−=

nss

nss fnTjnTgdtftjnTtnTgfG ππδδ 2exp)(2exp)()()(

o  Ts sampling period o  fs = 1/Ts sampling rate


3.2 Sampling Theorem

o  Given:

o  Claim: ∑∞

−∞=

−=m

ss mffGffG )()(δ

( )∑∞

−∞=

−=n

ss fnTjnTgfG πδ 2exp)()(

In this figure, fs = 2W .

3


3.2 Spectrum of Sampled Signal

dfffnjfL

fcf

fnjcfL

fmffGffL

s

s

f

fss

nn s

n

sm

ss

∫∑

∑

−

∞

−∞=

∞

−∞=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⇒

−=

2/

2/2exp)(1 where,2exp)(

, ExpansionSeries By Fourier

. period withperiodic isit that notice and ,)()(Let

ππ

∫ ∑

∫

−

∞

−∞=

−

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛ −=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

2/

2/

2/

2/

2exp)(

2exp)(1

s

s

s

s

f

fsm

s

f

fss

n

dfffnjmffG

dfffnjfL

fc

π

π⇒

Proof:

cn = G( f −mfs )exp − j 2πnfs

f⎛

⎝⎜

⎞

⎠⎟df

− fs /2

fs /2∫m=−∞

∞

∑ , s = f −mfs

= G(s)exp − j 2πnfs

(s+mfs )⎛

⎝⎜

⎞

⎠⎟ds

− fs /2−mfs

fs /2−mfs∫m=−∞

∞

∑

= G(s)exp − j 2πnfs

s⎛

⎝⎜

⎞

⎠⎟ds

− fs /2−mfs

fs /2−mfs∫m=−∞

∞

∑

= G(s)−∞

∞

∫ exp − j 2πnfs

s⎛

⎝⎜

⎞

⎠⎟ds

= g(−nTs )


( ) . where,2exp)(

2exp)()(

nmfmTjmTg

ffnjnTgfL

mss

n ss

−=−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∑

∑∞

−∞=

∞

−∞=

π

π⇒

Q.E.D.

4


3.2 First Important Conclusion from Sampling

o  Uniform sampling at the time domain results in a periodic spectrum with a period equal to the sampling rate.

∑∑∞

−∞=

∞

−∞=

−=⇒−=m

ssn

ss mffGffGnTtnTgtg )()()()()( δδ δ

In this figure, fs = 2W .


3.2 Reconstruction from Sampling

.2 Take Wfs = Ideal lowpass filter

.||for )(21)( WffGW

fG ≤= δ

5


3.2 Aliasing due to Sampling

.samples edundersamplby edreconstruc becannot )( ,2 When fGWfs <


o  A band-limited signal of finite energy with bandwidth W can be completely described by its samples of sampling rate fs ≥ 2W. n  2W is commonly referred to as the Nyquist rate.

o  How to reconstruct a band-limited signal from its samples?

3.2 Second Important Conclusion from Sampling

2W fs

6

( )

( )

[ ]

[ ]( ))(2sinc2 )(

)(2)(2sin)(2

)(2exp)(1

)2exp(2exp)(1

)2exp()(

)2exp()()(

ssn

s

s

s

ns

s

W

W sn

ss

W

Wn

sss

W

W

nTtWWTnTg

nTtWnTtWnTg

fW

dffnTtjnTgf

dfftjfnTjnTgf

dfftjfG

dfftjfGtg

−=

−−

=

−=

⎟⎠

⎞⎜⎝

⎛−=

=

=

∑

∑

∫∑

∫ ∑

∫

∫

∞

−∞=

∞

−∞=

−

∞

−∞=

−

∞

−∞=

−

∞

∞−

ππ

π

ππ

π

π

2WTs sinc[2W(t-nTs)] plays the role of an interpolation function for samples.


G(f) =

1

fsG�(f) for |f | W

G�(f) = L(f) = fs

1X

m=�1G(f �mfs)

=

1X

m=�1g(mTs) exp(�j2⇡mTsf)

See Slides 3-4 ~ 3-6


3.2 Band-Unlimited Signals

o  The signal encountered in practice is often not strictly band-limited.

o  Hence, there is always “aliasing” after sampling. o  To combat the effects of aliasing, a low-pass anti-aliasing

filter is used to attenuate the frequency components outside [-fs, fs].

o  In this case, the signal after passing the anti-aliasing filter is often treated as bandlimited with bandwidth fs/2 (i.e., fs = 2W). Hence,

⎥⎦

⎤⎢⎣

⎡−= ∑

∞

−∞=

nTtnTgtgsn

s sinc )()(

7


3.2 Interpolation in terms of Filtering

o  Observe that

is indeed a convolution between gδ(t) and sinc(t/Ts).

⎥⎦

⎤⎢⎣

⎡−= ∑

∞

−∞=

nTtnTgtgsn

s sinc )()(

∑ ∫

∫ ∑

∫

∞

−∞=

∞

∞−

∞

∞−

∞

−∞=

∞

∞−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎠

⎞⎜⎝

⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛

n sss

snss

ss

dTtnTnTg

dTtnTnTg

dTtg

Tttg

ττ

τδ

ττ

τδ

ττ

τδδ

sinc)()(

sinc)()(

sinc)(sinc*)(

∑∞

−∞=⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛

n ss

s

nTtnTg

Tttg sinc)(sinc*)(δ

(Continue from the previous slide.)

⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒

sTtth sinc)( filter) tion(interpolafilter tionReconstruc

)(rect)( fTTfH ss=⇒

2/sf− 2/sf

sf)(tgδ )(tg


)( fH

8


3.2 Physical Realization of Reconstruction Filter

o  An ideal lowpass filter is not physically realizable. o  Instead, we can use an anti-aliasing filter of bandwidth W,

and use a sampling rate fs > 2W. Then the spectrum of a reconstruction filter can be shaped like:

Signal spectrum with bandwidth W

Signal spectrum after sampling with fs > 2W

The physically realizable reconstruction filter

)(*)()()()()()(*)( idealidealrealizablerealizable thtgfHfGfHfGthtg δδδδ ≡≡=

Ideal filter of bandwidth fs/2.


9


3.3 Pulse-Amplitude Modulation (PAM)

o  PAM n  The amplitude of regularly spaced pulses is varied in

proportion to the corresponding sample values of a continuous message signal. Notably, the top of each pulse

is maintained flat. So this is PAM, not natural-sampling for which the message signal is directly multiplied by a periodic train of rectangular pulses.



o  The operation of generating a PAM modulated signal is often referred to as “sample and hold.”

o  This “sample and hold” process can also be analyzed through “filtering technique.”

)(*)()()()( thtmnTthnTmtsn

ss δ=−= ∑∞

−∞=

⎪⎩

⎪⎨

⎧

==

<<

=

otherwise,0,0,2/1

0,1)( where Ttt

Ttth .)()()( and ∑

∞

−∞=

−=n

ss nTtnTmtm δδ

10



o  By taking “filtering” standpoint, the spectrum of S(f) can be derived as:

n  M(f) is the message signal with bandwidth W (or having experienced an anti-aliasing filter of bandwidth W).

n  fs ≥ 2W.

)()(

)()(

)()()(

fHkffMf

fHkffMf

fHfMfS

kss

kss

∑

∑∞

−∞=

∞

−∞=

−=

⎟⎠

⎞⎜⎝

⎛−=

= δ



)()()(

)()()()(

)()()(

Equalizer FiltertionReconstruc

1||

fMfHfM

fHkffMffHfMf

fHkffMffS

ksss

kss

→→

−+=

−=

∑

∑

≥

∞

−∞=

)(1

))( of ],[ range (over the

fH

fMWW−

11


3.3 Feasibility of Equalizer Filter

o  The distortion of M(f) is due to M(f)H(f),

( ) ( )fTjfTTfHTttTt

th π−=⎪⎩

⎪⎨

⎧

==

<<

= expsinc)(or otherwise,0

,0,2/10,1

)( where

( )( )

⎪⎩

⎪⎨⎧ ≤=

=⇒otherwise0,

||,expsinc1

)(1

)( WffTjfTTfHfE π

Question: Is the above E(f) feasible or realizable?

.211 WfTT ss

>=>


( )⎪⎩

⎪⎨⎧ ≤

=otherwise0,

||,sinc1

)(~ WffTTfE

-1 -0.5 0.5 1

0.2

0.4

0.6

0.8

1)(~ fE

E.g., T = 1, W = 1/8.

This gives an equalizer:

)(~ fE ( )fTjTtπ

δ

expor )2/( +)(ti )(to

non-realizable! Why? A lowpass filter

)(1 to

Because "o1(t) = 0 for t < 0" does not imply "o(t) = 0 for t < 0."

12


3.3 Feasibility of Equalizer Filter

o  Causal

n  A reasonable assumption for a feasible linear filter system is that:

n  A necessary and sufficient condition for the above assumption to hold is that h(t) = 0 for t < 0.

)(th)(to)(ti

.0for 0)( have we,0for 0)( satisfying )(any For <=<= ttottiti

n  Simplified Proof:

∫∫ −=−=⇒⎩⎨⎧

<=

<= ∞

∞−

tdtihdtihto

ttitth

0)()()()()(

0for 0)(0for 0)(

ττττττ

0for 0)( <=⇒ tto

⎩⎨⎧

≥

<=>≠∫

−

∞− .0for ,1;0for ,0

)( take then,0 somefor 0)( Iftt

tiadttha

input! zero completely todueoutput nonzeroa be willthere

thatmeans which,0)()( ≠=−⇒ ∫−

∞−

adhao ττ

.0every for 0)( Therefore, >=∫−

∞−adh

aττ

.0for 0)(0every for 0 )( >=∂

∂⇒>= a

aaaa λ

λ

.0for 0)()(0every for 0)( >=−−=∂∂

⇒>= ∫∫−

∞−

−

∞−aahdh

aadh

aaττττ


13


3.3 Aperture Effect

o  The distortion of M(f) due to M(f)H(f)

is very similar to the distortion caused by the finite size of

the scanning aperture in television. So this is named the aperture effect.

o  If T/Ts ≦ 0.1, the amplitude distortion is less than 0.5%; hence, the equalizer may not be necessary.

( ) ( )fTjfTTfHTttTt

th π−=⎪⎩

⎪⎨

⎧

==

<<

= expsinc)(or otherwise,0

,0,2/10,1

)( where

-0.06 -0.04 -0.02 0.02 0.04 0.06

0.2

0.4

0.6

0.8

1


( )⎪⎩

⎪⎨⎧ ≤

=otherwise0,

||,sinc1

)(~ WffTTfE .211 and Wf

TT ss

>=>

( ) 04.0,10,1for otherwise0,

04.0||,sinc

1)(~

===⎪⎩

⎪⎨⎧ ≤

=⇒ WTTfffE s

)(~ fE 1.00264

14


3.3 Pulse-Amplitude Modulation

o  Final notes on PAM n  PAM is rather stringent in its system requirement, such

as short duration of pulse. n  Also, the noise performance of PAM may not be

sufficient for long distance transmission. n  Accordingly, PAM is often used as a mean of message

processing for time-division multiplexing, from which conversion to some other form of pulse modulation is subsequently made. Details will be discussed in Section 3.9.


3.4 Other Forms of Pulse Modulation

o  Pulse-Duration Modulation (or Pulse-Width Modulation) n  Samples of the message signal are used to vary the

duration of the pulses. o  Pulse-Position Modulation

n  The position of a pulse relative to its unmodulated time of occurrence is varied in accordance with the message signal.

15

Pulse trains

PDM

PPM



3.4 Other Forms of Pulse Modulation

o  Comparisons between PDM and PPM n  PPM is more power efficient because excessive pulse

duration consumes considerable power. o  Final note

n  It is expected that PPM is immune to additive noise, since additive noise only perturbs the amplitude of the pulses rather than the positions.

n  However, since the pulse cannot be made perfectly rectangular in practice (namely, there exists a non-zero transition time in pulse edge), the detection of pulse positions is somehow still affected by additive noise.

16


3.5 Bandwidth-Noise Trade-Off

o  PPM seems to be a better form for analog pulse modulation from noise performance standpoint. However, its noise performance is very similar to (analog) FM modulation as: n  Its figure of merit is proportional to the square of

transmission bandwidth (i.e., 1/T) normalized with respect to the message bandwidth (W).

n  There exists a threshold effect as SNR is reduced.

o  Question: Can we do better than the “square” law in figure-of-merit improvement? Answer: Yes, by means of Digital Communication, we can realize an “exponential” law!

)/ .,.( WBBeI Tn =

See Slide 2-162: figure-of-merit∝D2 =12BT ,Carson

W−1

⎛

⎝⎜

⎞

⎠⎟

2

=12Bn,Carson −1

⎛

⎝⎜

⎞

⎠⎟

2


3.6 Quantization Process

o  Transform the continuous-amplitude m = m(nTs) to discrete approximate amplitude v = v(nTs)

o  Such a discrete approximate is adequately good in the sense that any human ear or eye can detect only finite intensity differences.

17


3.6 Quantization Process

o  We may drop the time instance nTs for convenience, when the quantization process is memoryless and instantaneous (hence, the quantization at time nTs is not affected by earlier or later samples of the message signal.)

o  Types of quantization n  Uniform

o  Quantization step sizes are of equal length. n  Non-uniform

o  Quantization step sizes are not of equal length.


o  An alternative classification of quantization n  Midtread n  Midrise

midtread midrise

18


3.6 Quantization Noise

Uniform midtread quantizer



o  Define the quantization noise to be Q = M – V = M – g(M), where g( ) is the quantizer.

o  Let the message M be uniformly distributed in (–mmax, mmax). So M has zero mean.

o  Assume g( ) is symmetric and of midrise type; then, V = g(M) also has zero-mean, so does Q = M – V.

o  Then the step size of the quantizer is given by:

where L is the total number of representation levels.

Lmmax2

=Δmmax =1

L = 4

Δ =12

Example.

19



o  Assume that g( ) assigns the midpoint of each step interval to be the representation level. Then

{ }

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

Δ≥

Δ<≤

Δ−+

Δ

Δ−<

=⎭⎬⎫

⎩⎨⎧ ≤

Δ−Δ=≤

2,1

22,21

2,0

2)mod(PrPr

q

qq

q

qMqQ

⎭⎬⎫

⎩⎨⎧ Δ

<≤Δ

−⋅Δ

=22

1)( pdfOr qqfQ 1 mmax =1

L = 4

Δ =12

Example.



o  So, the output signal-to-noise ratio is equal to:

o  The transmission bandwidth of a quantization system is conceptually proportional to the number of bits required per sample, i.e., R = log2(L).

o  We then conclude that SNRO ∝ 4R, which increases exponentially with transmission bandwidth.

22max

2max22/

2/

2

32

121

1211 L

mP

LmPP

dqq

PSNRO =

⎟⎠

⎞⎜⎝

⎛=

Δ=

Δ

=

∫Δ

Δ−

20


Example 3.1 Sinusoidal Modulating Signal

o  Let m(t) = Am cos(2πfct). Then

L R SNRO (dB) 32 5 31.8 64 6 37.8 128 7 43.8 256 8 49.8

* Note that in this example, we assume a full-load quantizer, in which no quantization loss is encountered due to saturation.

P =

A2m2

and mmax

= Am

) SNRO =

3(Am2/2)A2

mL2

=

3

2

4

R= 10 log

10

(3/2) +R · 10 log10

(4) dB ⇡ (1.8 + 6R) dB



o  In the previous analysis of quantization error, we assume the quantizer assigns the mid-point of each step interval to be the representative level.

o  Questions: n  Can the quantization noise power be further reduced by

adjusting the representative levels? n  Can the quantization noise power be further reduced by

adopting a non-uniform quantizer?

21


Representation level

3.6 Optimality of Scalar Quantizers

v1 v2 vL-1 vL

o Let d(m, vk) be the distortion by representing m by vk. o Goal: To find {Ik} and {vk} such that the average distortion

D = E[d(M, g(M))] is minimized.

…

I1 I2 IL-1 IL … Partitions

),[1

AAIL

kk −=

=∪ Notably, interval Ik may not be

a “consecutive” single interval.


3.6 Optimality of Scalar Quantizers

o  Solution:

∑∫=

=L

k IMkIvIv

kkkkk

dmmfvmdD1

}{}{}{}{)(),(minminminmin

(I)  For fixed {vk}, determine the optimal {Ik}. (II) For fixed {Ik}, determine the optimal {vk}.

(I) If d(m, vk) ≦ d(m, vj), then m should be assigned to Ik rather than Ij.

{ }LjvmdvmdAAmI jkk ≤≤≤−∈=⇒ 1 allfor ),(),(:),[

22

(II) For fixed {Ik}, determine the optimal {vk}.

∑∫=

L

k IMkv

kk

dmmfvmd1

}{)(),(min

∫

∫∑∫

∂

∂=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂∂

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂∂

=

j

jk

IM

j

j

IMj

j

L

k IMk

j

dmmfvvmd

dmmfvmdv

dmmfvmdv

)(),(

)(),()(),( Since1

.0)(),(

:is optimal for the conditionnecessary a

=∂

∂∫jI

Mj

j

j

dmmfvvmd

v


Lloyd-Max algorithm is to repetitively apply (I) and (II) for the search of the optimal quantizer.


Example: Mean-Square Distortion

o  d(m, vk) = (m - vk)2

(I) { }interval. econsecutiva be should

1 allfor )()(:),[ 22 LjvmvmAAmI jkk ≤≤−≤−−∈=

v1 v2 vL-1 vL …

I1 I2 IL-1 IL …

Representation level

Partitions

23


Example: Mean-Square Distortion (II) :is optimal for the conditionnecessary A jv

]|[)(

)(1optimal, 1

1

+<≤==⇒

∫

∫+

+

jjm

m M

m

m M

j mMmMEdmmf

dmmmfv

j

j

j

j

Exercise: What is the best {mk} and {vk} if M is uniformly distributed over [-A,A).

Z mj+1

mj

@(m� vj)2

@vjfM (m)dM = �2

Z mj+1

mj

(m� vj)fM (m)dm = 0.

Hint: min{Ik}

min{vk}

D =1

2Amin{mk}

LX

k=1

Z mk+1

mk

✓m� mk +mk+1

2

◆2

dm.


3.7 Pulse-Code Modulation

(anti-alias)

24


3.7 Pulse-Code Modulation

o  Non-uniform quantizers used for telecommunication (ITU-T G.711) n  ITU-T G.711: Pulse Code Modulation (PCM) of Voice

Frequencies (1972) o  It consists of two laws: A-law (mainly used in

Europe) and µ-law (mainly used in US and Japan) n  This design helps to protect weak signal, which occurs

more frequently in, say, human voice.


3.7 Laws

o  Quantization Laws n  A-law

o  13-bit uniformly quantized o  Conversion to 8-bit code

n  µ-law o  14-bit uniformly quantized o  Conversion to 8-bit code.

n  These two are referred to as compression laws since they uses 8-bit to (lossily) represent 13-(or 14-)bit information.

25


3.7 A-law in G.711

o  A-law (A=87.6)

⎪⎪⎩

⎪⎪⎨

⎧

≤≤⎥⎦

⎤⎢⎣

⎡

++

≤+

=11,

)log(1|)|log(1)sgn(

1,)log(1

)(law-A

mAA

mAm

Amm

AA

mF

Linear mapping

Logarithmic mapping

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

output

input

)(law-A mF

m


26

-128 -112

-96 -80 -64 -48 -32

0

32 48 64 80 96

112 128

-4096 -2048 -1024 -512 -256

-128 -64

0

64 128

256

512 1024 2048 4096

outp

ut

input

)(law-A mF


13 bit uniform quantization

8 bit PCM code A piecewise linear approximation to the law.


Input Values Compressed Code Word

Chord Step

Bits:11 10 9 8 7 6 5 4 3 2 1 0 Bits: 6 5 4 3 2 1 0

0 0 0 0 0 0 0 a b c d x

0 0 0 0 0 0 1 a b c d x

0 0 0 0 0 1 a b c d x x

0 0 0 0 1 a b c d x x x

0 0 0 1 a b c d x x x x

0 0 1 a b c d x x x x x

0 1 a b c d x x x x x x

1 a b c d x x x x x x x

0 0 0 a b c d

0 0 1 a b c d

0 1 0 a b c d

0 1 1 a b c d

1 0 0 a b c d

1 0 1 a b c d

1 1 0 a b c d

1 1 1 a b c d

E.g. (3968)10 --> (1111,1000,0000)2-->(111,1111)2-->(127)10 E.g. (2176)10 -->(1000,1000,0000)2-->(111,0001)2-->(113)10

Compressor of A-law (assume nonnegative m)

27


Expander of A-law (assume nonnegative m)

Compressed Code Word Raised Output Values

Chord Step

Bits:6 5 4 3 2 1 0 Bits:11 10 9 8 7 6 5 4 3 2 1 0

0 0 0 a b c d

0 0 1 a b c d

0 1 0 a b c d

0 1 1 a b c d

1 0 0 a b c d

1 0 1 a b c d

1 1 0 a b c d

1 1 1 a b c d

0 0 0 0 0 0 0 a b c d 1

0 0 0 0 0 0 1 a b c d 1

0 0 0 0 0 1 a b c d 1 0

0 0 0 0 1 a b c d 1 0 0

0 0 0 1 a b c d 1 0 0 0

0 0 1 a b c d 1 0 0 0 0

0 1 a b c d 1 0 0 0 0 0

1 a b c d 1 0 0 0 0 0 0

E.g. (113)10 → (111, 0001)2 → (1000,1100, 0000)2 → (2240)10

In other words, (1001, 0000, 0000)2 + (1000,1000, 0000)2

2=

(2304)10 + (2176)10

2= (2240)10


3.7 µ-law in G.711

o µ-law (µ = 255)

n It is approximately linear at low m. n It is approximately logarithmic at large m.

.1for )log(1

)1log()sgn()(law- ≤

+

+= m

mmmF

µ

µµ

28

)(law- mFµ


-1 -0.8 -0.6 -0.4 -0.2

0 0.2 0.4 0.6 0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 m

)(law- mFµ

14 bit uniform quantization (213 = 8192)

8 bit PCM code


-128 -112

-96 -80 -64 -48 -32 -16

0 16 32 48 64 80 96

112 128

-8159 -4063 -2015 -991 -479

-223 -95 -31

0

31 95 223

479

991 2015 4063 8159

A piecewise linear approximation to the law.

29

Compressor of µ-law (assume nonnegative m) Raised Input Values Compressed Code Word

Chord Step

Bits:12 11 10 9 8 7 6 5 4 3 2 1 0 Bits: 6 5 4 3 2 1 0

0 0 0 0 0 0 0 1 a b c d x

0 0 0 0 0 0 1 a b c d x x

0 0 0 0 0 1 a b c d x x x

0 0 0 0 1 a b c d x x x x

0 0 0 1 a b c d x x x x x

0 0 1 a b c d x x x x x x

0 1 a b c d x x x x x x x

1 a b c d x x x x x x x x

0 0 0 a b c d

0 0 1 a b c d

0 1 0 a b c d

0 1 1 a b c d

1 0 0 a b c d

1 0 1 a b c d

1 1 0 a b c d

1 1 1 a b c d

Raised Input = Input + 33 = Input + 21H (For negative m, the raised input becomes (input – 33).) An additional 7th bit is used to indicate whether the input signal is positive (1) or negative (0).


Expander of µ-law (assume nonnegative m)


Compressed Code Word Raised Output Values

Chord Step

Bits:6 5 4 3 2 1 0 Bits:12 11 10 9 8 7 6 5 4 3 2 1 0

0 0 0 a b c d

0 0 1 a b c d

0 1 0 a b c d

0 1 1 a b c d

1 0 0 a b c d

1 0 1 a b c d

1 1 0 a b c d

1 1 1 a b c d

0 0 0 0 0 0 0 1 a b c d 1

0 0 0 0 0 0 1 a b c d 1 0

0 0 0 0 0 1 a b c d 1 0 0

0 0 0 0 1 a b c d 1 0 0 0

0 0 0 1 a b c d 1 0 0 0 0

0 0 1 a b c d 1 0 0 0 0 0

0 1 a b c d 1 0 0 0 0 0 0

1 a b c d 1 0 0 0 0 0 0 0

Output = Raised Output – 33 Note that the combination of a compressor and an expander is called a compander.

30

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

A-lawmu-law

Comparison of A-law and µ-law specified in G.711.



3.7 Coding

o  After the quantizer provides a symbol representing one of 256 possible levels (8 bits of information) at each sampled time, the encoder will transform the symbol (or several symbols) into a code character (or code word) that is suitable for transmission over a noisy channel.

o  Example. Binary code.

11100100 1 1 1 0 0 1 0

0 = change 1 = unchange 0

31


3.7 Coding

o  Example. Ternary code (Pseudo-binary code).

0 0 0 1 1 0 1

00011011→ B A

C

00011011→ACABBCBB

1

Through the help of coding, the receiver may be able to detect (or even correct) the transmission errors due to noise. For example, it is impossible to receive ABABBABB, since this is not a legitimate code word (character).


3.7 Coding

o  Example of error correcting code : Three-times repetition code (to protect Bluetooth packet header).

00011011→ 000,000,000,111,111,000,111,111

Then the so-called majority law can be applied at the receiver to correct one-bit error.

o Channel (error correcting) codes are designed to compensate the channel noise, while line codes are simply used as the electrical representation of a binary data stream over the electrical line.

32


3.7 Line Codes

(a)  Unipolar nonreturn-to-zero (NRZ) signaling

(b)  Polar nonreturn-to-zero (NRZ) signaling

(c)  Unipolar return-to-zero (RZ) signaling

(d)  Bipolar return-to-zero (BRZ) signaling

(e)  Split-phase (Manchester code)


3.7 Derivation of PSD

o From Slide 1-116, we obtain that the general formula for PSD is:

}.|{|)()( where)],()([21limPSD 2

*2 TttstsfSfSE

T TTT≤⋅==

∞→1

For a line coded signal, s(t) =1X

n=�1ang(t�nTb), where g(t) = 0 outside [0, Tb).

Hence, S(f) = G(f)1X

n=�1ane

�j2⇡fnTb and S2NTb(f) = G(f)N�1X

n=�N

ane�j2⇡fnTb .

) PSD = limN!11

2NTb|G(f)|2

1X

n=�1

N�1X

m=�N

E[ana⇤m]e�j2⇡f(n�m)Tb

!.

33

⎟⎠

⎞⎜⎝

⎛=

⎟⎠

⎞⎜⎝

⎛=

⎟⎠

⎞⎜⎝

⎛−=

⎟⎠

⎞⎜⎝

⎛=

−∞

−∞=

−

−=

−∞

−∞=∞→

−

−=

−−∞

−∞=∞→

∞

−∞=

−−−

−=∞→

∑

∑ ∑

∑ ∑

∑ ∑

b

b

b

b

fkTj

ka

b

N

Nm

fkTj

ka

bN

N

Nm

Tmnfj

na

bN

n

TmnfjN

Nmmn

bN

ekT

fG

ekNT

fG

emnNT

fG

eaaEfGNT

π

π

π

π

φ

φ

φ

22

122

1)(22

)(21

*2

)(1|)(|

)(21lim|)(|

)(21lim|)(|

][|)(|21limPSD


For i.i.d. {an},1

Tb

1X

k=�1�a(k)e

�j2⇡fkTb

!=

�2a

Tb+

µ2a

Tb

1X

k=�1e�j2⇡fkTb

=

�2a

Tb+

µ2a

T 2b

1X

k=�1�(f � k/Tb)(See Slide 3-4.)

(i.i.d. = independent and identically distributed)


3.7 Power Spectral of Line Codes

o  Unipolar nonreturn-to-zero (NRZ) signaling n  Also named on-off signaling. n  Disadvantage: Waste of power due to the non-zero-

mean nature (i.e., PSD does not approach zero at zero frequency).

⎪⎩

⎪⎨

⎧

⎩⎨⎧ <≤

=−=

∞−∞=∞

−∞=∑

otherwise,00,

)(

i.i.d., zero/one is }{ where,)()( b

nn

nbn TtA

tg

anTtgats

34



n  PSD of Unipolar NRZ

PSDU-NRZ = |G(f)|2 �2a

Tb+

µ2a

T 2b

1X

k=�1�(f � k/Tb)

!

= A2T 2b sinc

2(fTb)

�2a

Tb+

µ2a

T 2b

1X

k=�1�(f � k/Tb)

!

=A2Tb

4sinc2(fTb)

1 +

1

Tb

1X

k=�1�(f � k/Tb)

!

=A2Tb

4sinc2(fTb)

✓1 +

1

Tb�(f)

◆


o  Polar nonreturn-to-zero (NRZ) signaling n  The previous PSD of Unipolar NRZ suggests that a

zero-mean data sequence is preferred.

⎪⎩

⎪⎨

⎧

⎩⎨⎧ <≤

=

±

−=

∞−∞=∞

−∞=∑

otherwise,00,

)(

i.i.d., 1 is }{ where,)()( b

nn

nbn TtA

tg

anTtgats


PSDP-NRZ = |G(f)|2 �2a

Tb+

µ2a

T 2b

1X

k=�1�(f � k/Tb)

!

= A2Tbsinc2(fTb)

35



o  Unipolar return-to-zero (RZ) signaling n  An attractive feature of this line code is the presence of

delta functions at f = –1/Tb, 0, 1/Tb in the PSD, which can be used for bit-timing recovery at the receiver.

n  Disadvantage: It requires 3dB more power than polar return-to-zero signaling.

⎪⎩

⎪⎨

⎧

⎩⎨⎧ <≤

=−=

∞−∞=∞

−∞=∑

otherwise,02/0,

)(

i.i.d., zero/one is }{ where,)()( b

nn

nbn TtA

tg

anTtgats



n  PSD of Unipolar RZ

PSDU-RZ = |G(f)|2 �2a

Tb+

µ2a

T 2b

1X

k=�1�(f � k/Tb)

!

=A2T 2

b

4sinc2

✓fTb

2

◆ �2a

Tb+

µ2a

T 2b

1X

k=�1�(f � k/Tb)

!

=A2Tb

16sinc2

✓fTb

2

◆ 1 +

1

Tb

1X

k=�1�(f � k/Tb)

!

=A2Tb

16sinc2

✓fTb

2

◆ 1 +

1

Tb

1X

k=1�(f � k/Tb)

!

36



o  Bipolar return-to-zero (BRZ) signaling n  Also named alternate mark inversion (AMI) signaling n  No DC component and relatively insignificant low-

frequency components in PSD.

⎩⎨⎧ <≤

=−= ∑∞

−∞= otherwise,02/0,

)( where,)()( b

nbn

TtAtgnTtgats



n  PSD of BRZ o  {an} is no longer i.i.d.

.1for 0][

0161)1)(1(

161)1)(1(

161)1)(1(

161)1)(1(][

41

41)1(][

21

41)1(

41)1(

21)0(][

2

1

222

>=

=−−+−+−+=

−=−=

=++−+=

+

+

+

maaE

aaE

aaE

aE

mnn

nn

nn

n

!

37



PSDBRZ = |G(f)|2 1

Tb

1X

k=�1�a(k)e

�j2⇡fkTb

!

=

A2T 2b

4

sinc

2

✓fTb

2

◆· 1

Tb

✓�1

4

ej2⇡fTb+

1

2

� 1

4

e�j2⇡fTb

◆

=

A2T 2b

4

sinc

2

✓fTb

2

◆· 1

Tb

✓1

2

� 1

2

cos(2⇡fTb)

◆

=

A2Tb

4

sinc

2

✓fTb

2

◆sin

2(⇡fTb)



o  Split-phase (Manchester code) n  This signaling suppressed the DC component, and has

relatively insignificant low-frequency components, regardless of the signal statistics.

n  Notably, for P-NRZ and BRZ, the DC component is suppressed only when the signal has the right statistics.

⎪⎪

⎩

⎪⎪

⎨

⎧

⎪⎩

⎪⎨

⎧

<≤−

<≤

=

±

−=

∞−∞=

∞

−∞=∑

otherwise,02/,

2/0,)(

i.i.d., 1 is }{

where,)()(bb

b

nn

nbn TtTA

TtAtg

a

nTtgats

38



n  PSD of Manchester code

PSDManchester

= |G(f)|2 �2a

Tb+

µ2a

T 2b

1X

k=�1�(f � k/Tb)

!

= A2T 2b sinc

2

✓fTb

2

◆sin2

✓⇡fTb

2

◆ �2

Tb+

µ2a

T 2b

1X

k=�1�(f � k/Tb)

!

= A2Tbsinc2

✓fTb

2

◆sin2

✓⇡fTb

2

◆


Let Tb=1, and adjust A such that the total power of each line code is 1. This gives a fair comparison among line codes.

A =p2

A = 1

A = 2

A = 2

power= 1

power= 1

power= 1

power=

12+

12

power=

12+

12

PSDU-NRZ =1

2sinc2(f) +

1

2�(f)

PSDP-NRZ = sinc2(f)

PSDU-RZ =1

4sinc2

✓f

2

◆+

1

4

1X

k=�1sinc2

✓k

2

◆�(f � k)

PSDBRZ = sinc2✓f

2

◆sin2(⇡f)

PSDManchester = sinc2✓f

2

◆sin2

✓⇡f

2

◆A = 1

39


0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

U-NRZP-NRZ

U-RZBRZ

Manchester

2/1 π


3.7 Differential Encoding with Unipolar NRZ Line Coding

o  1 = no change and 0 = change.

ndno

dn = dn�1 � on = dn�1 on

40


3.7 Regeneration

o  Regenerative repeater for PCM system n  It can completely remove the distortion if the decision

making device makes the right decision (on 1 or 0).


3.7 Decoding & Filtering

o  After regenerating the received pulse at the last time, the receiver then decodes, and regenerates the original message signal (with acceptable quantization error).

o  Finally, a lowpass reconstruction filter whose cutoff frequency is equal to the message bandwidth W is applied at the end (to remove the unnecessary high-frequency components due to “quantization”).

41


3.8 Noise Consideration in PCM Systems

o  Two major noise sources in PCM systems n  (Message-independent) Channel noise n  (Message-dependent) Quantization noise

o  The quantization noise is often under designer’s control, and can be made negligible by taking adequate number of quantization levels.


3.8 Noise Consideration in PCM Systems

o  The main effect of channel noise is to introduce bit errors. n  Notably, the symbol error rate is quite different from

the bit error rate. n  A symbol error may be caused by one-bit error, or two-

bit error, or three-bit error, or …; so in general, one cannot derive the symbol error rate from the bit error rate (or vice versa) unless some special assumption is made.

n  Considering the reconstruction of original analog signal, a bit error in the most significant bit is more harmful than a bit error in the least significant bit.

42


3.8 Error Threshold

o  Eb/N0

n  Eb: Transmitted signal energy per information bit o  E.g., information bit is encoded using three-times

repetition code, in which each code bit is transmitted using one BPSK symbol with symbol energy Ec.

o  Then Eb = 3 Ec. n  N0: One-sided noise spectral density

o  The bit error rate (BER) is a function of Eb/N0 and transmission speed (and implicitly bandwidth, etc).


3.8 Error Threshold

o  Influence of Eb/N0 on BER at 105 bit per second (bps)

Eb/N0 (dB) BER About one bit error in every … 4.3 10-2 10-3 second 8.4 10-4 10-1 second 10.6 10-6 10 seconds 12.0 10-8 20 minutes 13.0 10-10 1 day 14.0 10-12 3 months

n  The usual requirement of BER in practice is 10-5.

43


3.8 Error Threshold

o  Error threshold n  The minimum Eb/N0 to achieves the required BER.

o  By knowing the error threshold, one can always add a regenerative repeater when Eb/N0 is about to drop below the threshold; hence, long-distance transmission becomes feasible. n  Unlike the analog transmission, distortion will

accumulate for long-distance transmission.


3.9 Time-Division Multiplexing

o  An important feature of sampling process is a conservation-of-time. n  In principle, the communication link is used only at the

sampling time instances. o  Hence, it may be feasible to put other message’s samples

between adjacent samples of this message on a time-shared basis.

o  This forms the time-division multiplex (TDM) system. n  A joint utilization of a common communication link by

a plurality of independent message sources.

44



o  The commutator (1) takes a narrow sample of each of the N input messages at a rate fs slightly higher than 2W, where W is the cutoff frequency of the anti-aliasing filter, and (2) interleaves these N samples inside the sampling interval Ts.



o  The price we pay for TDM is that N samples be squeezed in a time slot of duration Ts.

45



o  Synchronization is essential for a satisfactory operation of the TDM system. n  One possible procedure to synchronize the transmitter

and receiver clocks is to set aside a code element or pulse at the end of a frame, and to transmit this pulse every other frame only.


Example 3.2 The T1 System

o  T1 system n  Carries 24 64kbps voice channels with regenerative

repeaters spaced at approximately 2-km intervals. n  Each voice signal is essentially limited to a band from

300 to 3100 Hz. o  Anti-aliasing filter with W = 3.1 KHz o  Sampling rate = 8 KHz (> 2W = 6.2 KHz)

n  ITU G.711 µ-law is used with µ = 255. n  Each frame consists of 24 × 8 + 1 = 193 bits, where a

single bit is added at the end of the frame for the purpose of synchronization.

46


Example 3.2 The T1 System

n  In addition to the 193 bits per frame (i.e., 1.544 Megabits per second), a telephone system must also pass signaling information such as “dial pulses” and “on/off-hook.” o  The least significant bit of each voice channel is

deleted in every sixth frame, and a signaling bit is inserted in its place.

193 bit/frame⇥ 1

1 sample (from each of 24 voice channels)/frame

⇥ 8000 sample/sec = 1.544 Megabits/sec

(DS=Digital Signal)


3.10 Digital Multiplexers

o  The introduction of digital multiplexer enables us to combine digital signals of various natures, such as computer data, digitized voice signals, digitized facsimile and television signals.

47



o  The multiplexing of digital signals is accomplished by using a bit-by-bit interleaving procedure with a selector switch that sequentially takes a (or more) bit from each incoming line and then applies it to the high-speed common line.

(3 channels/frame)



o  Digital multiplexers are categorized into two major groups. 1.  1st Group: Multiplex digital computer data for TDM

transmission over public switched telephone network. n  Require the use of modem technology.

2.  2nd Group: Multiplex low-bit-rate digital voice data into high-bit-rate voice stream. n  Accommodate in the hierarchy that is varying from

one country to another. n  Usually, the hierarchy starts at 64 Kbps, named a

digital signal zero (DS0).

48


3.10 North American Digital TDM Hierarchy

o  The first level hierarchy n  Combine 24 DS0 to obtain a primary rate DS1 at 1.544

Mb/s (T1 transmission) o  The second-level multiplexer

n  Combine 4 DS1 to obtain a DS2 with rate 6.312 Mb/s o  The third-level multiplexer

n  Combine 7 DS2 to obtain a DS3 at 44.736 Mb/s o  The fourth-level multiplexer

n  Combine 6 DS3 to obtain a DS4 at 274.176 Mb/s o  The fifth-level multiplexer

n  Combine 2 DS4 to obtain a DS5 at 560.160 Mb/s



n  The combined bit rate is higher than the multiple of the incoming bit rates because of the addition of bit stuffing and control signals.

49



o  Basic problems involved in the design of multiplexing system n  Synchronization should be maintained to properly

recover the interleaved digital signals. n  Framing should be designed so that individual can be

identified at the receiver. n  Variation in the bit rates of incoming signals should be

considered in the design. o  A 0.01% variation in the propagation delay produced

by a 1℉ decrease in temperature will result in 100 fewer pulses in the cable of length 1000-km with each pulse occupying about 1 meter of the cable.



o  Synchronization and rate variation problems are resolved by bit stuffing.

o  Example 3.3. AT&T M12 (second-level multiplexer) n  24 control bits are stuffed, and separated by sequences

of 48 data bits (12 from each DS1 input).

50



Example 3.3 AT&T M12 Multiplexer

o  The control bits are labeled F, M, and C. n  Frame markers: In sequence of F0F1F0F1F0F1F0F1, where F0

= 0 and F1 = 1. n  Subframe markers: In sequence of M0M1M1M1, where M0 = 0

and M1 = 1. n  Stuffing indicators: In sequences of CI CI CI CII CII CII CIII CIII

CIII CIV CIV CIV, where all three bits of Cj equal 1’s indicate that a stuffing bit is added in the position of the first information bit associated with the first DS1 bit stream that follows the F1-control bit in the same subframe, and three 0’s in CjCjCj imply no stuffing. o  The receiver should use majority law to check whether a

stuffing bit is added.

51



o  These stuffed bits can be used to balance (or maintain) the nominal input bit rates and nominal output bit rates. n  S = nominal bit stuffing rate

o  The rate at which stuffing bits are inserted when both the input and output bit rates are at their nominal values.

n  fin = nominal input bit rate n  fout = nominal output bit rate n  M = number of bits in a frame n  L = number of information bits (input bits) for one input

stream in a frame


Example 3.3 AT&T M12 multiplexer

o  For M12 framing,

bits 288bits 1176244288

Mbps312.6 Mbps544.1

=

=+×=

=

=

LMff

out

in

ininout fLS

fLS

fM )1(1 framea of Duration

bit. stuffeda by replaced isbit One

−+−

==!"#

334601.01176312.6544.1288 =−=−=⇒ M

ffLSout

in

M

fout

= S4(L� 1)

4fin

+ (1� S)4L

4fin

L� 1

fin

= 185.88082902 µs M

fout

= 186.31178707 µs L

fin

= 186.52849741 µs

52



o  Allowable tolerances to maintain nominal output bit rates n  A sufficient condition for the existence of S such that

the nominal output bit rate can be matched.

⎥⎦

⎤⎢⎣

⎡−+

−≥≥⎥

⎦

⎤⎢⎣

⎡−+

−∈∈

ininS

outininS f

LSfLS

fM

fLS

fLS )1(1min)1(1max

]1,0[]1,0[

outinoutinoutin

fMLff

ML

fL

fM

fL 11 −

≥≥⇔−

≥≥⇔

54043.1312.61176287312.6

11762885458.1 =≥≥= inf



n  This results in an allowable tolerance range:

n  In terms of ppm (pulse per million pulses),

o  This tolerance is already much larger than the expected change in the bit rate of the incoming DS1 bit stream.

kbps 36735.51176/312.654043.15458.1 ==−

18.2312 and 8.11645458.1

10544.1

1054043.1

10 666

==⇒

+==

−

ppmppm

ppmppm

ba

ab

53


3.11 Virtues, Limitations, and Modifications of PCM

o  Virtues of PCM systems n  Robustness to channel noise and interference n  Efficient regeneration of coded signal along the transmission path n  Efficient exchange of increased channel bandwidth for improved

signal-to-noise ratio, obeying an exponential law. n  Uniform format for different kinds of baseband signal

transmission; hence, facilitate their integration in a common network.

n  Message sources are easily dropped or reinserted in a TDM system.

n  Secure communication through the use of encryption/decryption.


3.11 Virtues, Limitations, and Modifications of PCM

o  Two limitations of PCM system (in the past) n  Complexity n  Bandwidth

o  Nowadays, with the advance of VLSI technology, and with the availability of wideband communication channels (such as fiber) and compression technique (to reduce the bandwidth demand), the above two limitations are greatly released.

54


3.12 Delta Modulation

o  Delta Modulation (DM) n  The message is oversampled (at a rate much higher than

the Nyquist rate) to purposely increase the correlation between adjacent samples.

n  Then, the difference between adjacent samples is encoded instead of the sample value itself.


55


3.12 Math Analysis of Delta Modulation

Then. at time )( of ionapproximat DM thebe ][Let

).(][Let

sq

s

nTtmnmnTmnm =

]).1[][sgn(][ where

,][][]1[][

−−⋅Δ=

=+−= ∑−∞=

nmnmne

nenenmnm

qq

n

jqqqq

.}2/]1)/][[({ is wordcode ed transmittThe ∞−∞=+Δ nq ne

Chapter 3-110

o  The principle virtue of delta modulation is its simplicity. n  It only requires

the use of comparator, quantizer, and accumulator.

With bandwidth W of m(t)


]).1[][sgn(][ where

,][][]1[][

−−⋅Δ=

=+−= ∑−∞=

nmnmne

nenenmnm

qq

n

jqqqq

(m[n] = mq[n� 1] + e[n]

mq[n] = mq[n� 1] + eq[n]) mq[n]�m[n] = eq[n]� e[n] (See Slide 3-131)

56



o  Distortions due to delta modulation n  Slope overload distortion n  Granular noise




o  Slope overload distortion n  To eliminate the slope overload distortion, it requires

n  So, increasing step size Δ can reduce the slope-overload distortion.

n  Alternative solution is to use dynamic Δ. (Often, a delta modulation with fixed step size is referred to as a linear delta modulator due to its fixed slope, a basic function of linearity.)

condition) overload (slope )(maxdttdm

Ts≥

Δ

57



o  Granular noise n  mq[n] will hunt around a relatively flat segment of m(t). n  A remedy is to reduce the step size.

o  A tradeoff in step size is therefore resulted for slope overload distortion and granular noise.


3.12 Delta-Sigma Modulation

o  Delta-sigma modulation n  In fact, the delta modulation distortion can be reduced

by increasing the correlation between samples. n  This can be achieved by integrating the message signal

m(t) prior to delta modulation. n  The “integration” process is equivalent to a pre-

emphasis of the low-frequency content of the input signal.

58



n  A side benefit of “integration-before-delta-modulation,” which is named delta-sigma modulation, is that the receiver design is further simplified (at the expense of a more complex transmitter).

Move the accumulator to the transmitter.



A straightforward structure

Since integration is a linear operation, the two integrators before comparator can be combined into one after comparator.

59


3.12 Math Analysis of Delta-Sigma Modulation

]).1[][sgn(][ where],[]1[][ Then

. at time )()( of ionapproximat DM thebe ][Let

.)(][Let

−−⋅Δ=+−=

=

=

∫

∫

∞−

∞−

nininnnini

nTdmtini

dttmni

qqqqq

s

t

q

nTs

σσ

ττ

.}2/]1)/][[({ is wordcode ed transmittThe ∞−∞=+Δ nq nσ

,)()(]1[][]1[][][)1( s

nT

Tnqqq Ttmdttmninininin s

s

≈=−−≈−−= ∫ −σ

Since

we only need a lowpass filter to smooth out the received signal at the receiver end. (See the previous slide.)



o  Final notes n  Delta(-sigma) modulation trades channel bandwidth

(e.g., much higher sampling rate) for reduced system complexity (e.g., the receiver only demands a lowpass filter).

n  Can we trade increased system complexity for a reduced channel bandwidth? Yes, by means of prediction technique.

n  In Section 3.13, we will introduce the basics of prediction technique. Its application will be addressed in subsequent sections.

60


3.13 Linear Prediction

o  Consider a finite-duration impulse response (FIR) discrete-time filter, where p is the prediction order, with linear prediction

∑=

−=p

kk knxwnx

1

][][ˆ


3.13 Linear Prediction

o  Design objective n  To find the filter coefficient w1, w2, …, wp so as to

minimize index of performance J:

].[ˆ][][ where]],[[ 2 nxnxneneEJ −==

61

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+−=

−−+−−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

∑∑∑∑

∑∑∑

∑

== >=

= ==

=

p

kXk

p

k

p

kjXjk

p

kXkX

p

k

p

jjk

p

kk

p

kk

RwjkRwwkRwR

jnxknxEwwknxnxEwnxE

knxwnxEJ

1

2

11

1 11

2

2

1

]0[][2][2]0[

]][][[]][][[2]][[

][][

.)( function ationautocorrel withstatinoary be ]}[{Let kRnx X

0][2][2

]0[2][2][2][2

1

1

11

=−+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+−+−=

∂

∂

∑

∑∑

=

−

=+=

p

jXjX

Xi

i

kXk

p

ijXjX

i

jiRwiR

RwikRwjiRwiRJw


.1for ][][1

piiRjiRw X

p

jXj ≤≤=−∑

=


The above optimality equations are called the Wiener-Hopf equations for linear prediction. It can be rewritten in matrix form as:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−

−

][

]2[]1[

]0[]2[]1[

]2[]0[]1[]1[]1[]0[

2

1

pR

RR

w

ww

RpRpR

pRRRpRRR

X

X

X

pXXX

XXX

XXX

!!

"!#!!

……

or RXw = rX ⇒Optimal solution wo =RX−1rX

62


3.13 Toeplitz (Square) Matrix

o  Any square matrix of the form

is said to be Toeplitz. o  A Toeplitz matrix (such as RX) can be uniquely determined

by p elements, [a0, a1, …, ap-1].

pppp

p

p

aaa

aaaaaa

×−−

−

−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

021

201

110

!"#""

……


3.13 Linear Adaptive Predictor

o  The optimal w0 can only be obtained with the knowledge of autocorrelation function.

o  Question: What if the autocorrelation function is unknown? o  Answer: Use linear adaptive predictor.

63


3.13 Idea Behind Linear Adaptive Predictor

o  To minimize J, we should update wi toward the bottom of the J-bowel.

n  So when gi > 0, wi should be decreased. n  On the contrary, wi should be increased if gi < 0. n  Hence, we may define the update rule as:

where µ is a chosen constant step size, and ½ is included only for convenience of analysis.

ii w

Jg∂

∂≡

][21][ˆ]1[ˆ ngnwnw iii ⋅−=+ µ

o  gi[n] can be approximated by:

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−−=

−−+−−≈

−+−=∂∂≡

∑

∑

∑

=

=

=

p

jj

p

jj

p

jXjXii

jnxnwnxinx

inxjnxnwinxnx

jiRwiRwJng

1

1

1

][][ˆ][][2

][][][ˆ2][][2

)(2)(2/][


) wi[n+ 1] = wi[n] + µ · x[n� i]

0

@x[n]�

pX

j=1

wj [n]x[n� j]

1

A

= wi[n] + µ · x[n� i]e[n]

64


3.13 Structure of Linear Adaptive Predictor


3.13 Least Mean Square

o  The below pair results in the form of the popular least-mean-square (LMS) algorithm for linear adaptive prediction. 8

><

>:

wj [n+ 1] = wj [n] + µ · x[n� j]e[n]

e[n] = x[n]�pX

j=1

wj [n]x[n� j]

65


3.14 Differential Pulse-Code Modulation (DPCM)

o  Basic idea behind differential pulse-code modulation n  Adjacent samples are often found to exhibit a high

degree of correlation. n  If we can remove this adjacent redundancy before

encoding, a more efficient coded signal can be resulted. n  One way to remove the redundancy is to use linear

prediction. n  Specifically, we encode e[n] instead of m[n], where

e[n]=m[n]− m[n], where m[n] is the linear prediction of m[n].


3.14 DPCM

o  For DPCM, the quantization error is on e[n], rather on m[n] as for PCM.

o  So the quantization error q[n] is supposed to be smaller.

66


3.14 DPCM

o  Derive:

][][][ nqneneq +=

][][][][][ˆ

][][ˆ][

nqnmnqnenm

nenmnm qq

+=

++=

+=⇒

So we have the same relation between mq[n] and m[n] (as in Slide 3-110) but with smaller q[n].

][ˆ nm

][neq ][nmq


3.14 DPCM

o  Notes n  DM system can be treated as a special case of DPCM.

Prediction filter => single delay Quantizer => single-bit

67


3.14 DPCM

o Distortions due to DPCM n Slope overload distortion

o The input signal changes too rapidly for the prediction filter to track it.

n Granular noise


3.14 Processing Gain

o  The DPCM system can be described by:

o  So the output signal-to-noise ratio is:

o  We can re-write SNRO as:

][][][ nqnmnmq +=

]][[]][[

2

2

nqEnmESNRO =

QpO SNRGnqEneE

neEnmESNR ⋅==

]][[]][[

]][[]][[

2

2

2

2

error. prediction theis ][ˆ][][ where nmnmne −=

68



o  In terminologies,

⎪⎪⎩

⎪⎪⎨

⎧

=

=

ratio noise onquantizati tosignal ]][[]][[

gain processing ]][[]][[

2

2

2

2

nqEneESNR

neEnmEG

Q

p

Notably, SNRQ can be treated as the SNR for system of eq[n]= e[n]+ q[n].



o  Usually, the contribution of SNRQ to SNRO is fixed and limited. n  One additional bit in quantization will results in 6 dB

improvement. o  Gp is the processing gain due to a nice “prediction.”

n  The better the prediction is, the larger Gp is.

69


3.14 DPCM

o  Final notes on DPCM n  Comparing DPCM with PCM in the case of voice

signals, the improvement is around 4-11 dB, depending on the prediction order.

n  The greatest improvement occurs in going from no prediction to first-order prediction, with some additional gain resulting from increasing the prediction order up to 4 or 5, after which little additional gain is obtained.

n  For the same sampling rate (8KHz) and signal quality, DPCM may provide a saving of about 8~16 Kbps compared to standard PCM (64 Kpbs).


Speech Quality (Mean Opinion Scores)

Source: IEEE Communications Magazine, September 1997.

2 4 8 16 32 64 Unacceptable

Poor

Fair

Good

Excellent

Bit Rate (kb/s)

FS-1015

MELP 2.4 FS-1016

JDC2

G.723.1

GSM/2 JDC IS54 IS96

GSM

G.723.1 G.729 G.728 G.726 G.711

G.727 IS-641

PCM ADPCM

3.14 DPCM

IS = Interim Standard GSM = Global System for Mobile Communications JDC = Japanese Digital Cellular FS = Federal Standard MELP = Mixed-Excitation Linear Prediction

70


3.15 Adaptive Differential Pulse-Code Modulation

o  Adaptive prediction is used in DPCM. o  Can we also combine adaptive quantization into DPCM to

yield a comparably voice quality to PCM with 32 Kbps bit rate? The answer is YES from the previous figure. n  32 Kbps: 4 bits for one sample, and 8 KHz sampling

rate n  64 Kbps: 8 bits for one sample, and 8 KHz sampling

rate o  So, “adaptive” in ADPCM means being responsive to

changing level and spectrum of the input speech signal.


3.15 Adaptive Quantization

o  Adaptive quantization refers to a quantizer that operates with a time-varying step size Δ[n].

o  Δ[n] is adjusted according to the power of input sample m[n]. n  Power = variance, if m[n] is zero-mean.

n  In practice, we can only obtain an estimate of E[m2[n]].

]][[][ 2 nmEn ⋅=Δ φ

71


3.15 Adaptive Quantization

o  The estimate of E[m2[n]] can be done in two ways: n  Adaptive quantization with forward estimation (AQF)

o  Estimate based on unquantized samples of the input signals.

n  Adaptive quantization with backward estimation (AQB) o  Estimate based on quantized samples of the input

signals.


3.15 AQF

o  AQF is in principle a more accurate estimator. However it requires n  an additional buffer to store unquantized samples for the

learning period. n  explicit transmission of level information to the receiver

(the receiver, even without noise, only has the quantized samples).

n  a processing delay (around 16 ms for speech) due to buffering and other operations for AQF.

o  The above requirements can be relaxed by using AQB.

72


3.15 AQB

A possible drawback for a feedback system is its potential unstability. However, stability in this system can be guaranteed if mq[n] is bounded.


3.15 APF and APB

o  Likewise, the prediction approach used in ADPCM can be classified into: n  Adaptive prediction with forward estimation (APF)

o  Prediction based on unquantized samples of the input signals.

n  Adaptive prediction with backward estimation (APB) o  Prediction based on quantized samples of the input

signals. o  The pro and con of APF/APB is the same as AQF/AQB. o  APB/AQB are a preferred combination in practical

applications.

73


3.15 ADPCM

Adaptive prediction with backward estimation (APB).


3.16 Computer Experiment: Adaptive Delta Modulation

o  In this section, the simplest form of ADM modulation with AQB is simulated, namely, ADM with AQB.

o  Comparison with LDM (i.e., linear DM) where step size is fixed will also be performed.

][ne ][neq

]1[ −neq

This figure may be incorrect.

74



o  In this section, the simplest form of ADM modulation with AQB is simulated, namely, ADM with AQB.

o  Comparison with LDM (i.e., linear DM) where step size is fixed will also be performed.

][ne ][neq

]1[ −neq

I thus fix it in this slide.



⎪⎩

⎪⎨

⎧

Δ<−ΔΔ

Δ≥−Δ⎟⎟⎠

⎞⎜⎜⎝

⎛ −+×−Δ

=Δ

minmin

min

]1[ if,

]1[ if,][

]1[211]1[][

n

nnene

nnq

q

⎩⎨⎧

±

Δ

1. equalst output thaquantizer bit 1 theis ][ , iterationat size step theis ][

wherene

nn

q

Setting: m(t) =10sin 2π fs100

t⎛

⎝⎜

⎞

⎠⎟,ΔLDM =1 and Δmin =

18

75



LDM ADM

Observation: ADM can achieve a comparable performance of LDM with a much lower bit rate.

è

Fixed slope Exponentially decreasing slope

Exponentially increasing slope


3.17 MPEG Audio Coding Standard

o  The ADPCM and various voice coding techniques introduced above did not consider the human auditory perception.

o  In practice, a consideration on human auditory perception can further improve the system performance (from the human standpoint).

o  The MPEG-1 standard is capable of achieving transparent, “perceptually lossless” compression of stereophonic audio signals at high sampling rate. n  A human subjective test shows that a 6-to-1

compression ratio are “perceptually indistinguishable” to human.

76


3.17 Characteristics of Human Auditory System

o  Psychoacoustic characteristics of human auditory system n  Critical band

o  The inner ear will scale the power spectra of incoming signals non-linearly in the form of limited frequency bands called the critical bands.

o  Roughly, the inner ear can be modeled as 25 selective overlapping band-pass filters with bandwidth < 100Hz for the lowest audible frequencies, and up to 5kHz for the highest audible frequencies.



n  Auditory masking o  When a low-(power-)level signal (i.e., the maskee)

and a high-(power-)level signal (i.e., the masker) occur simultaneously in the same critical band, and are close to each other in frequency, the low-(power-)level signal will be made inaudible (i.e., masked) by the high-(power-)level signal, if the low-(power-)level one lies below a masking threshold.

77


SMR SNR for R-bit quantizer


o  Masking threshold is frequency-dependent.

NMR (noise-to-mask ratio) = SMR – SNR

(in the gray-color critical band)

Within a critical band, the quantization noise is inaudible as long as the NMR for the pertinent quantizer is negative.



78



o  Time-to-frequency mapping network n  Divide the audio signal into a proper number of subbands, which is

a compromise design for computational efficiency and perceptual performance.

o  Psychoacoustic model n  Analyze the spectral content of the input audio signal and thereby

compute the signal-to-mask ratio. o  Quantizer-coder

n  Decide how to apportion the available number of bits for the quantization of the subband signals.

o  Frame packing unit n  Assemble the quantized audio samples into a decodable bit stream.


3.18 Summary and Discussion

o  Sampling – transform analog waveform to discrete-time continuous wave n  Nyquist rate

o  Quantization – transform discrete-time continuous wave to discrete data. n  Human can only detect finite intensity difference.

o  PAM, PDM and PPM o  TDM (Time-Division Multiplexing) o  PCM, DM, DPCM, ADPCM o  Additional consideration in MPEG audio coding

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