10/3/03 EC2500MPF/FallFY04 42

II.B Baseband Transmission

Digital Baseband ReceptionMatched filter

DefinitionImplementation (correlator)

Application to digital baseband signalM-ary Baseband ReceptionBrief Review of Probability and Noise Concepts

Basic pdf definitionsWhite noiseNarrowband noise

Bit Error RateError probabilityThreshold definitionApplication to the binary matched filter detectorExamples

M-ary baseband PerformanceApplication to the CD Format

Speech rangeD/A converter issuesOversamplingNoise shaping & Dither effects

(Reception & Applications)

10/3/03 EC2500MPF/FallFY04 43

10) Digital Baseband Reception

• Goal: to recover s(t) from potentially noisy received signal

– use a filter to decrease the effect of noise

• Generic filter does not take advantage of known signal shape transmitted

better result obtained when using that information

“matched filter”

s(t)

�V

�VTb

t…

compareto zero

12 bn T� �

�� �� �

10/3/03 EC2500MPF/FallFY04 44

• Matched Filter

� Definition: a matched filter is a linear filter which minimizes the output signal to noise ratio (SNR) � at time T, where ��is definedas:

� �

� �

� � � �

� � � �

2020

22

2

j fT

n

s Tn t

S f H f e df

S f H f df

�

�

�

��

��

��

��

�

�

�

�

� Goal: find H(f) which minimizes �

� Proof: Use Schwartz’s inequality which states:

equality holds only when A(n) = KB*(x)K real constant

� � � � � � � �2

2 2A x B x dx A x dx B x dx�� � �

H (f)s(t) + n(t) s0(t) + n0(t)T

10/3/03 EC2500MPF/FallFY04 45

� � � �

� � � �

22

2

j fT

n

S f H f e df

S f H f df

�

�

��

��

��

��

�

�

�

� �

� �� � � �

� �

� �� � � �

22

2

222 2

j fTn

n

j fTn

n

S fdf H f S f e df

S f

S fdf H f S f e df

S f

�

�

�� ��

�� ��

�� ��

�� ��

� �

� �

� �

� �

� �

� �� � � �

� � � �

22

2

nn

n

S f dfH f S f df

S f

S f H f df�

��

��

��

��

�

� �

� �

�

� � � �� �

� �� � � �

22

2 2j fT j fTn

n

S fS f H f e df H f S f e df

S f� �

�� �

BA

� �

� �

2

n

S f dfS f

���

��

� � �

� �

� �� � � �* 2j fT

nn

S fKH f S f e

S f��

�

� �� �

� �

*2j fT

n

S fH f K e

S f��

� �

�max obtained when

10/3/03 EC2500MPF/FallFY04 46

� Example:

When n(t) is white noise �

� � � �

� � � �

* 2j fTH f KS f e

h t Ks T t

��� �

�

� �

� � 2n nS f ��

1

Tt

s(t)

t

h(t)

10/3/03 EC2500MPF/FallFY04 47

� Matched Filter Implementation (correlator)

� � � � � �� � � �

� � � �� � � �

� � � �� � � �

y t s t n t h t

s n h t d

s n s d

� � � �

� � �

��

��

��

��

� � �

� � �

� �

�

�

�

� � � �h Ks T� �� �

H (f)s(t) + n(t)

Ty(t)

�T

10/3/03 EC2500MPF/FallFY04 48

• Matched Filter Detector for Digital Baseband

...

Tb

s(t)

� �0

.bT

dt�y(T)s(t)

10/3/03 EC2500MPF/FallFY04 49

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10/3/03 EC2500MPF/FallFY04 51

• M-ary Baseband Reception

– We can transmit more than two symbols

– Example: 4-ary baseband communications

– How to apply binary result to M-ary case ?

10/3/03 EC2500MPF/FallFY04 52

11) Brief Review of Probability and Noise Concepts

• Probability distribution function F (x) =

• Probability density function f (x)

• Expected value E (x) =

• Variance 2x� �

• Basic pdfs

(1) Uniform pdf

f (x)

x

f (x) =

10/3/03 EC2500MPF/FallFY04 53

(2) Gaussian pdf

f (x)

x

f (x) =

10/3/03 EC2500MPF/FallFY04 54

x(n) is random with a constant PSD

• White Noise

Gx(f)

f

N0/2

• Narrowband Noise

– most communication systems contain bandpass filters

– white noise gets transformed into BP noise

– when noise band is small compared to center frequency fc

BP noise called narrowband noise

expressed as:

� � � � � � � � � �

� � 2

cos 2 sin 2

Re c

c c

j f t

w t x t f t y t f t

r t e �

� �� �

� �� � �

complex function

Quadraticnoisecomponents

10/3/03 EC2500MPF/FallFY04 55

� �

out

2V

P

E V n

�

�

�� �� �

�

w(n) v(n)BPF

� �20, wN �

–fL fL fH–fHf

H(f)white noise

10/3/03 EC2500MPF/FallFY04 56

12) Bit Error Rate

• Evaluate errors made during transmission of hits

• Errors occur when:

receive “1” when “0” is sentreceive “0” when “1” is sent

• How to decide if you receive a “1” or “0” when transmission is noisy ??

detection theory

1

PM PFA

1

0 0Send Receive

10/3/03 EC2500MPF/FallFY04 57

r(t)=s(t)+Kw(t)

N(0,�w2=1)

10/3/03 EC2500MPF/FallFY04 58

• f (w) =

• f (r|s0) =

• f (r|s1) =

For which range of “r” do you decide you sent a

“1” (s1) “0” (s0)

�0

r

• Assume you have additive white noise distortion atthe receiver

r(t)=si(t)+w(t), si(t)=s0 , s1

How to pick �0 ?

10/3/03 EC2500MPF/FallFY04 59

• Goal: To quantify likelihood of making an error in assigning bit values at the

receiver.

• Statistics of r(n)

– mean of r(n) ?

– r(n) deterministic ?random ?

Assume transmission is noisy

� � � � � �r n s n w n� �

receivedsignal

transmission noise(random Gaussian)

signalsent

� �20, wN �0

1

ss

���

10/3/03 EC2500MPF/FallFY04 60

• Notation

H1: receive a “1” (s1)

H0: receive a “0” (s0)

� �

� �1 1

0 0

P H s

P H s

�

�

• Correct decisions:

� �

� �

1 0

0 1

P H s

P H s

�

�

• Incorrect decisions:

eP �

• Overall probability of error:

• Pe when there is equal probability of sending s0 & s1

10/3/03 EC2500MPF/FallFY04 61

2

2

/ 2

0

1 1( ) erfc22 2

2erfc( ) 1 1 erf ( )

u

xx

u

xQ x e du

x e du x

�

�

�

�

�

� �� � � �

� �

� � � �

�

�

• Definitions: Q, erf, & erfc functions

• Assume s0=-V & s1=+V

0

1 0 0 1 012 ( | ) ( | ) ( | )2eP P H s P H s f r s dr

�

�

� � � �

10/3/03 EC2500MPF/FallFY04 62

BER for single sample detector

10/3/03 EC2500MPF/FallFY04 63

• How to select the threshold �0 ?

Maximum likelihood detector approach

Minimize the overall probability of error Pe

1 0 0 0 1 1( | ) ( ) ( | ) ( )eP P H s P s P H s P s� �

�0

r

20 1 0

01 1 0

In2

ws s PP s s

��

� ��� � � �

��

(More details in EC4570…)

10/3/03 EC2500MPF/FallFY04 64

• Application to Binary Matched Filter Detector

• Need statistics on y = y1 – y0

• pdf of y: ?

s1(t)

s0(t)

r(t)

y1

y0

y

Tb

compare tothreshold�

� �0

bTdt���

�

�

�

� �0

bTdt��

Is y random or deterministic?

r(t)=si(t)+w(t)

10/3/03 EC2500MPF/FallFY04 65

� � � �

� � � �� � � � � � � � � �

� � � � � � � �

1 10

1 1 00

1 10 0

or

b

b

b b

T

T

i i

T T

i

y r t s t dt

s t w t s t dt s t s t s t

s t s t dt w t s t dt

�

� � �

� �

�

�

� ��������

� � � �

0

1 0

y

y y t y t

�

� �

�

10/3/03 EC2500MPF/FallFY04 66

10/3/03 EC2500MPF/FallFY04 67

10/3/03 EC2500MPF/FallFY04 68

Bit error rate for matched filter detector

0 10

2 20 1

0 0

( ) ( )

1 ( ) ( )2

b

b b

T

T T

s t s t dt

E

E s t dt s t dt

� �

� �� �� �� �

� �

�

� �

10/3/03 EC2500MPF/FallFY04 69

• How to compute the threshold��0 ?

Recall for simple detector, the threshold was selected asthe mid point between the two means for basicproblem.How can we apply the result here ?

E[y|s0]=

E[y|s1]=

10/3/03 EC2500MPF/FallFY04 70

• Example

• Design a matched filter detector for the two signals.

• Find Pe when the additive white noise has apower P = 10–3 w/Hz.

T = 8 10–3 s

T

–1t

s0(t)

+1

Tt

s1(t)

+1

10/3/03 EC2500MPF/FallFY04 71

10/3/03 EC2500MPF/FallFY04 72

10/3/03 EC2500MPF/FallFY04 73

• Example

• Design a matched filter detector for the two signals.

• Find Pe when the additive white noise has a powerPe = 0.1 w/Hz.

t1

–1

s0(t)

+1 0.5 1t

s1(t)

+1

–1

sin(2�t)

10/3/03 EC2500MPF/FallFY04 74

10/3/03 EC2500MPF/FallFY04 75

• M-ary Baseband Performance

Assume we send M = 4 different levels(Bi= 0, A, 2A, 3A, i=0,…3)

r(t) ?� �0

bTdt��

Assume additive Gaussian noise r(t)=s(t)+n(t)

10/3/03 EC2500MPF/FallFY04 76

• P(error in receiving B2)=

10/3/03 EC2500MPF/FallFY04 77

2 2

20 0

P(error in receiving B ) 22 4

s sA T A TQ erfc

N N

� � � �� � � �� � �� � � �� � � �

• P(error in receiving B1)=

• P(error in receiving B0)=

• P(error in receiving B3)=

• Assume each error may occur as likely as theothers

Pe=

10/3/03 EC2500MPF/FallFY04 78

• Assume we send M different levels, computethe overall probability of error becomes:

10/3/03 EC2500MPF/FallFY04 79

� Dynamic range for audio signals Ref [3]

13) Application: Compact Disk

10/3/03 EC2500MPF/FallFY04 80

� Signal reproduction in CD Player

Digitaloversampling

filterX4

NoiseshaperH(f)Digital

input16 bits

44.1 KHz

x(n)y(n)

28 bits176.1 KHz

14 bits

14 bitsD/A LPF

–20 KHzf

xa(n)

20 KHz

88.2 KHzf

x(n)

44.1 KHz0

10/3/03 EC2500MPF/FallFY04 81

• Why use oversampling ?

• First, assume we don’t use oversampling.

x(r)z(t) y(t)LPF

Hr(j�)14 bitsD/A

001

anal

og o

utpu

t

010 011 100 101 110 111000

7654321

0

Input/outputrelationship fora unipolar D/A(3 bits)converter.

Ideal D/A Converter

t

x1(t)

t

z(t)

Practical D/A Converter

(p. 331)

x(n)x1(t) z(t)Hz(f)

hz(t)

tT

1

10/3/03 EC2500MPF/FallFY04 82

� D/A Converter Output Expression

� � � � � �

� � � � � �

1

1

k

z

x t x k t kT

z t x t h t

�� �

� �

�

� �

� �

� �

0

2

1 1

sin 22

T j tz

j T

j T

H e dt

ej

Te

�

�

�

�

�

�

�

�

�

�

�

� ��

� �� � �

� �

�

�s 2�s 3�s

�

|X1(j�)|

�s 2�s 3�s

�

|Z(j�)|

�s 2�s 3�s

�

|Hz(j�)|

� � � � � �1 zZ j X j H� � ��

2 /s T� ��

10/3/03 EC2500MPF/FallFY04 83

• Need for LPF filter ?

– to smooth out output steps

– to undo distortion added by D/A converter

�s 2�s

�

|Z(j�)|

�

|Hr(j�)|

�

Hr(j�)

10/3/03 EC2500MPF/FallFY04 84

• Oversampling in the Time Domain

Digitaloversampling

filterX4

NoiseshaperHs(f)16 bits

44.1 KHz

x(n)y(n)

176.1 KHz

14 bitsD/A LPF

a(n)

input to upsampler by 4

output to upsampler by 4

output of FIR filter

n�4321

010

n210

n�

001 010 � 001 000 000 000 010

x(n)y2(n)

y(n)� 4 FIR filter

10/3/03 EC2500MPF/FallFY04 85

f

Xa(f)

(KHz)

X(f)

f(KHz)132.3 176.488.244.10

Y1(f)

f(KHz)176.40

f

Y(f)

(KHz)

f

A(f)

(KHz)

without oversampling

with oversampling

Y1(f) after FIR filter

after analog LPF

� Advantages of Oversampling

10/3/03 EC2500MPF/FallFY04 86

• Example:

Assume1) you have an analog signal which spans [0 20KHz],2) the D/A converter has a sampling frequency fs=176.4KHz.Determine the characteristics (order and cutoff frequency) forthe anti-imaging Butterworth type filter which satisfy thefollowing specifications:

1) Image frequencies must be attenuated by at least 40dB2) Signal components may be altered by a maximum of0.5dB

� �1/ 22

1( )1 / n

c

H ff f

�� ��� �

10/3/03 EC2500MPF/FallFY04 87

10/3/03 EC2500MPF/FallFY04 88

– Goal: to decrease noise in the audio band

� Noise Shaping Filterno

ise

leve

l

noise shapingcharacteristic

noise levelwithout shaping

KHz

f20 88

10/3/03 EC2500MPF/FallFY04 89

�Dither

Figure 2.121 - Coding of Dithered SignalFigure 2.121 - Coding of Dithered Signal

Figure 2.122 - Fourier Transform of Dithered SignalFigure 2.122 - Fourier Transform of Dithered Signal

10/3/03 EC2500MPF/FallFY04 90

Poh1man Fig. 6-27 An example of noise shaping showing a 1kHz sinewave with -90 dB amplitude; measurements are madewith a 16 kHz lowpass filter.A. Original 20 bit recording.B. Truncated 16 bit signal.C. Dithered 16 bit signal.D. Noise shaping preserves information in lower 4 bits.

Ref [4,5]

�Dither & noise shaping effects

10/3/03 EC2500MPF/FallFY04 91

CD Section References:

[1] S. Mitra, Digital Signal Processing, McGraw-Hill, 1998.[2] K. Pohlmann, A. Red, Compact Disk Handbook, 2, 1992.[3] H. Nakajima and H. Ogawa, Compact Disk Technology, IOS Press, 1992.[4] B. Evans, Real Time Digital Signal Processing Lab, Fall 2003 (lecture 10 notes).[5] K. Pohlmann, Principles of Digital Audio, McGraw-Hill, 1995