pulse shaping and matched filters haitham hassanieh

ECE 463: Digital Communications Lab.

Lecture 3: Pulse Shaping and Matched FiltersHaitham Hassanieh

Previous Lecture:ü Up Conversion & Down Conversionü PAM vs QAM Spectral Efficiencyü Software Defined Radios

This Lecture:q Pulse Shaping Filtersq Matched Filtersq Symbol Timing Recoveryq Eye Diagrams

Digital Communication System

TX RX

1011010110011001 1011010110011001

Bits-to-SymbolsMapper

Bits

LPF BPF

PA

PLL

Mixer

Pulse Shaping

DACModulation (Encoding)

Symbols-to-Bits Mapper

Bits

LPFBPFLNA

PLL

Mixer

ChannelEqualization

&Synchronization

ADC Demodulation(Decoding)


LPF BPF

PLL

Mixer

LPFBPF

PLL

Mixer

Transmitter

Receiver

LPF BPFMixer

+90%

LPFBPF Mixer

90%


Bits

PA

Pulse Shaping

DAC

Modulation (Encoding)

DAC

ℜ'{}

ℑ+{}

LNA

ADC


Bits

ChannelEqualization

&Synchronization

Demodulation(Decoding)

ADC

+,

-.


TX RX

1011010110011001 1011010110011001


Bits

LPF BPF

PA

PLL

Mixer

Pulse Shaping



Bits

LPFBPFLNA

PLL

Mixer

ChannelEqualization

&Synchronization

ADC Demodulation(Decoding)


TX RX

1011010110011001 1011010110011001


Bits

LPF BPF

PA

PLL

Mixer

Pulse Shaping



Bits

ChannelEqualization

&Synchronization


LPFBPFLNA

PLL

Mixer

ADC


TX RX

1011010110011001 1011010110011001


Bits

LPF BPF

PA

PLL

Mixer

Pulse Shaping



Bits

ChannelEqualization

&Synchronization


LPFBPFLNA

PLL

Mixer

ADC MatchedFilter

Symbol TimingRecovery

PAM: Pulse Amplitude Modulation

−3 +3+1−1

00 011011

1011000110011001



-1,-3,+1,+3,-1,+1,-1

& ' :

) ' :

* + = - ) ' . + − '/0 12

3452

• /0: Symbol Time/ Baud Time

• 70 = 89:

: Symbol Rate or Baud Rate

• . + : pulse shape

Pulse Shaping

−3 +3+1−1

00 011011

1011000110011001



-1,-3,+1,+3,-1,+1,-1

& ' :

) ' :

* + = - ) ' . + − '/0 12

3452

• Simplest pulse shape: Rectangle

. + = Π 89:

/0+

1

++1

−1

−3

+3

. +

* +

Pulse Shaping: Rectangular Pulse

!"#

1

% # = Π ()* + , = !"sinc 1!",

,

!"

− 1!"

1!"


!"#

1

% # = Π ()* + , = !"sinc 1!",

,

!"

− 1!"

1!"

1/,


!"#

1

% # = Π ()* + , = !"sinc 1!",

,

− 1!"

1!"

1/,

,

− 1!"

1!"

+ , 4

≈ −1378Very wide bandwidth!Low spectral efficiency


!+1

−1

−3

+3

Satisfies the Nyquist Criterion For Inter-Symbol-Interference

Sampled Values: & ' = −1,−3,+1,+3,−1,+1,−1…& ' = ,['] No Inter-Symbol-Interference (ISI)

ISI: & ' = , ' + / 01 , ' − 1 + / 201 , ' − 2 +…

No ISIRectangular Pulse


!"#

1

% # = Π ()* + , = !"sinc 1!",

,

− 1!"

1!"

1/,

,

− 1!"

1!"

+ , 4

≈ −1378Very wide bandwidth!Low spectral efficiency No Inter-Symbol-Interference


!"#

1

% # = Π ()* + , = !"sinc 1!",

,

− 1!"

1!"

1/,

,

− 1!"

1!"

+ , 4

#

% # ∝ sinc 46()*


!

− 1$%

1$%

& ! '

(

) ( ∝ sinc '/012


!

− 1$%

1$%

& ! '

(

) ( ∝ sinc '/012

Infinite Responseà Impossible to realize in practice

Bandlimited in FrequencyNo ISI if sampling is alignedHigh ISI if sampling is not perfect

Pulse Shaping: Raised Cosine


!−1 − $2&'

1 − $2&'

( ! =

&' ! ≤ 1 − $2&'

&'2 1 + cos 0&'

$ ! − 1 − $2&' 1 − $

2&'≤ ! ≤ 1 + $

2&'

0 otherwise

&'

( !


!−1 − $2&'

1 − $2&'

( ! =

&' ! ≤ 1 − $2&'

&'2 1 + cos 0&'

$ ! − 1 − $2&' 1 − $

2&'≤ ! ≤ 1 + $

2&'

0 otherwise

&'

−1 + $2&'1 + $2&'

( !


!−1 − $2&'

1 − $2&'

( ! =

&' ! ≤ 1 − $2&'

&'2 1 + cos 0&'

$ ! − 1 − $2&' 1 − $

2&'≤ ! ≤ 1 + $

2&'

0 otherwise

&'

−1 + $2&'1 + $2&'

&'/2

12&'

−12&'

( !


! " =

$% " ≤ 1 − *2$%

$%2 1 + cos 0$%

* " − 1 − *2$% 1 − *

2$%≤ " ≤ 1 + *

2$%

0 otherwise

"−1 − *2$%

1 − *2$%

$%

−1 + *2$%1 + *2$%

$%/2

12$%

−12$%

! "

"−1 − *2$%

1 − *2$%

−1 + *2$%1 + *2$%

12$%

−12$%

! " 9


! " =

$% " ≤ 1 − *2$%

$%2 1 + cos 0$%

* " − 1 − *2$% 1 − *

2$%≤ " ≤ 1 + *

2$%

0 otherwise

"−1 − *2$%

1 − *2$%

$%

−1 + *2$%1 + *2$%

$%/2

12$%

−12$%

! "

"−1 − *2$%

1 − *2$%

−1 + *2$%1 + *2$%

12$%

−12$%

! " 9

−3;<cut-off


! " =

$% " ≤ 1 − *2$%

$%2 1 + cos 0$%

* " − 1 − *2$% 1 − *

2$%≤ " ≤ 1 + *

2$%

0 otherwise

"−1 − *2$%

1 − *2$%

$%

−1 + *2$%1 + *2$%

$%/2

12$%

−12$%

! "

"−1 − *2$%

1 − *2$%

−1 + *2$%1 + *2$%

12$%

−12$%

! " 9

−3;<cut-off

1$%


! " =

$% " ≤ 1 − *2$%

$%2 1 + cos 0$%

* " − 1 − *2$% 1 − *

2$%≤ " ≤ 1 + *

2$%

0 otherwise

"−1 − *2$%

1 − *2$%

$%

−1 + *2$%1 + *2$%

$%/2

12$%

−12$%

! "

"−1 − *2$%

1 − *2$%

−1 + *2$%1 + *2$%

12$%

−12$%

! " 9

−3;<cut-off

1$%

*2$%


!"

!"/2

12!"

−12!"

' (

(


!"

!"/2

12!"

−12!"

' (

(

) = 0.5

' ( =

!" ( ≤ 14!"

!"2 1 + cos 25!" ( − 52 1

4!"≤ ( ≤ 3

4!"

0 otherwise


!"

!"/2

12!"

−12!"

' (

(

) = 0.5) = 0.25

' ( =

!" ( ≤ 38!"

!"2 1 + cos 47!" ( − 372 3

8!"≤ ( ≤ 5

8!"

0 otherwise


!"

!"/2

12!"

−12!"

' (

(

) = 0.5) = 0.25

) = 1

' ( =

!"2 1 + cos 3!" ( ( ≤ 1

!"

0 otherwise


!"

!"/2

12!"

−12!"

' (

(

) = 0.5) = 0.25

) = 1

) = 0

' ( =

!" ( ≤ 12!"

0 otherwise


!"

!"/2

12!"

−12!"

' (

(

) = 0.5) = 0.25

) = 1

) = 0

' ( =

!" ( ≤ 1 − )2!"

!"2 1 + cos 4!"

) ( − 1 − )2!" 1 − )

2!"≤ ( ≤ 1 + )

2!"

0 otherwise

): Rolloff Factor↑ ) ∶ ↑ Bandwidth leakage↓ ) ∶ ↓ Bandwidth leakage


!"

!"/2

12!"

−12!"

' (

(

) = 0.5) = 0.25

) = 1

) = 0

' ( =

!" ( ≤ 1 − )2!"

!"2 1 + cos 4!"

) ( − 1 − )2!" 1 − )

2!"≤ ( ≤ 1 + )

2!"

0 otherwise

; < = sinc </!"cos 4)</!"1 − 4)?<?/!"?



!"

!"/2

12!"

−12!"

' (

(

) = 0.5) = 0.25

) = 1

) = 0

. / = sinc //!"cos 5)//!"1 − 4)8/8/!"8

/

. /



!"

!"/2

12!"

−12!"

' (

(

) = 0.5) = 0.25

) = 1

) = 0


2 3 = sinc 3/!"cos 9)3/!"1 − 4)<3</!"<

3

2 3


!"

!"/2

12!"

−12!"

' (

(

) = 0.5) = 0.25

) = 1

) = 0

): Rolloff Factor↑ ) ∶ ↑ Bandwidth leakage, ↓ Time Support, ↓Sidelobes à ↓ISI if sampling not aligned ↓ ) ∶ ↓ Bandwidth leakage, ↑ Time Support, ↑ Sidelobes à ↑ ISI if sampling not aligned

3 4 = sinc 4/!"cos :)4/!"1 − 4)<4</!"<

4

3 4


TX RX

1011010110011001 1011010110011001


Bits

LPF BPF

PA

PLL

Mixer

Pulse Shaping



Bits

ChannelEqualization

&Synchronization


LPFBPFLNA

PLL

Mixer

ADC MatchedFilter



TX RX

1011010110011001 1011010110011001

Pulse Shaping

MatchedFilter

Pulse Shaping and Matched Filtering

TX RX

1011010110011001 1011010110011001

Pulse Shaping

MatchedFilter


TX RX

1011010110011001 1011010110011001

MatchedFilter

ChannelPulse

Shaping


TX RX

1011010110011001 1011010110011001

MatchedFilter

Channel

= " # $ % & − $() *+

,-.+# $ 0 &

Pulse Shaping

Let us focus on a single symbol i.e., single $


TX RX

1011010110011001 1011010110011001

MatchedFilter

Channel

= " # ∗ % #

" # = " & ' # − &)*

" & + #

Pulse Shaping


TX RX

1011010110011001 1011010110011001

MatchedFilter

Channel

= " # ∗ %& #

" # = " ' ( # − '*+

" ' , #

Pulse Shaping


TX RX

1011010110011001 1011010110011001

MatchedFilter

Channel

! " = $ " + &(")

Consider Simple AWGN Channel. &(")is Gaussian noise.

Pulse Shaping

= * " ∗ ,- "* . $ "


TX RX

1011010110011001 1011010110011001

MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " = ) - 0 " − -23 ∗ +, " ∗ +/ " + & " ∗ +/ "


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " = ) - 0 " − -23 ∗ +, " ∗ +/ " + & " ∗ +/ "


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " = ) - 0 " − -23 ∗ +, " ∗ +/ " + & " ∗ +/ "

!.4 " !.5 "

674 8 = ) - 9:;<=>,?@ ⋅ B, 8 ⋅ B/ 8

!.4 " = ℱ:D 674 8 = E674 8 9;<=F@ G8

!.4 -23 = E674 8 9;<=>,?@ G8 = E) - 9:;<=>,?@B, 8 B/ 8 9;<=>,?@ G8


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " = ) - 0 " − -23 ∗ +, " ∗ +/ " + & " ∗ +/ "

!.4 " !.5 "

674 8 = ) - 9:;<=>,?@ ⋅ B, 8 ⋅ B/ 8

!.4 " = ℱ:D 674 8 = E674 8 9;<=F@ G8

!.4 -23 = E674 8 9;<=>,?@ G8 = ) - EB, 8 B/ 8 G8


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " = ) - 0 " − -23 ∗ +, " ∗ +/ " + & " ∗ +/ "

!.4 " !.5 "

!.4 -23 = ) - 67, 8 7/ 8 98

:;< = !.4 -23 >

? !.5 -23 >

? !.5 -23 > = ? & " ∗ +/(") >

= 6 @ 8 7/ 8 >98

= ;A2 6 7/ 8 >98


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " = ) - 0 " − -23 ∗ +, " ∗ +/ " + & " ∗ +/ "

!.4 " !.5 "

!.4 -23 = ) - 67, 8 7/ 8 98

:;< = !.4 -23 >

? !.5 -23 >

? !.5 -23 > = ;@2 6 7/ 8 >98

= ) - > ∫7, 8 7/ 8 >

;@2 ∫ 7/ 8 >

GOAL: Find 7/ 8 that maximizes the SNR


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

012 = ) - 4 ∫6, 7 6/ 7 4

182 ∫ 6/ 7 4


:6, 7 6/ 74≤ : 6, 7 4 ⋅ : 6/ 7 4

Cauchy-Schwarz Inequality :

= : 6, 7 4 ⋅ : 6/ 7 4 =76/ 7 = > ⋅ 6,∗ 7


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

012 = ) - 4 ∫6, 7 6/ 7 4

182 ∫ 6/ 7 4


SNR is maximized when: 6/ 7 = : ⋅ 6,∗ 7 +/ " = : ⋅ +,∗ −"

+/ " matches +, " … hence, the name matched filter !

012 =) - 4 ∫ 6, 7 4 ∫ 6/ 7 4

182 ∫ 6/ 7 4

= 2 ) - 4

18


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " = ) - 0 " − -23 ∗ +, " ∗ +/ " + & " ∗ +/ "


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " = ) - 0 " − -23 ∗ +, " ∗ +,∗ −" + & " ∗ +,∗ −"

+ " should satisfy Nyquist Criterion for ISI

• Let + " be a raised cosine• What is +, " ?

4 5 = ℱ +(") = ℱ +, " ∗ +,∗ −" = 4, 5 ⋅ 4,∗ 5 = 4, 5 8 = 49: 5

4, 5 = 49: 5 Use square-root of raised cosine filter = 4;99: 5


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " = ) - 0 " − -23 ∗ +, " ∗ +,∗ −" + & " ∗ +,∗ −"

+ " should satisfy Nyquist Criterion for ISI

• Let + " be a raised cosine• What is +, " ?Square-root of raised cosine filter: 45667 8 = 467 8

+5667 " = − 123

sin 1 − = >"23 + 4="23 cos 1 + = >"

23>"23 1 − 4="

23B

Symbol Timing Recovery

MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " = ) - 0 " − -23 ∗ +, " ∗ +,∗ −" + & " ∗ +,∗ −"

+ "


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " = ) - + " − -12


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " =0) - + " − -234

Sample signal at " = 523

!. 523 =0) - + 523 − -234

+ " satisfies Nyquist à + 523 − -23 = 60 - ≠ 51 - = 5

= )[5] (NO ISI)


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " =0) - + " − -234

In practice, we have sampling offset: Sample signal at " = 523 + 6

!. 523 + 6 =0) - + 523 − -23 + 64

≠ )[5] (ISI)

= ) 5 + 6 + 0 ) - + 523 − -23 + 64:;

ISI<1


!

" !


!

" ! " ! − $%


!

" ! " ! − 2%&


!

" ! " ! − 3%&


!

" ! " ! − $%&


!

" ! " ! − $%&

0 %& 2%& 3%& 4%& 5%&


!

" ! " ! − $%&

0 %& 2%& 3%& 4%& 5%&

,%&,%& + .

Must correct timing offset to avoid ISI!

Power is maximum when . = 0


MatchedFilter

Channel

! " = $ " + &(")

Pulse Shaping

= ) " ∗ +, ") - $ " !. " = ! " ∗ +/(")

!. " =0) - + " − -234

In practice, we have sampling offset: Sample signal at " = 523 + 6

Compute: 7 8 = !. " + 8 9


Find: 8; = argmax7 8Shift samples by :8;

MatchedFilter

Symbol Sync

AB;

8;

Eye Diagram

Qualitative tool to evaluate: • Quality of Pulse• ISI• Distortion• Noise

Generated by fragmenting the received signal !(#)into overlapping fragments of length 2'(and overlaying all the fragments.

Eye Diagram

Eye Diagram: Raised Cosine ! = 1

$ %

& %

Eye Diagram: Raised Cosine ! = 0.5

& '

( '

Eye Diagram: Rectangle

! "

# "

Eye DiagramRaised Cosine ! = 1 Raised Cosine ! = 0.5

Rectangular

Eye Diagram

! "

# "

Eye Diagram

Eye Diagram: 4PAM

10 20 30 40 50 60-6

-4

-2

0

2

4

6Eye diagram for sinc pulse shape

10 20 30 40 50 60-6

-4

-2

0

2

4

6Eye diagram for SRRC (0.5) pulse shape

Definitions & Variables• ![#]: Bit steam

• &[#]: Symbols

• #: Symbol index at TX

• ': Sampling index at RX

• ((*): Baseband Signal

• ,(*): Pulse Shape

• -(.): Frequency Spectrum of pulse

• /[#]: Sampled values at the receiver.

• 01: Symbol Time

• 21: Symbol Rate

• Π : Rectangle Function

• sinc : Sinc Function

• 8: Roll-off Factor of Raised Cosine

• ∗:Complex Conjugate

• :Magnitude

• ; :Expectation

• ℱ : Fourier Transform

• ℱ=> : Inverse Fourier Transform

• ,? * : Transmitter Pulse

• ,@ * : Receiver Pulse

• A * : Additive Gaussian Noise

• -? . : Spectrum of TX pulse

• -@ . : Spectrum of RX pulse

• -BC . : Spectrum of raised cosine pulse

• -DBBC . : Spectrum of square root raised cosine.

• E . : Frequency Spectrum of Noise

• FG: Energy of noise spectrum

• H(*): Received Signal

• HI(*): Received Signal aIer Matched Filter

• HID(*): Signal component of HI(*)

• HIJ(*): Noise component of HI(*)

• KLD(.): Frequency Spectrum of HID(*)

• M: Sampling offset

• N[O]: Magnitude of pulse samples

• OP: Sample index of pulse peak

• QRP : Z-transform representing delay of OP samples

pulse shaping and matched filters haitham hassanieh

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