1

Discrete Hilbert Transform

7th April 2007Digital Signal Processing IIslamic University of Gaza

2

Overview

Hilbert Transforms Discrete Hilbert Transforms DHT in Periodic/Finite Length

sequences DHT in Band pass Sampling

3

Transforms

Laplace Transforms Time domain s-plane

Fourier Transforms (FT/DTFT/DFT) Time domain frequency domain

Z- Transforms Time domain Z domain ( delay domain )

Hilbert Transforms For Causal sequences relates the Real Part of FT

to the Imaginary Part FT

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Why Hilbert Transforms?

Fourier Transforms require complete knowledge of both Real and Imaginary parts of the magnitude and phase for all frequencies in the range –π < ω < π

Hilbert Transforms applied to causal signals takes advantage of the fact that Real sequences have Symmetric Fourier transforms.

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Because of the possible singularity

at x=t, the integral is considered

as a Cauchy Principal value

Analog Hilbert Transforms

dt

tx

tgxH

)(1)(

t

t

dttx

tgdttx

tgxH

)(1)(1lim)(

0

)(1)( xgF

xFxHF

The Hilbert Transform of the

function g(t) is defined as

The forms of the

Hilbert Transform are

dt

t

txgxH

)(1)(

dt

tx

tgxH

)(1)(

So the Hilbert transform is a Convolution

)(1

)( xgx

xH

)(1

wsignix

F

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A note on Symmetry

For real signals we have the following Fourier transforms relationships

• Any complex signal can be decomposed into parts having • Conjugate Symmetry ( even for real signals)• Conjugate Anti-Symmetry (odd for real signals)

)()()( nxnxnx oe

)()()( nXnXnX oe

)()0()()(2)(

)()0()()(2)(

)()(2

1)(

)()(2

1)(

nxnunxnx

nxnunxnx

nxnxnx

nxnxnx

oo

ee

o

e

(1)

(2)

(3)

(4)

(5)

(6))()(

)()(

)()()(

jIo

jRe

jwI

jR

eFnf

eFnf

ejFeFnf

7

A note on Symmetry …

x[n]

xe[n]

xo[n]

x[-n]

n

8

Problem1 )2cos(1)( jR eX

1) Find Xi(w)

)2sin()(

)sin(2

1

2

1)(

)2(2

1)2(

2

1)(

)2sin()(

)2sin()2cos(11)(

)2()()(

)2(2

1)2(

2

1)()(

2

1

2

11)(

22

2

22

jwI

jjI

o

jwI

jjw

e

jjjwR

eX

jeejX

nnnx

aliter

eX

jeeX

nnnx

nnnnx

eeeX

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Problem2 1cos21

)cos(1)(

2

jR eX

2) Find X(z)

zz

zX

nunx

nununx

zzzX

polesthebetweenROCzz

zzzX

zz

zzzX

ee

eeeX

eX

n

nne

R

R

R

jj

jjj

R

jR

1

1

1

1

21

1

2

2

1

1)(

][)(

][2

1][

2

1)(

1

1

1

1

2

1)(

2)1)(1(

))(2/(1)(

)(1

))(2/(1)(

1)(1

))(2/(1)(

1cos21

)cos(1)(

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Derivation of Hilbert Transform Relationships

deXeX

deXx

xdeXj

deXeX

ejXeXeX

jkeU

ekeU

xdeUeXeX

jR

jwI

jR

jR

jR

jR

jI

jR

j

K

K

j

K

Kj

j

wjjR

j

2cot)(

2

1)(

)(2

1]0[

]0[2

cot)(2

)(2

1)(

)()()(

2cot22

1)2()(

1

1)2()(

]0[)()(1

)( )(

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The Hilbert Transform Relationships

deXxeX

deXeX

jI

jwR

jR

jwI

2cot)(

2

1]0[)(

2cot)(

2

1)(

The above equations are called discrete Hilbert Transform Relationships hold for real and imaginary parts of the Fourier transform of a causal stable real sequence.

deXxeX

deXeX

jI

jwR

jR

jwI

2cot)(

2

1]0[)(

2cot)(

2

1)(

deXdeXeX j

Rj

Rjw

I 2cot)(

2cot)(

2

1)( lim

0

Where P is Cauchy principle value

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Note: A periodic sequence cannot be casual in the sense

used before, but we will define a “periodically causal” sequence

Henceforth we assume N is even

)1,...(1,0][][2

1][

)1,...(1,0][][2

1][

)1,...(1,0][][][

~~~

~~~

~~~

Nnnxnxnx

Nnnxnxnx

Nnnxnxnx

oeo

oee

oe

Definitions:

Periodic Sequences

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][][][

1,...,1)2/(0

1)2/,...(2,12

2/,01

][

1,...,1)2/(0

1)2/,...(2,1][2][

1,...,1)2/(0

2/,0][

1)2/,...(2,1][2

][

~~~

~

~~

~

~

~

nunxnx

NNn

Nn

Nn

nu

NNn

Nnnxnx

NNn

Nnnx

Nnnx

nx

Ne

N

o

e

e

Periodic Sequences …

14

x~[n]

x~[-n]

n

x~o[n]

x~e[n]

Periodic Sequences …

15

1. Compute x~e[n] from X~

R[k] using DFS synthesis equation2. Compute x~[n] from x~

e[n]3. Compute X~[k] from x~[n] using DFS analysis equation

]2/[)1(]0[][][1

][

][][1

][

][][1

][][

0

)/cot(2][

0

)/cot(2

0

][

][][1

][][][

~~1

0

~~~

1

0

~~~

1

0

~~~~

~

~

1

0

~~~~~

NxxmkVmXjN

kXSimilarly

mkVmXN

kXj

mkVmXN

kXkX

evenk

oddkNkjkV

evenk

oddkNkj

kN

kU

mkUmXN

kXjkXkX

kN

m

NIR

N

m

NRI

N

m

NRR

N

N

N

m

NRIR

Periodic Sequences …

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Finite Length Sequences

It is possible to apply the transformations derived if we can visualize a finite length sequence as one period of a periodic sequence.

For all time domain equations replace x~(n) with x(n) For freq domain equations ---

otherwise

NnNxxmkVmjXNkX

otherwise

NnmkVmXNkjX

kN

mNI

R

N

mNR

I

0

10]2/[)1(]0[][][1

][

0

10][][1

][

1

0

1

0

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Problem 3

N=4, XR[k]=[ 2 3 4 3 ], Find XI[k]

Method 1 V4[k]=[ … 0 -2j 0 2j … ]

jXI[k]=[ 0 j 0 –j ]

Method 2 xe[n]=[ 3 -1/2 0 -1/2 ]

xo[n]=[ 0 -1/2 0 -1/2 ]

jXI[k]=[ 0 j 0 –j ]

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Relationships between Magnitude and Phase

We obtain a relationship between Magnitude and phase by imposing causality on a sequence x^(n) derived from x(n)

CepstrumComplexeXjeXeX

eXnx

eeXeXnx

jjj

jF

eXjjjF j

)(arg()(log)(ˆ

)(ˆ][ˆ

)()(][ ))(arg(

The fact that the minimum phase condition ( X(z) has all poles and zeros inside the unit circle) guarantees causality of the complex cepstrum.

deXx

deXPxeX

deXPeX

j

jj

jj

)(log2

1]0[ˆ

2cot)(arg(

2

1]0[ˆ)(log

2cot)(log

2

1)(arg(

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Complex Sequences

Useful in useful in representation of bandpass signals Fourier transform is zero in 2nd half of each period. Z-Transform is zero on the bottom half The signal called an analytic signal (as in continuous time signal

theory)

00

0)(2)(

00

0)(2)(

)()(2

1)(

)()(2

1)(

][][][

00)(

*

*

jwijw

jwrjw

jwjwjwi

jwjwjwr

ir

jw

ejXeX

eXeX

eXeXejX

eXeXeX

njxnxnx

eX

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Complex Sequences …

)()()()(

1)(

0

0)(

)()()(

0)(

0)()(

jwi

jwjwijw

jwr

jw

jwr

jwjwi

jwr

jwrjw

i

eXeHeXeH

eX

j

jeH

rTransformeHilbert

eXeHeX

ejX

ejXeX

Note:

Such a system is also called a 90º phase shifter.

-xr[n] can also be obtained form a xi[n] using a 90º phase shifter

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Complex Sequences …

mir

mri

njnj

mxmnhnx

mxmnhnx

n

nn

nnh

djedjenh

][][][

][][][

00

0)2/(sin2

][

2

1

2

1][

2

0

0

Hilbert

Transformer

Xr[n] Xr[n]

Xi[n]

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)()()(

)()(2

1)(

)()(2

1)(

)()(

][][][][

00)(

][][][

*

*

)(

jr

jji

jjji

jjjr

jj

irnj

j

ir

eSeHeS

eSeSejS

eSeSeS

eXeS

njsnsenxns

eX

njxnxnx

c

c

Representation of Bandpass Signals

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Representation of Bandpass Signals …

sr[n]

sin(wcn)

xr[n] si[n]

])[sin(][)cos(][)sin(][][

])[cos(][)sin(][)cos(][][

][][][][

][

][arctan][

][][][

][][

2

122

nnnAnnxnnxns

nnnAnnxnnxns

enAenjxnxns

nx

nxn

nxnxnA

enAnx

ccicri

ccicrr

nnjnjir

i

r

ir

nj

cc

Hilbert

TransformerX

+

X

cos(wcn)

Hilbert

Transformer

+

+-Hilbert

Transformer

X

X

+

sin(wcn)

cos(wcn)

xr[n]+

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Bandpass Sampling

1

0

/)2( )(1

)(

][][][M

k

Mkjjd

ir

eSM

eS

njsnsns

C/D

Hilbert

Transformer

↓M

↓MT

Sr[n]=Sc[nT]

Sid[n]

Srd[n]

Si[n]

Sc(t)

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Bandpass Sampling …

Reconstruction of the real bandpass signal involves

1. Expand the complex signal by a factor M

2. Filter the signal using an ideal bandpass filter

3. Obtain Sr[n]=Re{se[n]*h[n]}

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Concluding Remarks

Relations between Real and Imaginary part of Fourier transforms for causal signal were investigated

Hilbert transform relations for periodic sequences that satisfy a modified causality constraint

When minimum phase condition is satisfied logarithm of magnitude and the phase of the Fourier transform are a Hilbert transform pair

Application of complex analytic signals to the efficient sampling of bandpass signals were discussed

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References

Discrete Time Signal Processing, 2nd Edition. © 1999 Chapter 11 pages 755-800, Alan V Oppenheim, Ronald W Schafer with John R Buck