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
4
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.
5
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
6
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
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
9
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)(
10
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
)( )(
11
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
12
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
13
][][][
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 …
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 …
16
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
17
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 ]
18
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(
19
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
20
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
21
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]
22
)()()(
)()(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
23
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]+
24
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)
25
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]}
26
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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