1 Digital Signal Processing, Fall 2010 Lecture 3: Sampling and reconstruction, transform anal sis of LTI s stems Zheng-Hua Tan transform analysis of LTI systems Digital Signal Processing, III, Zheng-Hua Tan 1 Department of Electronic Systems Aalborg University, Denmark [email protected]Course at a glance Discrete-time signals and systems MM1 System Fourier transform and Z-transform Filt d i MM2 Sampling and reconstruction MM3 System analysis Digital Signal Processing, III, Zheng-Hua Tan 2 DFT/FFT Filter design MM5 MM4
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Digital Signal Processing, Fall 2010
Lecture 3: Sampling and reconstruction, transform anal sis of LTI s stems
Zheng-Hua Tan
transform analysis of LTI systems
Digital Signal Processing, III, Zheng-Hua Tan1
Department of Electronic Systems Aalborg University, Denmark
Part I: sampling and reconstruction Periodic sampling
F d i i f h li Frequency domain representation of the sampling
Reconstruction
Part II: system analysis
Digital Signal Processing, III, Zheng-Hua Tan3
Periodic sampling
From continuous-time to discrete-time)(txc ][nx
nnTxnx c ),(][
Sampling period
Sampling frequencyT
Tf
T
s
s
/2
/1
Digital Signal Processing, III, Zheng-Hua Tan4
3
Two stages
Mathematically Impulse train modulator
Conversion of the impulse
In practice?
Conversion of the impulse train to a sequence
nc
cs
n
nTttx
tstxtx
nTtts
)()(
)()()(
)( )(
Digital Signal Processing, III, Zheng-Hua Tan5
n
c nTtnTx )()(
) )()()( ( dtxtx cc
nnTxnx c ),(][
Periodic sampling
Tow-stage representation Strictly a mathematical representation that is
convenient for gaining insight into sampling inconvenient for gaining insight into sampling in both the time and frequency domains.
Physical implementation is different.
a continuous-time signal, an impulse train, zero except at nT
a discrete-time sequence time normalization
)(txs
][nx
Digital Signal Processing, III, Zheng-Hua Tan6
a discrete-time sequence, time normalization, no explicit information about sampling rate
Many-to-many in general not invertible
][nx
4
Part I-B: Freq. domain represent.
Part I: sampling and reconstruction Periodic sampling
F d i i f h li Frequency domain representation of the sampling
Reconstruction
Part II: system analysis
Digital Signal Processing, III, Zheng-Hua Tan7
Frequency-domain representation
From xc(t) to xs(t)
skT
jSnTtts )(2
)( )( )(
The Fourier transform of a periodic impulse train is a periodic impulse train.
nc
ksc
nc
ccs
ks
n
nTtnTx
kjXT
nTttx
jSjXjXtstxtx
Tj
)()(
))((1
)()(
)(*)(2
1)( )()()(
)()()()(
s
?
Digital Signal Processing, III, Zheng-Hua Tan8
The Fourier transform of xs(t) consists of periodic repetition of the Fourier transform of xc(t).
5
Frequency-domain
ksc
ks
kjXT
jX
kT
jS
))((1
)(
)(2
)(
s
Digital Signal Processing, III, Zheng-Hua Tan9
NsNNs 2or
Recovery
)()()( srr jXjHjX
Ideal lowpass filter with gain
T and cutoff frequency
)()(
)(
jXjX cr
NscN
c
Digital Signal Processing, III, Zheng-Hua Tan10
6
Aliasing distortion
Due to the overlap among the copies of , due to
not recoverable by lowpass filtering
)(c jX
Ns 2
)( jX not recoverable by lowpass filtering)(c jX
Digital Signal Processing, III, Zheng-Hua Tan11
Aliasing – an example
ttxc 0cos)(
a 1a.1
b.1
k
sc kjXT
jX ))((1
)(s
Digital Signal Processing, III, Zheng-Hua Tan12
ttxr 0cos)(
ttx sr )cos()( 0b.2
a.2
7
Nyquist sampling theorem
Given bandlimited signal with
Nc jX ||for ,0)(
)(txc
Then is uniquely determined by its samples
If
is called Nyquist frequencyN
nnTxnx c ),(][
Ns T 2
2
)(txc
Digital Signal Processing, III, Zheng-Hua Tan13
is called Nyquist frequency
is called Nyquist rateN
N2
Fourier transform of x[n]
From to ][ )( nxtxs
cs nTtnTxtx )()()(
From to
By taking continuous-time Fourier transform of xs(t)
)( )( js eXjX
Tnjcs enTxjX )()(
n
nnTxnx c ),(][
))()(( dtetxjX tj
Digital Signal Processing, III, Zheng-Hua Tan14
By taking discrete-time Fourier transform of x[n]
n
cs enxj )()(
n
njj enxeX ][)( )(|)()( TjT
js eXeXjX
) )()( ( dtetxjX
8
Fourier transform of x[n]
)(|)()( TjT
js eXeXjX
k
sc kjXT
jX ))((1
)( s
From Slide 14,
From Slide 8,
i.e. is simply a frequency-scaled version of
kc
kc
Tksc
T
sj
T
kjX
TT
k
TjX
T
kjXT
jXeX
)2
(1
))2
((1
|))((1
|)()(
k
)( jX s)( jeX
So,
Digital Signal Processing, III, Zheng-Hua Tan15
p y q ywith
retains a spacing between samples equal to the sampling period T while always has unity space.
T
][nx)(txs
Sampling and reconstruction of Sin Signal
)cos())3/2cos(()4000cos()(][
aliasing no 12000/26000/1
4000)4000cos()(
0
0
nnnTnTxnx
TT
ttx
c
s
c
)4000()4000()()( jXtx cc
)())(()()(][ 0c
TTjXjXeX Tj frequency normalized with )/(|)()( s/s
k
sc kjXT
jX ))((1
)( s
Digital Signal Processing, III, Zheng-Hua Tan16
)16000cos()(
about How
ttxc
9
Part I-C: Reconstruction
Part I: sampling and reconstruction Periodic sampling
F d i i f h li Frequency domain representation of the sampling
Reconstruction
Part II: system analysis
Digital Signal Processing, III, Zheng-Hua Tan17
Requirement for reconstruction
On the basis of the sampling theorem, samples represent the signal exactly when:
B dli it d i l Bandlimited signal
Enough sampling frequency
+ knowledge of the sampling period recover the signal
Digital Signal Processing, III, Zheng-Hua Tan18
10
Reconstruction steps
Given x[n] and T, the impulse train is
cs nTtnxnTtnTxtx )(][)()()(
(1)
i.e. the nth sample is associated with the impulse at t=nT.
The impulse train is filtered by an ideal lowpass CT filter with impulse response
nn
cs )(][)()()(
(2)
)(thr )( jHr
Digital Signal Processing, III, Zheng-Hua Tan19
n
rr nTthnxtx )()()(
)()()( Tjrr eXjHjX
Ideal lowpass filter
Tsc /2/
asfrequncy cutoff chooseCommonly
Digital Signal Processing, III, Zheng-Hua Tan20
Tt
Ttthr /
)/sin()(
11
Ideal lowpass filter interpolation
CT signal
Modulated impulse train
Digital Signal Processing, III, Zheng-Hua Tan21
n
r TnTt
TnTtnxtx
/)(
)/)(sin(][)(
Ideal discrete-to-continuous-time converter
Digital Signal Processing, III, Zheng-Hua Tan22
12
discrete-to-continuous-time converter
“Practical DACs do not output a sequence of dirac impulses (that, if ideally low-pass filtered, result in the original signal before sampling) but instead output a sequence of piecewise constant values or rectangular pulses”rectangular pulses
Digital Signal Processing, III, Zheng-Hua Tan23
From http://en.wikipedia.org/wiki/Digital-to-analog_converter.
Ideally sampled signal. Piecewise constant signal typical of a practical DAC output.
Applications
Digital Signal Processing, III, Zheng-Hua Tan24
13
Part II System analysis
Part I: sampling and reconstruction
Part II: system analysis Frequency response
System functions
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
Digital Signal Processing, III, Zheng-Hua Tan25
System analysis
Three domains
Time domain: impulse response, convolution sum
Frequency domain: frequency response
z-transform: system function)()()( jjj eHeXeY
k
knhkxnhnxny ][][][*][][
Digital Signal Processing, III, Zheng-Hua Tan26
LTI system is completed characterized by …
)()()( zHzXzY
14
Part II-A: Frequency response
Part I: sampling and reconstruction
Part II: system analysis Frequency response
System functions
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
Digital Signal Processing, III, Zheng-Hua Tan27
Frequency response
Relationship btw Fourier transforms of input and output
)()()( jjj eHeXeY
In polar form
Magnitude magnitude response, gain, distortion
|)(||)(||)(| jjj eHeXeY
)()()( eHeXeY
Digital Signal Processing, III, Zheng-Hua Tan28
Phase phase response, phase shift, distortion
)()()( jjj eHeXeY
15
Ideal lowpass filter – an example
Frequency response
||0
,|| ,1)( cjeH
Frequency selective filter Impulse response
Noncausal, cannot be implemented!
|| ,0 c
nn
nnh c
lp ,sin
][
0 ,0][?
nnh
Digital Signal Processing, III, Zheng-Hua Tan29
p How to make a noncausal system causal?
In general, any noncausal FIR system can be made causal by cascading it with a sufficiently long delay!
But ideal lowpass filter is an IIR system!
Phase distortion and delay
Ideal delay system][][ did nnnh Delay distortion
Ideal lowpass filter with linear phase
1|)(| jid eH
dnjjid eeH )(
||,)( dj
id neH Linear phase distortion
Digital Signal Processing, III, Zheng-Hua Tan30
| ,0
,|| ,)(
c
cnj
jlp
deeH
nnn
nnnh
d
dclp ,
)(
)(sin][
Ideal lowpass filter isalways noncausal!
16
Group delay
A measure of the linearity of the phase
Concerning the phase distortion on a narrowbandsignalsignal
For this input with spectrum only around w0, phase effect can be approximated around w0 as the linear approximation (though in reality maybe nonlinear)
)cos(][][ 0nnsnx
)( j neH
0 w0
Digital Signal Processing, III, Zheng-Hua Tan31
and the output is approximately
Group delay
0)( dneH
))(cos(][|)(|][ 000 dd
j nnnnseHny
)]}({arg[)]([
jj eH
d
deHgrd
An example of group delay
Figure 5.1, 5.2, 5.3
Digital Signal Processing, III, Zheng-Hua Tan32
17
An example of group delay
Digital Signal Processing, III, Zheng-Hua Tan33
Part II-B: System functions
Part I: sampling and reconstruction
Part II: system analysis Frequency response
System functions
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
Digital Signal Processing, III, Zheng-Hua Tan34
18
System function of LCCDE systems
Linear constant-coefficient difference equation
M
m
N
kk mnxbknya
00
][][
z-transform format
N
M
m
mmzb
X
zYzH 0
)(
)()(
M
m
mm
N
k
kk zXzbzYza
00
)()(
mk 00
Digital Signal Processing, III, Zheng-Hua Tan35
N
kk
M
mm
N
k
kk
zd
zc
a
b
zazX
1
1
1
1
0
0
0
)1(
)1(
)(
)(
k
k
m
m
dzz
zd
zcz
zc
at pole a 0at zero a
r denominato in the )1(
0at pole a at zero a
numerator in the )1(
1
1
Stability and causality
Stable h[n] absolutely summable
H( ) h ROC i l di h i i l H(z) has a ROC including the unit circle
Causal h[n] right side sequence
H(z) has a ROC being outside the outermost pole
Digital Signal Processing, III, Zheng-Hua Tan36
19
Inverse systems
Many systems have inverses, specially systems with rational system functions
M
zc 1)1(
][][*][][
)(
1)(
1)()()(
nnhnhng
zHzH
zHzHzG
i
i
i
M
N
kk
i
N
kk
mm
zc
zd
b
azH
zd
zc
a
bzH
1
1
1
0
0
1
1
1
0
0
)1(
)1(
)()(
)1(
)1(
)()(
Digital Signal Processing, III, Zheng-Hua Tan37
Poles become zeros and vice versa.
ROC: must have overlap btw the two for the sake of G(z).
m
mzc1
)1(
Example
11
1
1
901901
9.0||,9.01
5.01)(
zz
zzH
So,
1
1
11
1
5.01
9.0
5.01
1
5.01
9.01)(
z
z
zz
zzHi
]1[)5.0(9.0][)5.0(][
5.0||1
nununh
znn
i
Digital Signal Processing, III, Zheng-Hua Tan38
20
Part II-C: All-pass systems
Part I: sampling and reconstruction
Part II: system analysis Frequency response
System functions
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
Digital Signal Processing, III, Zheng-Hua Tan39
All-pass systems
Consider the following stable system function
1
*1
1)(
az
azzHap
ll t f hi h th1|)(| jH
j
jj
j
jj
ap
ae
eae
ae
aeeH
1
1
1)(
*
*
Digital Signal Processing, III, Zheng-Hua Tan40
all-pass system: for which the frequency response magnitude is a constant.
1|)(| jap eH
21
Example: First-order all-pass system
P275 Example 5.13
Digital Signal Processing, III, Zheng-Hua Tan41
Part II-D: Minimum-phase systems
Part I: sampling and reconstruction
Part II: system analysis Frequency response
System functions
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
Digital Signal Processing, III, Zheng-Hua Tan42
22
Minimum-phase systems
Magnitude does not uniquely characterize the system
St bl d l l i id it i l Stable and causal poles inside unit circle, no restriction on zeros
Zeros are also inside unit circle inverse system is also stable and causal (in many situations, we need inverse systems!)
such systems are called minimum-phase
Digital Signal Processing, III, Zheng-Hua Tan43
such systems are called minimum-phase systems (explanation to follow): are stable and causal and have stable and causal inverses
Minimum-phase and all-pass decomposition
Any rational system function can be expressed as:
S ppose H( ) has one ero o tside the nit circle at)()()( min zHzHzH ap
Suppose H(z) has one zero outside the unit circle at
1
*11
1
*11
1)1)((
))(()(
cz
czczzH
czzHzH
1||,/1 * ccz
Digital Signal Processing, III, Zheng-Hua Tan44
minimum-phase all-pass
23
Frequency response compensation
When the distortion system is not minimum-phase system:
)()()( HHH 1)()()( min zHzHzH apdd )(
1)(
min zHzH
dc
)()()()( zHzHzHzG apcd
Digital Signal Processing, III, Zheng-Hua Tan45
Frequency response magnitude is compensated
Phase response is the phase of the all-pass
Properties of minimum-phase systems
From minimum-phase and all-pass decomposition
)()()( min zHzHzH ap
From the previous figure, the continuous-phasecurve of an all-pass system is negative for
So change from minimum-phase to non-minimum-phase (+all pass phase) always decreases the
0
)](arg[)](arg[)](arg[ min j
apjj eHeHeH
Digital Signal Processing, III, Zheng-Hua Tan46
phase (+all-pass phase) always decreases the continuous phase or increases the negative of the phase (called the phase-lag function). Minimum-phase is more precisely called minimum phase-lag system
24
Part II-E: GLP systems
Part I: sampling and reconstruction
Part II: system analysis Frequency response
System functions
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
Digital Signal Processing, III, Zheng-Hua Tan47
Design a system with non-zero phase
System design sometimes desires Constant frequency response magnitude
Z h h ibl Zero phase, when not possible accept phase distortion, in particular linear phase since
it only introduce time shift
Nonlinear phase will change the shape of the input signal though having constant magnitude response
Digital Signal Processing, III, Zheng-Hua Tan48
25
Ideal delay
1|)(| jeH
||,)( jjid eeH
)(
)(sin][
n
nnhid
1|)(| jid eH
)]([
||,)(j
id
jid
eHgrd
eH
h
phaselinear with lowpass Ideal
Digital Signal Processing, III, Zheng-Hua Tan49
][][ did nnnh
dnwhen
)(
)(sin][
d
dclp nn
nnnh
Generalized linear phase
Linear phase filters
|)(|)( jjj eeHeH
Generalized linear phase filters
constantsrealareand
, offunction real a is )(
)()(
j
jjjj
eA
eeAeH
Digital Signal Processing, III, Zheng-Hua Tan50
constants real are and
26
Summary
Part I: sampling and reconstruction Periodic sampling