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2-1 Fourier Transform for Discrete-time Signals and Systems
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Discussions
◼ What is the impulse response? How to obtain it?
◼ How to obtain the output of an LTI system given the input?
◼ How about if the impulse response is an IIR?
◼ What domain are we talking about?
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Frequency Representation
Red
Orange
Yellow
Green
Blue
Indigo
Violet
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The World in Frequency Domain
◼ What you hear…
Frequency or spectrum analysis & process
◼ What you see…
The World in Frequency Domain
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Examples of frequency range
Signals related to communications:
Radio broadcast, shortwave radio signals, radar, satellite communications, space communications, microwave, Infrared, Ultraviolet, Gamma rays and X rays… They all have their frequency range, respectively.
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Examples of frequency range
Frequency Range
3G：1880MHz-1900MHz & 2010MHz-2025MHz
4G：1880-1900MHz & 2320-2370MHz & 2575-2635MHz
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Example of Frequency Analysis
◼ Frequency and time in terms of Fourier Transform
1822
Fourier
Laplace Lagrange
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Frequency Analysis
◼ Continuous-Time Periodic SignalsExamples: square waves, sinusoids…
Fourier Series
−=
=k
tjk
kectx 0)(
Linear weighted sum of sinusoids or complex exponentials
−−
=0
0
0)(2
0
dtetxc
tjk
k
Analysis Synthesis
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◼ Continuous-Time Aperiodic Signals
Fourier Transform
dejXtx tj
−= )(
2
1)( dtetxjX tj −
−= )()(
Analysis Synthesis
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◼ Discrete-Time Periodic Signals
Fourier Series
−
=
=1
0
2
][N
k
knN
j
kecnx
Analysis Synthesis
−
=
−
=1
0
2
][1 N
k
knN
j
k enxN
c
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◼ Discrete-Time Aperiodic Signals
Fourier Transform
deeXnx njj
−= )(2
1][
Analysis Synthesis
nj
n
j enxeX −
−=
= ][)(
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Discussions
◼ Is Fourier Transform the only way to represent signals in the frequency domain?
◼ Why is Fourier Transform designed that way?
◼ Does any signal or system have its Fourier Transform?
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Eigenfunctions for LTI systems
◼ Eigenfunction:In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that
returns from the operator exactly as is, except for a multiplicative
scaling factor.
fAf =
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◼ Eigenfunctions for LTI systems:
Sinusoidal or componential sequences
If we apply a sinusoidal sequence input to an LTI system, the
output will be sinusoidal with the same frequency as the input, while the
][n
][n
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njenx =][][ny][nx
][nh
)(][][ knj
k
ekhny −
−=
=
kj
k
nj ekhe −
−=
= ][
kj
k
j ekheH −
−=
= ][)(
If define
njj eeH )(=
EigenfunctionEigenvalue
Frequency Response:
Related to the frequency!!!
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◼ The frequency response is complex
)()()( j
I
j
R
j ejHeHeH +=
))(arg()()( jeHjjj eeHeH =
Rectangular Form – real and imaginary parts
Polar Form – magnitude and phase parts
[ ] [ ]nx n a u n=
=
−=0
)(n
njnj eaeX
=
−=0
)(n
njae
jae−=
-1
11aif
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◼ The frequency response is periodic
kj
k
j ekheH −
−=
= ][)(
kjkjkjkj eeee −−−+− == 2)2(
kj
k
j ekheH )2()2( ][)( +−
−=
+ =
)()(
)()(
)2(
)2(
jrj
jj
eHeH
eHeH
=
=
+
+
period 2
r:
integer
0 2
−
Depict it
◼ The frequency response is continuous
◼
deeXnx njj
−= )(2
1][ nj
n
j enxeX −
−=
= ][)(
Discrete in Time
Continuous in Frequency
2
of multiple odd
tocolse0 2
−
of multipleeven
20toclose or
Frequency
high
low
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◼ Example 4.1: frequency response of the ideal delay system
][][ dnnxny −=
][ny][nx}{T
][][ dnnnh −=d
dd
njj
njnjnnj
eeH
eeeny
−
−−
=
==
)(
][)(
njenx =][
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Example 4.2: Ideal frequency-selective filters
)( jhp eH
−c
c−
1
)( jlp eH
−c
c−
1
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)( jbs eH
−a
1
ba−b−
)( jbp eH
−a
1
ba−b−
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◼ E.g. 1 Frequency response
of the moving-average system (p44)
−=
−++
=2
1
][1
1][
21
M
Mk
knxMM
ny
−
++=
−++
= −=
otherwise
MnMMM
knMM
nhM
Mk
,0
,1
1
][1
1][
21
21
21
2
1
Smooth out rapid variations
Lowpass filtering
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2
1
1 2
1 2
1 2
1,1
1[ ] [ ]1
0,
M
k M
M n MM Mh n n k
M Motherwise
=−
−
+ += − = + +
2
1 1 2
1( )
1
Mj j n
n M
H e eM M
−
=−
=+ +
2 1( )1 2
1 2
sin[ ( 1) / 2]1( )
1 sin( / 2)
j M Mj M MH e e
M M
− −+ +=
+ +
Periodic
Lowpass filtering
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Steady-state and transient responses
◼ Suddenly applied complex exponential inputs: suddenly applied at an arbitrary time (n = 0 here)
][][ nuenx nj=
=−
=
njkjn
k
eekh
n
ny
0
][
0,0
][
Causal LTI
+=
==
−100
)()()(nkk
n
k
0
[ ] [ ]
'
j k
k
y n e h n k
k n k
=
= −
= −
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njkj
nk
njkj
k
eekheekhny
−
= −
+=
−
=
10
][][][
Steady-state response
transient response
njkj
k
ss eekhny
= −
=
0
][][
njj
ss eeHny )(][ =
njkj
nk
t eekhny
= −
+=
1
][][
=
+=
01
][][][knk
t khkhny
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FIR
IIR
+=
−
+=
=11
][][][nk
njkj
nk
t kheekhny
1
)(][][
−
==
Mnfor
eeHnyny njj
ss
1. If the samples of the impulse response approach
zero with increasing n, so
does the transient response
2. For a stable system, the transient response dies out
when n approaches infinity.
Mnforexcept
nh
=
0
0][
=
0
][][k
t khny
Bounded
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Representation of Sequences by Fourier Transforms
◼ Many sequences can be represented by
Fourier Integral
deeXnx njj
−= )(2
1][
Analysis –
Inverse Fourier Transform
Synthesis –
Fourier Transform
nj
n
j enxeX −
−=
= ][)(
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)()()( j
I
j
R
j ejXeXeX +=
))(arg()()( jeXjjj eeXeX =
Rectangular Form – real and imaginary parts
Polar Form – magnitude and phase parts
Magnitude spectrum
or amplitude spectrum
Phase spectrum
Restricted in the range of
−
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Frequency response of LTI systems
deeHnh njj
−= )(2
1][
][ny][nx][nh
nj
n
j enheH −
−=
= ][)(
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◼ Does any signal or system have its Fourier Transform?
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◼ Convergence of the infinite sum
Conditions for Fourier Transform
deeXnx njj
−= )(2
1][
nj
n
j enxeX −
−=
= ][)(
allforeX j )(Convergence
Sufficient condition: x[n] is absolutely summable
Proof
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nj
n
j enxeX −
−=
= ][)(
−=
n
nx ][
−=
−n
njenx ][
−=
)(
][
j
n
eX
nxif
Since a stable sequence is absolutely summable, all stable sequences have Fourier transforms.
And any FIR system is stable and has the Fourier transform
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Absolute summability is a sufficient condition for the existence of a Fourier transform representation.
And it also guarantees uniform convergence.
If some sequences are not absolutely summable, but are square summable, such sequences can be represented as Fourier transform if the condition of uniform convergence is relaxed.
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−=n
nx2
][
−=
−
−=
−
=
=
M
Mn
njjM
n
njj
enxeX
enxeX
][)(
][)(
0)()(lim2
=−−→
deXeX j
M
j
M
Square summable
The absolute error
may not approach zero at each value of , as
but the total energy in the error does.
)()( j
M
j eXeX −
→M
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Example: Square-summability for the ideal lowpass filter
1, ,( )
0, ,
cj
lp
c
H e
=
)( jlp eH
−c
c−
1
1[ ]
2
1 1( )
2 2
sin,
c
c
cc c
c
j n
lp
j n j nj
c
h n e d
e e ejn jn
nn
n
−
−
−
=
= = −
= −
sin j nc
n
ne
n
−
=−
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Square-summability for the ideal lowpass filter (p52)
sin( )
Nj j nc
N
n N
nH e e
n
−
=−
=
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sin( )
Mj j mc
M
m M
mH e e
m
−
=−
=
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Fourier transform for some special sequences
−=−
−=
−=
−=
++−
=
+−==
+−==
+==
rj
j
r k
kk
j
k
nj
k
r
jnj
r
j
re
eUnu
raeXeanx
reXenx
reXnx
k
)2(1
1)(][
)2(2)(][
)2(2)(][
)2(2)(1][
00
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have aperiodic spectra
have spectra
◼ Periodic signals have (Fourier series)
have continuous spectra
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Exercise: Does the following signal has a Fourier transform?
[ ] [ ]nx n a u n=
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Conclusions
◼ Frequency-domain representation for systems and sequences, Fourier transform
◼ Next lectures: Symmetry Properties of The Fourier Transform, Fourier Transform Theorems, Discrete Fourier Series, Properties of the Discrete Fourier Series
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Assignment
◼ Preparation for the next lecture:
◼ Solve problems 2.8, 2.11
◼ Watch the movie of “Interstellar”
End of lecture 4
Thanks!
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