4-1 representation of periodic sequences: discrete fourier ...nwpu-dsp.com/lecture_notes/4- 1...
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4-1 Representation of Periodic Sequences: Discrete Fourier Series
Uniform convergence of the Fourier transform requires: the sequence to be absolutely summable.
Mean-square convergence of the Fourier transform requires: the sequence to be square summable.
Periodic sequences are neither absolutely summable nor square summable.
Periodic sequences:
Discrete-Time Periodic Signals
- period
Discrete Fourier Series
(DFS)
Sum of harmonically related complex exponential sequences: with frequencies integer multiples of the fundamental frequency
associated with the periodic sequence.N
2
Harmonically related periodic complex exponentials
- period
knN
j
k ene2
][
knN
jN
k
ekXN
nx21
0
][~1
][~
The symbol of tilde denotes “periodic”
periodic?
][ Nnek
Continuous-Time Periodic SignalsExamples: square waves, sinusoids…
Fourier Series
k
tjk
kectx 0)(
0
0
0)(2
0
dtetxc
tjk
k
Analysis Synthesis
0
2=
T
1
0
2
][~1
][~N
k
knN
j
ekXN
nx
1
0
2
][~][~ N
n
knN
j
enxkX
What’s the main difference between the Fourier Series of
discrete-time signals and continuous-time signals?
A continuous-time periodic signal generally requires infinitely many harmonically related complex exponentials.
A discrete-time periodic signal with period Nrequires only
to represent itself.
？
1
0
21 N
n
knN
j
eN
Orthogonality of the complex exponentials
otherwise
mNrke
N
N
n
nrkN
j
,0
,11 1
0
)(2
21 1( )2
0 0
1 1=
N Nj k r nj mnN
n n
k r mN e eN N
2 2 ( )1 ( )
2( )0
1 1 1=
1
j k r nN j k r nN
j k r nn N
ek r mN e
N Ne
1
0
rnN
j
e2
1
0
)(21
0
1
0
2
][~1
][~N
n
nrkN
jN
k
N
n
rnN
j
ekXN
enx
1
0
1
0
)(2
1][
~N
k
N
n
nrkN
j
eN
kX
otherwise
mNrke
N
N
n
nrkN
j
,0
,11 1
0
)(2
][~
][~1
0
2
rXenxN
n
rnN
j
knN
jN
k
ekXN
nx21
0
][~1
][~
1
0
)(2
][~][~ N
n
nNkN
j
enxNkX
nNN
jN
n
knN
j
eenx 21
0
2
][~
][~
kX
The Fourier series coefficients of a periodic sequence is periodic.
Discrete Fourier series (DFS)
1
0
2
][~1
][~N
k
knN
j
ekXN
nx
1
0
2
][~][~ N
n
knN
j
enxkX
kn
N
N
k
WkXN
nx
1
0
][~1
][~
1
0
][~][~ N
n
kn
NWnxkX
Discrete Fourier series (DFS)
knN
jkn
N eW2
Analysis
Synthesis
][~
][~ kXnx DFS
E.g. 1 Find the DFS of a periodic impulse train
otherwise
rNnrNnnx
r ,0
,1][][~
10][][~ Nnnnx
01
0
][][~
N
N
n
kn
N WWnkX
knN
jkn
N eW2
1
21 1
0 0
1 1[ ]
N N j knkn N
N
k k
x n W eN N
E.g. 2 Find the DFS of a periodic rectangular impulse train
N
4 4(2 /10)
10
0 0
[ ] kn j kn
n n
X k W e
10
5(4 /10)10
10
1 sin( / 2)
1 sin( /10)
kj k kn
k
W ke
W k
Figure 8.2 Magnitude and phase of the Fourier series coefficients of the sequence of Figure 8.1.
Properties of the DFS
Linearity
][~
][~11 kXnx DFS
][~
][~
][~][~2121 kXbkXanxbnxa DFS
][~
][~22 kXnx DFS
If
then
Shift of a sequence
][~
][~ kXnx DFS
][~
][~ kXWmnx km
N
DFS
If
then
][~
][~ lkXnxW DFSkl
N
Nmmwhere
WWkm
N
km
N
mod1
1
knN
jkn
N eW2
1
0
][~][~ N
n
kn
NWmnxkX
Duality
][~
][~ kXnx DFS
][~][~
kxNnX DFS
If
then
kn
N
N
k
WkXN
nx
1
0
][~1
][~
1
0
][~][~ N
n
kn
NWnxkX
DFS of a periodic impulse train
otherwise
rNnrNnnx
r ,0
,1][][~
10][][~ Nnnnx
01
0
][][~
N
N
n
kn
N WWnkX
knN
jkn
N eW2
1
1
0
21
0
11][~
N
k
knN
jN
k
kn
N eN
WN
nx
otherwise
mNrke
N
N
n
nrkN
j
,0
,11 1
0
)(2
E.g. 3 Duality in the DFS
otherwise
rNnrNnnx
r ,0
,1][][~
1][~
kX
knN
jkn
N eW2
,[ ] [ ]
0,r
N k rNY k N k rN
otherwise
1][1
][~ 01
0
N
N
k
kn
N WWkNN
ny ( )X k 1
[ ] [ ] [- ] [ ] [ ]Y k Nx k Nx k y n X n
Symmetry properties
Similar to the case of the aperiodic sequence. Summarized as
properties 9-17 in Table 8.1.
Table 8.1 SUMMARY OF PROPERTIES OF THE DFS
Table 8.1 (continued) SUMMARY OF PROPERTIES OF THE DFS
Periodic convolution][
~][~
11 kXnx DFS ][~
][~22 kXnx DFS If
][~][~][~213 nxnxnx
1
0
213 ][~
][~1
][~ N
l
lkXlXN
kX
][~
][~
][~
213 kXkXkX
1
0
213 ][~][~][~N
m
mnxmxnx
][~
][~22 kXnx DFS
][~
][~
][~
213 kXkXkX
1
0
213 ][~][~][~N
m
mnxmxnx][
~][~
11 kXnx DFS
Key 1The sum is over the finite interval, instead of an infinite one.
Key 2The values in the convolution interval repeat periodically. (see e.g. 8.4)
kn
N
N
k
WkXN
nx
1
0
][~1
][~
1
0
][~][~ N
n
kn
NWnxkX
Discrete Fourier series (DFS)
knN
jkn
N eW2
Analysis
Synthesis
][~
][~ kXnx DFS
The DFS can be considered as a sequence of finite length, or as a periodic sequence defined for all k.
Fourier Transform of Periodic Signals
Part II
The Fourier Transform of Periodic Signals
Periodic sequences: not absolutely summable, not square summable.
Usually represented in the frequency domain by a discrete sum of complex exponentials.
DFS can also be extended to a train of impulses in the frequency domain.
Within the Framework of Fourier transform…
N
kkX
NeX
k
j
2][
~2)(
~
Fourier Series
Since the period of is N and the impulses are
spaced at integer multiples of , has the necessary periodicity with period . (N samples in the range of )
)(~ jeXN2
2
A function of : a Fourier transform representation
][~
kX
2
Inverse Fourier Transform
)2(0 N
N
kkX
NeX
k
j
2][
~2)(
~
deeX njj
2
0)(
~
2
1
deN
kkX
N
nj
k
2
0
2][
~2
2
1
deN
kkX
N
nj
k
2
0
2][
~1)2( Nk
1
2
0
1[ ]
Nj N kn
k
X k eN
][~
][~ kXnx DFS
N
kkX
NeX
k
j
2][
~2)(
~
21
0
1[ ] [ ]
N j knN
k
x n X k eN
1
0
2
][~][~ N
n
knN
j
enxkX
][~ nxThe inverse Fourier transform of the impulse train given above is the original periodic signal, , as desired.
Formally, the Fourier transform of a periodic sequence does not converge. However, by introducing the impulses, periodic sequences can be included in the general framework of Fourier transform analysis.
Why the Fourier transform since DFS representation already well represents the periodic sequences?
-------leading to simpler or more compact expressions and simplified analysis
E.g. 4 The Fourier Transform of a Periodic Impulse Train
r
rNnnp ][][~
kallforkP 1][~
)(~ jeP
N
kkX
NeX
k
j
2][
~2)(
~
N
k
Nk
22
Basis for interpreting the relationship between a periodic signal and a finite-length signal.
E.g. 5 Find the DFS of a periodic impulse train
otherwise
rNnrNnnx
r ,0
,1][][~
10][][~ Nnnnx
01
0
][][~
N
N
n
kn
N WWnkX
knN
jkn
N eW2
1
][~][ nxvsnx
][~][][~ npnxnx
r
rNnnp ][][~
r
rNnnx ][][
r
rNnx ][
)(~ jeX )(
~)( jj ePeX
)(~ jeP
N
k
Nk
22
kN
jkNj eXeXkX
2
2 )()(][~
)(~
)( jj ePeX
N
k
NeX
k
j
22)(
N
keX
Nk
j
2)(
2
N
k
2
N
keX
Nk
kNj
2)(
2 2
)(~ jeX
The DFS of the periodic sequence can be considered as equally spaced samples of the Fourier transform of the finite-length sequence, which can be obtained by extracting one period of the periodic sequence.
kN
jkNj eXeXkX
2
2 )()(][~
Conclusions
Discrete Fourier Series, Properties of the Discrete Fourier Series, The Fourier Transform of Periodic Signals
Next lecture:
Discrete Fourier Transform
Assignment
Preparation for the next lecture：DFT
Solve the following problems:
8.3, 8.4，8.56
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