Discrete-Time Fourier Methods
M. J. Roberts - All Rights Reserved.
Edited by Dr. Robert Akl
1
Discrete-Time Fourier Series Concept
A signal can be represented as a linear combination of sinusoids.
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2
Discrete-Time Fourier Series Concept
The relationship between complex and real sinusoids
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3
Discrete-Time Fourier Series Concept
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4
Discrete-Time Fourier Series Concept
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5
Discrete-Time Fourier Series Concept
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6
Discrete-Time Fourier Series Concept
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7
Discrete-Time Fourier Series Concept
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8
Discrete-Time Fourier Series Concept
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9
Discrete-Time Fourier Series Concept
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10
The Discrete-Time Fourier Series
The discrete-time Fourier series (DTFS) is similar to the CTFS.
A periodic discrete-time signal can be expressed as
x[n] =
å
k= N
c x [ k ] e j 2 p kn/ N
1 n0 + N -1
- j 2 p kn/ N
e
n
x
cx [ k ] =
]
[
å
N n=n0
where c x [ k ] is the harmonic function, N is any period of x [ n ]
and the notation,
å
means a summation over any range of
k= N
consecutive k’s exactly N in length.
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11
The Discrete Fourier Transform
The discrete Fourier transform (DFT) is almost identical to the DTFS.
A periodic discrete-time signal can be expressed as
1
x[n] =
N
å
k= N
X [ k ] e j 2 p kn/N
X[ k ] =
n0 + N -1
å
n=n0
x [ n ] e- j 2p kn/ N
where X [ k ] is the DFT harmonic function, N is any period of x [ n ]
and the notation,
å
means a summation over any range of
k= N
consecutive k’s exactly N in length. The main difference between the
DTFS and the DFT is the location of the 1/N term. So X [ k ] = N c x [ k ] .
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12
The Discrete Fourier Transform
Because the DTFS and DFT are so similar, and because the DFT is
so widely used in digital signal processing (DSP), we will concentrate
on the DFT realizing we can always form the DTFS from
c x [ k ] = X [ k ] / N.
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13
The Discrete Fourier Transform
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DFT Example
Find the DFT harmonic function for
(
)
x éë n ùû = u éë n ùû - u éë n - 3ùû * d 5 éë n ùû
using its fundamental period as the representation time.
X éë k ùû =
å
x éë n ùû e- j 2p kn/ N
n= N
4
(
)
= å u éë n ùû - u éë n - 3ùû * d 5 éë n ùû e- j 2p kn/5
n=0
2
= åe
- j 2 p kn/5
n=0
2
= åe
n=0
- j 2 p kn/5
1- e- j6p k /5 e- j3p k /5 e j3p k /5 - e- j3p k /5
=
= - jp k /5 ´ jp k /5 - jp k /5
- j 2p k /5
1- e
e
e
-e
=e
- j 2 p k /5
(
) = 3e
sin (p k / 5)
sin 3p k / 5
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- j 2 p k /5
(
)
drcl k / 5,3
15
The DFT Harmonic Function
1
We know that x [ n ] =
N
å
k= N
X [ k ] e j 2 p kn/N so we can find x [ n ]
from its harmonic function. But how do we find the harmonic
function from x [ n ] ? We use the principle of orthogonality like
we did with the CTFS except that now the orthogonality is in
discrete time instead of continuous time.
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16
The DFT Harmonic Function
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The DFT Harmonic Function
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18
The DFT Harmonic Function
Below is a set of complex sinusoids for N = 8. They form a
set of basis vectors. Notice that the k = 7 complex sinusoid
rotates counterclockwise through 7 cycles but appears to rotate
clockwise through one cycle. The k = 7 complex sinusoid is
exactly the same as the k = -1 complex sinusoid. This must be
true because the DFT is periodic with period N.
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19
The DFT Harmonic Function
The projection of a real vector x in the direction of another
real vector y is
xT y
p= T y
y y
If p = 0, x and y are orthogonal. If the vectors are complexvalued
xH y
p= H y
y y
where the x H is the complex-conjugate transpose of x. xT y
and x H y are the dot product of x and y.
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20
The DFT Harmonic Function
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21
The DFT Harmonic Function
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22
The DFT Harmonic Function
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23
The DFT Harmonic Function
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24
The DFT Harmonic Function
The most common definition of the DFT is
N -1
X[ k ] = å x [ n] e
n=0
- j 2 p kn/N
,
1
x[n] =
N
å
k= N
X [ k ] e j 2 p kn/N
Here the beginning point for x [ n ] is taken as n0 = 0 . This is
the form of the DFT that is implemented in practically all computer
languages.
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25
Convergence of the DFT
• The DFT converges exactly with a finite
number of terms. It does not have a “Gibbs
phenomenon” in the same sense that the
CTFS does
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26
The Discrete Fourier Transform
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27
DFT Properties
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DFT Properties
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DFT Properties
It can be shown (and is in the text) that if x [ n ] is an even
function, X [ k ] is purely real and if x [ n ] is an odd function
X [ k ] is purely imaginary.
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30
The Dirichlet Function
The functional form
sin ( p Nt )
N sin (p t )
appears often in discrete-time
signal analysis and is given the
special name Dirichlet function.
That is
drcl ( t, N ) =
sin (p Nt )
N sin ( p t )
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31
DFT Pairs
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The Fast Fourier Transform
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The Fast Fourier Transform
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34
Generalizing the DFT for
Aperiodic Signals
Pulse Train
This periodic rectangular-wave signal is analogous to the
continuous-time periodic rectangular-wave signal used to
illustrate the transition from the CTFS to the CTFT.
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35
Generalizing the DFT for
Aperiodic Signals
DFT of
Pulse Train
As the period of the
rectangular wave
increases, the period of
the DFT increases
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36
Generalizing the DFT for
Aperiodic Signals
Normalized
DFT of
Pulse Train
By plotting versus k / N 0
instead of k, the period
of the normalized DFT
stays at one.
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37
Generalizing the DFT for
Aperiodic Signals
The normalized DFT approaches this limit as the
period approaches infinity.
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38
Definition of the Discrete-Time
Fourier Transform (DTFT)
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39
The Discrete-Time Fourier
Transform
The function e- jW appears in the forward DTFT raised to the nth power.
It is periodic in W with fundamental period 2p . n is an integer. Therefore
e- jWn is periodic with fundamental period 2p / n and 2p is also a period
of e- jWn . The forward DTFT is
¥
( ) = å x[n] e
X e
jW
- jWn
n=-¥
a weighted summation of functions of the form e- jWn , all of which repeat with
( )
every 2p change in W. Therefore X e jW is always periodic in W with period
2p . This also implies that X ( F ) is always periodic in F with period 1.
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40
DTFT Pairs
We can begin a table of DTFT pairs directly from the definition.
(There is a more extensive table in the text.)
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41
The Generalized DTFT
By generalizing the CTFT to include transform that have impulses
we were able to find CTFT's of some important practical functions.
The same is true of the DTFT. The DTFT of a constant
X( F ) =
¥
å
n=-¥
¥
Ae- j 2p Fn = A å e- j 2 p Fn
n=-¥
does not converge. The CTFT of a constant turned out to be an
impulse. Since the DTFT must be periodic that cannot the the
transform of a constant in discrete time. Instead the transform must
be a periodic impulse.
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42
The Generalized DTFT
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The Generalized DTFT
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Forward DTFT Example
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Forward DTFT Example
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Forward DTFT Example
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More DTFT Pairs
We can now extend the table of DTFT pairs.
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48
DTFT Properties
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DTFT Properties
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50
DTFT Properties
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DTFT Properties
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DTFT Properties
Time scaling in discrete time is quite different from time
scaling in continuous-time. Let z [ n ] = x [ an ] . If a is not an
integer some values of z [ n ] are undefined and a DTFT cannot
be found for it. If a is
an integer greater than
one, some values of x [ n ]
will not appear in z [ n ]
because of decimation and
there cannot be a unique
relationship between their
DTFT’s
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53
DTFT Properties
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54
DTFT Properties
In the time domain, the response of a system is the convolution
of the excitation with the impulse response
y[n] = x[n] * h[n]
In the frequency domain the response of a system is the product
of the excitation and the frequency response of the system
( )
( ) ( )
Y e jW = X e jW H e jW
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55
DTFT Properties
Find the signal energy of x [ n ] = (1 / 5 ) sinc ( n /100 ) . The
straightforward way of finding signal energy is directly from
the definition E x =
¥
å x[n]
2
.
n=-¥
Ex =
¥
å (1 / 5 ) sinc ( n /100 )
n=-¥
2
¥
= (1 / 25 ) å sinc 2 ( n /100 )
n=-¥
In this case we run into difficulty because we don't know how
to sum this series.
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56
DTFT Properties
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Transform Method Comparisons
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Transform Method Comparisons
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Transform Method Comparisons
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Transform Method Comparisons
Using the equivalence property of the impulse and fact that both
e j 2 p F and d1 ( F ) have a fundamental period of one,
é jp /6 d1 ( F - 1 /12 ) - jp /6 d1 ( F + 1 /12 ) ù
Y ( F ) = (1 / 2 ) ê e
+e
jp /6
- jp /6
e
0.9
e
- 0.9 úû
ë
Finding a common denominator and simplifying,
d1 ( F - 1 /12 ) (1 - 0.9e jp /6 ) + d1 ( F + 1 /12 ) (1 - 0.9e- jp /6 )
Y ( F ) = (1 / 2 )
1.81 - 1.8 cos (p / 6 )
Y ( F ) = 0.4391 éëd1 ( F - 1 /12 ) + d1 ( F + 1 /12 ) ùû
+ j0.8957 éëd1 ( F + 1 /12 ) - d1 ( F - 1 /12 ) ùû
y [ n ] = 0.8782 cos ( 2p n /12 ) + 1.7914 sin ( 2p n /12 )
y [ n ] = 1.995 cos ( 2p n /12 - 1.115 )
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61
Transform Method Comparisons
The DFT can often be used to find the DTFT of a signal. The
DTFT is defined by X ( F ) =
¥
å
n=-¥
x [ n ] e- j 2 p Fn and the DFT
N -1
is defined by X [ k ] = å x [ n ] e- j 2 p kn/ N . If the signal x [ n ] is causal
n=0
and time limited, the summation in the DTFT is over a finite
range of n values beginning with 0 and we can set the value of
N by letting N - 1 be the last value of n needed to cover that finite
N -1
range. Then X ( F ) = å x [ n ] e- j 2 p Fn . Now let F ® k / N yielding
n=0
N -1
X ( k / N ) = å x [ n ] e- j 2 p kn/N = X [ k ]
n=0
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62
Transform Method Comparisons
The result
N -1
X ( k / N ) = å x [ n ] e- j 2 p kn/ N = X [ k ]
n=0
is the DTFT of x [ n ] at a discrete set of frequencies F = k / N or
W = 2p k / N. If that resolution in frequency is not sufficient, N
can be made larger by augmenting the previous set of x [ n ] values
with zeros. That reduces the space between frequency points
thereby increasing the resolution. This technique is called zero
padding.
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63
Transform Method Comparisons
We can also use the DFT to approximate the inverse DTFT.
The inverse DTFT is defined by x [ n ] = ò X ( F ) e j 2 p Fn dF
1
1 N -1
and the inverse DFT is defined by x [ n ] = å X [ k ] e j 2 p kn/ N .
N k =0
We can approximate the inverse DTFT by
N -1 ( k +1) / N
x[n] @ å
k =0
ò
k/N
N -1
( k +1) / N
k =0
k/N
X ( k / N ) e j 2 p Fn dF = å X ( k / N )
ò
e j 2 p Fn dF
e j 2 p ( k +1)n/ N - e j 2 p k / N e j 2 p n/N - 1 N -1
x[n] @ å X( k / N )
=
X ( k / N ) e j 2 p kn/ N
å
j2p n
j2p n k =0
k =0
N -1
x[n] @ e
jp n/ N
1 N -1
sinc ( n / N ) å X ( k / N ) e j 2 p kn/ N
N k =0
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64
Transform Method Comparisons
For n << N,
1 N -1
x [ n ] @ å X ( k / N ) e j 2 p kn/ N
N k =0
This is the inverse DFT with X [ k ] = X ( k / N ) .
Use this result to find the inverse DTFT of
X ( F ) = éë rect ( 50 ( F - 1 / 4 ) ) + rect ( 50 ( F + 1 / 4 ) ) ùû * d1 ( F )
with the inverse DFT.
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65
Transform Method Comparisons
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Transform Method Comparisons
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67
The Four Fourier Methods
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68
Relations Among Fourier Methods
Multiplication-Convolution Duality
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69
Relations Among Fourier Methods
Parseval’s Theorem
Discrete Frequency
1
Continuous Time
T0
Discrete Time
ò
T0
å
n= N
x ( t ) dt =
2
x[n]
¥
å
X[ k ]
å
X[ k ]
k=-¥
2
1
=
N
k= N
Continuous Frequency
2
¥
ò
x ( t ) dt =
2
-¥
2
¥
å
n=-¥
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¥
ò
X ( f ) df
2
-¥
x [ n ] = ò X ( F ) dF
2
2
1
70
Relations Among Fourier Methods
Time and Frequency Shifting
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71
Relations Among Fourier Methods
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72