Chapter7

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Discrete-Time Fourier Methods

Discrete-Time Fourier Series Concept A signal can be represented as a linear combination of sinusoids.

Discrete-Time Fourier Series Concept The relationship between complex and real sinusoids

Discrete-Time Fourier Series Concept

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 =

= å

c x [ ]

e j

2 p

c x = 1

N n

0 +

1 å x [ ]

2 p

where c x and the notation,

= å

consecutive

’s exactly

in length.

[ ] means a summation over any range of

The Discrete Fourier Transform The discrete Fourier transform (DFT) is almost identical to the DTFS.

A periodic discrete-time signal can be expressed as x where X

= 1

N k

= å

X [ ]

e j

2 p

X =

0 +

1 å

0 x [ ]

2 p

[ ] is the DFT harmonic function,

is any period of x and the notation,

= å

means a summation over any range of [ ] consecutive k’s exactly

in length. The main difference between the DTFS and the DFT is the location of the 1/

term. So X =

c x [ ] .

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 = X /

The Discrete Fourier Transform

DFT Example Find the DFT harmonic function for x ( u u éé

3 éé ) * d 5 using its fundamental period as the representation time.

= é

n e

2 p

= é 4

= 0 ( u u éé

3 éé ) * d 5

n e

2 p

/5 = = é 2

= 0 é 2

= 0

2 p

/5 = =

1 1 -

6 p

3 p

/5 p

/5 =

2 p

/5 sin 3 p

sin ( ( p

k e

/5 / 5 ) / 5 ) = é 3

e j

3 p

e j

2 p

/5 -

drcl (

3 p

/ 5,3 )

The DFT Harmonic Function We know that x = 1

N k

= å

X [ ]

e j

2 p

so we can find x from its harmonic function. But how do we find the harmonic function from x [ ] ? 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.

The DFT Harmonic Function

The DFT Harmonic Function Below is a set of complex sinusoids for

= 8. They form a set of

basis

vectors. Notice that the

= 7 complex sinusoid rotates counterclockwise through 7 cycles but appears to rotate clockwise through one cycle. The

= exactly the same as the

7 complex sinusoid is = 1 complex sinusoid. This must be true because the DFT is periodic with period

The DFT Harmonic Function The projection of a real vector

in the direction of another real vector

y y

= 0,

and

are orthogonal. If the vectors are complex valued

y y

where the

is the complex-conjugate transpose of

and

are the

dot product

and

The DFT Harmonic Function

The DFT Harmonic Function The most common definition of the DFT is X =

1 å

= 0 x [ ]

2 p

, x Here the beginning point for x = 1

N k

= å

[ ] is taken as

0 X [ ]

e j

2 p

= 0 . This is the form of the DFT that is implemented in practically all computer languages.

Convergence of the DFT • The DFT converges exactly with a finite number of terms. It does not have a “ Gibbs phenomenon CTFS does ” in the same sense that the

The Discrete Fourier Transform

DFT Properties

DFT Properties It can be shown (and is in the text) that if x function, X X [ ] is purely real and if x [ ] is purely imaginary.

[ ] is an even [ ] is an odd function

The Dirichlet Function The functional form sin (

sin p

( ) ) appears often in discrete-time signal analysis and is given the special name

Dirichlet

function. That is drcl = sin (

sin p

( ) )

DFT Pairs

The Fast Fourier Transform

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.

Generalizing the DFT for Aperiodic Signals DFT of Pulse Train As the period of the rectangular wave increases, the period of the DFT increases

Generalizing the DFT for Aperiodic Signals

Normalized

DFT of Pulse Train By plotting versus

0 instead of

, the period of the normalized DFT stays at one.

Generalizing the DFT for Aperiodic Signals The normalized DFT approaches this limit as the period approaches infinity.

Definition of the Discrete-Time Fourier Transform (DTFT)

The Discrete-Time Fourier Transform The function

W appears in the forward DTFT raised to the

th power. It is periodic in W with fundamental period 2 p .

is an integer. Therefore

is periodic with fundamental period 2 p /

and 2 p is also a period of

. The forward DTFT is X = é é

=-é x [ ]

a weighted summation of functions of the form

every 2 p change in W . Therefore X ( )

, all of which repeat with is always periodic in W with period 2 p . This also implies that X ( ) is always periodic in

with period 1.

DTFT Pairs We can begin a table of DTFT pairs directly from the definition. (There is a more extensive table in the text.)

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 = ¥ å

=-¥

2 p

¥ å

=-¥

2 p

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.

The Generalized DTFT

Forward DTFT Example

More DTFT Pairs We can now extend the table of DTFT pairs.

DTFT Properties

DTFT Properties Time scaling in discrete time is quite different from time scaling in continuous-time. Let z integer some values of z = x [ ] . If

is not an [ ] are undefined and a DTFT cannot be found for it. If

is an integer greater than one, some values of x will not appear in z [ ] [ ] because of decimation and there cannot be a unique relationship between their DTFT’s

DTFT Properties

DTFT Properties In the time domain, the response of a system is the convolution of the excitation with the impulse response y = x [ ] * h In the frequency domain the response of a system is the product of the excitation and the frequency response of the system Y ( ) = X ( ) H

DTFT Properties Find the signal energy of x = ( ) sinc (

/ 100 ) . The straightforward way of finding signal energy is directly from the definition

E x E x

= ¥ å

=-¥ x 2 .

= ¥ å

=-¥ ( ) sinc (

/ 100 ) 2 = ( 1 / 25 ) ¥ å

=-¥ sinc 2 (

/ 100 ) In this case we run into difficulty because we don't know how to sum this series.

DTFT Properties

Transform Method Comparisons

Transform Method Comparisons Using the equivalence property of the impulse and fact that both

e j

2 p

Y and d 1 = ( ) have a fundamental period of one, é

e j

p /6 éé d 1 (

F e j

p /6 1 / 12 0.9

) +

p /6 d 1

(

j F

p /6 + 1 / 12 0.9

) é éé Finding a common denominator and simplifying, Y Y y = = ( ) d 0.4391

[ ] = + y [ ] 1 = ( éé d 1

( 1 / 12

F j

0.8957

) 1 / 12 éé d 1 ( (

( ( ) 1 + 0.9

e j

p /6 d 1 + 1 / 12 p

/ 12 ) 1.81

1.8 cos + (

+ 1 / 12 ) d 1 p

/ 12 ) ( +

d 1 1 / 12 1.7914 sin 2 1.115

) ( ( ) éé

p + 1 / 12 / 6 ) ) éé p

/ 12 ) ) ( 1 0.9

p /6 )

Transform Method Comparisons The DFT can often be used to find the DTFT of a signal. The DTFT is defined by X is defined by X = = é é

=-é x [ ]

2 p

and the DFT

1 é

= 0 x [ ]

2 p

. If the signal x [ ] is causal and time limited, the summation in the DTFT is over a finite range of

values beginning with 0 and we can set the value of

by letting

1 be the last value of

needed to cover that finite range. Then X X (

) = =

1 é

= 0 x [ ]

2 p

. Now let

F N

1 é

= 0 x [ ]

2 p

= X ®

yielding

Transform Method Comparisons The result X (

is the DTFT of x /

) =

1 é x [ ]

2 p

= X

n n

= 0 [ ] at a discrete set of frequencies

W = 2 p

/ =

. If that resolution in frequency is not sufficient,

can be made larger by augmenting the previous set of x [ ] values with zeros. That reduces the space between frequency points thereby increasing the resolution. This technique is called

zero padding

Transform Method Comparisons We can also use the DFT to approximate the inverse DTFT.

The inverse DTFT is defined by x and the inverse DFT is defined by x = = ò 1 X ( )

e j

2 p

N N

1 å

= 0

X [ ]

e j

2 p

We can approximate the inverse DTFT by x x @ @

1 å

= 0 ( ) /

X (

1 å

= 0 X (

)

e j

2 p

Fn dF

1 å

= 0

)

e j

2 p ( )

2 p

n e j

2 p

= X (

) ( ) /

N e j

2 p

N j

2 p

1 å

= 0

e j

2 p

X (

dF N

)

e j

2 p

x @

e j

sinc (

) 1

N N

1 å

= 0 X (

)

e j

2 p

Transform Method Comparisons For

, x @ 1

N N

1 é

= 0 X (

)

e j

2 p

This is the inverse DFT with X = X (

) .

Use this result to find the inverse DTFT of X ( ) = éé ( (

1 / 4 ) ) + (

+ 1 / 4 ) ) éé* d 1 with the inverse DFT.

Transform Method Comparisons

The Four Fourier Methods

Relations Among Fourier Methods

Multiplication-Convolution Duality

Relations Among Fourier Methods

Parseval

’

s Theorem

Continuous Time Discrete Time 1

0 ò

0 Discrete Frequency x 2

= ¥ å

=-¥ X

= å

x 2 = 1

N k

= å

X 2 2 Continuous Frequency ¥ -¥ ò ¥ å

=-¥ x x 2

2 = = ¥ -¥ ò ò 1 X X 2

Relations Among Fourier Methods

Time and Frequency Shifting

Relations Among Fourier Methods

Chapter7

Transcript Chapter7

Discrete-Time Fourier Methods

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