Lecture 06 - DFT windiwing.pdf

8/14/2019 Lecture 06 - DFT windiwing.pdf

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Lecture 6: Windowing of DFT

Instructor:

Dr. Gleb V. Tches lavski

Contact: [email protected]

Office Hours: Room 2030

Class web site:

http://ee.lamar.edu/gleb/dsp/index.htm

ELEN 5346/4304 DSP and Filter Design Fall 2008

by Steve Higgins

2

The idea of windowing

( )Recall: j j nnX e x e

= (6.2.1)- not really practicaln=

What if we observe aninfinitely longsequence over a finitelength time window?


0 D-1Than we dont see therest of the signal.

Let D is the number of samples (data points) observed.


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Windowing; what it causes

R

n n nWe specify y W x = (6.3.1)- a windowed view of xn

where:1, 0,1,..., 1

0

R

n

n DW

otherwise

= =

(6.3.2)

yn is a D- sequence (finite length), therefore, we can evaluate its DFT.

( ) ( ) ( )( )1j R j j

ny Y e W e X e d

= (6.3.3)


periodic convolution property

( )j

Y e

Therefore, is a dispersed/convoluted/obscured view of ( )j

X e

( ) 2 , 0,1,..., 1k

j

n kkDFT

P

y Y Y e k P =

= = zero-padded length

(6.3.4)

4

Windowing; what it causes (cont)

( )0 0a sinusoid 2 ( 2 )j n j

nDTFT

l

Let x e X e l = = + (6.4.1)

( ) ( ) ( )0( )01

2 ( 2 )2

jj R j R

l

Y e W e l d W e

= + =If 0 [-, ]

(6.4.2)

0

2

, 0,1,... 1

kj

R P

kFrequency sampling Y W e k P

= =

(6.4.3)


( ) 02 2 2

11

2

1 1 10 2 2 2

sin1 2

1sin

2

D D Dj j j

Dj DDj

j nR j R

n jj j jn

De e e

eW e W e e

ee e e

=

= = = =

(6.4.4)


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Leakage

( ) 0 2 , 1,2,...R j zl

W e lD

= = = (6.5.1)

( )

( )

sin / 20;

sin / 2

DWhen D

Let D = 100

(6.5.2)

40

60

80

100

Spectrum of a rectangular window


-0.3 -0.2 -0.1 0 0.1 0.2 0.3-40

-20

0

20

2

D

4

D

2

D

4

D

This is what causes so calledleakage observation offrequency components that donot exist in the spectrum of thesignal Due to observation only!

6

Frequency sampling

80

100

DFT means frequencysam lin !

80

100

-20

0

20

40

60

2

D

4

D

2

D

4

D

-20

0

20

40

60

We can only observe specificfrequency components of thesinc curve, and, therefore, cansee only the specific frequencycomponents of our signal!

-


-0.3 -0.2 -0.1 0 0.1 0.2 0.3-40

-0.3 -0.2 -0.1 0 0.1 0.2 0.3-40

of observed frequency samples.

BUT! our sinusoid leaves at 0 and this is the only frequency where we expect it tobe! Therefore, by smart choice of P we can observe the signal ONLY at 0 and atthe zero-crossings, i.e. at

0

2, 0,1, ...l l

P

= = (6.5.1)


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Characteristics of window functions

Windows characteristics

101

20

lg|Y|

Peak Side Lobelevel (PSL), dB

Side Lobe Roll-off (SLR),dB/octave


-0.3 -0.2 -0.1 0 0.1 0.2 0.310

0

Main lobe width(MLW)

at xx dB

rectangular window4

D

8

Characteristics of window functions

Windows characteristics

101

20

lg|Y|

We want:

MLW narrow for betterspectral resolution PSL lower to have lessmasking for nearby


-0.3 -0.2 -0.1 0 0.1 0.2 0.310

0

SLR better (faster) tohave less masking for faraway components


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Multiple sinusoids

( )an input is a mixtureof sinusoids: m m

j n

n m

m

Usually x A e +

= (6.9.1)

1 1Remember : cos( )

2 2m m m mj n j j n j

m mn e e e e + = +

Assume for simplicity that m= 0 for every m.

( )

2

( )

2 ( 2 )

mm

j

m m

m l

j kjj R R P

m k m

X e A l

Y e A W e Y A W e

= +

= =

(6.9.2)

(6.9.3)

(6.9.4)


m m

As a consequence, for two equal amplitude sinusoids, we will observe 2

peaks of |Yk| about half the time (depending on relative phase) when thefrequency difference is:

1 2 2 D = = - Rayleigh limit of frequencyresolution related to the window!

(6.9.5)

10

Multiple sinusoids

3535

Two sinusoids at the Rayleigh limit

0

5

10

15

20

25

30

Magnitude(dB)

5

10

15

20

25

30

Magnitude(dB)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10

-5

Normalized Frequency (rad/sample)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-5

0

Normalized Frequency (rad/sample)

Phase shift 0o Phase shift 90o


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Multiple sinusoids

Phase shift 180o 30

40

-30

-20

-10

0

10

20

Magnitude(dB)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-40


Two sisoids are NOT resolved! They appear as a single peak in thefrequency domain!

12

Multiple sinusoids

40

)

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20

0

20

Normalized Frequency ( rad/sample)

Magnitude(dB

-20

0

20

gnitude(dB)

-10

-5

0

5

10

15

20

25

30

Magnitude(dB)


DFTs of two sinusoids ofdifferent frequencies anddifferent magnitudes.

DFT of their sum

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-40


Ma

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-15



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Multiple sinusoids

Masking 30

40S

1

S

one signal isobscured byanother inthe freq.domain

-10

0

10

20

Magnitude(dB)

S1+S

2

0

0.5

1

1.5

1+S2


e nee abetter window!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-40

-30

-20


0 20 40 60 80 100

-1.5

-1

-0.5S=

Time, samples

( )

( )

1

2

sin 3

0.1sin 3 5

s n

s n D

=

= +

14

Multiple sinusoids

Alternative 40

S1

von Hannwindow

Two sins canbe resolved!!

-60

-40

-20

0

Magnitude(dB)

S2

S1+S

2


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-120

-100

-80



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Multiple sinusoids

40

Rectangular

i

-60

-40

-20

0

Magnitude(dB)

iwo w n owscomparison


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-120

-100

-80


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Different DFT windows

Rectangular (Boxcar) window

1, 0,1,... 1nW n D= = (6.14.1)



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von Hann (Hanning) window

1 cos sin , 0,1,... 12 1 1

nW n DD D

= = (6.15.1)


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Hamming window

0.54 0.46cos , 0,1,... 11

nW n DD

= = (6.16.1)



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Blackman window

0.42 0.5cos 0.08cos , 0,1,... 11 1

nW n DD D

= + = (6.17.1)


20


Kaiser window2

0

21 1

1

nI

D

( )0, 0,1,... 1K

nW n DI

= = (6.18.1)



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Different windows comparison

Window MLW (2/D) PSL (dB) SLR (dB/oct)Rectangular 0.9 -13 -6

Hanning 2 -31 -18

Hamming 2 -41 -6


Blackman 3 -57 -18

MLW narrow for better spectral resolution PSL lower to have less masking from nearby components SLR better (faster) to have less masking from far away components

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Summary

Windowing of a simple waveform, like cos(0t) causes itsFourier transform to have non-zero values (commonly calledleakage) at frequencies other than 0. It tends to be worst(highest) near 0 and least at frequencies farthest from 0.If there are two sinusoids, with different frequencies, leakagecan interfere with the ability to distinguish them spectrally. Iftheir frequencies are dissimilar, then the leakage interfereswhen one sinusoid is much smaller in amplitude than the


other. That is, its spectral component can be hidden by theleakage from the larger component. But when thefrequencies are near each other, the leakage can besufficient to interfere even when the sinusoids are equalstrength; that is, they become unresolvable.


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Conclusions

Selection of a window function for DFT must be donebased on the application. A rectangular window is thechoice when a high frequency resolution is desired.However, this type of window may a be a bed pick ifwe expect signals in a wide dynamic range.


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Lecture 06 - DFT windiwing.pdf