outline lecture, week 10 the discrete fourier transform ii · lecture, week 10 the discrete fourier...

Lecture, week 10The Discrete Fourier Transform II

Week 10, INF3190/4190

Andreas Austeng

Department of Informatics, University of Oslo

October 2019

AA, IN3190/4190 (Ifi/UiO) Lecture, week 10 Oct. 2019 1 / 21

Outline

Outline

1 Spectral smoothingSpectral smoothingDiscrete time windowsWindow resolution

2 Using the DFTLinear convolution of long sequencesBandlimited interpolation


Spectral smoothing

Outline




Spectral smoothing

Finite length data (I)

If we define a window w [n] =

(1, n 2 0, 1, . . . , N � 10 otherwise

and a periodic signal xp[n], n 2 �1 . . .1,then the product x [n] = xp[n] w [n]describes a time limited version of xp[n].Taking the DFT of x [n] might cause problems as:

I Spectral leakage (when N 6= kTs, k 2 N, Ts being the period of xp[n])I Smoothing


Spectral smoothing

Finite length data (II)

Effect of rectangularwindowing on the spectrumof a sinousoidal signal.x [n] = xp[n] w [n], i.e.X (e|!) = Xp(e|!) ⇤W (e|!).


Spectral smoothing

Finite length data (III)Different parts of the periodic signal xp[n] = 0.5 sin(2⇡F0n):

�20 0 20 40 60�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Am

plitud

e

N = 3FsF0

�20 0 20 40 60�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Am

plitud

e

N = 3FsF0

-5

0 100 200 300 400 500�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Am

plitud

e

N = 11FsF0

-5

DFT)

DFT)

DFT)

-Fs2

-F0 0 F0 Fs2

0

14

12

Mag

nitu

de

N = 3FsF0

-Fs2

-F0 0 F0 Fs2

0

14

12

Mag

nitu

de

N = 11FsF0

-5

-Fs2

-F0 0 F0 Fs2

0

14

12

Mag

nitu

de

N = 11FsF0

-5


Spectral smoothing

Finite length data (IV)Different parts of the periodic signal xp[n] = 0.5 sin(2⇡F0n) + 0.5 sin(2⇡F1n):

�20 0 20 40 60�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Am

plitud

e

N = 3FsF0

�20 0 20 40 60�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Am

plitud

e

N = 3FsF0

-5

0 100 200 300 400 500�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Am

plitud

e

N = 11FsF0

-5

DFT)

DFT)

DFT)

-Fs2

-F1-F0 0 F0F1 Fs2

0

14

12

Mag

nitu

de

FDFT {x[n]} = X[k]

-Fs2

-F1-F0 0 F0F1 Fs2

0

14

12

Mag

nitu

de

FDFT {x[n]} = X[k]

-Fs2

-F1-F0 0 F0F1 Fs2

0

14

12

Mag

nitu

de

FDFT {x[n]} = X[k]


Spectral smoothing Spectral smoothing

The spectrum of a time limited window function

A window w [n] =

(1, n 2 0, 1, . . . , N � 10 otherwise

has the DTFT W (⌦) = sinN⌦/2sin⌦/2 e�|(N�1)⌦/2

> N = 14;> omega = l inspace(�pi , p i ,512) ;> W = ( s in (N*omega / 2 ) . / s in (omega / 2 ) ) . . .

. * exp(�j * (N�1)*omega / 2 ) ;> p l o t (omega , r e a l (W ) ) ;

> h = fvtool (ones (N , 1 ) /N , 1 , . . .' f requencyrange ' , ' [�pi , p i ) ' ) ;

> p r i n t �dpng WindowedSin


Spectral smoothing Discrete time windows

Response, typical windows; quality metrics

Quality metrics, windows;I Maks verdi; Peak value (P)I Maks sidelobe value, Peak Sidelobe Level (PSL)I -3dB width (W3), ⇠

p2/2 P

I -6dB width (W6), ⇠ 0.5P

I Width, 1st zero (WM )I Width to reach max-SL level (WS)



Some typicalwindows (I)

The different windowsgive compromisesbetween resolution andleakage



Some typical windows (II)



Finite length data – rectangular windowDifferent parts of the periodic signal xp[n] = 0.5 sin(2⇡F0n) + 0.5 sin(2⇡F1n):

�20 0 20 40 60�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Am

plitud

e

N = 3FsF0

�20 0 20 40 60�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Am

plitud

eN = 3Fs

F0-5

0 100 200 300 400 500�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Am

plitud

e

N = 11FsF0

-5

DFT)

DFT)

DFT)

-Fs2

-F1-F0 0 F0F1 Fs2

0

14

12

Mag

nitu

de

FDFT {x[n]} = X[k]

-Fs2

-F1-F0 0 F0F1 Fs2

0

14

12

Mag

nitu

de

FDFT {x[n]} = X[k]

-Fs2

-F1-F0 0 F0F1 Fs2

0

14

12

Mag

nitu

de

FDFT {x[n]} = X[k]



Finite length data – Hamming windowDifferent parts of the periodic signal xp[n] = 0.5 sin(2⇡F0n) + 0.5 sin(2⇡F1n):

�20 0 20 40 60�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Am

plitud

e

N = 3FsF0

�20 0 20 40 60�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Am

plitud

e

N = 3FsF0

-5

0 100 200 300 400 500�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Am

plitud

e

N = 11FsF0

-5

DFT)

DFT)

DFT)

-Fs2

-F1-F0 0 F0F1 Fs2

0

14

12

Mag

nitu

de

FDFT {x[n]} = X[k]

-Fs2

-F1-F0 0 F0F1 Fs2

0

14

12

Mag

nitu

de

FDFT {x[n]} = X[k]

-Fs2

-F1-F0 0 F0F1 Fs2

0

14

12

Mag

nitu

de

FDFT {x[n]} = X[k]


Spectral smoothing Window resolution

Windows, resolution (I)How close can two frequencies be if we want to be able to separate them?

Frequency resolution: �f = WMFs, where WM is window dependent and gives distance to 1stzero, and Fs is the sampling frequency.

Eksempel:

Given:

x(t) = A1 cos 2⇡f0t + A2 cos 2⇡(f0 + �f )t

A1 = A2 = 1 f0 = 30 Hz Fs = 128 Hz

(a) What is the smallest resolvable �f using a Boxcar (rectangular) and Hanning (vonHann) window when:

i N = 256, Zero-padded to length 2048ii N = 512, Zero-padded to length 2048iii N = 256, Zero-padded to length 4096



Windows, resolution (II)Answer to "(a) What is the smallest resolvable �f using a Boxcar and Hanning, when"

i N = 256, Zero-padded to length 2048) Boxcar: �F = 2

NFs = 2

256 128Hz = 1Hz

) Hanning: �F = 4N

Fs = 4256 128Hz = 2Hz

ii N = 512, Zero-padded to length 2048) Boxcar: �F = 2

NFs = 2

512 128Hz = 0.5Hz


Fs = 4512 128Hz = 1Hz

iii N = 256, Zero-padded to length 4096) Boxcar: �F = 2

NFs = 2

256 128Hz = 1Hz


Fs = 4256 128Hz = 2Hz

Boxcar gives the best resolution, but has the highest sidelobes / spectral leakage.

The resolution improves when N is increased.

Zero-padding has no effect..

Hanning gives half the resolution, but has less leakage.



Windows,resolution (III)


Using the DFT

Outline




Using the DFT Linear convolution of long sequences

Convolution of long sequences

Given h[n] of length N1, and x [n] of length N2.1 Zero-pad h[n] and x [n] to length L � N1 + N2 � 1.2 Compute L-point DFT of h[n] and x [n].3 Perform multiplication: Y [k ] = H[k ] X [k ].4 Find inverse of Y [k ]! y [n] = h[n] ⇤ x [n].

Sometime, this is not practical:I Long sequences require much storage, large computational load, and long time (we

have to wait for last sample).Solution: Block-convolution. Two types:

I Overlap-addI Overlap-save



Overlap-add

1 Split x [n] into m sequences xi [n] oflength M.

2 Compute convolution of xi [n] and h[n](of length N1 + M � 1).

I May be done using DFTs3 Add all Xi [k ]

X [k ] =M�1Pi=0

Xi [k ]



Overlap-save

Let x [n] be the long sequence:1 Zero-pad x [n] with N � 1 zeros in front.2 Divide into k over-lapping segments

(N � 1 elements overlapp) of lengthM + N � 1 (larger than length of h[n]).

3 Zero-pad h[n] to length M + N � 1.4 Compute the circular convolution of

each part and cut overlapping region.


Using the DFT Bandlimited interpolation

Interpolation in frequency, from length N to N ⇥M

1 x [n]DFT! X [k ]

2 Zero-pad X [k ] til Xi [k ] as follows:

Xi [k ] =

({X [0], . . . ,X [N/2 � 1], . . . (M � 1)N ”zeros” . . .X [N/2], . . . ,X [N � 1] If N odd{X [0], . . . , 0.5X [N/2], . . . (M � 1)N � 1 ”zeros” . . . 0.5X [N/2], . . . ,X [N � 1] if N even

3 xi [n]DFT Xi [k ]

Equivalentto zero-pad x [n] between samples, perform DFT, LP-filter result, and IDFT.


outline lecture, week 10 the discrete fourier transform ii · lecture, week 10 the discrete fourier...

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