•Three shorter Midterms: Digital Signal Processingee123/sp14/Notes/Lecture09_Spect...Based on...

M. Lustig, EECS UC Berkeley

EE123Digital Signal Processing

Lecture 9

based on slides by J.M. Kahn1


Announcements

• Lab 01 part I and II posted will post III today or tomorrow

• Lab-bash Tuesday 2-3pm 521 Cory • Three shorter Midterms:

– 02/26 in class– 04/02 in class– 04/30 (or 28 TBD) in class– 05/05 or 05/06 (TBD) project presentations.

• Posters and demos

2


Announcements

• Last time: –Frequency analysis with DFT–Windowing

• Today:– Continue – Effect of zero-padding– Start Short-time Fourier Transform

3

Windows Properties

These are characteristic of the window type

Window Main-lobe Sidelobe �s

Sidelobe �20 log10

�s

Rect4⇡

M + 10.09 21

Bartlett8⇡

M + 10.05 26

Hann8⇡

M + 10.0063 44

Hamming8⇡

M + 10.0022 53

Blackman12⇡

M + 10.0002 74

Most of these (Bartlett, Hann, Hamming) have a transition widththat is twice that of the rect window.

Warning: Always check what’s the definition of M

Adapted from A Course In Digital Signal Processing by Boaz Porat, Wiley, 1997

Miki Lustig UCB. Based on Course Notes by J.M Kahn Spring 2014, EE123 Digital Signal Processing

4

Windows Examples

Here we consider several examples. As before, the sampling rate is⌦s

/2⇡ = 1/T = 20 Hz.Rectangular Window, L = 32

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

n

w[n

]

Rectangular Window, L = 32

-20 -10 0 10 200

5

10

15

20

25

30

35

40

/2 (Hz)

|W(ejT)|

DTFT of Rectangular Window

0 5 10 15 20 25 30

-1.5

-1

-0.5

0

0.5

1

1.5

n

v[n]

Sampled, Windowed Signal, Rectangular Window, L = 32

-20 -10 0 10 200

5

10

15

20

/2 (Hz)

|V(ejT)|

DTFT of Sampled, Windowed Signal


Ω

ωT

ωT

5

Windows Examples

Triangular Window, L = 32

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

n

w[n

]

Triangular Window, L = 32

-20 -10 0 10 200

5

10

15

20

/2 (Hz)

|W(ejT)|

DTFT of Triangular Window

0 5 10 15 20 25 30

-1.5

-1

-0.5

0

0.5

1

1.5

n

v[n]

Sampled, Windowed Signal, Triangular Window, L = 32

-20 -10 0 10 200

2

4

6

8

/2 (Hz)

|V(ejT)|



ωT

ωT

6

Windows Examples

Hamming Window, L = 32

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

n

w[n

]


-20 -10 0 10 200

5

10

15

20

/2 (Hz)

|W(ejT)|

DTFT of Hamming Window

0 5 10 15 20 25 30

-1.5

-1

-0.5

0

0.5

1

1.5

n

v[n]

Sampled, Windowed Signal, Hamming Window, L = 32

-20 -10 0 10 200

2

4

6

8

10

/2 (Hz)

|V(ejT)|



ωT

ωT

7


Windows Examples


0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

nw

[n]


-20 -10 0 10 200

5

10

15

20

25

30

35

40

/2 (Hz)

|W(ejT)|

DTFT of Hamming Window

0 10 20 30 40 50 60

-1.5

-1

-0.5

0

0.5

1

1.5

n

v[n]

Sampled, Windowed Signal, Hamming Window, L = 64

-20 -10 0 10 200

5

10

15

20

/2 (Hz)

|V(ejT)|



ωT

ωT

8


Optimal Window: Kaiser

• Minimum main-lobe width for a given side-lobe energy %

• Window is parametrized with L and β– β determines side-lobe level– L determines main-lobe width

Rsidelobes

|H(ej!)|2d!R ⇡�⇡ |H(ej!)|2d!

OS Eq 10.12

9


Exampley = sin(2⇡0.1992n) + 0.005 sin(2⇡0.25n) | 0 n < 128

10


Example

11

Zero-Padding

In preparation for taking an N-point DFT, we may zero-padthe windowed block of signal samples to a block length N � L:

(v [n] 0 n L� 1

0 L n N � 1

This zero-padding has no e↵ect on the DTFT of v [n], sincethe DTFT is computed by summing over �1 < n < 1.

E↵ect of Zero Padding

We take the N-point DFT of the zero-padded v [n], to obtainthe block of N spectral samples:

V [k], 0 k N � 1


12

Zero-Padding

Consider the DTFT of the zero-padded v [n]. Since thezero-padded v [n] is of length N, its DTFT can be written:

V (e j!) =N�1X

n=0

v [n]e�jn!, �1 < ! < 1

The N-point DFT of v [n] is given by:

V [k] =N�1X

n=0

v [n]W kn

N

=N�1X

n=0

v [n]e�j(2⇡/N)nk , 0 k N � 1

We see that V [k] corresponds to the samples of V (e j!):

V [k] = V (e j!)��!=k

2⇡N

, 0 k N � 1

To obtain samples at more closely spaced frequencies, wezero-pad v [n] to longer block length N. The spectrum is thesame, we just have more samples.


13

Frequency Analysis with DFT

Note that the ordering of the DFT samples is unusual.

V [k] =N�1X

n=0

v [n]W nk

N

The DC sample of the DFT is k = 0

V [0] =N�1X

n=0

v [n]W 0n

N

=N�1X

n=0

v [n]

The positive frequencies are the first N/2 samplesThe first N/2 negative frequencies are circularly shifted

((�k))N

= N � k

so they are the last N/2 samples. (Use fftshift to reorder)


14

Frequency Analysis with DFT Examples:

Hamming Window, L = 32, N = 32

0 5 10 15 20 25 30

-1.5

-1

-0.5

0

0.5

1

1.5

n

Zero

-Padded v

[n]

Sampled, Windowed Signal, Hamming Window, L = 32, Zero-Padded to N = 32

0 5 10 15 20 25 300

2

4

6

8

10

k

|V[k

]|

N-Point DFT of Sampled, Windowed, Zero-Padded Signal

0 5 10 15 200

2

4

6

8

10

/2 (Hz)

|V[k

]|,

|V(ejT)|

Spectrum of Sampled, Windowed, Zero-Padded Signal

|V(ej T)||V[k ]|,

k = k2 /NT


ωT

15

Frequency Analysis with DFT Examples:

Hamming Window, L = 32, Zero-Padded to N = 64

0 10 20 30 40 50 60

-1.5

-1

-0.5

0

0.5

1

1.5

n

Zero

-Padded v

[n]

Sampled, Windowed Signal, Hamming Window, L = 32, Zero-Padded to N = 64

0 10 20 30 40 50 600

2

4

6

8

10

k

|V[k

]|

N-Point DFT of Sampled, Windowed, Zero-Padded Signal

0 5 10 15 200

2

4

6

8

10

/2 (Hz)

|V[k

]|,

|V(ejT)|

Spectrum of Sampled, Windowed, Zero-Padded Signal

|V(ej T)||V[k ]|,

k = k2 /NT


ωT

16

000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000 00000000000 000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000

Rect window

iDFT20

17

000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000 00000000000 000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000

iDFT200

18

http://www.neuroradiologycases.com

A 40 yo pt with a history of lower limb weakness referred for mri screening of brain and whole spine for cord. MRI sagittal T2 screening of dorsal region shows a faint uniform linear high signal at the center of the cord. The signal abnormality likely to represent:

(1) Cord demyelination.(2) Syrinx (spinal cord disease).(3) Artifact.

Answer : Its an artifact, known as truncation or Gibbs artifact

19

Frequency Analysis with DFT

Length of window determines spectral resolution

Type of window determines side-lobe amplitude.(Some windows have better tradeo↵ betweenresolution-sidelobe)

Zero-padding approximates the DTFT better. Does notintroduce new information!


20

Potential Problems and Solutions

Potential Problems and Solutions

Problem Possible Solutions

1. Spectral error a. Filter signal to reduce frequency content above ⌦

s

/2 = ⇡/T .

from aliasing Ch.4 b. Increase sampling frequency ⌦

s

= 2⇡/T .

2. Insu�cient frequency a. Increase L

resolution. b. Use window having narrow main lobe.

3. Spectral error a. Use window having low side lobes.

from leakage b. Increase L

4. Missing features a. Increase L,

due to spectral sampling. b. Increase N by zero-padding v [n] to length N > L.


21

iSpectrum Example

22


Discrete Transforms (Finite)

• DFT is only one out of a LARGE class of transforms

• Used for:–Analysis–Compression–Denoising–Detection–Recognition–Approximation (Sparse)

Sparse representation has been one of the hottest research topics in the last 15 years in sp

23

• Spectrum of a bird chirping– Interesting,.... but...– Does not tell the whole story– No temporal information!


Example: Bird Chirp

Play Sound!

0 0.5 1 1.5 2 2.5

x 104

0

100

200

300

400

500

600

Hz

Spectrum of a bird chirp

No temporal information!

Miki Lustig UCB. Based on Course Notes by J.M Kahn Fall 2011, EE123 Digital Signal Processing

Example of spectral analysis

x[n]

n

24

• To get temporal information, use part of the signal around every time point

• Mapping from 1D ⇒ 2D, n discrete, w cont.

• Simply slide a window and compute DTFTM. Lustig, EECS UC Berkeley

Time Dependent Fourier Transform

X[n,!) =1X

m=�1x[n+m]w[m]e�j!m

*Also called Short-time Fourier Transform (STFT)

25

• To get temporal information, use part of the signal around every time point


Time Dependent Fourier Transform

X[n,!) =1X

m=�1x[n+m]w[m]e�j!m

*Also called Short-time Fourier Transform (STFT)

26


Spectrogram

Time, s

Fre

quency, H

z

2 4 6 8 10 12 14 16 180

1000

2000

3000

4000

0 500 1000 1500 2000 2500 3000 3500 4000 45000

5

10

15

20

25

30

Frequency, Hz

0 500 1000 1500 2000 2500 3000 3500 4000 45000

2

4

6

8

10

12

Frequency, Hz

0 500 1000 1500 2000 2500 3000 3500 4000 45000

5

10

15

20

25

30

Frequency, Hz

0 500 1000 1500 2000 2500 3000 3500 4000 45000

2

4

6

8

10

12

Frequency, Hz

27

Xr[k] =L�1X

m=0

x[rR+m]w[m]e�j2⇡km/N


Discrete Time Dependent FT

• L - Window length• R - Jump of samples • N - DFT length

• Tradeoff between time and frequency resolution

28

t

!

�t · �! � 1

2

�t

�!


Heisenberg Boxes

• Time-Frequency uncertainty principle http://www.jonasclaesson.com

29

�! =2⇡

N

�t = N

�! ·�t = 2⇡


DFT

X[k] =N�1X

n=0

x[n]e�j2⇡kn/N

!

tone DFT coefficient

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