1

Discrete Fourier Transform

2

Multiply element-by-element

3

Cumulative sum shows:

4

2 signals of same frequency and phase

5

Multiply element-by-element

6

Non-zero cumulative sum

7

Same frequency but /2 phase difference

8

Element-by element product with both sine and cosine waves

9

Cumulative sums

10

Wave: partly sine, partly cosine

11

Element-by-element multiplication

12

Cumulative sum

13

dftsimp2demo(f, fs, timelen, amp)dftsimp2demo(200, 1000, 0.02, 1)

0 100 200 300 400 500 600 700 800 900 10000

2

4

6

8

10

12

14

dftsimp2demo(f, fs, timelen, amp)dftsimp2demo(200, 1000, 0.05, 1)

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

30

15

dftsimp2demo(f, fs, timelen, amp)dftsimp2demo(200, 10000, 0.05, 1)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

50

100

150

200

250

300

160 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-8

-6

-4

-2

0

2

4

6

8

dftcomplex2demo(f1, f2, fs, timelen, a1, a2)dftcomplex2demo(200, 400, 10000, 0.02, 5, 4)

170 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

100

200

300

400

500

600

dftcomplex2demo(f1, f2, fs, timelen, a1, a2)dftcomplex2demo(200, 400, 10000, 0.02, 5, 4)

18

dftspeech2demo(wavfile, timelen)dftspeech2demo('atest.wav', 0.04)

2 2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045

x 104

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

19

dftspeech2demo(wavfile, timelen)dftspeech2demo('atest.wav', 0.04)

0 2000 4000 6000 8000 10000 120000

5

10

15

20

25

20

Use dB scale and frequencies to Fs /2

0 1000 2000 3000 4000 5000 6000-40

-30

-20

-10

0

10

20

30

21

dftspeech2demo(wavfile, timelen)dftspeech2demo(‘itest.wav', 0.04)

2 2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045

x 104

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

22

dftspeech2demo(wavfile, timelen)dftspeech2demo(‘itest.wav', 0.04)

0 1000 2000 3000 4000 5000 6000-50

-40

-30

-20

-10

0

10

20

30

40

23

DFT Procedure• Given the window (frame) length, decide the

base frequency• Multiply by sine wave at each multiple of base

frequency• Multiply by cosine wave at each multiple of

base frequency• Calculate magnitude and phase spectra using

but....,component cosine

component sinetan

component cosinecomponent sine

1

22

X

X

24

Complex Exponential• Given the window (frame) length, decide the

base frequency• Multiply by sine wave at each multiple of base

frequency• Multiply by cosine wave at each multiple of

base frequency• Calculate magnitude and phase spectra using

but....,component cosine

component sinetan

component cosinecomponent sine

1

22

X

X

25

Compact Formulae• DFT

1,,2,1,02 1

0

2

NkexN

kX

N

n

Nknjn

• IDFT

1,,2,1,021 1

0

2

NneN

kX

Nx

N

k

Nknjn