1 Discrete Fourier Transform
Discrete Fourier Transform
Mar 21, 2016
Discrete Fourier Transform. Multiply element-by-element. Cumulative sum shows:. 2 signals of same frequency and phase. Multiply element-by-element. Non-zero cumulative sum. Same frequency but /2 phase difference. Element-by element product with both sine and cosine waves. - PowerPoint PPT Presentation
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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
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Non-zero cumulative sum
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Same frequency but /2 phase difference
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Element-by element product with both sine and cosine waves
9
Cumulative sums
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Wave: partly sine, partly cosine
11
Element-by-element multiplication
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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
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dftsimp2demo(f, fs, timelen, amp)dftsimp2demo(200, 1000, 0.05, 1)
0 100 200 300 400 500 600 700 800 900 10000
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dftsimp2demo(f, fs, timelen, amp)dftsimp2demo(200, 10000, 0.05, 1)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
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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
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dftcomplex2demo(f1, f2, fs, timelen, a1, a2)dftcomplex2demo(200, 400, 10000, 0.02, 5, 4)
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dftcomplex2demo(f1, f2, fs, timelen, a1, a2)dftcomplex2demo(200, 400, 10000, 0.02, 5, 4)
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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
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dftspeech2demo(wavfile, timelen)dftspeech2demo('atest.wav', 0.04)
0 2000 4000 6000 8000 10000 120000
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Use dB scale and frequencies to Fs /2
0 1000 2000 3000 4000 5000 6000-40
-30
-20
-10
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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
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dftspeech2demo(wavfile, timelen)dftspeech2demo(‘itest.wav', 0.04)
0 1000 2000 3000 4000 5000 6000-50
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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 cosinecomponent 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 cosinecomponent sinetan
component cosinecomponent sine
1
22
X
X
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Compact Formulae• DFT
1,,2,1,02 1
0
2
NkexNkX
N
n
Nknjn
• IDFT
1,,2,1,021 1
0
2
NneNkX
Nx
N
k
Nknjn
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