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Chapter 4 Fourier Transform of Discrete-Time Signals 2 nd lecture Mon. June 17, 2013. 4.1 Discrete-Time Fourier Transform. Continuous-time F.T . from Chapt . 3:. CTFT. Now, the input is x[n]. Define Discrete-time F.T .:. DTFT. Inverse DTFT (4.1.3, p. 175). Eq. 4.27:. - PowerPoint PPT Presentation
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Chapter 4Fourier Transform of Discrete-
Time Signals2nd lecture
Mon. June 17, 2013
Chapter 4Fourier Transform of Discrete-
Time Signals2nd lecture
Mon. June 17, 2013
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• Continuous-time F.T. from Chapt. 3:
4.1 Discrete-Time Fourier Transform4.1 Discrete-Time Fourier Transform
• Now, the input is x[n]. Define Discrete-time F.T.:
CTFT
DTFT
( ) [ ] j nX x n e
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• Eq. 4.27:
Inverse DTFT (4.1.3, p. 175)Inverse DTFT (4.1.3, p. 175)
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1[ ] ( )
2jnx n X e d
Perform integration over any 2 interval.
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Example 4.3 – Rectangular Pulse
Example 4.3 – Rectangular Pulse
1, ... [ ]
0, all other qn q q
p nn
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Example 4.3 – Rectangular Pulse, cont’d
Example 4.3 – Rectangular Pulse, cont’d
Even signal, so DTFT is purely real. Use the def. (Eq. 4.2) and follow Ex. 4.1 to get:
( 1)( )
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q j q j qj n
jq
e eP e
e
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Example 4.3 – Rectangular Pulse, cont’d
Example 4.3 – Rectangular Pulse, cont’d
MATLAB code* to plot magnitude:
*This script can be found on the class website with the filename dtft_pulse.m
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Example 4.3 – Rectangular Pulse, cont’d
Example 4.3 – Rectangular Pulse, cont’d
Can plot from to , or from 0 to 2 .
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Sect. 4.2 – Discrete Fourier Transform (DFT / FFT)
Sect. 4.2 – Discrete Fourier Transform (DFT / FFT)
This is arguably the most important result in all of signal processing and modern communication.
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Frequency DomainFrequency Domain
“Hello”“Hello”
FrequencyFrequency
Volt
age
Volt
age
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Discrete Fourier Transform Discrete Fourier Transform
Need to store the transform in computer memory & files.
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[ ] , 0,1,..., 1N
j kn Nk
n
X x n e k N
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1[ ] , 0,1,..., 1
Nj kn N
kk
x n X e n NN
Inverse DFT:
DFT:
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Discrete Fourier Transform Discrete Fourier Transform
N-point DFT is computed using the FFT algorithm.
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0
[ ] , 0,1,..., 1N
j kn Nk
n
X x n e k N
MATLAB:
DFT:
n=1:1024;x=(-.7).^n;xf=fft(x);stem(xf)
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Discrete Fourier Transform Discrete Fourier Transform N-point DFT:
N is always 2n in practice. Common values are 1024, 4096.
First point is k = 0, last point is k = N-1, center point is N/2.
Magnitude is symmetric around N/2:|X(N-1)|=|X(1)|, |X(N-2)|=|x(2)|, …
