Unformatted text preview: in seconds. In the Fourier transform an analog signal is multiplied by the Fourier kerand integrated with respect to time. In the DFT the integration is replaced by
nel
summation and the Fourier Kernel by
. Since does not have units, has units of
radians. 0 0 ©&
( ' % &
( ' % The DFT maps a discrete signal into the frequency domain. So far, the frequency is
a continuous variable, but let us assume that one wants to discretized in the same
way we have discretized the temporal variable . The limits of are given by
,
remember that is an angular frequency. If the time series is a points time series, we
can discretized the frequency axis as follows: 0 31)
¥ 20('& 0 § 4 0 B E "" )B)
GFD4) 44" #CA' § 7 @981 ¢§ 60
)437 5 (3.5) Now we can deﬁne the DFT as follows:
B E 4 " )B)
H&
§ ¥5
" RQP) 4" " #IA' § 7 ) )(' ¥% ¡ ¨ © ¨46 ¡0 (3.6) 4 ¥ T10¢ 5¡
¨©
&
&
" 4" © e ih¨© qgIdcX ' % © e ih3© dcX ' % B
"
b X
g
© &'
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 Winter '12
 Sacchi
 Digital Signal Processing, DFT

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