Fourier Series

1
Fourier Series
2
Jean Baptiste Joseph Fourier(French)(17631830)
3
Discrete Fourier Transform

• ??????,??????????(Discrete Fourier Transform,
  DFT)????????????????????????
• ??????????????????,?????????,???????????

4
Fast Fourier Transform

• FFT (Fast Fourier Transform),???????????
• FFT????
• ??????????
• ????????????????
• ????(Sampling rate)???????????2 ????
• ????N ???2k????

5
Fourier Series

• ????(periodic)?????????????(amplitude),????(freque
  ncy)?
• ??(sinusoidal)??(harmonic)
• ?
• ??(cosinusoidal)??(harmonic)
• ???(composition)

6
Fourier Series

• ?????(Fourier Series)???????????????????,?????????
  ?????????
• ????x(t) ???????????

7
???
8
????(harmonic )
9
???????
10
Adding harmonics
11
????
12
Fourier Series

• ????
• eif cosf isinf
• ?????????????

13
Fourier Series

• ????????????,?????

14
Discrete Fourier Transform (DFT)

• ??????,??????????(Discrete Fourier Transform,
  DFT)????????????????????????
• ??????????????????,?????????,???????????

15
Discrete Fourier Transform (DFT)

• ????????????,????T??,???N ?????????
• DFT ???

16
Discrete Fourier Transform (DFT)

• ??m??????m???,n ??????n ???
• X(m) ?????m?????,x(n) ?????n ?????

17
Fast Fourier Transform

• FFT (Fast Fourier Transform),???????????
• FFT????
• ??????????
• ????????????????
• ????(Sampling rate)???????????2 ????
• ????N ???2k????

18
?????????

• A complex nth root of unity is a complex number z
  such that zn 1.
• ?n e 2? i / n principal n th root of unity.
• e i t cos t i sin t.
• i2 -1.
• There are exactly n roots of unity ?nk, k 0,
  1, . . . , n-1.

?2 i
?1
?3
?0 1
?4 -1
?7
?5
?6 -i
19
?????????

• ?n2 ?n/2
• ?nnk ?nk